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OOSTIUBirTIOllS m COSIOMISY iSt TEE FTSDAMEIiTAL FBOBLEMS OF eHtUWT
THE GASES IN ROCKS
BOLLDT THOMAS CHAHBBBLER
WASHINGTON. D. 0.
r THE OARiraatK Iott i tpt i om or Washhtotok
1S06
OABNEGIB INSTITUTION OP WASHINGTON
Publication No. 106
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rRBss OF T B.'iapraooW*ooi»?AW
• • •
• • •
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• • • •
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"•• • • • •
• • • ••
THE GASES IN ROCKS.
It has been known for a long time from microscopical studies that
some minerals inclose minute cavities which contain both liquid and gas-
eous matter. For a much shorter period it has been known that various
igneous rocks, when exposed to red heat in a vacuum, evolve several times
their volume of gas of quite variable composition. Since these gases occur
in proportions entirely different from those of the constituents of the air,
it has not seemed probable that they were derived directly from our
present atmosphere, unless the rocks manifest some power of selective
absorption not now understood. The apparent difficulties involved in this
conception have suggested that some earlier atmosphere was rich in those
gases. This involves a hypothesis relative to the changes through which
the atmosphere has passed, and leads on to a theory of its origin and that
of the earth itself. An alternative hypothesis regards these gases, not as
the products absorbed by a molten earth from its surrounding gaseous
envelope, but as entrapped in the body of the earth during its supposed
accretion, and hence that they are a source from which accessions to our
present atmosphere might be derived.
A study of these gases in the rocks has seemed, therefore, to give
promise of results of some value to atmospheric problems and, perhaps, to
those of cosmogony. Because of this, it appeared advisable to determine
more widely the range and the distribution of these gases, their relations
to other geologic phenomena, and the states in which the gases, or gas-
producing substances, exist in the rocks. The desirability of supplementary
work will become more evident when it is noted that, while a considerable
number of investigators have analyzed the gases in rocks, as will appear in
the following historical statement, nearly all have contented themselves
with a few determinations, and that even a full compilation of all such
results leaves much to be desired from a geological point of view.
HISTORICAL SKETCH.
As early as 1818 the attention of Sir David Brewster was called to the
subject of inclosed water by the explosion of a crystal of topaz when
heated to redness ; but his studies were not published until 1826. In the
mean time, Sir Humphry Davy opened the cavities in a few crystals and
examined chemically the imprisoned liquid and gas.* Piercing a cavity in
several cases suddenly caused the inclosed gas-bubble to contract to from
one-sixth to one-tenth of its original volume. The gas was thought to be
pure nitrogen. The basaltic rock from the neighborhood of Vicence con-
»Sir Humphry Davy, Plul. Trans. 1822, Pt. n, pp. 367-376 : Ann. de Chim. et Phys.,
t. 21 (1822), pp. 132-143.
3
4 THE OASES IN ROCKS.
tained gas (supposedly nitrogen) in a still more rarefied state, as its density
was 60 to 70 times less than that of the atmospheric air. Upon perforating
a cavity in a quartz crystal from Dauphin^, an almost perfect vacuum was
discovered. Davy regarded the rarefied condition of the inclusions in the
crystals as strong evidence that the waters and gases did not penetrate
the crystals at ordinary temperatures and pressures. This he believed a
decisive argument in favor of the Huttonian, or Plutonian, school. How-
ever, a crystal from Brazil gave a very different result; an immediate
expansion to a volume 10 to 12 times greater than the original followed the
opening of a cavity. The composition of the gas was not determined. The
existence of compressed gas in the same sort of cavities seems adverse to the
conclusions which Davy based upon his earlier experiments, but he sought
to explain the difference by supposing the crust to have been formed under
a pressure more than sufficient to balance the expansion due to the heat.
Brewster^ attacked the problem by observing the temperature at which
the inclosed liquid passed over into the gaseous state. A number of tests
showed this to range from 74^ to 84^ F. When raised to this temperature
the vacuity always reappeared. Brewster interpreted as follows:
The existence of a fluid which entirely fillB the cavities of crystalB at a temp»Bture
varying from 74® to 84® may be hdd as a proof that these crystals were formed at the
ordinary temperature of the atmosphere.
For thirty years after Brewster the field was neglected until, in 1868,
Simmler' reviewed Brewster's work in the light of advancing scientific
knowledge. Studying the liquid inclusions in quartz, topaz, ameth3rst,
garnet, and other minerals, he arrived at the conclusion that the power of
expansion of the liquid in these inclusions showed it to be carbon dioxide.
Some years later Sorby, continuing the researches along the lines suggested
by Brewster and Simmler, found that the amount of expansion of liquid
carbon dioxide from 0^ C. to 30^ C. corresponded closely to that observed
in the liquid of the sapphires with which he experimented.' In these sap-
phires it was noted that the liquid disappeared when warmed to approxi-
mately 30^ G. As the critical temperature for carbon dioxide, above which
no amount of pressure will condense it to a liquid, is 30.92^ C. (87.7^ F.),
there remained little room for doubt that the gas was largely carbon
dioxide. Sorby remarked that this gas '' might have been inclosed, either
as a highly dilated liquid or as a highly compressed gas; but since the
other ^ liquid has deposited crystals which dissolve on the application of
heat, it seems most probable that the water was caught up in a liquid
state, sometimes, perhaps, holding a considerable amount of carbon dioxide
in solution as a gas.''
In the same year Vogelsang and Geissler' heated quartz crystals and,
passing an electric spark through the gas thus liberated, examined its
1 Sir David Brewster, Trans. Roy. Soc. Edinburgh, vol. 10 (1826), pp. 1-41 ; Edin. Jour,
of Science, vol. 6. pp. 153-156.
» R. T. Simmler, Pqgg. Ann. , vol. 106 (1858), pp. 460-466.
»H. C. Sorby and PTJ. Butler, Proc. Roy. Soc., vol. 17 (1869), pp. 291-303. Earlier
papers by Sorby appeared as follows : Phil. Mag., 4th series, vol. 15, p. 152 ; Quart. Jour.
Geol. Soc., vol. 14 (1858), p. 453 ; Proc. Roy. &c., vol. 13 (1864), p. 333.
^SiJine water.
* Vogelsang and Geissler, Pogg. Ann., vol. 137 (1869), pp. 5^75.
HISTORICAL SKETCH. 5
spectrum, which was found to show the presence of much carbon dioxide,
together with water and a very weak trace of hydrogen. The presence
of the hydrogen line, however, the authors were inclined to attribute to
water-vapor.
Further researches upon the critical point of the gas in mineral cavities,
carried on by Hartley,* yielded results varying from 26° to 34® C. The
lowering of the temperature he ascribed to the presence of some incon-
densable gas, perhaps nitrogen, while he believed that the raising of the
critical point observed in some of the quartz specimens was due to hydro-
chloric acid.
Forster* and Hawes' investigated smoky quartz, the former distilling
from the Tiefengletscher crystals a brown fluid of an empyreumatic odor,
giving reactions for ammonia and carbonic acid, from which he concluded
that the coloring matter of smoky quartz was due to a nitrogenous hydro-
carbon, decomposable by heat; the latter made a microscopic study of the
liquid carbon dioxide in the bubbles of the cavities.
Investigations which have opened up a broader field were begun by
Graham^ in 1867 upon the Lenarto meteoric iron. By submitting a strip
of the iron to a red heat in a vacuum for 35 minutes he obtained 5.38 cubic
centimeters of gas from 5.78 cubic centimeters of the metal. Heated for an
additional 100 minutes, there were evolved 9.52 cubic centimeters of gas
having the following composition: H2, 85.68; 00,4.46; C02,none; N2,9.86.
As this meteorite yielded about three times its volume of gas, and
since "it has been found difficult, on trial, to impregnate malleable iron
with more than an equal volume of hydrogen, under the pressure of our
atmosphere,'' Graham drew the inference that this meteorite came from a
body having a dense atmosphere of hydrogen gas. By the same process
Mallet* extracted 3.17 volumes of gas from a Virginia meteorite. His
results were in accordance with those of Graham: H2, 35.83; CO, 38.33;
CO2, 9.75; N2, 16.09.
Wohler* heated to redness some of the metallic granules from the iron
basalt of Ovifak, Greenland, obtaining more than 100 volumes of gas which
burned with a bluish flame (mostly carbon monoxide mixed with a little
of the dioxide). His results, however, were vitiated by having used an
iron combustion tube.
Pursuing the method adopted by Graham and Mallet, A. W. Wright ^
conducted a series of experiments on meteorites, which have remained to
the present day the source of most of our knowledge of the gas content of
these interesting bodies. Wright's chief contribution lies in his two tables
showing that there is a marked difference between the gas contents of the
iron and stony types of meteorites; for while, in the former, hydrogen is
* W. N. Hartley, Jour. Chem. Soc. (1876), vol. 2, pp. 237-250.
*A. Foreter, Pogg. Ann., 143 (1871), pp. 173-194.
»G. W. Hawes, Aioi. Jour. Science, vol. 21 (1881), pp. 203-209.
*TboB. Graham, Proc. Roy. Soc., vol. 16 (1867), p. 602.
•J. W. Mallet. Plroc. Roy. Soc, vol. 20 (1872), pp. 366-370.
•F. WdWer, Fogg. Ann., 146 (1872), pp. 297-302.
» A. W. Wnght, Am. Jour. Science, vol. 9 (1876), pp. 294-302 and 469-460 ; vol. 11
(1876), pp. 263-282; vol. 12 (1876), pp. 166-176.
6
THE OASES IN ROCKS.
the most abundant gas, carbon dioxide is the most characteristic con-
stituent of the latter. His analyses are given in table 1.
Table 1.
Meteorite.
Iron meteorites :
Tazewell County, Tenn
Shingle Springs, Gal
Arva, Hungary
Texas
Dickson County, Tenn.
Stony meteorites :
Guernsey, Ohio
Pultusk, roland
Pamallee, India
Weston, Conn
Iowa County, la
COi.
14.40
13.64
12.56
8.59
13.30
59.88
60.29
81.02
80.78
35.44
CO.
41.23
12.47
67.71
14.62
15.30
4.40
4.35
1.74
2.20
1.80
OH,
2,05
3.61
2.08
1.63
0.0
H,.
42.66
68.81
18.19
76.79
71.40
31.89
29.50
13.59
13.06
57.88
N,.
1.71
5.08
1.54
1.78
2.25
1.57
2.33
4.88
VoL
3.17
0.97
47.13
1.29
2.2
2.99
1.75
2.63
3.49
2.50
This table shows that in the iron meteorites carbon dioxide in no case
constituted more than 15 per cent of the gas evolved, while in every case
but one the quantity of carbon monoxide was considerably greater. In
the stony meteorites carbon monoxide is low, while carbon dioxide is, in
the majority of analyses, much the most abundant gas. Hydrogen is more
important in the iron meteorites than in the stony.
The same experimenter determined also the gases given off by the same
meteorite at different temperatures. His figures for the Iowa County
meteorite are shown in table 2.
Table 2.
Gas.
AtlW>.
At 250°.
Below
red heat
At low
red heat
At full
redhaat
Carbon dioxide
95.46
.00
4.54
0.00
92.32
1.82
5.86
0.00
42.27
5.11
48.06
4.56
35.82
0.49
58.51
5.18
5.56
0.00
87.53
6.91
Carbon monoxide
Hydrocren
Nitroflfen
Total
100.00
100.00
100.00
100.00
100.00
The progressive decrease in the percentage of carbon dioxide and the
corresponding increase of hydrogen with the elevation of the temperature
are striking. His inquiries into other phases of the problem will be deferred
until the discussion of principles, where it will be possible to treat each
factor to better advantage, in its proper relation to the whole subject.
Several years later Wright applied his method of gas extraction and
reliable quantitative analysis to the gases in smoky quartz/ which here-
tofore had been subjected chiefly to qualitative microscopical studies.
However, only one determination was made — that of a crystal from
Branchville, Connecticut, which yielded a small quantity of gas of the
following composition: Carbon dioxide, 98.33; nitrogen, 1.67; hydrogen
sulphide, sulphur dioxide, ammonia, fluorine, and chlorine, trace.
^Wright, Am. Jour. Science, vol. 21 (1881), pp. 209-216.
HISTORICAL SKETCH.
Wright regarded the fluorine and chlorine as being combined, and the
ammonia as probably existing together with some of the carbon dioxide in
the form of ammonium carbonate. The amount of water obtained, calcu-
lated as vapor, was slightly more than twice the volume of carbon dioxide.
Following Wright, Sir James Dewar,* in collaboration with Mr. Ansdell,
made several more analyses of the meteoritic gases, and then, in an
endeavor to discover the source and significance of these gases, directed a
series of experiments upon the theory that graphite might be the retentive
or generative constituent. Their analyses of the gases from graphites and
from the matrix from which graphites have come revealed moderately
high volumes. ^^^ 3^
Material.
Sp.gr.
Vol.
CO,.
CO.
Hf
CH4.
N«.
Celestial graphite
Borrodale graphite
Siberian graphite
Cevlon flnraDiiite
2.26
2.86
2.05
2.25
1.64
2.45
2.59
7.25
2.60
2.55
0.22
7.26
5.32
IJ27
91.81
36.40
57.41
66.60
50.79
82.38
94.72
• • • •
7.77
6.16
14.80
3.16
2.38
0.81
2.50
22.2
10.25
7.40
2.50
13.61
2.21
5.40
26.11
20.83
3.70
39.53
0.47
0.61
0.1
6.66
4.16
4.50
3.49
1.20
1.40
Unknown graphite
Gneiss
Feldsnar
Because the quantity of gas yielded by these specimens of graphite was
so considerable, Dewar proceeded to ascertain whether graphite could
absorb the different gases when allowed to stand in each of them for 12
hours. His experiments with the celestial graphite which had previously
been deprived of its gases indicated that little or no absorption had taken
place. "It is therefore evident," sa3rs Dewar, "that the large quantities
of gas occluded in celestial meteorites can not be explained by any special
absorptive power of this variety of carbon." Attempts to split up the
hydrogen-producing compound with strong nitric acid and also to wash
out, with ether, the possible carbonaceous source of the methane, appeared
to show that the hydrogen existed in a very stable compound, and that,
while the ether lessened the quantity of methane which the graphite after-
wards furnished, it did not dissolve out all the carbonaceous compounds
present, or else that the marsh-gas was subsequently formed during the
heating of the material.
Dewar's analyses of gases from stony meteorites, which are in accord
with Wright's results, are given in table 4.
Table 4.
Meteorite.
8p.gr.
Vol.
2.51
3.54
1.94
0.55
CO,.
63.15
66.12
64.50
39.50
CO.
28.48
18.14
22.94
25.4
CH4.
N,.
Dhurmsala, India
Pultusk
3.175
3.718
3.67
2.50
1.31
5.40
3.90
18.50
3.9
7.65
4.41
• • • •
1.31
2.69
3.67
16.60
M0C8
Pomice stone
An analysis of the gas extracted from the Orgueil meteorite revealed
much sulphur dioxide, which Professor Dewar believed to have been derived
1 Dewar and Ansdell, Ploe. Roy. Inst., vol. 11 (1884-1886), p. 332 and pp. 541-552.
8 THE QA8ES IN ROCKS.
from the decomposition of sulphate of iron. In all, 57.87 volumes were
obtained: CO2, 12.77; CO, 1.96; CH4, 1.50; N2, 0.56; SO2, 83. Leaving
out the SO2, 9.8 volumes remain, as follows: CO2, 76.05; CO, 11.67; CH4,
8.93; N2, 3.33. This analysis, the most remarkable of the series, though
Dewar does not mention the fact, shows a complete absence of hydrogen
(an uncommon phenomenon), while the percentage of marsh-gas is unusu-
ally high. There is, however, reason to suspect that there was hydrogen
liberated, but that it was oxidized to water by the action of the iron com-
pound, following the decomposition of the sulphate.
In 1888 W. F. Hillebrand^ discovered that the mineral uraninite when
treated with acids slowly disengaged bubbles of gas. As the result of a
well-selected series of tests this appeared, in the light of the chemical
knowledge of that day, to be nitrogen. Trials with different varieties of
the mineral revealed a rather significant relation between the percentage of
uranyl and this gas. The greater the amount of the oxide, the more gas
obtained.
Several years later. Sir William Ramsay's scepticism was aroused when
his attention was called to the paper by Hillebrand, for he hesitated to be-
lieve that free nitrogen could be produced by treating any substance with
sulphuric acid. To test the case, he decomposed cleveite with this acid,
obtaining little nitrogen, but some 20 cubic centimeters of argon, which
the spectroscope showed to be mixed with some other gas.' A brilliant
yellow line which appeared in this spectrum coincided exactly with D3, the
so-called " helium " line, first discovered in the spectrum of the chromo-
sphere of the sun by Sir Norman Lockyer in 1868. This was the first real
acquaintance with helium, until then known only as a h3rpothetical sub-
stance existing in the sun. Lockyer immediately became interested in this
discovery of helium in a terrestrial mineral, and attempted to prove that
it was not a single gas, but a compound or a mixture of gases, basing his
contention upon various strange lines in the spectrum.'
Ramsay, continuing his study of the gas from cleveite, perceived what
had been previously overlooked, namely, that hydrogen generally was more
abundant than helium — in one case amounting to 80 per cent of the total
gas. The hypothesis that this hydrogen might have been formed by the
breaking up of an unstable hydride, the form in which Ramsay thought
the helium should be evolved, if it were derived from combination with the
uranium or yttrium of the mineral, was put to the test, with the result
that the evidence pointed strongly against the theory.* A series of miner-
als powdered and fused with potassium acid sulphate were found to yield
gas, sometimes helium,* but oftener hydrogen and the oxides of carbon.*
In 1896 W. A. Tilden made an attempt to determine the condition in
which helium and the associated gases exist in minerals. Argon and helium
were of particular interest, for Tilden believed that these two elements will
» W. F. HiUebrand, BuU. 78, U. 8. G. S., pp. 43-79.
* Sir William Ramsay, Proc. Rojr. Soc., vol. 68 (189 , , ,^
' Sir J. N. Lockyer, a series of six short papers in Proc. Roy. Soc., vols. 58, 59, and 60.
68 (1896), pp. 66^7.
* Ramsay, Procr Roy. Soc, vol. 68, pp. fil-^9.
* Ramsay, Proc. Roy. Soc., vol. 69, j>p. 325-330.
* Ramsay and Travers, Proc. Roy. Sens., vol. 60, pp. 442-448.
HISTOBICAL SKETCH.
9
not be found to enter into combination at such temperatures as are ordi-
narily attainable. In his own words:
It also appears improbable that in the minerals from which the mixture of gases con-
taining helium has been extracted this element exists in a state of ordinary chemical com-
bination, for, on treating the mineral with acids, no compound of heb'um with hydrogen has
yet been observed, and the components of the minerals are of a kind which are commonly
regarded as chemically saturated.^
The minerals monazite and cleveite were found to yield gas at low tem-
peratures (60° and 110°, respectively), carbon dioxide appearing first. The
monazite heated to 130° to 140° gave gas which, for the first time, showed
the D3 line, indicating the presence of helium. Between 140° and 250°
there was obtained carbon dioxide with about one-fourth of its volume of
a gas rich in helium. At higher temperatures up to 446° (boiling sulphur)
there was less gas evolved. Cleveite behaved in a similar way. Studies
on the absorption of helium by cleveite demonstrated that the mineral
does not absorb this gas at the ordinary pressure, although placed in a
helium atmosphere for nine weeks. But under pressure of 2.5 and 7 atmos-
pheres, Tilden believed that he obtained an appreciable absorption. A
trial with the Peterhead granite, which contained no helium in the first
place, proved that the granite would absorb none of the gas whatever,
even aided by a pressure of 7 atmospheres.
The finding of hydrogen as well as carbon dioxide in this Peterhead
granite^ led Tilden to investigate the gases inclosed in crystalline rocks.'
His five complete analyses are as given in table 5.
Tablk 6.
Bock.
Granite, 8kye
Gabbro, liwd
Fyroxene ^eiss, Ceylon
GneisB, Senngapatam . . ,
Basalt, Antrim
CO,.
23.60
6.50
77.72
31.62
32.08
CO.
6.45
2.16
8.06
5.36
20.08
CH4.
3.02
2.03
0.56
0.51
10.00
H,.
61.68
88.42
12.49
61.93
36.15
N,.
5.13
1.90
1.16
0.56
1.61
In addition to these analyses, 25 carbon-dioxide determinations were
made.
Tilden believed the gas to be " wholly inclosed in cavities which are
visible in thin sections of the rock when viewed under the microscope.
* * * To account for the large proportion of hydrogen and carbon
dioxide in these gases, it is only necessary to suppose that the rock inclos-
ing them was crystallized in an atmosphere rich in carbon dioxide and
steam, which had been, or were at the same time, in contact with some
easily oxidizable substance, at a moderately high temperature. Of the
substances capable of so acting, carbon, a metal, or a protoxide of a metal
present themselves as the most probable." Hydrogen and carbon monox-
» W. A. Tilden, Proc. Roy. Soc., vol. 69 (1896), p. 218.
' Wright's two analyses, showing that trap rodcs yield much the same gases as meteor-
ites, also served to call attention to this field for investigation.
•Tikien, Chem. News, vol. 76 (1897), pp. 169-170.
10
THE OA8ES IN ROCKS.
ide might then be produced by the reducing action of metallic iron or fer-
rous oxide upon steam and carbon dioxide at high temperature, according
to the reactions —
3Fe + 4H2O = FesOi + 4H«
3Fe + 4CO2 = FeaOi + 4C0
The origin of the marsh-gas is assigned in this paper to the action of
water at high temperature upon metallic carbides, or similar compounds,
in the earth's interior, as suggested by Mendel6ef * and the more recent
studies of Moissan.^
A year after the publication of Tilden's article, criticism of his paper,
and in fact of the work of all previous investigators in this line, was made
by M. W. Travers, who undertook to prove that the different gases, not
excluding even argon and heUum, did not exist in the gaseous state in min-
erals, but were formed by chemical interaction between the non-gaseous
materials in the combustion-tube.' The key to his position lay in the two
reversible reactions —
3FeO + H2O = FejOi + H2
BFeO + CO2 = FesOi + CO
His table revealed a certain relation between the hydrogen and carbon
monoxide produced, and the quantity of ferrous oxide and water present
in the mineral. It is shown in table 6. The figures for FeO and H2O refer
to the percentages in the rock; the gases are expressed in cubic centimeters
per gram of rock.
Table 6.
Mineral.
Chlorite, Moravia
Serpentine. Zennatt. .
Gabbro, Isle of 8kye .
Mica, Westchester, Pa.
Foliated talc, Tyrol . .
Feldspar, Petern^ui. .
FeO.
HsO.
H,.
CO.
CO,.
10.6
4.6
2.180
0.494
0.123
2.7
9.5
0.800
None.
None.
6.1
1.6
0.490
None.
None.
1.4
0.13
• • • •
0.06
0.150
0.4
4.5
• • • •
0.04
0.070
2.1
1.00
• • • •
0.214
1.201
Four of these (including the mica of meta-sedimentary origin) were
secondary minerals whose gas may have been produced entirely by chem-
ical reactions in the tube, without having very great bearing upon the
problem of the gas-content of primary minerals and rocks which have not
undergone extensive weathering and alteration. The only rock tried, the
gabbro, may be pointed out as unique in yielding only hydrogen without
either of the oxides of carbon or nitrogen.
Armand Gautier/ in 1901, came to the conclusion that the gases which
he obtained from several igneous rocks did not escape from inclusions, for
the most part, but were products of chemical reactions at raised tempera-
tures. A small quantity of gas was obtained by heating granite powder,
moistened with pure water, up to 300° in a vacuum. By heating the same
' Mendel^f, Prin. of Chem., transl. of £[amensky and Greenaway, vol. 1, pp. 364-365.
HISTORICAL SKETCH.
11
granite powder together with a mixture of two parts of sirupy phosphoric
acid and one part of water to only 100^, he received more than 10 times as
much gas as was evolved at 300^ without the acid, or about 1.5 volumes.
Table 7 comprises Gautier's analyses of the gases expelled at red heat.
Table 7.
Rock.
Granite, Vire I
Granite, Vire II
Granite, Vire III
Granitoid porphyry, Esterel
Ophite, Villeiianqae I
Ophite, Yillefranqae II
(n>hite, Villefranqne III. . . .
Lherzolite, Lherz
H«R.
COt.
CO.
CH4.
H,.
N,.
Vol.
Trace
14.80
4.93
2.24
77.30
0.83
lAv.
1.71
8.98
5.12
1.09
82.80
0.42
6.9
0.69
14.42
5.50
1.99
76.80
0.40
0.00
59.25
4.20
2.53
31.09
2.10
7.6
3.44
28.10
3.91
1.40
63.28
0.05
1 Av.
6.56
30.66
4.45
0.66
58.90
0.13
7.60
0.46
35.71
4.85
1.99
56.29
0.68
11.85
78.35
1.99
0.01
7.34
Trace
16.7
The r61e played by these gases in vulcanism, and their connection with
thermal waters, is discussed in more recent papers.^ Following Gautier,
Huttner^ showed by a series of experiments that when a stream of dry car-
bon dioxide is passed over a rock powder at a temperature of 800^, carbon
monoxide results, owing to a reduction of the dioxide, as this investigator
believes, by some of the hydrogen given ofif from the rock. As the miner-
als orthite and gadolinite yielded no carbon monoxide, though abundant
hydrogen, when gelatinized in hydrochloric acid, he came to the conclusion
that this gas does not exist in rocks.
In 1905 there appeared a paper by Albert Brun,* ''Quelques Recherches
sur le Volcanisme," based upon studies of lavas from Vesuvius, Stromboli,
and other Mediterranean volcanoes. While no complete gas analyses were
undertaken, much experimentation was done, covering the expulsion of
gases and vapors at or near the fusion point of the lavas. This author ex-
presses the opinion that it is the liquefaction of the rock which produces
the gases, these being engendered by chemical bodies contained within the
lava itselif. The gases recognized are nitrogen and its derivative ammonia,
chlorine with derived hydrochloric acid, and hydrocarbons. The nitrogen
is assigned to nitrides, and the ammonia to reactions between nitrides and
hydrocarbons, while a dissociation of chlorides furnishes free chlorine which
may take hydrogen from hydrocarbons to form hydrochloric acid. A
source for hydrogen and carbon dioxide is recognized in hydrocarbon com-
pounds, though Brun was less interested in the gases expelled below the
melting-point of the lava.
^Gautier, Comptes Rendus, vol. 132, pp. 740-746 and 932-938; Economic Geology,
vol. 1 (1906), pp. 688-697.
>K. Hattner, Zeitschrift fOr Anorip. Cbem., 43 (1905), pp. 8-13.
'A. Brun, Archives des Sciences pnys. et naturelles, Geneve, 1905.
12 THB GA8ES IN ROCKS.
METHOD OF PROCEDURE.
To obtain the gases for these investigations, the general methods of
Graham, Mallet, and Wright were adopted, though the details of the appa-
ratus were modified in many particulars. The gas is extracted from the
rock material which has been finely pulverized, by heating the powder in a
vacuum. For this purpose an apparatus consisting of a combustion-tube
connected with a mercury-pump capable of producing and maintaining a
vacuum of a fraction of a millimeter pressure is required. Simplicity bmng
desirable in order to insure the uniform working of the pump in the pres-
ence of corrosive gases, such as hydrogen sulphide and sulphur dioxide,
which attack and befoul the mercury, the most elementary tjrpe of Topler*
pump was used in these experiments.
To the receiving end of the pump a long, horizontal, calcium chloride
drying-tube is fused. The ideal method would be to seal the combustion-
tube containing the rock powder directly to the free end of this drying-
tube. But inasmuch as the pump and drying-tube are both constructed
of soft glass, whereas the tube in which the high-temperature combustions
are to be made must, of necessity, consist of the most refractory glass, which
can not be readily united to the fusible glass, one break in the system is
imavoidable. This is made at the end of the drying-tube, which is groimd
so as to receive a tightly fitting hollow stopper of the same hard, blue
Jena composition tubing as the combustion-tube. A 5-millimeter tube of
blue Jena glass joins the combustion-tube to the stopper, and is taken of
sufficient length to allow of repeated cutting and resealing to successive
tubes, as they become useless from slow deformation imder the combined
influence of high temperature and vacuum.
The capillary exhaust-tube of the pump, dipping under mercury in
a trough, is bent upward at its lower extremity, so as to deliver the gas
expelled from the pump directly into the receptacle designed for holding
it. For this purpose, a separatory fimnel of about 125 cubic centimeters
capacity, held by a clamp in an inverted position over the mercury trough,
proved most serviceable.
In making an analysis, the rock specimen is first reduced to a powder
of sufficient fineness to pass through a sieve of 30 meshes to the inch. A
portion of this powder, roughly estimated to approach the maximum
quantity which can with safety be placed in the combustion-tube, is then
weighed and carefully poured into the tube through the hollow stopper,
which, on account of its shape, serves as a funnel. Because the rock-dust
in falling becomes somewhat packed, the tube must afterwards be held in
a horizontal position, and gently shaken or tapped, to establish a free pas-
sageway for the gases, extending the entire length of the tube; otherwise,
upon attempting to exhaust, preparatory to heating, the air entrapped
in the powder, having no avenue of ready escape, will expand so rapidly
as to force some of the material into the drying-tube.
Thus carefully filled, the tube is placed in the combustion-furnace,
which stands upon a table of height such that the stopper end of the com-
^ DeBcribed by Travers, A study of gases, pp. 5-10.
METHOD OF PBOGEDUBE. 13
bustion-tube meets approximately the ground end of the calcium chloride
tube coming from the pump. The pump itself, installed upon a specially
constructed table resting on jacks, can be raised or lowered, or tilted at a
slight angle in any direction necessary to enable the stopper protruding
from the furnace to fit exactly into the drying-tube. As the whole appa-
ratus is now rigid glass from end to end, care is required in fitting the two
parts together, lest there be strain suiScient to cause serious fracture.
To prevent leakage during the extraction of the gas, the groimd-^ass
connection (the only source of leakage) is completely incased in a thick
coating of parafl^e.^
The air in the apparatus is now pumped ofiF imtil the exhaustion can
be carried no further, at which point the pressure may be in the neighbor-
hood of 0.01 millimeter. If allowed to stand for several days this vacuum
remains entirely without change. When ready, the burners in the furnace
are lighted, the separating funnel in which the gases are to be collected is
filled with mercury, and the evolution of the gas is under way. As fast as
the gases are liberated by the heat they are pumped over into the collect-
ing-funnel — a process usually requiring about 3 or 4 hours before the last
traces of gas have been expelled.
ANALYSIS OF THE GAS.
After constant temperature has been established in the room, the gas
is drawn from the receiver into a Lunge nitrometer and the carbon dioxide
and hydrogen sulphide absorbed by the introduction of a cubic centimeter
of 30 per cent potassium hydroxide solution. The remaining gas is trans-
ferred to a gas-burette, filled with water, after which the remainder of the
analysis is carried on according to the method described by Hempel.'
From the potassium hydroxide solution the amount of hydrogen sulphide
absorbed is determined by titration with N/100 iodine solution. If it be
desired to measure the quantity of helium and argon, the gas remaining
after the removal of all the constituents, except nitrogen and these inert
gases, is passed over metallic calcium heated to redness.* This absorbs the
nitrogen, leaving only helium and argon, which are examined spectro-
scopically.
^ Whenever rocks containixig a large proportion of quarts are teeted, it is neoessarv
to substitute a porcelain tube, since (luarts scratches glass, causing it to crack when hMted.
The connections are readily made tight with paraflfine.
' Hempel, Methods of gas auafysis : Technical method.
* Travers, A study of gases, p. 102.
THE OASES IN BOCKS.
THE ANALYSES.
The analyses are numbered in table 8 in the order in which they were
made, and therefore furnish a chronologic&l account of these investiga-
tions. At the commencement of these studies, it being deemed advisable
to make a rapid survey of the field and establish the range of the phenome-
non, in order to direct later experiments more intelligently, less attention
was given to securing the most trustworthy method of extracting the gas.
Twenty-two analyses were made during the preliminary trial stage, before the
apparatus was overhauled and sealed glass connections substituted for nibljer
tubing. As it was difficult to prevent a slight leakage of air into the tubes of
the ori^nal apparatus, the first 22 analyses are characterized by a higher
percentage of nitrogen than those made afterwards under more favorable
conditions. Hydrogen sulphide was not determined in the first 17 analyses.
Table S gives the percentage of each gas in the total volume of gas; and
the volume of each gas (at 0° and 760 mm. pressure) per unit-volume of rock.
Taslb 8.
1
1
1
1
1
^
^
-i
Sperlmini Ko. sod nmub.
^
1
1
u
ii
Ii
1
1
j
2
£
s
s
H
1
S
1. Meafum-emlnca vfhile BreniU!. mUier
P.ct.
1».20
4.SH
4.85
MM
7.00
100
low i» quaru.
Vol.
1.44
[pldgHr phenocrygla.
B. KMW»5n fcbia from Mcs.bi diBlriot,
100
p'ct
(KI.70
i33
.10
31 .M
.:|
8.66
Minn. Fn>m oldtrt known scrCea ol
Vol."
-.67
1104
LsliB Buperioc iMion, hlehly tarboii-
ated. (SiKTiraen 108%. sTide IWfifi, U.
8- G. 8.) Prom V. B, Van UIpb,
P. c(.
H.71
SM
2.8S
7I.S8
lUO
100
Gnelii Irom cod lact of Kcemtln greeo-
Vol.
3.08
Honeuid lAurentlan KnuilCcDurRat
ftirtKB, Out Fram C, R. Van BiK.
P.CL
B1.SS
1.31
100
rram acuart Weller.
.3&
10J3
B. Fortin ihalo. Iths™. N. Y. Prom Dr.
WSlcr. INKroKPii omltWd froio analy-
22.ra
liw
33!ki
100
i;<7
.83
va
3.80
it* becmnn of I«ikBKc of air Into tube.)
7. IIIUOOTlte, Canada^ nmltiial taken from
P. ct.
100
klMKC dab ol mica lu Walkcc Uiueum
8. Poumam mndstone, Bjusboo.WlB. Treat-
Vol.
«.70
.13
,01
JO
3S,X<
Vol."
.38
I.W
fl. Pik™"po^™K™iIiw, f™ ni^We-road
P. et
»,24
iTO
acre
Vol.
;37
M
Eiained. yullowJuh-pink gniifte cdhbII-
8ao?-««lDwl™lnkwr ■"iSiUo o" the
peak proper has bteii inlraded.
10. Rhyollle. Uarble Mounlsin on north
iilOT«« of San Fnuu'lsni Psak*, Arii.
P.ct
M.1B
IfilU
7H
6.S1
J6.60
100
Vol.
ioi
11. Keweenawan dfabaw, Irom 1 milp oM
P. Cl.
•■?
S.t3
78.14
6.or.
100
at IJriwer Junpilon, Foik Co., Wis. (a
M
.25
12. ftuarliilc scbiil, Follpfted 2 mileawralh-
wanolBaraboo. Wis,
.03
.J6
IS. AndeRllu. silver (,Tcek Ba«lQ.Oui»s- Co.,
Colo,
P. CL
Vol.
?)
se-M
7.ei
W.91
28.20
H. Coanw pon^m f"™ dump pUeoI Ban
Pedro MfncdursT Co., Colo.
■('I
Vol.
.05
.02
■^
.01
:m
TBB ANALYSES.
Tabli 8 — Oontjnued.
a|«slunNo.udnaufti.
P.ct.
onoL
H,8.
00^
™.
CH^
H,
H*
T^
U. Oftbbro dlorile Irora Bboul 13,000 tt cle™-
P.ct.
22.01
6.H
a.0B
«£.»
B.07
m
tloD on souili «ide a! M t, Hoedels. Colo,
Vol.
Aa
.10
.»<
MS
.14
1.S1
IB. CcsiB&greJaed gabbra dlorlle, Bummlt of
P. Et,
(')
loos
1.02
8.8S
100
•ec. % DolDlh, Hlnn. SpeeUnen 160^
BUaB«02,D.B.G.e- ThUtoonenlothe
,'4
izM
.12
.01
1.70
7t.'sfi
S.'V7
1.21
Vol
.13
jn
2.08
.at
t.«t
Bt Looto RIvra aeria (HMO tt thick)
wUcb b the lowot member Dl the Ke-
-H%JEBSs'
P.ct.
0.M
UM
AM
ST.SO
Vol
.41
m
L4&
^40
2.S0
hMdtodtmWedihale. '
"Ui MMdve emnite porphm Horw Race,
P.ct
Sl.»
10«
M
S4.1*
100
Vol
■.SI
•TS
iM
.22
t.8B
•n na&Kralnod gnelm. buqd rock-mu u
P.ct
M
KM
&n
.70
S.4S
100
„ r6.'l; JGESS.g'^.SSiS'"-
Vol
M
IM
.a
.02
2J0
DUVuNm. IDand^. Specimen 11(M.
SUde 5701, V. a ti. a. CollecUon 0. H.
P.ot.
M
14.7S
»
tiM
LIS
.08
LS7
J»
».V
wmumi.
Ulch. Bpedmen 14302. aOdiinH. from
P.ct.
M.11
20.15
100
Vol
LW
.IS
.74
ZM
C B. Vui HIM.
». SchlMirlttacbloillohLBUckHIlIi.e.Dia.
Docrlbed by Von Hte, Bull. 0. 8. A.,
P.ct.
n.ia
2.H
LIO
82.BS
1.K
100
VOL
M
.10
.04
.06
S.7J
__ BiienM of bslbolllhlc gianile.
P.ct.
S.IS
t.8i
S7J1
6.1U
100
Obtained Iram BliUie « Oo., nuuble
VOL
M
.10
.06
.01
.ss
.04
.66
Vol.
M
.00
M
3LU
2.90
5.48
.00
0.1S
2.00
.10
*:S
1:51
1.02
.02
100
'%
100
4.7S
n.9e
ZM
n.ii
4.S2
M
'v£
■ss
33
1.74
«fi.86
4JI2
iM
lOO
P. CL
M
le.oT
B.OT
ZZl
S7.SS
100
VoL
.00
M
jn
sa
.OS
M
.91
P.ct
.0«
S.M
tM
SLA
S.1S
100
Vol.
.19
l.W
.oe
2.n
■^rS*SS-r^:S
P.CL
n.w
11.41
i.a
40.02
t.as
100
VoL
....
.00
.87
.70
SkS^^'^=^
i; Fc^Oh 8.001 FeO,4.«4jOBO,ej!2;HgO,U«;Na,0,tJS:
I Carbonated.
>No.l7. An&lfiila(byBtn»iK): B10b49JS; AliOkll
E,a 1.81: lW).l-Mi 100.80. ^^
•Nal9. Aiuayili(^BLBnILei,p.lU): B10i,M.8S; A1,0» 26.40; re,0).l.tti hO, L6B i CW), 8.W ; HsO.LSS;
,0.s.ee: KM.i
n.l8:00,,J
m collected fr
D Inlnnlria of giuilte-gneln lata fcraembme, which hu been odled
■jupa iiuiuuBn br BiDOk^ the gimnilfrfnelM beliw placed at the lop of the Upper ElUDiilan. BnlletlD 82. p. 2S.
TtuavrisihowasgmdatloDframmaalTepoiphyTl&Knnlte loTcr; fli]»-baiMle(IaiKlr~ "- "••-• ■■ •—
o[ the Intmdre dike. 4adnB(teialmoAliItpe—*^' "--"■ -■-■ — '---
t™. No.21i««lmllartoNo.aO,boln)tooUl .,
tieuigmore like ftdU)rlte In comixieltlaa. nM low gea-Tdiime of No. 20 In oomparlnn with the two ot
mem Is peihape lo be explained by tlu bet that It & a much mora ponxu rock, cmmbUDK (MuUI; to a flne poinlM
Te poqthyrlilc Kianlte lo rer; floe-buMled apelM. Ho. 19 l> fnm the oenler
bitp«TOptIbIr on both aide* Into the IbMMTalned snelM of which No. 20 tt the
DO the ddei, bat to banded. Thto rack & more Baiic than trplcal gianiic,
It&ami
«iof CHb^and
M Utile N, pneent.
THE QABB8 IN BOCKS.
Tablb 8 — Ctmtinued.
or'Toi.
H*
CO,
00.
CH..
H,.
»►
TdW.
P.ot.
007
87 JO
LW
a«a
».71
Loe
100
lesIon. apeclmsn luee.SUdaG74f. Db-
Vol,
'.(a
XI.07
JH
.ao
Uil
J3
a.7s
BCrtbed bj Wllltams. Bull, ffit, pp, ea-7fl.
Ad eilremelr Kllered gabbro, ooolaln-
ins calclte. teilclte, cblorlU, uid lao-
Mich. Specimen UTlTWleBSei.D. a
P.ot.
07
2S.S1
8.0J
ZK
5B.sa
1^
100
Vol
:oo
.SG
.15
.07
i.n
2.W
O.8. rrom C. R. Van HIM.
P.M.
.N
a2.S8
180
■US
L7S
g?SS"SrES'A'SS.'»;
Vol.
.18
aD.08
LIS
ioB
10.10
jn
SZOB
elcen nil ifIilh i.anvJor nns hesUsL
i-rom C, B, VBn illsB.
P.Ct
M
T.C6
LM
1.77
87.88
IJO
100
..oS£2SMf».n|
Vol.
.01
.»
.08
M
1.18
JM
XM
P.Ct.
v.m
cut
100
Vol.
XO
.07
M
"•°e.W£."¥SsK.Sri°-
P.Ct
.07
*708
*.7*
L57
UM
ILli
100
Vol.
.CO
i!u
LOS
1«8
P.Ct.
.10
10.88
4.85
1.66
77.20
5J7
100
DeKalbOa.,a& Fram & W. UcCalllc.
Vol.
.00
M
.oe
.tn
.80
M
.71
IB. Omnc reddish Archeu Kiaalle. BIk
P.Ct.
1.10
1.81
I.BI
100
fssjferi.°rjS5..«ss
VOL
M
1.M
ottboclaJie form largecry91s.lt: queru
!■ abundant, but bintlte Is mther spubc.
DeKrlbedlnGoal.ofUinn.,vi>l.&,p.H14.
lb. OrtonTllle Kmnltc, bintlie crystnln u[ loat
P.cL
B3.40
iM
LtO
100
epwlmen stpanited (rom (luanx oiid
gravicj' solution.
ii«
jn
M
.a
1U7
1
1
i
1
S
1
P
1
1
|2
r
11
i
1
i
1
1
1
1
iL Bftmllton ihale, Nemom's euUon,
P,ct
80.M
21.17
4,r.i
f.SB
100
IB mile* W. ol IfuhTlUe, Tenn.
Proni um AugoaU T. Hinlock.
Vol
19.38
:m.iq
1.38
3j!o3
M.86
g^fyffikss^
Kttll; eilned wi&ln the rock In
tbe Eueooi etate can not be staled :
mp^trf It probeblj came from de-
ent Heavy brown tan aim pro-
149. Oll-rw* from !!Sd-iInc mine n««
P.M.
ON
mis
iM
4.00
S6.98
11,1;
2.18
PlatWrtllcWta. From H.F. Bain.
Vol
S.M
e'is
10.1!
*-H
S.!iO
fi7.«
Quebec PminP. D. Adanu.
P.rt
M.Vi
!fl1
ijl
100
i78
aioE
:o9
.06
.07
6M
Fe,0,&jei0i0,UGiUS0,llJ«iHa,0.2M: EALM;
1K0.S3. All«]nl«:810t,«BJ)6: JU,Ot.3t.7S:raO,B.«5; Fe,0
HA7.SS; COhO.«; 100.% (R. B. Rlgn BiiU.83,p. 78.)
) Probably onlf » muUI put of pi wH leallr irannt In the I
what • bltmntnoui ihale may yield.
■Ametamorpluaeditotaof tbeOienTtlleieilea. BefeieiiM: Am. Jonr. BcL, rol. EO,p.SS. SeacrlbediaaflDa-
Bimlned nrnetlferoai allllmarlte gneia containing mneh qoafti and orthoolaae. Graphfte and pvrlte ftlao pnent,
MUlnB the gnelM to weather to » Very mitr oolor. Analyil* by V. V. Bnnt, ot UoOlIl OnlvenUr : So. U. BtO^
aiM;Tlp„!.06; AlfO>,J9.n |_Pe,0| and FeO, LOO ; Fed* Ut ; XnO, tiWM : OaO.OJSi KgO.LSIi ilifi,Q.n;Kfi,
SelM to weather to a very
8: AlfOt, 19.731 Pr " "'
iM 1 BiO Xlgnicko). 1.82 1 WM.
THE ANALTSES.
Table 8— Continued.
U. Nephelite (yenite, north ol HouDtBln Lake,
Tp. of Uethuen. Ontulo. F. t>. Adaitu.
ol Igneona origin.
4S. Inm-bwrluB buKlt, OvUak, DlMo Iiluid,
GreoDluia.
«. lUgnetlte nad Snaka River bed. Idaho.
trnpuilllea b; magDet.
47. Andcdle, Bod UonnMio, HW. ol San Fnn-
daoo PealoL Ariz. OcillMted b; W. W.
Atirood. lb. Arlliiir Tk^lor dacrlbei
thlt took u ui andcalte porphRr wbon
■Toinid-mis {H p. ot. ol ue wbole) oon-
lliti of 48 p. ct. pjTOzaie. n p. ot pUglo-
claav (tobnuioctte), and U p. ct mu-
nKlta. Tha phmooiTiti an llnw^ou
fddqiH, angfta, bonibleode, and mas-
nstlte. Oocnn aa Inegnlai blocki ee-
UMiied In ttn tuff or TOlnnto biMcia ol
which Red Honntaln la bnllt.
IS. FnraraiB cr7aUl& Red Ifonntaln, Arii.
Collected by W. W. Atwood. Tbese
cryat&ti, noglng In ilie bum ■ beui lo a,
■mall marble, were louod looae on Ihe
-. _....iangle, Colo. Spec
Ho. UH V7&. Q. a. A kv& flow bolong-
' r to tha Inleimiediale seiica. From
). KccACUt
dark^^soined nwk, with tbe
, —Tlalble to the naked, eye.
ConUtna UHle or no leldipai and eom-
jandlTel]' little nebhellne. Olivine and
•Dills an the mon important romtltii-
enU. The pnaenee of mellllte IndlcaCea
'amoiiDtoI Ikoelnmapn*. Age
k ol Shonkln
(Ft Bentoo Folio, p. 1} m » nwk at —
■yeolle familr Terr iloh In auslle, con-
talntng aooeiMr* oIlTbM and blMk mlea.
White the chief lJabt«ol(ired aoutltDent
Ii orthoelaae, nepnellta and aoOallte are
aga. Collected
dunbtlDlI} pi
ceant In TmrrlDK mnnm
'e of proliabir Eocene
W. a. Weed.
I. Thenllte, frotn tbe laccolltee on Upper
Bhleld River RuId, Craiy Mountafue.
Mont. U. S. HBtloQBl Museum Ko.TaiSK.
Collected bf W. H. Weed. Described
(Uttla Belt UouDtaina Fidlo, p. 4) aa a
datk-frav taaaltio lock eonunon] j occoi-
tlDg in abEal*, bat mtlj In dike*. Poi^
ph^Mc 1 1 J ilala of anglte fonn the moat
proainent phenooryM, atoagh large
plalaa of brown mlea an oommon. Ool-
oilen pan D( the ground-maM I> a gnnu<
lar miiton of nepbellte — ' " ■"-
H. Qnuti lyenlte porphyry, amnmll of En-
flnaet Honnlaln, Bllverton Quadmiiglc,
C^ No. ST3S V. S. O. 8. From Dr.
Deacilbed (SUTBrton Folio, p. 11)
1^
THE OASES IN BOCKS.
Table 8— Continued,
Specimen Ho. and rmatka.
P. ct
orToL
H^.
OOt
00.
CH»
B.,
N^
T^tal.
Eltber an intr^Te or i«(dng apon anda»
KIdgn. Alteration doe to Intnuloo of
joggtj^(8e.«i»ly.laNo.8i.) From
». OviMlleraM inelM. Dwwin'i Faiia, Tp,
of Hawdeo^ebe^nobablTcrf ie<fi-
EC nS^Srade nenile. Cape' Oi^betb,' He.
talni nnmenrai &r£ qnarta crntels and
dear uldlnea. embedded In alight gny
'ySt
F.CT.
P, ct.
Vol.
0.M
IS.IJ
.04
P)
O
"1
"3
,07
2Z.U
sill
»,41
.01
.oa
1.IB
.IX
M
4.M
.M
.01
.to
SZJO
O0.ST
7.M
-OS
2.(6
.03
1.3S
'S
1L28
.at
.08
i"
lOO
.27
.„...».. „,™„.
1
1
i
e
h
|l
1
L
js
IE
ga
i
1
1
1
forks ol Deep Creek on lidge from Mt.
Boffner, a mifa from lower contact, TeU-
urldc QoadrauBle, Colo. No. 2SK U. a
p. cL
p. ct.
Vol.
W;
P, ct.
.08
Z
.02
M
'.m
.M
:22
.00
•Siii
.M
(i)
70.39
.'A
:o6
.«
.06
,11
,'u
.51
1
IISJ
1.10
s,i;
"3
i7.i;
.06
1.88
100
;%
loo'
.go
4.8S
lOO'
.00
62. Andealle. Kodta Hills, Cutter Co.. Colo..
from summit of small hill east at spring
on ROBiU Road. Dike faciei oC Fifnglo
from CherUey, Quebec. F. D. Adanu.
6*. Aodcslte, Uparl Islands.
Harburn Tp., Ontario. Dr. Adams re-
gards this as almost cettalnly an altered
Sediment, lor It occurs In beds inleiBtratl-
pyille and conlalna giaphlie, Aaalyaia
showslttobeatehamose-
06. Gnel« ol igneous origin lot as, R. 9.
WollasloQ Yp-, OnUrio. F. D. Adams,
ton Tp, dniario. Probably ol eedimea-
loty origin. F.D. Adanu.
•s
ulis
.03
,oa
M.3
-1
tist
.06
K*iO. B.OT : HtO. o.es ; FfOi, ai
THE ANALYSES.
Tabu 8 — Continuml.
No. uid rfDurki.
eg. Amphibolic. U&Tirell'i CttMiag, Glamor-
gKH Tp., OnlHrio. A hlgblr altwed llmc-
Mone. Dr. Aduiu.
t. Blcc iwk, CuudMn Pid£c R. R. 0,2S mile
■ul at 8iidbiU7, Ont&iio. Probkbly im
kltand NdliaaDt ol Hun>nI*D age. Dr.
TV. Andedle. QimDlie 1
Utah. AbKcollteiii
CE.LetUi.
71. Vdn qoaiU, Oiuille UonnMln. Iron Co,
VXtiL Aaaodued wltb Iron ore whlcl
bw dereloped mloag the contact of tbi
UeeoUtauMlntnideaUmeitoiie. Aoooid
Ins to Lelth It tni premcubly decided
fioD the andMlle (Na 70).
L PbODoIlIetnch^te, cut DUtot Bull Honn-
UlD, PIke'B rtmk Qnubuigle. Colo. Ka
ausu.ao.a OcKrlbedbyCnwIFlke'i
Fenk Folio, p. S) u ■ demtk gnildKTeen
iDck with tabol&r orxau ol Hnldliie,
which give it B typical potptyritle t«x-
74. Dlotllk Dtet Ue fheet. FeoobMOl Bov
- ■ " No. 11S7. From E. 8.
. _. , T lanow* of ihe
Bmboo, AblanuiB, Wli. Hard, tndu-
nted while nnditone, not lu from Ihe
contact with the qnaittlls.
■7$. Patadam BaudMone, Ablemani. WlA
S. Fennbui red nndnone, OaMen ol the
Ooda. Mar Oolontdo CIV, Colo.
). Topu qi — ' ■■ — °-'- —
X Bt. PaUc ouidatoDe, Minn.h.h. Cieek. be-
low the lall^ MtDDOipiiU^ Hlim. A Kjfl.
roDBrtatblT while, coanefnlned nnd-
HDoe. bAw heallns in the tube the
I. Bt-PMerandtfone, Ulnneliaha. The sand
Na SO, pvlmlKd and heated a second
82. ftoartilW. taloDglni
urdon Tp., Queb
Indkateil that tlier «ero fonned from
aqueota aolatbm. Eatlrely tmniparent
and wlIboQt Tldble luclaHona.
- I, Indian TiaU Bldse, U Plata
'e.0olo.(aabeetlnl' -
nanandfemle mloenli, aod mora aoclte
than hornblende : feldapara much Bericl-
tlied BDd obecuied by chlorite, epldole.
which WBi DuMe to ocape might not 11111 remain within tti
ji tbe (reah laud, moit of the gaa In the Snt trial apparently c*
THE QABEB IN BOCKB.
TablzS
■On
Itl
nw
)d
F.et
B*
oo»
00.
CH..
H>.
Mf
<V
■MA
too. Hiob. Ftom A, C. lAne. TbSt
pleoe Game from a depth of G24 feet. mMB-
^^ alonsa^ 4^^tlui >>°^>^
a»elmeD ai Noa. 88 and 87.
P.ct
Vol.
i
p. Cl.
.M
.00
i3M
1.81
«.»
87.07
38.80
2.10
.W
a.47
oioi
•:S
1S.M
M
IT.U
2.98
1.33
J
.01
2.48
I,S7
.00
4.fi2
.01
0O.»
XB4
.00
8.M
.01
42.98
2.60
88.98
M
'S
"1
liio
.«
Uh2S
1.78
.10
7.23
100
Jl
<
•^'SSSSSIS
SpHtmeu No. and nmarki.
1
i
1
P
li
6
1
f
X
i
i
1
1
9S. PiWhblends. Brnver Co.. Colo, From
'94. CarnoU'te, Coiomdo, from H. N. McCoy.
Uranyl vanadate, probably with some
From C. K, teitb.
H. OraneHU rook, Mvwbi dMrlpt, Uinii. A
From Pr. LellH.
97. Mi<«tKiu» guartiiw. I'lnta MtB.. Utah.
rrom Dr. t^iih.
P.CL
vSi.
0-00
.17
....
:67
1
.23
.24
.12
2
H
78
«
8!
:27
!23
2.4fl
6.a
38.48
IM
Z"
8.02
100
6,»
100
100
1.42
_ . le from bydrocarboni (oinble In alcohol or etber, thla material, after
being verr DnelT pulrerlted, waa dlgeated with alcohol (free from organic Imporlttea) for 20 houn : then with tat-
free etlier IOT4t houit. It wai then thoraughly waihed with ethet on a Alter which bad prerlouily beea treated
Ih the aame tat-free etber. Afterward! dried at 100° Id aD oveo.
'To get rid ofallcarbonatea the powder wBi treated with Goncenlntednitrfaaold for 88 httan. Unchgaawaa
renoB. including a eopiona evolution ol nitric oxide. The powder waewaebed until all ti ■ --■'
.—. .^ 1....... '-)d In an air-bath at 1180. This malerlal he ~ ' "" ""
lUrte add (or three days In a vacuum. Thi. , —
rbid by barium chloride, and then dried In aD oi
u maintained between tbeee two oombnrtloni.
THE ANALYSES.
Tablb 8 — Continued.
se. Quartiite, tUb HIU, ccai WBaaan, Wle.
From ekmuGl WDldmBn. Famoua for
IB s" babbles. Powdered on sii aa-
vU umI meuillli: iron Ihui introiluoed
removed aa auDpletelr u poolble wlib
199. QiBLrdle, Rib Hill. WIb. game rpeelmen
uNo. W. GTanules uwil lasuso ol tine
powder. CnuDbleH rend ily Into grao 111 efl,
vhicb were tresled wiUi boHlng by-
dnchlorfai acid Eo iBmove uiy iron
wblcb mlsht IwTe oome (rom tbeinvil.
UO. AmUwon^PTi l tllMoninnartt, Cmgo. neM
OtwueTNAW amtti AloL No.^N. B.
W.; Jlo.liaiU. o( W. Dr. Ulth. Re-
dnMd to ■ eouM pcwder on an ■nrll
■Dd tmtad Willi bolllDg dilate hydro-
cblorloacld wbleh removed — '— ■-
' " ' '"it being eareti
u toorv Hiiely :
lOL Bay) faom penuUle dike. New EngUnd,
WUksr HnraiD CoUecUon. UMerlal
far tlita mnaljvii t&kcn from ■ tnavlve
^trfl, g bwbM In dluKiter, u tranrau-
■DtHwliidow.BlBBuid witboat Tlnble
InfflnJmnoi luumritla. InRtaad ol be-
liiC pnlTeriMd ae mMetU wu mod In
tte Km ol amBll tnwmenta wbleh were
wadiadwtth bolllDgliydroobloilc acid.
Alter beatfog, the Inaapareut Ing-
lOIo. Am 7t b^Swied to eipet all the gaa,
the Taenam wu nulnlalned oTcniigbl,
and the material healed 4 hours maie
oexldaT.
coDtalnlDK many eryitals
- "heaterfleld, Uaffi. Wall
Walker
1C8. Qoarti Inno pegmatite vebi, Jonaa Fall).
Baltbnora, uZ Ka. am, CqIt. ol Wli.
aoDeotloii. Dr. Leltb. medmen con-
talned ■ lane eintal ol mlciaolfiie over
3 tncbea In lenstli. Indicating oondltknii
favotoble to crntalllialloiL Nomlcro-
eUne mm used tot the analyah, howcTer.
II pGomatlte* contain mocb nt. It wai
tboiiKbl this one ahould yield a good
Tolume.
IM. Albite, Olbb'i mica mine, Yancey Co., N,
Ids. Quarta from Miocene lava. Iron Co., Ulah.
No.*661*U. S.U.S. From E. C. Hanler.
lot. Allegan meteoiile, a Btony aerolite which
ItS at Allegan. Hlcta., /ntr 10, ISW, and
waa dog oat of the aand eaXl hot, within
t mlnntea of Ita (all. From tJ. 8.
XaHoul MDamm thraogb Q. F. Hei^
ML DtMrlbed bf HcnOl and eiokee
(?>[iB.WaaliiD>tan Aoademy ol Sdeooea,
vol. », ML n-a)- Before etttiaatlnK
lliegM,ine imwdered meteoric material
wu heated in a vacuam at lJin° 'nr 3
boon In the presence of phi
ehcm^llvfomhli
ed
n to enable the dry-
rb all moliture not
■TheToIomeof carbon dioxide being much neater In the case of the giannlea Chan In the finely induced
powder atinngly niggeats that much of this gu 1* mechanically incloaed in caviUe* within the quartiite, and
OKBiiei when the gnnulea aie putrerited. Tbla nilninn waa itrenglhened by a alight cncking noiw which
eame fmm within the tube aa aoon u heat wu applied. The gu came o9 wltb a nub wlien the tube wu heated.
'Ko.108. Chemical analnia by Dr. H. N. Stokes: MetalilcpArt, ,!S.<Mpercent.,ufollowa: Fe,21.0ll; Cii,,Dl;
in.lA;Co,.U. Btiony part, n.M pet cent., u follow! : SIO,, M.K : TIC^.IH ; PA,. 17; jLWt.2M; Ctfi^ .69: FeO.
t-tl; r^&M; MnO, .Q; CaO, I.TS; UgO. 2t.W; KtO, .23; Nb,0, .M; IV> at Ol?, JM; above 110°, Jt; lOO.OOv
THE GASES IN ROCKS.
Table 8 — Oonoludnd.
epKl<HIlMO.IUldt*B»ri»
«^.
H^
SO,.
CO,,
CQ.
CH^
Hf.
Bf
WH»
T»U
■107. Eitacado metoorlle. fell near Eetacado,
Tex.,lal8Ki. De«!rlh«lbyK.B.Ho*»rd
(Aid. Jout. Scl., vol. 2i (l4oe), pp. 65-60).
Kept iTi K Tacuum at onlluary iewpen-
P.ct.
«.»
aa.<T
at.si
8.W
8ej»
LN
WO
Vol.
.00
M
.n
.01
.u
houn: then bated U Iso" lor 6 hoon.
and BUawed to remHn In vmouo (or 19
(ramTolQW. Men™.
P.oL
M
4i.Hl
iB.ii
l.tt
..-,
Vol.
.00
10JS7
H,fl7
4,»
M.43
SS.W
VOL
'if
a,J5
lim
.33
i Third detannioBUon
100*
LBS
m
'.oa
L3i
jn
*;S
"109. Irotiorp, Iron Co,, Utah. No, , U, 3 G 6
kud ■□Iphsle :
Flnl portion (gu whloh oune off In
IP.cL
fil.S5
a-M
Z1.3&
.m
too
MmlnQioi).
Second portion jgM which cameoff In
Third portion (material be«W3honni
Vol.
'vSt
Pet.
JH.J6
io.a
4.47
W.12
83,28
.IB
ZLB6
100
5Sl :"
.33
■OB
.40
more).
111). Basaltic IbtB. Rllanea, Hawaii, CaKcade
Vol.
P.ct.
m
1.7a
70.12
21.41
S.»
2,49
2.01
~It»
oflHW. A rather poiou* U\-», having a
Vol.
.60
,JS
.02
.»
.a ,S»iS»SS,u„.™.
P.cL
B.IM
2.S3
LH
of April. 1306, ooUected by F. B, Taylor.
Vol.
.03
:!i
,05
,01
:oi
.01
.4*
Uarcb !», IKT. From •oath aide of a
quarrrintiDntor the ohurch In Boko
Bpeclinen cune from 10 ft. below the lop
and ^ IL fram the bottom of Ihe Bow.
which had been blamed at tbla point.
'lUL Frenh lava, Vauvliu ume flow »i last:
p.ct
1S.U
a.*t
7.M
3.S3
LEO
l.tl
0.5 kUomtWj BE, oi church where No.
.05
.01
.01
Ill was collected. About 8 ft below
iiiirrace and 2 II from bottom ol bed.
Collected March 30. 1907, by F. Taylor.
1
'No. 107. Chemical analysli by J. U. DaviM>n : re.14.S8; Nl. l.SO; Co. .!«; 8,1.37: P, -15; BIOi, 3&.B1: FeO.
16.63; Uia 22.74: UiO,3.M; Alk)., S.«0; Na,0, 2.07: K,0, ,32 ; 100.96,
* HO. 108. Average ol IS anjayaea compiled by Fairiiiglon ( Pub. Field Colombian Uiueam No. 130, pp. 82-84):
Pe, W.U; Nl.7.90:^, 0.<S; P, oi4: B 0.14; SI, 0.02; HlK., 0.67: W.16.
'lion bonngiand fUlnnwere uaed, butwftb cheae there waa included a little nut, wblch adhered to themeUl
when It wai wluidrawn wiih a magnet.
« The ma,terlal naed In tbla delerminatkin coDBlited of bright borlngi csrctollT (reed frotn rtut > pocket of which
■—*»■""•-'- -^- —."«'"*"«• 4.. j-iiu— n..f (.. ...i*....#»....t. ■»» —..-'■-^ 1q diawliu oat tike metallic borings
._ .^ , ,..th themetu In the ccmboMlon tube.
d to much leas than In the fliM detetmlnatlon. Mo flUnga were uaed.
iHoa boringi from the inleilor of the apedmen weracairefnlljrnadeby Wm. Gaectoar A Co..
Botemiao-hutniment maken. There waa no vUble nut adhering to theae boifnga, which were then woAed twice,
with a magnet, wlthoat any fmpuritlei bedDg left behind. The white paper on wnkh thli (iteration wae perlonned
faOedloBhow the illghleat dlaooloiBllan, aach al It bad done In the two pnvloua detennlnatloni. Anuoal the
material waa healed at 100° for S houn, m the preaence of PfOt. and then allowed to Mand In the —"■••'" M_i-h>
.. . ,_. — ,«. — » .^. wai perfect at the «id of thlattme, Che flrattw., —
olvea when heat waa applied, were not kept, dnca
. at the tnd of thlattme, Che Brat two pnmplngi ol gaa
. >■— waa applied, were not kept, dnca It wH dstrad
lranend<Hu dect thepraaenccol a little lion
Aeompailaon M the , _- ,
nut will have upon the gaota emdved from k metallic meteorite. Much of thla gaa la doubtlen derived
caibonale and the hydimted oxide of Iron, aa will be explained under the topic of gaa due to cbemlcal
Oimt care la therefore neceMur in making gaa analjili ol Iron meteorila to avoid an* contamination ol nuk
• The men atriklng leatnre ol theee analyaea Ii the nnnaual amount of aulphur dioxide, which Indicates ai
Olldlied condition ol the ore,
' The odor of anJphur dioxide wia very prominent In Ihe gaa obtained bom then two Veauvlan lavaa.
THE ANALT8E6.
GROUPINGS AND CLASSIFICATIONS OF ANALYSES.
As the volumes and relative proportions of the gases found in the fore-
going analyses vary within wide limits, the nature of this variation can
best be shown by grouping the results. To make these tables as complete
as possible, not only the results of the present studies, but all the available
analyses of other investigators, have been included in the lists. Except
In the case of four of the five analyses by Tllden, relative to which suffi-
cient data are not ^ven, all of the figures in these tables refer to volumes
of gas per volume of rock. Previous investigators have usually given the
total volume of gas and the percentages of each constituent. From these
I have calculated the volumes for each individual gaa.
Tabu e.—AnoJyiM elaasified by gnmpi
ofnxkt.
No.
19
X
31
i
%
s
s
s
u
»
Buck nai loulltj.
H,S. CO,.
-r--
a,.
%
Tool
AuItM.
OraniUiand gneUm r/ ignema orisin.
*i8
4.60
1
ts
1
1
.23
':S
.06
.11
3.02
•i
iio
s
.03
3
.03
1.60
■s
.38
1
.96
S.1S
:»
!32
.08
ioa
.'(*
.04
1
i
.u
11
3.'3§
.76
Tilden.
Oftutler.
ChiSberiln.
Do
Do
Do
1
S
Sgj>nn.
i
Do
1
■nidon.
0*titier.
Do
11 w
Sse^arS-SS,,::::;:::::
It
tr
M
OslnbT Hub snnlH, oeorgU
Blone HoudC^ gTBuIte, OeoiglK
OnonrUle e»nlte, lUnn
OmiiitB porphjTT bowlder. New York. . . .
Average of IS uukljMi , . ,
■newaniuarcmp.
NephoUle irenile, OnlArio
abonkln[te. Hlgbwood Uountkloa, Hoot.
lasSSS'ia?'"-"::::;::
ir;
1^
0.29
O.Cft
:o5
1,SS
2.22
H-tA
sa
.03
"1
The oalibTOJlionte ffronp.
8S£S:li?3-^"""' *^ ":!:::::::::
~^
J
1
a.ifl
'.0!
'.oi
;ao
:os
.91
1:S
A
1
1.10
1
.OS
1.3S
:st
3.2e
XI
1.78
SffiSiS^S^u'Sf*^.:::::;;:::
,«7
SfjS«Sf-':^
Dlorlle,Paiol»ootB»T.Me.
S0.08
'■&
O.S0
a.Bn
11
.11
.1»
.03
4.7n
DIabaKiandbatalU.
iS
Keireenswan ai«b»e, Wtooooiln
THB QAfiBB IN BOCXB.
Tasia 9. — Analyf daarified by group* o/ rodcM. — Oontiiuwd.
Mo.
BockMdloclil^.
a^,
06,.
OD.
™,.
Hf
»►
ToBL
k,miv^
1
U
m
13
i
<g
se
J
71
89
lOD
ffi
90
6B
W
W
mibtun oRil doHlte-Cont.
CM
s
<l.25
1
1
1
■.All
3.15
3.M
LOB
.02
.01
.01
cot
■is
!06
.06
.m
M
IS!
Lao
S.BS
J6
.42
.62
Do
Co
S
Do
Do
Do
1
Do
a,«nb-lln.
CtambecUn.
Do
TUd(m.
g
Do
Do
Do
Do
Do
S
caarnVSi
tr.
S
.H
.Is
.21
1
1
.01
:!i
.00
.OB
:s3
.08
.16
!oz
.06
.06
.01
7^
J
AnOtiOa.
AndcHlle, Red Uomttaln, AHl
It.
tr!
tr.
AodMlt^Oi^M Mountain. Dt.li
Pboi»illHtncbrte.I1ka'iPnfe
An(l(Mlte.nmuiiltQlCMiaba
ATcncool Taoalrm
Rhjollle, Marble Mountain, ArlMO* . . .
RbiroUle Tltropbyre. Telluride
SehltU.
1.M
.22
.07
.IS
.OB
.06
.08
.01
:g
.OS
.27
.13
.01
.01
=
.ra
.on
!io
^
li;
S
.87
1109
a.73
GSTb.
.S2
.06
.06
S
.a
■H
7J7
.BO
.n
OMim porphyry, Onray Co., Colo.
Topai qiuirt»porpli]TT. Saxmj
.12
.07
1
.09
.12
.00
.OS
■1
tr.
.05
.(B
M
'.e&
Smokf qaarti. Btancbvllle, COQQ.
I
StsiSStS&S'iST':
.oi
as
.la
1.S5
:!!
carb.
!,!
a
la
:o9
.16
.07
t
o.Se
.01
s
M
I2.«
S1.9S
l.SO
.06
.02
Lia
!04
.10
.04
.OS
.03
.07
.H
.02
.09
.06
■«
_M
n.N
M.tB
1.73
6.10
2!39
•:i
.79
4J8
3.22
Ppaxo^fnelH, Cef km (p.ct.)
Alt«TedJDnUidclliale,ColDT«<]o
*tr.'
-SO
II.
tl.
Si
a
tr.
IS
'
1
J2
IS
.BT
=
=
.05
1^
.06
3^
THE ANALTBEB.
Table 0. — Ano^Ms doMifUd by gnrnp* of roeka. — Concluded.
IS
i
u
a
s
BockindlDsllI;.
"•■■
CO..
CO.
CHh.
H.-
N..
Tuna,
AulTO.
SAoItl.i
....
10.48
'.US
11
0.23
^
4S.6S
dumberlln.
Do
1
SSBS^::;;:::
1:S
Avenweof flret3«nalT»M
SuiieHlut,paiTdBradmnd aiteUMi'.'.'.'.
ti.'
to.
tr.
tr.
tr.
1
376
1
M
.18
M
'.U
n
-01
Jil
.23
.24
1
)1
.84
s
^«P^;::
.l!M
.11
.17
.»
Table 10. — Variou* minerab.
Ko.
Hluml and locality.
H..3.
CO,.
m. Bt
CH,.
N,.
H.t
TO--
Auirn.
i
so
s
IM
nbtanr
•1
■s
am
12,45
if
.06
]!33
1
0.01
.04
.OS
.01
:<K
.OS
1
S
M
1
.17
1.27
2!eo
S.flS
L77
i
6.W
"1
:o7
Dewu.
Do
Do
Klwhln Md
S
1
Do
Chloriu, ZopUn, MonTia
!u
FcldjpM, FeteihtBdEmiilU ....
!2S
.18
'.IS
.24
:o«
-10
1
.06
i;45
KsrsffiSStiiia.
CL32
iml'.
1
tr!
PltrtW«n<l^Be»TBrCo?, a>]i^'.'.
.02
IJd
.ig
.M
M
.ce
.03
3.18
It should, perhaps, be stated that in making this and other averages
of aiial3'Be8, in those cases where, on account of excessive carbonation, no
figures are ^ven for carbon dioxide, the average amount of this gas cal-
culated from the other analjnses ia assumed to be present in those rocks
marked "carbonated." This addition is added to the average total and
makes this figure slightly greater than the average of the column which it
foots. The same method has been used for carbon monoxide in the three of
Travers's analyses where carbon monoxide and hydrogen are put together.
THE QABE8 IN BOCKS.
Tabu II.— Sloi^r «
K..
Meltarifa!.
H^
t-o.
cx>.
CH,.
!»►
N,.
T.»,.
A.-I5*.
IDS
107
1.80
toe
ti
i.ia
.OS
its
.IB
.as
.06
a
■fa
.20
1
.oa
.so
.0«
1
in
.OS
.09
-SS
tr.
198
l.Tt
Its
3.49
if
67.87
.49
.M
8,80
T
Do
onroeu 1
Allegan, Mfeh
b.
4.00
Avenge of 19 uuljni
The figures for the Orguell meteorite which yielded euch a remarkable
amount of aulphur dioxide make the average for the sulphur gases an abnor-
mal one. The presence of this gas in quantity must mean that the meteor-
ite has suffered much from weathering and oxidation subsequent to its
fall. Considerable troilite has passed into iron sulphate which has been
decomposed by the heat of the combustion-furnace.
Omitting the sulphur dioxide of this specimen, the average total volume
of gas from stony meteorites is reduced to 4.80 times the volume of the
meteoritic material.
Table 12.— 7
N».
Mctaorlts.
11^.
(.X),.
CO.
CH.,
H,.
N.
Toul,
An.l/«.
10»
I.™.rtr.
1
%
IJS
?1
1
,10
2.8G
1,85
1.
.S3
I'S' "m
T»
,7a
L«
:"
ZSB
A»ai»ge onjiling Arvii QiBleo'rite
,67 1 M
1 Flight, PMl.TtMl». So
17a (
18«).
yptat-
4Mu>
dp.M
«.
Methane was determined in only two of these analyses. In these two
it averaged 0.10 volume; but in order to make the figures consistent in the
table, it was necessary to average these as if the eight other meteorites
yielded no marsh-gas, though it is highly probable that this gas was present
and has been included in the figures given for hydrogen.
The unusual amount of gas from the Arva specimen recalls the be-
havior of the Toluca meteorite,' which, at the first attempt, produced 24.42
volumes of gas, owing to the presence of a small quantity of iron rust, but
whose pure metal evolved only 1.85 volumes. An average, omitting the
Arva, is therefore made.
> Ante, p. 22.
THE ANALYSES.
27
AVERAQES OF THE GROUPS.
Table 13. — Igneous rocks.
Order.
1
2
S
4
b
6
7
8
Type of rock.
Basic schists
DIabMes and basalts
Gabbroe and diorites
Oranitet and gneisses —
Andesltes
Syenites
Rhyolites
Miscellaneoas poipliyries
No. of
analy-
2
14
11
19
7
4
4
2
HsS.
0.00
.19
.02
.00
.00
.00
.00
.00
COs.
4.06
8.96
2.31
1.47
1.86
.18
.69
.32
CO.
0.19
.44
.13
.22
.18
.07
.06
.06
CH^.
0.06
.12
.07
.05
.06
.05
.02
.04
3.44
2.54
2.09
1.36
.20
.91
.06
.33
N,.
0.13
.11
.11
.09
.09
.04
.06
.04
Total.
7.87
7.36
4.73
3.19
2.39
1.26
.87
.79
The general averages bring out the fact that, while rocks of each group
may vary considerably among themselves, each group as a whole fits into
a logical place in relation to the other groups. The established order
appears to be, most gas from those rocks which contain the greatest pro-
portion of ferromagnesian minerals. Though much influenced by other
conditions, such as relative age and nature of the igneous mass, the general
deduction may be made that the volume of gas obtained from rocks
varies, in a rough way, in proportion to the percentage of ferromagnesian
minerals present. Diabases, basalts, and basic schists take first rank in
the quantity of gas evolved. Next to them appear diorites and gabbros
which are also near the basic end, but formed under different conditions.
Andesites are out of their place in this list, as they take precedence over
granites in the proportion of ferromagnesian minerals, but these andesites
were all either of Tertiary or Recent age, whereas most of the granites came
from Pre-Cambrian formations, and, as the next table will show, ancient
igneous rocks yield more gas than modern ones. The rhyolites, which com-
bine a scarcity of basic minerals with Tertiary age, foot the list.
It is to be noted that the rank of a type of rock on the basis of an
individual gas does not in all cases correspond to its rank for some other
gas, or in respect to total volumes. The andesites tested gave more carbon
dioxide than either the granites or the syenites, though both of these types
greatly surpassed the andesites in the matter of hydrogen. But this in-
volves another factor: in deep-seated rocks, hydrogen and carbon dioxide
are of about equal importance; in surface flows, carbon dioxide predomi-
nates. Though carbon monoxide and methane are somewhat variable,
the minor gases generally increase or decrease with the total volumes.
Table 14. — Rocks of sedimentary origin.
Ord«r.
Type of rock.
No. of
aiukly-
BM.
Sul-
phur
gases.
00,.
00.
CH4.
H,.
N,.
Total.
1
2
S
flhales (non-bitomlnoDS)
Metamorphosed sediments
Sandstones and quartsites
8
13
12
0.00
.57
.02
3.72
.77
.29
0.45
.22
.11
0.11
.06
.02
0.97
1.52
.17
0.18
.06
.08
5.43
3.18
.69
Among sedimentary rocks, sandstones and quartzites yield less gas
than shales, while the metamorphic group, comprising both altered shales
and sandstones, together with modified limestones, take an intermediate
position, though they surpass shales in hydrogen and the sulphur gases.
28
THB GA8S8 IN BOCK8.
Tabls l6,—MdecrU69.
Order.
1
2
Ttp* of meteorite.
Stony
Withoat 80| of OigoeU
Iron
Neglecting Anra
No. of
Aoaly-
12
12
10
9
SdI-
phar
4.00
.00
.00
.00
00^
8.77
8.77
.78
.21
00.
0.24
.24
8.80
.87
CH4.
0.20
.20
.02
.02
Hf
0.50
.50
2.86
L67
Hf
ao»
.00
.80
.24
TbteL
8J0
4.80
7.96
2J8
A comparison of the two types of meteorites indicates that carbon
dioxide is much more important in the gas from stony specimens than in
that from the metallic bodies, but that iron meteorites yield several times
as much carbon monoxide and hydrogen as do the stones. Sufficient data
are not at hand to permit a comparison of the amount of marsh-gas from
these two t3rpes; nitrogen, however, appears to come in greater volume from
the iron meteorites.
ANALTBES CLASSIFIED BY THB AGE OF THE BOCKS.^
Table 16. — Igneous rocks.
Order.
1
2
8
4
Age.
Aichean
Proterocoie
Tertiary
Recent Utsb
Total Pre<;ambrian
Grand total
No. of
analy-
7
8
18
5
28
51
HsS.
0.03
.00
.00
.03
.02
.01
GO*.
7.44
1.85
1.20
.41
2.76
2.16
CO.
0.85
.81
.18
.07
.23
.18
OH4.
0.07
.07
.06
.01
.06
.05
H^
8.70
2.06
.68
.06
2.12
1.86
0.21
.16
.07
.08
.12
TolaL
11.80
4.47
1.96
6J1
8J5
In addition to those rocks which could be classed either as Archean or
Proterozoic, there were others which could only be called Pre-Cambrian;
they are included under the head of Total Pre-Cambrian.
The rapid and steady decline in the quantity of every gas, in passing
down the columns from the Archean through the Proterozoic and Tertiary
to Recent lavas, is very striking. These differences may be due to a com-
bination of causes. The older rocks may yield more gas than the recent,
owing to metasomatic changes which have been slowly taking place within
the rocks. If this be so, the analyses indicate that this process is progress-
ing at an exceedingly slow rate. Or the early magmas may have been more
highly charged with gas, some of which has escaped as they were worked
over and over and brought to the surface in later times. Both of these
processes have probably been operative.
Table 17. — Sedimentary and meUi-sedimerUafy rocks.
Order.
Age of rocki.
No. of
anitly-
soj*
C0|.
CO.
OU4.
H,.
N,.
TolaL
1
2
8
17
10
1
0.45
.00
.00
0.68
1.84
carb.
0.17
.25
.15
0.04
.05
.04
1.82
.41
.45
0.06
.18
.04
2.71
2.28
.68
PaleoEoic
Mesozoic
Total
28
.27
.08
.20
.04
.06
.06
2.48
1 In this claasification of analyses by the age of the rocks, and in the following one
based on granularity, only my own analyses have been used.
THB ANALTBS8.
29
Age appears to make less difference in the gas evolved from sedimentary
or metansedimentary rocks than it does in the case of igneous rocks. All
of the Proterozoic specimens were of metamorphic types, while only one
of the Paleozoic sediments had been metamorphosed. The Mesozoic repre-
sentative was a Jurassic shale altered by an intrusive. The unusual amount
of sulphur gas in the Proterozoic list is due to two weathered rocks which
contained iron sulphate. However, even with these omitted, the hydrogen
sulphide is abnormally high in the rocks of this age. One of the Paleozoic
shales was so calcareous as to yield 9.28 volumes of carbon dioxide, which
accounts for the large quantity of this gas. The two bituminous shales
(analyses 41 and 42) are not included in these averages, since their exces-
sive volume of gas from organic sources would so influence the figures as
to disguise some of the characteristics of the other rocks.
ANALYSES CLASSIFIED BT THE GRANULABITT OF THE ROCKS.
Table 18. — Igneous rocks.
Order.
1
2
8
4
OmnnUrity.
Fine-grained
Medium-grained
Coane-grained
Varioos porphyries ( mostly Ter-
tiary)
No. of
analy-
ses.
22
18
11
H^.
0.02
.01
.01
.00
00,
2.75
2.87
.40
.41
00.
0.81
.17
.10
.07
CH4.
0.06
.05
.04
.04
H,.
1.68
1.41
1.20
.22
N,.
0.12
.10
.08
.05
Total.
4.94
4.11
1.88
.79
From this table it would appear that the fine-grained rocks g^ve off
more gas than those of coarser granularity. One of the reasons for this
difference probably lies in the fact that metasomatic changes are favored
in fine-grained rocks, whose crystals, being smaller, afford more numerous
junction-planes between the crystals, through which solutions more readily
traverse the rock than in the coarse-grained varieties. Among other
changes, hydration and carbonation should alter fine-grained rocks more
effectively than coarse-grained ones.
Fineness of grain in igneous rocks usually means that the lava cooled
rajndly, and this would hinder the escape of the inclosed gas. But in the
process of slow crystallization, such as produces large crystals and coarse
texture, much more of the gas would be likely to be crowded out of the
growing crystals. However, as a general rule, fine-grained igneous rocks
are siurface flows, while coarse-texture types were formed at some depth
below the surface, and hence a larger proportion of whatever gas was
expelled from the rapidly cooling lavas would be more likely to escape
altogether than would be the case with the gas which was excluded from
growing crystals in deeper horizons, as in bathylithic intrusions, where final
escape was difiScult. In this problem of granularity, as in the matter of age,
the quantities of gas evolved are probably determined by a combination of
complex factors rather than by any sin^e cause.
RESULTS AT DIFFERENT TEMPERATURES.
The different gases are not all expelled from rock material at the same
temperature, nor are they evolved at the same rate. In general, hydro-
gen sulphide and carbon dioxide are not only the first gases to appear, but
30
THB GASES IN BOCKS.
they are more rapidly g^ven off than the others. Carbon monoxide follows
the dioxide as the temperature is raised, and generally increases in relative
importance, as the latter begins to subside, toward the end of the com-
bustion. Hydrogen and marsh-gas are most conspicuous at high temper-
atures, and hence attain higher percentages in the last half of the gas than
in the first portion. Nitrogen appears to be disengaged with much diflScuIty,
requiring considerable time at an elevated temperature. These general
facts may be graphically represented by plotting the curves based on the
experiments with the Baltimore gneiss.^ (See fig. 1.)
6.62
0>
E
o
>
1.54
100 200
300 400 500
Temperature
600
700
800*850*
Fio. 1. — ^Plot of ounres representiiig volume of eaeh gas per volume of roek
obtained at different temperaturee from Baftimore gneiae.
Nitrogen is omitted from this diagram owing to an imfortunate leak-
age of air during a part of the experiment, which was sufiScient to vitiate
the results for this gas.
^ For tables, see pp. 36-37.
ABSORPTION. 31
ABSORPTION,
To determine how much gas might be reabsorbed after being expelled by
heat, it was thought desirable to use a rock capable of producing a large
volume of gas. For this purpose a diabase from Nahant, Massachusetts,
which yielded 13.9 volumes of gas, was selected. This material was heated
at full blast until the gas evolution had practically ceased, which required
about four hours; 182 cubic centimeters of gas were obtained. After allow-
ing the powder to remain in the vacuum overnight, it was removed and
still more finely pulverized in an agate mortar. It was then submitted to
forced heat, yielding an additional 20 cubic centimeters of gas in six hours.
On the third day the powder gave up but 1 cubic centimeter in four
hours. As practically all the gas available under these conditions was now
removed, the heat was turned off, and 132.01 cubic centimeters of this gas
(at 27.0^ and 758 millimeters) immediately introduced into the combustion-
tube, which was allowed to cool. At the end of 43 hours 101.84 cubic
centimeters (at 20.0^ and 750 millimeters) remained to be pumped out.
This being equivalent to 103.73 cubic centimeters at 27.0^ and 758 milli-
meters, leaves 28.28 cubic centimeters as the volume of gas absorbed by
the powder. The material in the tube was now heated for 2) hours, but
only 3.47 cubic centimeters could be extracted before the gas evolution
ceased. Of this, carbon dioxide constituted more than 85 per cent. There
still remain 24.81 cubic centimeters lost in the operation — ^a loss which is
probably to be attributed to the oxidation of that quantity of hydrogen
to water by ferric oxide, while the tube was cooling. This water-vapor
being removed by the calcium chloride drying-tube, hydrogen could not
be again freed by the reverse reaction when the tube was reheated. The
carbon dioxide may be explained b}"^ carbonation of iron or calcium and
the subsequent decomposition of these carbonates when heated the second
time.
In order to ascertain how much absorption there might be at ordinary
temperatures, 72 cubic centimeters of the remaining gas, from which the
carbon dioxide had been removed, since carbonation is a recognized proc-
ess, was allowed to stand in the tubes for eight days. At the end of this
time no appreciable quantity of the gas had been absorbed. From this
and the preceding experiment, it is quite evident that while rock material
may take up certain gases while cooling from a higher temperature under
special conditions, at ordinary temperatures absorption, if it goes on at
all, takes place very slowly. Reversible chemical reactions undoubtedly
play an important part in such absorption as takes place under changing
temperatures.
Professor Dewar experimented with celestial graphite to ascertain its
absorbing power for certain gases. After exhausting the graphite of its
gases, dry carbon dioxide was drawn through the tube for twelve hours
at ordinary temperatures. The tube was then heated and about 1.1 vol*
umes of gas, containing 98.4 per cent carbon dioxide, pumped off. The
graphite on the first heating had given 7.25 volumes of gas, of which 91.8
per cent was carbon dioxide. Dry marsh-gas was next passed over the
32
THB GA8X8 IN B0GK8.
powder for twelve hours; upon heating, only 0.9 volume, containing 94.1
per cent carbonic acid, was obtained. The same experiment repeated with
hydrogen gave only 0.17 volume, in which carbon dioxide reached 95 per
cent.^ From these figures it would seem that absorption is not very itor
portant. The steadily decreasing volumes of gas with each successive
heating show the difiSculty with which the gas is expelled, for apparently
it is liberated more readily after an interval of time than if reheated imr
mediately. Hence, unless the material used be completely deprived of its
gas, there is always a danger in assigning to absorption what may, in reality,
be only the last portions of the original gas.
Wright used another method in testing the hypothesis that the gas
obtained from meteorites has been derived from our atmosphere by a
process of absorption. He believed that if the gas be due to absorption
from the earth's atmosphere, a meteorite should have stored up more of
it after being exposed for a considerable period than shortly after its fall.
His original analysis of the gas from a meteorite which fell in Iowa Coxmty,
Iowa, on February 12, 1875, was made a short time after its fall. A year
later, to extract the same quantity of gas from another fragment of the
same meteorite required not only a longer time than in the first analysis,
but more intense heating as well.' if any difference actually existed, a
loss rather than a gain was indicated in this interval.
To test the effect of air exposure on a rock powder which had previously
been heated until the gas evolution had completely ceased, the exhausted
powders of my investigation were kept stored in paper bags, and several
of them were reheated after intervals of some months. Two analjrses of
the iron basalt from Ovifak, Greenland, made 10 months apart, were as
follows:
Table 19.
1
•
Analysis
No. 45.
Analysis after
10 months.
Hydrogen solphide
Carbon dioxide
0.00
3.74
1.74
.17
2.24
.16
8.05 vols.
0.00
1.18
.30
.03
.04
.21
1.76 vols.
Carbon monoxide
Methane
Hydrofiren
Nitrogen
Total
This basalt yielded about one-third as much carbon dioxide after the
interval as it did when originally heated, but the hydrogen in the second
portion of gas was almost a negligible quantity.
A second test was made mth a chloritoid schist from the Black Hills,
after an interval of more than a year.
> Sir James Dewar, Proc. Roy. Inst., vol. 11 (1886), p. 647.
* Wright, A. W., Am. Jour. Sd., vol. 11 (1876), p. 262.
ABSORPTION.
33
Table 20.
Analysis
No. 28.
Analysis
a year later.
Hydrogen sulphide
Our bon dioxiae
0.00
.46
.10
.04
3.07
.05
3.72 vols.
0.00
.29
.21
.08
2.11
• • • •
2.69 vols.
Carbon monoxide
Methane r . . .
Hvdroffen
■■-*j7 ^■**'^^**"* ••••••••4,. »..••••
Nitrogen
Total
In this case, hydrogen has been restored somewhat more completely
than carbon dioxide. Both of them amount to approximately two-thirds
of the original volume of these gases.
A third test was made with amphibolite, after an interval of four months.
Table 21.
Analysis
No. 94.
Analysis four
months latex.
Hydrogen salphide .' . . .
Carbon dioxiae
0.00
2.23
1.10
.10
2.84
.13
6.40 vols.
0.00
.64
.34
.12
1.86
.09
3.05 vols.
Carbon monoxide
Methane
Hvdrofien
Nitrosen
Total
The recovery is here more marked in the case of hydrogen than in that
of carbon dioxide.
A foarth test was made with Keweenawan diabase from Houghton,
Michigan, after an interval of six months.
Table 22.
Analysis
No. 85.
Analysis 6
months later.
Hydrogen sulphide
Carbon dioxiae
0.00
1.31
.09
.09
2.34
.05
3.88 vols.
0.00
1.33
.08
.03
.43
.05
1.92 vols.
Carbon monoxide
Methane
Hvdrofien
Nitrogen
Total
After reposing six months in a paper bag, this diabase gave as much
carbon dioxide, when heated, as it had in the first combustion; but less
than one-fifth as much hydrogen was evolved on the second heating.
It is clear that an interval of time partially restores the gas-producing
properties of these rock powders. For this phenomenon, there are two
possible explanations. Either the first heating does not expel all of the
gas contained in the rock, which, by some sort of diffusion or molecular
3
34
THB GASES IN BOCKS.
rearrangement, gradually prepares itself to come off when again heated, or
else the rock powder absorbs gases from the atmosphere. If the carbon
dioxide were derived from the decomposition, at high temperatures, of a
carbonate such as that of calcium, the oxide of calcium thus produced
would be likely to capture carbon dioxide from the air, though perhaps
this would be a slow process in a paper bag where the circulation of w' was
comparatively limited. Also, if the hydrogen came from chemical reactions
between ferrous salts and water combined in hydrated minerals, the atmo-
sphere might have restored to these minerals some of the water which they
lost when first heated. It was thought that rehydration, if combined
water be a vital factor in the production of hydrogen, could be more readily
effected by placing the exhausted powder in water for a few days than by
wrapping it up in a paper bag for as many months.
Accordingly, the Keweenawan diabase powder (No. 85) which ori^n-
ally gave 3.88 volumes, and after six months 1.92 volumes, was heated a
third time (a week later) with the evolution of very little gas. This powder,
after cooling in the vacuum, was taken out of the combustion-tube and
immediately placed in a flask filled with freshly distilled water. A stopper
being fitted into the flask, it was allowed to stand for 66 hours. At the
end of this time, the water was poured off, the powder quickly, but thor-
oughly, dried and put into the combustion-tube. When heated, this powder
gave off 0.79 volume of gas; but instead of being largely hydrogen, 67.72
per cent of this was carbon dioxide. Hydrogen amounted to only 14.69 per
cent, while carbon monoxide reached 15.06 per cent. An analysis of this
gas gave:
^ ^ Tabi^ 23.
Percentages.
Volumes.
Carbon dioxide
67.72
15.06
.19
14.69
2.34
100.00
0.63
.12
.00
.12
.02
.79
Carbon monoxide
Methane
Hvdroeen
Nitrogen
Total
This carbon dioxide could not have come from the air, but must have
existed within the material and must have withstood three successive
heatings in the combustion-tube. From a comparison of these figures
with the two previous analyses of the gas from this material, what is true
of the carbon dioxide would appear to be true of the hydrogen as well.
This experiment favors the conclusion, that the gas which is obtained from
a rock powder by a second heating after a period of time, is not due to a
process of selective absorption from the atmosphere, but rather to changes
which have been slowly taking place within the powder itself.
However, the results of these experiments upon the absorption of gas
by rock powders at ordinary temperatures and pressures can not throw
much light upon the source of the gases, or how they came to be embodied
in the rocks, since the conditions under which the rocks were formed must
have been very different. While high temperatures, in general, tend to
STATES OF THE GASES. 35
expel the gaseous constituents of the rocks, high pressures would have the
effect of promoting absorption. Moreover, it is possible that molten
lavas might absorb, or dissolve, certain gases without an increase of pres-
sure. But the testimony of volcanic gases and of the scoriaceous surfaces
of lava flows favors the idea that gases and vapors are constantly being
boiled out of molten lavas whenever exposed under the ordinary atmo-
spheric pressure. Lavas give off gas rather than absorb it, at the earth's
surface; however, at considerable depths below the surface the action may
be entirely different. If the conception be entertained that the earth's
interior is, for the most part, solid with only threads of liquid lava here
and there, the question for this solid portion would be one of the ability
of great pressure to cause a solid to absorb gases. This need not be further
dwelt upon, since most of the igneous rocks which are accessible have been
in the liquid state at some time. In the case of the threads of liquid magma
there is reason to suppose that gas, if it could be brought into contact
with this lava, would become incorporated in it owing to the great pres-
sure. But this does not explain the original source of the gases, nor how
they can be brought in contact with the liquid rock under the prevailing
conditions of temperature and pressure.
STATES IN WHICH THE GASES EXIST IN ROCKS.
In order to explain the immediate source of the gases obtained by
heating rock material in vacuo, three different hypotheses naturally pre-
sent themselves. The simplest of these is to suppose the gases to exist in
minute cavities or pores, having been entrapped within the rock during the
process of solidification. This supposition is suggested and supported by
the observation that microscopic slides of some minerals, notably quartz
and topaz, reveal numerous small gas-bubbles. But while there is evidence
that some gas is thus held in cavities, there is equally strong evidence to
show that the greater part of it can not be attributed to this source.
To escape the diflSculties encountered by the first hypothesis, appeal
is made to the imperfectly understood property of some of the elements
to "occlude," or dissolve within their mass, certain gases. It is remem-
bered that under the proper conditions palladium will occlude 900 times
its own volume of hydrogen, and that the same gas is also absorbed, in
lesser degree, by other metals, particularly platinum and iron, while silver
has a similar affinity for oxygen. This principle applied to igneous rocks
as a hypothetical source of their gases becomes at once a more difficult
proposition to prove or disprove.
The third hypothesis, more conservative than either of the others,
assumes that these gases do not exist in the rocks in the uncombined, or
gaseous state, but are produced in the combustion-tube by chemical re-
actions at high temperature. The oxides of carbon and sulphur are assigned
to the decomposition of carbonates and sulphates; methane to organic
matter present, carbides, or to high temperature reactions between hydro-
gen and the carbon gases; sulphureted hydrogen to sulphides; nitrogen
to nitrides; while hydrogen is liberated from steam by the action of metallic
iron or ferrous salt.
36
THE QABES IN BOCKS.
GASES IN CAVITIES.
The studies of Brewster, Davy, Sorby, Hartley, and others, have
established the presence of gas, generally carbon dioxide, though sometimes
nitrogen, in the minute cavities of certain crystals. This has been widely
known to geologists, and hence, when it was discovered that many crystal-
line rocks 3deld gas upon heating in vacuo, it was natural to suppose that
the gas came from cavities. Such was the view taken by Tilden.* But
while microscopical investigations indicated that carbon dioxide consti-
tutes more than 90 per cent of the gaseous matter inclosed in these cavi-
ties, and hydrogen is not found in more than traces, the latter gas is the
most important constituent of the mixture derived from rocks by heat.
In addition to this, the observation that those rocks which are not known
to contain many gas cavities produced several times as much gas as the
cavernous quartzes also suggested that the bulk of the gas, at least, could
not be attributed to inclosure in cavities. Moreover, basic rocks were
found to be more productive than acidic, whereas it had generally been
supposed that the latter, owing to their greater viscosity, should entrap
more gas and vapor than the more fluid basic lavas.
The suspicion that the gas did not come from cavities in any large
degree was strengthened by the observation that the composition of the
gas varied according to the temperature to which the rock powder was
heated. If the gas comes from cavities, its liberation should commence
with a slight rise of temperature and should continue more or less steadily,
as the heat increases, until the expansive force of the gas opens up most
of the pores. Since all gases expand equally, one should burst its con-
fines as soon as another, and a sample of gas obtained at any given
temperature should not differ very widely in composition from that evolved
at any other.
Neglecting hydrogen sulphide and nitrogen, the character of the gas
obtained at various temperatures from Baltimore gneiss ' is shown by the
following table:
Tablb 24.
Gas.
At96CP,
At4480.
At 540°.
AtOOOO.
AtSOOO.
At850».
Carbon dioxide
Carbon monoxide
Methane
93.7
6.3
0.0
0.0
37.5
4.2
25.0
33.3
27.0
2.4
1.8
68.8
13.6
2.3
2.5
81.6
19.3
2.8
2.1
75.8
0.0
1.1
3.4
95.5
Uydroffen
Total
100.00
.03
100.00
.03
100.00
1.28
100.00
1.40
100.00
5.30
100.00
.94
Volumes
Or, combining the separate analyses so that each figure represents the
percentage of the total gas obtained up to the specified temperature, the
result is as shown in table 25:
» Tilden, Chem. News, vol. 76 (1897), p. 169; Proc. Roy. Soc., vol. 64 (1897), p. 463.
' Material of analysis, No. 28.
STATES OF THE GASES.
37
TlBLE 25.
Gm.
Carbon dioxide . .
Oarbon monoxide
Methane
Hydrogen
Total
Volamee ..
200(0 360°
100.00
.03
20Pto4A&°
93.7
69.7
6.3
5.3
0.0
10.7
0.0
14.3
100.00
.06
20° to 540°
100.00
1.34
20Pto&OfP,20PU)8X]OP
30.1
20.2
2.5
2.4
2.4
2.5
65.0
74.9
100.00
2.74
100.00
8.04
20010850°
19.5
17.5
2.7
2.6
2.2
2.3
75.6
77.6
100.00
8.98
Carbon dioxide thus appeared first, constituting 93 per cent of the gas
evolved at 360® C, while hydrogen was not present in a measurable quan-
tity. On the other hand, at the highest temperature used (850®) hydrogen
amounted to 95 per cent of the total and carbon dioxide was entirely
wanting. The steady decrease in the proportion of carbon dioxide with
the elevation of the temperature, and the proportionate increase in the
value of the hydrogen, are striking. The minor constituents, carbon
monoxide and methane, underwent some variations, but did not change
so radically. The former came off at the lower temperature, but declined at
full red heat. I>Htrogen, it appears from other experiments, does not appear
in the gases obtained by moderate heating, but increases steadily in impor-
tance when the heat is carried higher. It is the last gas to be liberated.
The complete table, expressing the volumes of each gas per unit volume
of gneiss, follows:
^ Table 26.
Temperature.
HtS.
C0|.
CO.
CH4.
H,.
Ty>tal.
100®. boiliniF water
0.00
.00
trace
trace
.42
.18
.01
.00
0.00
.00
.03
.01
.21
.17
1.12
.00
0.00
.00
tr.
tr.
.02
.03
.16
.01
0.00
.00
.00
.01
.02
.03
.12
.03
0.00
.00
.00
.01
.54
1.02
4.39
.86
0.00
.00
.03
.03
1.28
1.40
5.30
.94
218®, boiling naphthalene
360®, boiling anthracene
448®, boiline solphar
540®, metal oath
600®, dnll red heat
800®. foil blast
850®, forced heat
Total
.61
1.54
.22
.20
6.82
8.98
These results are graphically represented in the curves of figure 1.
The fact that little gas could be obtained below 450® is in itself a strong
argument against the hypothesis that the gases come from pores, and
there also seems no way in which the behavior of the gases, as set forth by
these curves, can be consistently fitted into that theory.
TESTIMONY OF THE METEORITES.
Meteorites have alread}' been subjected to investigation of this sort,
though not with this purpose in mind. Mallet divided the gas which he
extracted from the meteoric iron of Augusta Coxmty, Virginia, into three
portions;* his results have been reduced by Wright* to the figures given
in table 27:
» Mallet, Proc. Roy. Soc., vol. 20, p. 367. * Wright, Am. Jour. Sci., vol. 2, p. 261.
38
THB OASES IN ROCKS.
Table 27.
OOt.
CO.
H,.
N^
Beginning . , , , - t
16.09
4.28
8.69
80.74
46.12
47.00
42.62
43.64
18.36
11.66
6.01
36.96
Middle
End
The analyses of meteorites by Wright show that, in all cases, carbon
dioxide reached a higher percentage in the gas evolved at 500® than it
did in that obtained at red heat, and that the reverse of this was true of
hydrogen in the stony meteorites. In the iron meteorites, however, two
analyses indicated a marked fall in hydrogen with the increase of heat, while
the other two were characterized by an increase. Wright's figures for the
meteorite from Guernsey County, Ohio, illustrate the continuous decrease
in the percentage of carbon dioxide: At 100®, 95.92 p. ct.; at 250®, 86.36 p.
ct. ; at 500®, 82.28 p. ct. ; incipient red heat, 33.55 p. ct. ; red heat, 19.16 p. ct.
The volume of gas obtained at each temperature is only stated for
500® and red heat. These show that up to 500®, 2.06 volumes were evolved,
and that above this point only 0.93 volume was received. From this it
appears that the diminishing percentages of carbon dioxide above 500®
represent an absolute slackening of the output of that gas, as well as an
apparent decrease due to the greater evolution of hydrogen. It might
be argued that in this case, where gas was produced at only 100®, the
cavities contributed the carbon dioxide, yielding it early and then slacken-
ing, as would be expected; but even if this be admitted, the hydrogen
manifestly can not be ascribed to that source. Tending in a measure to
support this view is the work of Sorby,* who has shown that olivine crystals
in the meteorites of Aussun and Parnallee, when examined under the micro-
scope, contain numerous small cavities filled with gas, similar to those which
have been observed in many terrestrial minerals.
In his earlier paper, Wright expressed the opinion that the gases were
partly condensed upon the particles of iron and partly absorbed within
them. Later he took the position that while some gas may be condensed
upon the fine particles of iron, a large part of the carbon dioxide, and prob-
ably also of the other gases, is mechanically imprisoned in the substance of
the meteorite. This view, which does not seem to be in accord with his
researches at different temperatures, he bases largely upon a single experi-
ment. Material from the Iowa meteorite was finely pulverized and the
iron grains separated from the non-metallic powder. A third portion
consisted of coarse fragments of the meteorite. The three portions heated
for the same time gave the following results:
Tablb 28.
00«+C0.
H^
N,.
Volume!.
Powder
66.96
38.72
48.07
80.96
69.38
60.93
2.08
1.90
1.00
0.97 1..g
0.61 / ^-^^
1.87
Iron
Fragments
» Sorby, Proc. Roy. Soc., vol. 13 (1864), pp. 333-334.
STATES OF THE GASES.
39
The greater volume of gas from the fragments was taken to indicate
that a portion of the gas was lost in the process of pulverization. An
analysb of these figures reveals the fact that the difference in volume
was chiefly due to the deficiency of the combined portions in hydrogen,
instead of carbon dioxide, and that while there was also a slight loss of
the latter gas, there was a decided gain in nitrogen.
Returning to the rocks, Tilden ^ is authority for the
statement that it does not make much difference in the
quantity of gas evolved, whether the material be taken
in chunks or in a fine powder. Instead of abandoning
the idea of cavities, he believed them to be very minute.
But this is approaching an alternative hypothesis; if
the reduction of the cavities is carried far enough — to
intermolecular spaces — practical occlusion is the result.
Another objection to the theory of mechanically-re-
tained gases apparently exists in the slowness with which
the gas is liberated when the material is heated. Usually
about three hours and often a very much longer time is
required to expel the gas. Unless the gas from cavities
be assumed to escape by diffusion through the walls of
the inclosing mineral, instead of violently bursting its
confines, there is no reason why it should not come off
with a rush when the combustion-tube is heated rapidly
to redness. Some rocks, generally those yielding a mod-
erate quantity of gas in which carbon dioxide is the
principal constituent, often give up their gas quickly —
mostly within the first 60 to 90 minutes, although the
generation continues for a longer time, before ceasing
altogether. But other varieties of rock, particularly
those noted for greater volumes, in which the percent-
age of hydrogen runs high, emit gas slowly and steadily
for three or four hours.
These considerations led me to try a series of experi-
ments which should show how much gas actually could
be obtained from the opening of cavities alone. For this
purpose a crusher was devised (fig. 2), capable of pulver-
izing a rock specimen in a complete vacuum. Adopt- Fio.2.-Appar»tu8for
ing the principle of the familiar steel mortar, this was ^^j^rro&BpttimmB
constructed in three pieces. The cylindrical cup in
which the rock material is crushed possesses an internal diameter of 7
centimeters and a depth of 9 centimeters. The walls are purposely
made thick and strong and the bottom is protected from the abrasion of
hard minerals by inserting a disk of hardened steel. Inserted in the walls
is a stopcock through which the apparatus is to be exhausted and the
gases later pumped out. A circular steel cap, or cover, provided with six
screws, whose sockets are depressed in the top of the cylinder, is intended
to make the chamber of the mortar air-tight. In the center of the cap-
» Tilden, Chem. NewB, vol. 75 (1897), p. 169; Proc. Roy. Soc., vol. 64 (1897), p. 453.
40 THE GASES IN ROCKS.
piece is a hole large enough to permit the ready movement of the piston-
shaft. Around this hole on the upper side there is welded a short piece of
steel tubing which is to guide the piston-rod and serve as a place of attach-
ment for the rubber tubing in which the shaft of the piston is incased.
The piston is a shaft 50 centimeters in length, 2.2 centimeters in thickness^
to which is attached a head piece of hardened steel which will fit snu^y
into the cylinder. Near the upper end of the piston is a cross-bar serving
as a handle, and also a flange to which the rubber tube is to be fitted.
When ready to put together, the piston-shaft is incased in a 14nch
tube of pure rubber, 45 centimeters long, which is tightly fitted and wired
to the flange near the end of the rod, whereupon the other end of the shaft
is slipped through the hole in the cover-piece, and the piston-head affixed.
The lower end of the rubber tube is wired to the steel tube of the cover-
piece which, after the rock specimen has been placed in the cylinder, is
fitted with a rubber washer and screwed as tightly as possible to the cylinder.
The rubber tube is taken of length sufficient to allow the head of the piston
to touch the bottom of the cylinder; by pulling upward on the handle the
rubber wrinkles and folds upon itself, affording ample play to the piston.
The stopcock is connected with the mercury-pump and the cylinder of
the crusher exhausted, after which vigorous strokes delivered at the end of
the pbton with a heavy mallet crush the rock, thus opening the gas cavities.
Whatever gas is liberated, is pumped into the receiver and analyzed in the
ordinary way.
RESULTS.
Of the first rock tested, a basalt from the Faroe Islands, 42 grams
were crushed finely enough to pass through a 30-mesh sieve, besides several
times as much, less completely pulverized. In all, less than 0.1 cubic
centimeter of gas was obtained, which may be considered as practically no
gas at all, since this small quantity is within the leaking possibilities of the
apparatus.
A slightly scoriaceous basalt from Hawaii produced about 0.1 cubic
centimeter of gas, which appeared to be largely air. No carbon dioxide
could be detected. Of this basalt, 18.3 grams passed through the sieve.
15.73 grams of vein quartz from Utah (No. 71 of the analyses) gave no
trace of gas.
In an effort to demonstrate conclusively that the lack of gas liberated
by crushing these lavas was not due to defective apparatus, a glass bulb
of measured capacity, filled with air, was broken in the crusher in place of
the rock ordinarily used. The result showed that gas introduced into the
crusher can be extracted without sensible change in volume. As diffusion
through the rubber tubing was considered a possible, though not very
probable, source of error, a further trial was made, using hydrogen, lightest
and most active among the gases, in order to put the apparatus to as
severe a test as possible. The purity of this hydrogen had previously been
established by analysis. The bulb broken, the gas was pumped off and
exploded with air. The observed shrinkage agreed, within the limit of
error, with the amount of hydrogen calculated to have been contained
within the bulb.
STATES OF THB GASES. 41
Being desirous of finding some specimen which would yield gas when
crushed in this manner, I procured some crystals of cavernous quartz
from Porretta, Italy, in which several of the cavities exceeded a millimeter
in diameter. 5.91 grams were crushed to sufficient fineness to pass through
the sieve, and 61.66 grams were partially crushed. 0.08 cubic centimeter
of carbon dioxide was obtained, which, supposing that it all came from the
5.91 grams, would be equivalent to only 0.03 of the volume of the quartz.
An analysis showed also a little methane and some nitrogen, but the amount
of gas available was too small for the determination to be of any value.
The result of this last test agrees with the microscopic studies of the
early investigators. Carbon dioxide exists in the cavities of quartz, but
its volume, compared with the volume of the inclosing mineral, is small.
Microscopical observations seem to show that gas cavities occur almost
exclusively in a certain set of minerals which combine hardness usually with
imperfect cleavage, namely, quartz, topaz, garnet, spinel, beryl, chrysoberyl,
corundum in the form of rubies, sapphires, and emeralds, and diamond.
These are minerals which, once they had inclosed gas, would hold it, even
under great pressure.
GASES DUE TO CHEMICAL REACHONS.
HYDROGEN.
The double series of iron salts, ferrous and ferric, together with the
intermediate ferroso-ferric compounds, reacting with oxidizing or reducing
agents, undergo various reversible reactions whose possibilities are great.
When steam is passed over metallic iron or ferrous oxide at a red heat, it
is decomposed, giving up oxygen to the iron, and at the same time pro-
ducing free hydrogen. The reactions may be written:
Fe + HjO = FeO + H, 3FeO + HjO = Fefi^ + H,
Hydrogen is produced in this way most rapidly at temperatures about
500®. Stromeyer is authority for the statement that the breaking up of
water begins at 150° but takes place very slowly; at 200° somewhat more
rapidly; at 360° the process requires several hours; at 860° it is complete
in less than one hour; while near the melting-point of iron several minutes
are sufficient.^
The authorities agree that ferric oxide is not formed in this process;
the magnetic oxide, FegO^ is the final product of the action of a current
of steam upon ferrous oxide.^
But these reactions are completely reversible. According to Gay-
Lussac, magnetite is reduced to the metal by hydrogen at every tempera-
ture between 400° and the highest degree of heat obtainable in the com-
bustion-furnace, particularly at the same temperature at which steam is
split up by glowing iron.' Siewert states that ferric oxide (from the oxa-
late) is not altered by hydrogen at 270° to 280°; between 280° and 300°
* Stromeyer, Pogg. Ann., vol. 9, p. 476.
'Among others, Kegnault, Ann. de Chim. et Phys., vol. 62, p. 348.
* Gay-Lussac, Ann. de Chim. et Phys., vol. 1, p. 33.
42 THE OASES IN ROCKS.
it is reduced to ferrous oxide, and when heated above 300^, to the metal.^
The more recent studies of Moissan give different figures ;* ^^,0, is reduced
by hydrogen at 300° to Fe^O^ in 30 minutes; at 500"* to FeO in 20 minutes;
at 600° to 700° to metallic iron.
If the hydrogen or water-vapor produced by these reactions is not
removed, the process continues only until a condition of equilibrium is
established. In extracting the gases from rocks, the products of these
reactions were rapidly removed, so that final equilibrium was probably
never attained. Hence, in these experiments the direction in which thd
reaction will proceed depends upon whether there is ferrous oxide and
water, ox ferric oxide and hydrogen, most abundantly stored in the rock.
Ferrous and ferric salts behave, in general, like the oxides.
Since most igneous rocks contain ferrous as well as ferric salts, the
possibility that, when heated in the presence of steam, hydrogen will be
produced, must always be taken into account. In terrestrial rocks water
of constitution is generally present and often is not expelled below a bright
red heat. Thus, a rock containing a ferrous compound in appreciable
amount, together with water of crystallization, a portion of which is re-
tained up to red heat, will be in a condition to furnish hydrogen upon the
application of heat.
In general, the analyses show that the greater the amount of iron present
in the rock, the more hydrogen may be expected. This may be the result
of chemical action, or a selective occlusion of hydrogen manifested by iron
and its compounds. Magnetite, being the end product of the reaction of
water upon iron, can not produce hydrogen by this chemical interaction,
though it might possess the occlusive properties of iron compouinds. Anal-
ysis of the black sand from the bed of the Snake River, Idaho,' indicates
that iron in the form of magnetite does not yield much hydrogen. How-
ever, these figures have no great significance, for, even though an abundance
of hydrogen existed in the ore, either occluded or mechanically imprisoned,
the magnetite would, at red heat, quickly oxidize it to water, with the
exception of a small portion of free hydrogen maintained by the reverse
reaction. The analyses show that basic diabases and basalts yield the
most gas, while acidic rhyolites give but little. These are also among the
maximum and minimum iron-bearing lavas. But the difference in hydro-
gen is much greater proportionately than the difference in ferrous salts.
Table 13 ^ also shows that andesites, which are nearer the basic end of the
scale than the acidic, do not greatly exceed the rhyolites in hydrogen.
The difference between the two types of rocks, acidic and basic, in point
of volume of the individual gases, while somewhat more conspicuous in
the case of hydrogen, is generally true of the other gases as well.
Endeavoring to prove that the hydrogen obtained by heating minerals
came entirely from chemical reactions, Travers experimented with the
secondary mineral chlorite, calculating how much ferrous iron should have
been oxidized to give the quantity of hydrogen and carbon monoxide
evolved.* This he found to agree closely with the difference in amount of
* Siewert, Jahresbericht d. Chem., 1864, p. 266. * Ante, p. 27.
* Moissan, Comptes Rendus, vol. 84, p. 1296. » Travers, Proc. Roy. Soc., vol. 64, p. 132.
' Analysis No. 46.
STATES OF THB OASES.
43
ferrous iron present before and after heating. Another test with feldspar
from the Peterhead granite not showing correspondence, seemed to Travers
to be explained by the presence of both metallic iron and ferrous oxide in
the feldspar. The presence of a considerable amount of metallic iron in
a feldspar which crystallized from an acidic magma containing an excess
of ffllica is quite unusual. This feldspar treated with dilute sulphuric
acid yielded about four volumes of hydrogen.
Against the theory that the hydrogen was largely derived from the
action of water-vapor on ferrous compounds, may be placed the very
marked change in color which the rock undergoes during the process of
heating. I have observed that whenever a rock powder, before being
placed in the tube, possesses an orange, brownish, or reddish tint due to
ferric oxide, the combustion invariably alters the tone to a greenish gray.
This suggests a reduction of ferric oxide to ferrous oxide, a process con-
suming hydrogen. In order to test this question, a specimen of bright-red
Permian sandstone from the Garden of the Gods near Colorado Springs^
was powdered. These Red Beds are supposed to consist of thoroughly
oxidized material; this opinion was partially confirmed by chemical tests
which gave a weak reaction for ferrous iron, but indicated much ferric.
After heating, the brick-red sand had become dull gray-green in color. The
gray sand from the combustion-tube gave a stronger reaction for fer-
rous iron. Later, a quantitative determination of the ferrous iron present
before and after heating was undertaken. Equal weights of the two sands
were boiled with strong sulphuric acid ' for two hours and then allowed to
stand overnight. In each case the solution was effected in an atmosphere
of carbon dioxide to prevent oxidation by oxygen from the air. The two
solutions were then titrated with potassium permanganate solution. 3.09
grams red sand required 1.97 cubic centimeters N/10 EMn04; 3.09 grams
gray sand required 2.52 cubic centimeters N/10 KMn04. ^-^^ cubic centi-
meter N/10 EMn04 is equivalent to 0.015 gram of iron, which is the
weight of the metal reduced from the ferric to the ferrous state. For the
total weight of sand used in the gas analysis (85 grams), the increase in
ferrous iron should be 0.423 gram, which would correspond to an oxidation
of approximately 85 cubic centimeters of hydrogen. Yet both hydrogen
and carbon monoxide were obtained from this sandstone in considerable
quantities.
Table 29.
Percent.
Volumes.
Hydrogen sulphide
Gurbon dioxide
0.05
carbonated
S0.69
6.27
27.28
5.71
100.00
0.00
.71
.07
.32
.06
1.16
Carbon monoxide
Methane
Hydnxsen
Nitrogen
Total
' Analysis No. 78.
' 3 parts cone, acid to 1 part water.
44
THE GA8S8 IN BOCKS.
In order to ascertain the quantitative effect of the presence of ferric
oxide in moderate amount, 0.77 gram of pure Fe,0, was mixed with 20.04
grams of diabase powder, tinting thb latter a reddish brown. An analysis
of the resulting gas and of the original diabase gave the figures shown in
the following table:
Table 30.
Retalting gaa.
Original <UAbMe.i
Percent.
Volumes.
Per cent.
yoliiine&
Hydrogen sulphide
Carbon dioxiae
0.06
72.99
3.90
.87
20.79
1.39
0.00
7.08
.38
.09
2.01
.13
Hydrogen sulphide
Carbon dioxide
0.04
61^
2.47
1.32
83.69
1.23
0.00
8.51
.34
.18
4.68
.17
Carbon monoxide
Methane
Carbon monoxide
Methane
Hydroeen
Hvdrocen
Nitroffen
Nitrosen
Total
Total
100.00
9.69
100.00
13.88
1 Analysia No. 86.
A comparison of these results shows that, while the yield of hydrogen
was diminished by the ferric oxide to less than half of what it would have
been, the carbon monoxide was not affected. The ferric oxide apparently
only went down to a state of equilibrium, and was not in suj£cient quantity
to offset the copious evolution of hydrogen from the diabase. The brown
color, however, was replaced by green.
To get rid of the iron, and particularly ferrous iron, material from the
same diabase specimen was treated with concentrated nitric acid for 66
hours. Much gas came off at first, nitric oxide, perhaps from the action
of the acid on pjrrite, being very conspicuous. The powder, washed repeat-
edly on a filter until all the acid had been removed, was dried in an oven
overnight and then heated at 115^ in an air-bath for half an hour. Two
and a half hours at red heat, in vacuo, then expelled only 0.23 volume of
gas from the diabase powder. Its composition is given in table 31.
Table 81.
Percent
Volumes.
Hydrogen sulphide
Carbon dioxide
0.00
25.23
20.15
6.36
21.21
27.05
100.00
0.00
.06
.06
.01
.06
.06
.23
Carbon monoxide
Methane
Hydroffen
Nitrogen
Total
A similar test was made with dilute sulphuric acid, in a vacuum. In
this experiment, the gas driven off by the acid during the first 2 J hours
was collected and analyzed. Table 32 shows this to have been chiefly
carbon dioxide.
STATES OF THB OASES.
45
Table 32.
Percent
Voliimei.
Hydrogen sulphide
Carbon dioxide
0.00
98.10
.03
.25
1.62
100.00
0.00
6.44
.00
.02
.10
6.66
Carbon monoxide )
Methane /
Uydroeen
Nitrogen
Total
As a precautionary measure, to avoid the introduction of any metallic
iron in the process of pulverization, the diabase was reduced to a powder
in an agate mortar. The brass sieve was not used. Hence this hydrogen
did not come from any action of the acid upon a metal introduced during
the manipulations.
This powder, after remaining in a vacuum with an excess of sulphuric
acid for three days, was washed thoroughly on a Gooch filter until the last
traces of calcium sulphate had been removed. After drying for an hour
at 125^, the powder was placed in the combustion-tube and heated to
redness. The sulphuric acid left more gas in the rock than the nitric.
Tablb 33.
Percent
Volumes.
Hydrogen Bolphide
Cflffoon dioxide ^ . x ^ ....... a
13.30
38.19
9.01
3.96
33.80
1.76
100.00
0.21
.62
.14
.06
.64
.03
1.60
Carbon monoxide
Methane ............ t t t r x r
Hydrosen
Nitrosen
Total
From these experiments it would appear that acids remove the critical
gas-producing factors without liberating a notable amount of any gas
except carbon dioxide. Whether hydrogen may not pass into solution
with the iron, without being freed, is a question which naturally arises,
but the balance of chemical opinion is against this supposition.
Professor Dewar digested celestial graphite in strong nitric acid for
several hours and, after washing and drying, found that with heat it gave
exactly the same amouint of hydrogen as before treating with the acid.
This would suggest that, in the case of celestial graphite, the hydrogen
was not connected with iron, but existed in some very stable form.^
If all the hydrogen was produced by the reaction of water on ferrous
salts, it would seem as if the volume obtained should bear a direct relation
to the quantity of these two critical constituents present in the rock.
To throw light on this matter, two rocks of the same origin, but of different
chemical composition, presented the most favorable line of attack. An
intrusive andesite and a specimen of vein quartz derived from the mag-
^ Dewar, Proc. Roy. Inst., vol. 11, p. 560.
46
THB GA8BS IN ROCK8.
matic waters of the intrusion, kindly furnished by Dr. G. K. Leith, were
used to illustrate this point.^ Though containing very different quantities
of ferrous compounds, they 3delded identical volumes of methane, hydrogen,
and nitrogen. These analyses are given in table 34.
Table 34.
Andesite.
Veinqnarti.
Percent
Volumes.
Per cent.
VdameB.
Hydrogen sulphide
Gfixbon dioxiae
0.03
77.50
4.76
.96
15.36
1.42
0.00
2.66
.16
.03
.63
.06
Hydrogen sulphide
Cw*bon dioxide
0.00
13.03
11.26
4.00
64.40
6.41
0.00
.11
.09
.03
.53
.06
Carbon monoxide
Methane
Carbon monoxide
Methane
Hvdroflren
Uvdrofien
Nitroflren
Nitrocen
Total
Total
100.00
3.43
100.00
.81
The great excess of carbon dioxide in the andesite is assigned to car-
bonation of that lava subsequent to its formation — a process to which the
quartz would not be susceptible.
The observation that comparatively pure quartz yielded half a volume
of hydrogen suggested a quantitative analysis to determine the amount
of iron actually contained in hydrogen-producing quartz. For the purpose
quartz from Orange, New South Wales, was selected. 102.72 grams of the
quartz yielded 4.81 cubic centimeters of hydrogen at 0° and 760 millimeters.'
After the gas had been extracted, two different portions of the exhausted
mineral were digested with aqua regia — one of them boiled for an hour,
the other being allowed to stand during several days, and occasionally
warmed to the boiling-point. The acid may be considered to have dis-
solved all the iron from which gas could have escaped. To make the case
certain, all of the iron detected has been supposed to have existed in the
quartz as ferrous oxide, although some of it undoubtedly occurred in the
form of ferric compounds. The iron was weighed as FojOj.
First determination:
22.22 gms. quartz contained 0.0015 gm. Fe,Oa
102.72 gms. quartz would contain 00693 gm. Fe,Oa
102.72 gms. quartz would contain 00485 gm. Fe
Fe (as FeO) required to give 1 c.c. hydrogen 00748 gm.
Maximum amount hvdrogen from reaction 65 c.c.
Hydrogen actually obtained (at 0^ and 760 mm.) 4.81 c.c.
Hydrogen not from this reaction 4.16 c.c.
Second determination:
52.02 gms. quartz contained 0.0042 gm. Fe,0,
102.72 gms. quartz would contain 00829 gm. Fe,0,
102.72 gms. quartz would contain 00580 gm. Fe
Fe (as FeO) required to give 1 c.c. hydrogen 00748 gm.
Maximum amount hydrogen from reaction 77 c.c.
Amount of hydrogen actually obtained 4.81 c.c.
Hydrogen not from this reaction 4.04 c.c.
*■ Analyses Nos. 70 and 71.
* Analysis No. 100.
STATES OF THE GASES. 47
According to these two determinations, this quartz evolved respectively
7.4 or 6.2 times as much hydrogen as could have been generated by the
reaction 3FeO+HjO = Fe^O^ + H,.
If the iron existed as pyrite, four times as much hydrogen as could
come from ferrous oxide might have been produced in accordance with
the equation
3FeS,+4H,0 = Fe30, + 4Hj+6S
On the basis of this equation the excess of hydrogen from the quartz
is much reduced.
Fint determination:
102.72 gma, quartz contain 0.00485 gm. Fe
Fe Tas FeS,} required to give 1 c.c. hydrogen 00187 gm.
Hyorogen poesible from reaction 2.60 c.c.
Hydrogen actually obtained 4.81 c.c.
Hydrogen not from this reaction 2.21 c.c.
Second determination:
102.72 toDB. quartz contain 0.00580 gm. Fe
Fe Tas FeS,) required to give 1 c.c. hydrogen 00187 gm.
Hyorogen possible from reaction 3.08 c.c.
Hydrogen actually obtained 4.81 c.c.
Hydrogen not from this reaction 1.73 c.c.
These computations assume not only that all the iron in the quartz
was combined as pjrrite; and that it was completely oxidized to magnetite,
but that the hydrogen sulphide produced was entirely dissociated into
hydrogen and sidphur. But the iodine titration in the gas analysis revealed
0.36 cubic centimeter (at 0^ and 760 millimeters) of sulphur gas whose
odor was that of hydrogen sulphide rather than sidphur dioxide. If this
were H^S, it would diminish the amount of hydrogen which could have come
from the reaction by 0.36 cubic centimeter; if, however, it were sulphur
dioxide the volume of possible hydrogen would be swelled by 0.72 cubic
centimeter in accordance with the reaction
S+2H,0 = S0,+2H,
But as there was certainly much more hydrogen sulphide than sulphur
dioxide absorbed by the potassium hydroxide solution, it will be safe to
balance the possible SO, formed, by the £[,8 undissociated, and ignore
these corrections, which would probably reduce, rather than increase, the
quantity of hydrogen which might result from pjrrite.
If the iron had all been locked up in the mineral chalcopyrite (CuFeS,)
the hydrogen might be accounted for, but chemical tests failed to detect
the copper which this supposition would require. Just how much hydrogen
might be expected from iron nitride (Fe,N) is not certain, since, in the
presence of superheated steam, the nitrogen is more likely to unite with
hydrogen and come off as ammonia rather than as free nitrogen, and
ammonia is not dissociated short of the electric spark. That most of the
iron in the quartz is in the form of a nitride is highly improbable. Iron
carbide also woidd not 3deld sufficient hydrogen.
Another mineral apparently containing very little iron, but which
3delded considerable hydrogen, was the beryl of analyses 101 and 101a.
48 THB GASES IN BOCKS.
Though as transparent as window-glass, one volume of this beryl con-
tributed 0.31 volume of hydrogen. A determination of its accessible iron
was made by pursuing the same method as was used for the quarts. The
results were:
35.00 gms. beryl contained 0.0003 gm. Fe^,
127.52 gms. beryl would contain 00109 gm. Fe,0,
127.52 sms. beryl would contain 00076 gm. Fe
Fe (as FeO) required to give 1 c.c. hydrogen 00748 gm.
Maximum amount hydrogen from reaction 0.10 c.c.
Hydrogen actually obtained (0^ and 760 mm.) 14.S9 c.c.
Hydrogen not from this reaction 14.79 c.c.
This beryl expelled nearly 150 times as much hydrogen as can be
assigned to the interaction of steam and ferrous oxide under the most
generous assumptions. The actual hydrogen is 37 times the maximum
quantity possible from this weight of iron, either as pjrrite or in the metallic
state. Here is a very declared case demonstrating the inadequacy of chem-
ical reactions involving iron to generate the hydrogen obtained.
Heated in a closed tube with a limited amount of air, beryl is known
to give up a small quantity of water which, in some varieties of the mineral,
may reach 2 per cent. The question whether the excess of hydrogen over
that possible from reactions between water and iron coidd have arisen
from the dissociation of this water is easily answered. The recent researches
of Nemst upon the dissociation of steam indicate that, at temperatuies
below 2000^ C, the process takes place only to a very limited extent.
At 1124^ C, which is somewhat above the point to which the beryl was
heated, only 0.0078 per cent of the total steam can be dissociated.^ At
this temperature, 127 grams of beryl containing 2 per cent of water should,
on the basis of Nemst's figures, yield 0.24 cubic centimeter of hydrogen,
provided the gas was quickly cooled. Hence only a small portion of the
hydrogen can be attributed to the dissociation of water present in the
mineral.
To the question of the importance of ferrous salts in the production of
hydrogen, it is possible that meteorites, which have usually been regarded
as free from water, can add testimony of some value. Though it is true
that in freshly fallen specimens hydrous minerals have not yet been recog-
nized,^ nevertheless, the researches of Graham, Mallet, Wright, and Dewar,
besides my analyses of the Allegan, Estacado, and Toluca meteorites,
have shown that these bodies, when heated, give off much gas, rich in
hydrogen. If these meteorites really contained no water, either original
or by absorption from the earth's atmosphere, the hydrogen obtained
from them can not be attributed to the decomposition of water; it must
have been held within the mass of each meteorite, either entrapped or
occluded.^ But in several instances, at least, the investigators have stated
that a certain quantity of water was driven off, though perhaps this came
from weathered aerolites. The chemical analysis of the Allegan meteorite,
1 Nemst, Chem. Central-Blatt, 1905, 2, p. 290.
» Farrington, Jour, of GeoL, vol. 9 (1901), p. 632.
s It is to be remembered that a few meteorites have been found to contain hydro-
carbons, from which hydrogen mieht arise, but the presence of these hydrocarbons from
inorgamc sources is more remarkable than that of hydrogen itself.
STATES OF THE GASES. 49
which was dug up while still hot, gave Stokes 0.25 per cent of water. ^
Perhaps this was moisture absorbed from the air by deliquescent com-
pounds, such as lawrencite; still, on the other hand, there appears no
reason, at the present time, why a part of this water should not be a pri-
mary constituent of the meteorite. This uncertainty points out the desir-
ability of further, and more critical, studies upon the composition and
properties of meteorites, before attempting to base an argument upon the
absence of water in these bodies.
Other possible sources of hydrogen are hydrogen sulphide, hydro-
carbons, and the products of radioactivity. As the decomposition of
sulphureted hydrogen has already been mentioned, and is also treated
under the head of that gas, it need not be discussed here. Hydrocarbons
can only be represented in small quantities in igneous rocks, and should
produce more methane than free hydrogen. Unless the analysis shows
much marsh-gas, hydrogen from this source must be unimportant.
CARBON DIOXIDE.
The carbonates of most metals are decomposed by heat with the liber-
ation of carbon dioxide. On this account the determination of the carbon
dioxide yielded by rocks which have undergone much carbonation is of
little value. Many rocks which appear to be perfectly fresh have neverthe-
less suffered slight carbonation while in the zone of weathering, and thus
possess carbon dioxide in a combined state ready to be evolved when suffi-
ciently heated. This carbon dioxide from the non-gaseous constituents
of the rock embarrasses the determination of the free gas, since there is no
way of separating the carbonic acid from these different sources.
The degree of heat necessary to decompose carbonates throws some
light on the question. Erdmann and Marchand state that already at
400^ traces of carbon dioxide are given off from calcium carbonate.' The
studies of Debray show that at the boiling-points of mercury and sulphur,
350® and 448° respectively, the development of CO, from calcite in vacuo
is inappreciable.' The same investigator found that at 860° calcite gives
up carbonic anhydride until a pressure of 85 millimeters is reached, when the
action ceases. At 1040° the pressure may rise to 520 millimeters before
the evolution of gas is stopped. In the presence of carbon dioxide at the
ordinary atmospheric pressure, calcite retains all of its optical and other
properties unaltered, even at 1040°. Carbon dioxide from calcium carbonate
is thus not of any quantitative importance below 450°. In general, most
of the carbonic acid from the rocks is expelled at temperatures above
450°. But considerable CO, often appears before the heat reaches 400°,
as is shown by the Baltimore gneiss. Perhaps this gas may be assigned to
ferrous carbonate. Iron carbonate would be expected to decompose more
readily than calcium carbonate, though I have been unable to discover at
what temperature the process commences.
» Ante, p. 21.
' Erdmann and Marchand, cited by Gmelin-Kraut, Anorg. Chem., 2, p. 354.
' Debray, Gomptes Rendus, vol. 64, p. 603.
50
THE GASES IN BOCKS.
When a finely powdered igneous rock is treated with hydrochloric
acid and gently warmed, a few small bubbles of carbon dioxide usually
are seen to rise to the surface of the acid. This gas comes from the action
of the acid upon small quantities of carbonate present in the rock. To
test the quantitative importance of this action and to discover whether
other gases are freed by acid, 25.13 grams of diabase from Nahant, Massa-
chusetts,^ were placed in a flask connected with the mercury-pump, and the
air removed. Dilute sulphuric acid was introduced into the flask through
a dropping funnel. The gas developed in the cold during the fiirst 2} hours
was found to have the following composition:
Table 35.
Percent
Volmnet.
Hydrogen salphide
Carbon dioxide
0.00
98.10
.03
.25
1.62
100.00
• • • •
6.44
.00
.02
.10
6.56
Methane
Hydroeen
Nitrogen
Total
Practically all of the carbon dioxide thus set free is to be assigned to
a carbonate.
The apparatus was allowed to stand for three days, during which time
more gas came off. At the end of this period, the powder was washed,
dried, and then submitted to the ordinary process of heating in the tube.
Of the gas received, 38.19 per cent, or 0.62 volume per volume of rock,
was carbon dioxide. Powder from the same specimen of diabase, not
treated with acid, yielded 8.51 volumes of carbonic anhydride in the com-
bustion-tube. This amounted to 61.25 per cent of the total gas.'
While carbon dioxide, both gaseous and liquid, occurs in minute cavities
in certain minerals and rocks, and while rocks also, doubtless, contain
some of this gas in a state of occlusion, it seems probable, on account of
the wide dissemination of carbonates in small quantities through the
accessible rocks near the earth's surface, that the greater part of the carbon
dioxide obtained by the method of heating rock material in vacuo is derived
from the decomposition of carbonates in the combustion-tube. It may
be assumed that more of the carbonates in igneous rocks are secondary
than primary.'^ But though a knowledge of this immediate source of much
of the carbon dioxide in the rocks does not lead far toward the elucidation
of the problem of the ultimate source of this gas, it imposes no restrictions
upon the more comprehensive view that the carbonic acid which is now
locked up in the rocks chemically, as a result of weathering and carbona-
tion, was given to the atmosphere and hydrosphere originally from the
magmas themselves.
CARBON MONOXIDE.
Metallic iron and ferrous salts reduce carbon dioxide to monoxide
under practically the same conditions that they liberate hydrogen from
water-vapor.
1 Analysis No. 88.
' Analysis No. 86.
STATES OF THE OASES. 51
3FeO + CO, = FesO^ + CO
While this action commences below 400^, it takes place slowly, and it is
chiefly at higher temperatures that it becomes of quantitative importance.
As in the case of hydrogen and water- vapor, this reaction is reversible,
the direction in which it will proceed depending upon the proportions of
the substances present. Either metallic iron or ferric oxide, heated in a
mixture of equal parts of carbon monoxide and carbon dioxide, produces
ferrous oxide.^ Siderite at red heat passes into a magnetic oxide ^ith the
formation of both carbonic acid and carbonic oxide. According to
Dobereiner this reaction takes place as follows:^
SFeCO, = SFeO.FeA + 4C0, + CO
Glasson,' however, says that 4FeO.Fej08 results, at first giving two parts
of CO3 and one of CO, but that later the proportion changes to five parts
of CO3 and one of GO.
It is, therefore, the normal thing for a rock containing carbon dioxide
(whether occluded, or in cavities, or a carbonate) and iron in the ferrous
condition to generate carbon monoxide on the application of heat. In
this connection it may be noted that carbon monoxide rises very conspic-
uously in relative importance whenever there is metallic iron present in
the material tested. The iron-bearing basalt of Ovifak, Greenland, gave
21.63 per cent of this gas compared with 46.50 per cent of the dioxide,**
the Allegan meteorite, 38.61 per cent of CO and 41.74 per cent of CO,;*
while the Estacado meteorite developed 29.31 per cent monoxide and only
28.47 per cent dioxide.* These were specimens of stony material contain-
ing grains of metallic iron. Quite different is the Toluca iron meteorite,
whose nearly pure metal evolved 71.05 per cent carbonic oxide with but
6.40 per cent carbonic anhydride.' Wright's figures for iron meteorites
are equally noted for high percentages of carbon monoxide.'
However, there are two other chemical sources for carbon monoxide,
one of which is, perhaps, especially applicable to iron meteorites. It is
known that the carbides of chromium and iron, when heated with the
oxides of these metals, produce carbonic oxide.* As these meteorites often
contain considerable carbon, some of it perhaps as a carbide, scrupulous
care is always necessary in preparing the metal for the analysis, to avoid
introducing any rust from the oxidized exterior of the mass.
The other principle must always be operative in the combustion-tube.
Boudouard has shown that at the temperatures of the combustion-furnace
hydrogen reduces carbon dioxide, forming carbon monoxide at the expense
of both hydrogen and the dioxide.'
» Wright and Luff, Jour. Chem. Soc., vol. 33 (1878), p. 604.
' Cited by GmeUn-Kraut, Anorg. Chem., vol. 3, p. 319.
' Analysis No. 45.
* Analysis No. 106.
» Analysis No. 107.
* Analysis No. 108.
^ See p. 6.
' Borchers and McMillan, Electric Smelting and Refining, p. 545.
•O. Boudouard, Chem. Central-Blatt (1901), 1, p. 1360.
52 THE GA8E8 IN BOCKS.
COj+H,=CO + HjO
Equal volumes of hydrogen and carbon dioxide heated at 850^ for
one hour gave CO, 44.3 per cent, CO 8.3, H, 42.0, and H^O 5.4 per cent.
Heated for three hours under the same conditions, the proportion of carbon
monoxide rose to 18 per cent. When rocks are heated for analysis, the
gas is usually pumped off at short intervals, and this reaction, because of
its slowness, becomes less important. Htlttner has appealed to thb reaction
to explain the presence of carbon monoxide in minerals.
But metallic iron also has a penchant for absorbing carbon monoxide
at the proper temperature. This process is usually called occlusion, and
may perhaps partake of the nature of a combination in which the gas
temporarily unites with the iron as iron carbonyl, Fe(C0)4,* an unstable
compound readily giving up carbonic oxide. It seems likely that a portion
of the carbon monoxide developed from these irons, particularly those of
meteoritic origin, actually exists in the iron as monoxide, and that not all
of it has been formed by reduction of the dioxide.
SULPHXTR DIOXIDE.
Certain rocks, when heated, disengaged sulphur dioxide in considerable
quantities.' These were ferruginous rocks of rusty appearance, generally
metamorphosed pyritiferous shales which had undergone much weather-
ing. By oxidation, the original pyrite had been partially converted into
ferrous sulphate (Fe804) and basic ferric sulphate (Fe^jO,), both of which
were decomposed by the heat of the combustion-furnace.
2FeS04 = Fe A + SO, + SO, Fe,S ,0, = FejOi + 280,
The sidphur trioxide was reduced to the dioxide either by hydrogen sul-
phide, hydrogen, ferrous oxide, or sidphur.
SO,+H,S = SOj+HjO + S
It has been my observation that whenever sulphur dioxide was evolved
a slight sublimate of sulphur collected toward the cool end of the tube.
This may have been derived from the reaction above, or from hydrogen
sulphide and sidphur dioxide, coming from ferrous disulphide and sulphate,
respectively, and which can not exist together.
2H,S + SO,=2HjO + 3S
The sulphur dioxide obtained in the study of rocks is all assigned to
these reactions, though it is not impossible that this compound may occur
in small quantities, as a gas or a liquid, imprisoned in minute cavities.
HYDROGEN SULPHIDE.
When iron pjnites (FeS,) is heated in a stream of hydrogen, ferrous
sulphide (FeS) and free sulphur result." Though no hydrogen sulphide
^ Fe and CO also exist feebly united in other proportions, as iron pentacarfoonyl,
Fe(0O)f, and heptacarbonyl, FeCCO)^.
' Analyses Nos. 43, 65, 93, and 109.
* Rose, Pogg. Ann., vol. 6, p. 533.
STATES OF THE OASES. 53
IS formed in this manner, that gas is produced when pyrite is decomposed
by steam at high temperatures.^
FeSa + HjO = FeO + H,S + S
As pyrite is frequently present in igneous rocks which generally evolve
water-vapor upon the application of heat; the limited quantities of hydro-
gen sulphide obtained may be explained in this way. But unless the
hydrogen sulphide be removed, this process can proceed only to a certain
point, for, according to Berzelius, iron disulphide is formed when FeCO,,
Fcj04, Fe,Os, or Fe(OH), is heated with hydrogen sulphide to temperatures
between 100® and red heat.' At red heat, a current of dry hydrogen sul-
phide completely converts Fe,04 into FesS4 in two hours, while a still
farther increase of temperature results in the formation of FeS and a deposit
of sulphur.'
An inspection of the analyses shows that sulphureted hydrogen is
rarely obtained in large amounts from igneous rocks. An average of 75
analyses from a wide range of rocks (but omitting bituminous shales)
gave 0.59 per cent of this gas. But this figure is not a good working aver-
age, since it has been much influenced by the high sulphide percentage
of a few individuals. Deducting the five highest of these, the remaining
70 analyses give an average of 0.27 per cent of hydrogen sulphide. In 19
cases out of the 75, this gas was entirely lacking.
While it is probable that not much of this gas was given off from the
rock material in the first place, a portion of it doubtless disappeared before
passing through the pump into the gas-receiver. At the temperature of
the combustion-furnace, hydrogen sulphide is apt to be partially dissociated
into its elements, thus swelling the already large volume of hydrogen pres-
ent.^ Gautier states that sulphur heated in a tube filled with hydrogen
sulphide causes the decomposition of the gas with the result that its sul-
phur is added to the free sulphur, while hydrogen, nearly pure, remains.'
When the rock has been considerably weathered, and some of the pyrite
oxidized into iron sulphate, so that, in addition to hydrogen sulphide,
sulphur dioxide is disengaged, the former gas will be partially or com-
pletely decomposed, depending upon the relative proportions of the two
gases.
2H,S+SO,=2H,0+3S
The bituminous shale from Newsom's Station, near Nashville, Ten-
nessee,' yielded sulphureted hydrogen to the extent of 30.94 per cent of
the total gas, which is equivalent to the unusual amount of 29.38 volumes
of hydrogen sulphide from one volume of shale. A specimen of the well-
^ With an excess of steam the reaction goes further: 3FeS,+4H,0 « Fefit+SHfi
-f38+H,.
' Beraelius, cited by Graham-Otto, Anorg. Chem., 4, p. 718.
* Sidot, Chem. Gentral-Blatt, vol. 40 (1S69), p. 1038.
* See p. 47.
* Gautier, Comptes Rendus, vol. 132 (1901), p. 189. When jpvrite is heated either
in a vacuum, or in a stream of dry carbon dioxide, FerSg and free sulpnur result (Berzelius,
Rammekberg^ cited in Gmelin-Kraut, Anorg. Chem., 3, p. 335).
* Analysis No. 41.
54 THE OASES IN BOCKS.
known ''oil rock" from a lead and zinc mine near Platteville, Wisconsin,
produced 6.79 per cent hydrogen sulphide, corresponding to 3.90 volumes
per volume of rock.* How prolific a source of hydrogen sulphide the organic
matter in certain shales may be, is indicated by these two experiments.
If a shale of this sort, undergoing extensive metamorphism, did not lose
all of its sulphur compounds during the transforming process, the meta-
morphic product might still be distinguished by a high content of hydrogen
sulphide. Perhaps the specimen of Baltimore gneiss obtained from Spring
Mill, on the Schuylkill River,* from which 4.91 per cent, or 0.30 volume,
of sulphureted hydrogen was extracted, may have been derived from such
a shale. Other sulphur compounds in small amounts have been noted
in the gases from rocks. The potassium hydroxide solution in the Lunge
nitrometer, after having absorbed whatever hydrogen sulphide and carbon
dioxide there may have been in the gas under analysis, frequently emits
an odor suggesting a mercaptan. When air is let into the pump and tubes,
after the removal of the gas for analysis, and then pumped out, it usually
is charged with odors of more or less offensive nature. These suggest
that other complex reactions prevail at the high temperatures employed
in extracting the gas. Gautier detected a trace of ammonium sulphocyanide
in the gas from a granitoid porphyry from Esterel.'
METHANE.
Moissan believed that the hydrocarbons of the petroleum type which
occur in the earth's crust were, in many cases, derived from the action of
water upon metallic carbides in the deep interior.^ His important researches
upon carbides form the experimental basis for the hypothesis that the
methane obtained by heating igneous rocks has resulted from these com-
pounds. Even with cold water, the carbides of barium, strontium, calcium,
and lithium give pure acetylene, while under the same conditions aluminum
and beryllium carbides generate pure methane.
CaC, + 2H,0 « Ca(OH), + C^H, A1,C, + 12H,0 = 4A1(0H), + 3CH,
The carbides of the rarer metals, cerium, lanthanum, yttrium, and
thorium, yield various mixtures of acetylene and marsh-gas; from manga-
nese carbide, marsh-gas and hydrogen result. But the most remarkable
of the carbides is that of uranium, which with water at ordinary tempera-
tures produces (in addition to a gaseous mixture of methane, hydrogen,
and ethylene) both liquid and solid hydrocarbons. Under ordinary condi-
tions water does not decompose the carbides of molybdenum, tungsten,
chromium, or iron.
These reactions suggest two alternative hypotheses to explain the
occurrence of methane in the gas obtained from igneous rocks. The most
limited of these supposes the marsh-gas to be produced from a carbide
' AnalyBis No. 42.
' Analysis No. 28.
* Gautier, Comptes Rendus, vol. 132, pp. 61-62.
* Moissan, Proc. Roy. Soc., vol. 60 (1897), pp. 166-160.
STATES OF THE OASES. 55
in the combustion-tube. Such a carbide must have withstood the action
of water, both magmatic and meteoric, ever since the solidification of the
rock. The other hypothesis seeks to avoid this difficulty by postulating
carbides in the very hot rocks where the hydrogen and oxygen may not be
combined as water; then, at a later stage, it allows water to decompose
the carbides wdth the evolution of marsh-gas, which is retained within the
rock. In this case the gas itself would exist in the rock specimen tested.
It is to be noted that several of these carbides, including that of the
widespread element calcium and the less stable sodium and potassium
compounds,^ give acetylene when decomposed by water. In none of the
rocks examined, with one exception, has acetylene been detected. This
may possibly eliminate calcium carbide from the gas-contributing com-
pounds in the rocks. The absence of acetylene also carries with it some
slight evidence against carbides in general, since calcium plays a very
important r61e in rock evolution, and it is not likely that acetylene, if
formed, would pass into methane. However, it is not impossible that
aluminum carbide, which yields methane with water, may exist in the
earth's crust while calcium carbide is lacking. According to F. W. Clarke,'
aluminum constitutes 8.16 per cent of the solid crust of the earth, while
iron and calcium comprise 4.64 and 3.50 per cent, respectively. Aluminum
also forms very stable compounds in nature. Moreover, aluminum oxide
fused in the electric furnace with calcium carbide gives yellow crystals of
aluminum carbide.' Perhaps at high temperatures iron carbide might
be decomposed by steam with the formation of marsh-gas.
Besides carbides, organic matter suggests itself as a possible source of
the methane. This organic matter may have been either (1) accidentally
introduced into the combustion-tube, or (2) have been incorporated in the
rocks from life which inhabited the earth during the later stages of growth,
as outlined by the planetesimal hypothesis. The first possibility may be
practically dismissed, since great care was exercised to avoid the intro-
duction of any foreign matter with the rock powder. If dependent upon
such accidental conditions, this gas would only occasionally be present.
Under the planetesimal hypothesis, life may have existed long before the
growth of the planet was completed and its present size attained. Organic
deposits buried in sedimentary beds which have since undergone exten-
sive metamorphism should furnish marsh-gas. These rocks, worked over
and reworked by the volcanic activity in Archean times, might perhaps
account for the widespread occurrence of this gas. Formed in this way,
it may be retained in the rocks as a free or occluded gas, since it is very
stable at high temperatures.
Other theoretical sources of methane are high-temperature reactions
within the combustion-tube, in which hydrogen and the oxides of carbon
participate. Brodie produced 6 per cent of marsh-gas by submitting
* Moiflsan, Jour. Chem. Soc., vol. 64, 2, p. 332.
» F. W. Clarke, Bull. 168 U. S. G. S. (1900), p. 15.
* Moissan, Gomptes Rendus, 125 (1897), pp. 839-^4; Jour. Chem. Soc., vol. 64
(1898), 2, p. 161.
56 THB OASES IK ROCKS.
approximately equal volumes of hydrogen and carbon monoxide to the
action of electricity for five hours in an induction-tube.* From his results
he expressed this reaction by the equation,
C0+3H,=CH,+H,0
Though it is not safe to assume that this reaction will take place at
the temperatures of the combustion-furnace, the observation that marsh-gas
is obtained when a rock powder, exhausted of its gases, is exposed to the
air for a few months, and reheated, possibly points toward some reaction
of this nature.
NITROGEN.
If the nitrogen which is obtained from heating igneous rock powders
in vacuo is derived from some chemical compounds decomposable at red
heat, a metallic nitride at once suggests itself as the most probable form in
which the nitrogen would occur. Iron nitride may be taken as the type for
discussion. While different nitrides of iron, having the compositions of
FejNj, FejN, and FejNj, have been described by some authors, a compre-
hensive study, by Fowler, of this formerly little-known compoimd, forces
the conclusion that there exists ' only one iron nitride, FesN. This com-
pound may be prepared by the action of ammonia either upon ferrous
chloride, or finely divided iron. While the action between ferrous chloride
and ammonia commences near the melting-point of lead (327°), a tem-
perature of 600° appears to be necessary for the production of iron nitride
in quantity.' The nitride can also be produced at 850° to 900°, but this
is probably the highest limit of the reaction.*
This nitride is very soluble in dilute acids, giving ammonia. When
heated to redness in hydrogen, ammonia results. At 200° it is oxidized
in the air to ferric oxide, abandoning nitrogen, which does not appear to
be oxidized. At 100° steam causes a slight evolution of ammonia. Accord-
ing to Fowler the temperature of decomposition of iron nitride in an inert
gas (nitrogen) must certainly be above 600°.
Silvestri has found iron nitride coating some of the fumarole deposits
of Etna,* and Boussingault * recognized nitrogen in the Lenarto meteor-
ite by certain tests which led him to believe that it existed there as a
metallic nitride. Silvestri discovered that at red heat the nitride from
Etna was decomposed, delivering up its nitrogen. His experiments show-
ing that iron nitride may be prepared artificially, by igniting the ordinary
lava in a current of ammonium chloride vapor, probably illustrate what
takes place in the fumaroles. But this nitride, which derives its nitrogen
from ammonia or ammonium salts, in no way requires the existence of
nitrides within the magma itself, except as a possible source for the nitro-
gen which unites with hydrogen to form the ammonia. There is danger
in transferring the characteristics of fumarole deposits, which are formed
' Sir B. C. Brodie, Proc. Roy. Soc., vol. 21 (1873), p. 246.
» Fowler, Jour. Chem. Soc, vol. 79 (1901), pp. 285-299.
» Fowler, Chem. News, vol. 82 (1900), p. 246.
* Beilby and Henderson, Jour. Chem. Soc., vol. 79, p. 1246.
» Silvestri, Pogg. Ann., vol. 167 (1876), pp. 166-172.
* Boussingault, Comptes Rendus, vol. 63 (1861), pp. 78-79.
STATES OF THE OASES. 57
in limited quantities under the quite exceptional conditions of abundant
currents of free gases and very active hot vapors, to the main magmas.
If the nitrogen obtained from rock powders be derived from a nitride,
it should be accompanied by ammonia, since, in the presence of hydrogen
or water-vapor, it is this gas, rather than free nitrogen, which is given off.
Tests made with Nessler's solution show that ammonia is one of the gases
extracted from rocks, though always appearing in limited amounts. In
the process ordinarily employed for extracting the gas, it is absorbed by
the calcium chloride drying-tube. Ammonia is scarcely to be considered
as a source of free nitrogen, since this compound is only dissociated at the
temperature of the electric spark.
Whether all of the free nitrogen can be assigned to the decomposition
of iron nitride may be tested with the quartz from New South Wales.^ Sup-
posing all of the iron in this quartz to have existed as iron nitride, and
to have been completely decomposed without the production of any am-
monia, the analysis still shows an excess of nitrogen over what could have
been produced in this way. The reaction may be taken as
2Fe,N = 4Fe + N,
102.72 gmB. quarti contained 0.0058 gm. Fe
F^ (as PeaN) required to give 1 c.c. nitrogen 0100 gm.
Nitrogen poonble from reaction 58 c.c.
introgen actually obtained (0^ and 760 mm.) 86 o.o.
Ezoeaa of nitrogen 28 o.o.
A duplicate determination of the iron in this weight of quartz gave only
0.0048 gram; on this basis, the excess of nitrogen would be still greater.
It is highly improbable that all of the iron in this quartz was combined as
a nitride. Some of it was unquestionably pyrite. To ascertain how much
of this nitrogen can be ascribed to atmospheric air adhering to the tubes,
as well as to leakage during the process of extraction, a blank combustion
was resorted to. The empty combustion-tube was kept at bright-yellow
heat for the length of time which was required to expel the gas from the
quartz. 0.15 cubic centimeter of gas was collected in the receiver when
the tube was exhausted by the pump. Adhesion of air to the quartz itself
might be supposed to increase this figure, though the material used for this
analysis was not the usual fine powder, but small fragments which would
be less liable to entrap air. In general, while iron nitride is to be accepted
as a possible source of this nitrogen, it is inadequate to produce the quanti-
ties of this gas determined by analysis. The presence of other metallic
nitrides in this comparatively pure quartz does not seem likely. Silicon
nitride, however, may be present and might possibly contribute a portion
of the nitrogen.
OOCLUDED GASES.
Though occlusion is a phenomenon but imperfectly understood, there
appear to be three different ways in which it is manifested. In the first
of these the absorption seems to be dependent upon porosity. An example
of this is charcoal, one variety of which absorbs 172 volumes of ammonia,
^ AnalysiB No. 100.
58 THB OASES IN BOCKS.
165 volumes of hydrogen chloride; 97 volumes of carbon dioxide, and 2
volumes of hydrogen.* The aflBnity of molten silver for oxygen illustrates
another phase of absorption often classed as occlusion. It has long been
known that silver absorbs 22 times its own volume of oxygen when melted,
but gives up most of this gas, often with violence, as it solidifies.' This
is properly a solution of a gas in a liquid, and not in a solid, as in the case
of true occlusion. The third type is the absorption of gases by compact
metals, either on their surface or within their mass, such as the occlusion
of hydrogen by palladium, platinum, and iron. This is, in the main, inde-
pendent of porosity.
Hydrogen is absorbed by these metals at ordinary temperatures, but
is only given off at higher temperatures. This principle was demonstrated
by Graham, who placed a thin plate of palladium, charged with hydrogen,
in a vacuum and observed that at the end of two months the vacuum was
still perfect. No hydrogen had vaporized in the cold, but on the applica-
tion of a heat of 100^ and upwards, 333 volumes of gas were evolved from
the metal.' The degree of heat required to expel hydrogen absorbed by
platinum and iron was found to be little short of redness, although the
gas had entered the metal at a low temperature. Another series of experi-
ments by the same investigator showed that, to be occluded by palladium
and even by iron, hydrogen does not need to be applied under sensible
pressure, but on the contrary, when highly rarefied, it is still freely absorbed
by these metals. These results have been confirmed by Mond, Ramsay,
and Shields,^ who found that platinum black at very low pressures absorbed
a certain quantity of hydrogen. On increasing the pressure of the hydro-
gen up to about 200 to 300 millimeters, a further quantity was absorbed,
but beyond this point an increase of pressure had comparatively little
effect. These investigators regarded 1 10 volumes as the amount of hydro-
gen really occluded by platinum black, although 310 volumes were actually
absorbed.
Experiments indicate that the quantity of hydrogen occluded depends
greatly upon the condition of the metal. When chemically reduced, cobalt
may occlude 59 to 153 volumes, nickel 17 to 18, and iron 9 to 19 volumes.*
Though common iron wire occludes only 0.46 volume of hydrogen,* this
same metal, ^when electrolytically deposited, may absorb nearly 250 vol-
umes of this gas.' The maximum quantity of hydrogen occluded by any
metal, so far as recorded, is 982 volumes absorbed by freshly precipitated
palladium.* Dumas has shown that aluminum heated in vacuo to 1400°
gives off more than its own volume of gas, consisting chiefly of hydrogen
with a little carbon monoxide, but without traces of carbon dioxide, oxygen,
or nitrogen.* Under the same conditions, magnesium rapidly expels 1.5
Barker, Textbook of Physics, p. 183.
Chimie Min^rale, Moissan, t. 1, p. 203.
Graham, Chemical and Physical Researches, pp. 283-290.
Proc. Roy. Soc., vol. 68 (1895), pp. 24^24S.
Chimie Min^rale, Moissan. t. l,p. 51.
Graham, Chemicied and Pnysical Researches, p. 279.
Cailletet, Llnstitut, Nouv. S6r., Ann. 3, p. 44.
Graham, Chemical and Physical Researchas, p. 287.
Dimias, Comptes Rendus, 90 (1880), p. 1027.
STATES OF THE OASES.
69
volumes of nearly pure hydrogen. Many other metals behave similarly.
Non-metallic substances appear to possess this property in a lesser degree.
Porcelain occludes hydrogen, whether because of its porosity or solvent
qualities is not certain. Quartz is said to be penetrable, at high tempera-
tures, by the gases from the oxyhydrogen flame,^ which points towards a
form of occlusion.
In addition to hydrogen, other gases are occluded. Litharge, when in
the molten condition, dissolves hydrogen, carbon monoxide, and nitrogen,
of which it retains a portion on solidifying.' Cast iron, on cooling, retains
4.15 volumes of carbon monoxide,' which perhaps may be due to the for-
mation of iron carbonyl, Fe(G0)4, or similar unstable compounds.
Analyses show that whenever metallic iron is present in notable quan-
tities carbonic oxide becomes an important constituent of the gas evolved.
The following analyses of the gases from various types of iron indicate
the proportions of hydrogen, carbon monoxide, carbon dioxide, and nitro-
gen which this metal may absorb, given in percentages of the total gas
content:
Tablb 36.
Iron.
White, carhonaceous, cast iron
Mild steel
Ordinary gray charcoal iron
Gray coke iron
Steel
Bessemer steel before adding spiegel —
Bessemer steel after adding spiegel
Open-hearth steel
Capola pig iron
Horseshoe nail, heated 2 hours
Same, heated 2 hoars more
Analyst.
Troost &
Hautefeoille'
Parry
Cailletet
Do
Troost &
Haatefeoille
Muller «
Do
Do
Do
Graham'
Do
H..
CO.
74.07
1
1
16.76
62.6
24.3 •
38.60
49.20
32.70
67.90
CO,.
3.69
16.66
22.27
63.65
88.8
0.7
77.0
• • • •
67.8
2.2
83.3
2.5
35.0
50.3
21.0
68.0
2.27
7.7
i Cited by Cohen, Meteoiltenkunde, p. 181.
iCited by Lane, Bull. Geol. Soc., yol. 5 (1894), p. 264.
Whatever may prove to be the ultimate significance of occlusion, and in
whatever condition these gases are stored in the iron, whether it be in the
nature of a solution, as Mendel6ef has suggested, or as definite compounds —
hydrides, nitrides, and carbonyls — the fact remains that these gases exist
within the metal and are in many respects similar to the gases locked up in
iron meteorites. Fresh iron borings from the interior of a metallic meteorite
have usually been assumed to be free from any hydration or carbonation
from terrestrial agencies, and so have been held to contain true meteoritic
gases. Some question respecting this belief has arisen from certain analyses
which, as we have seen, indicate secondary action.^ In addition to this
the gases actually received in the laboratory may not represent the original
^ Poynting and Thomson, Properties of Matter, p. 204.
' Le Blanc and Cailletet, cited by Violle, Cours de Physique, t. 1, p. 922.
* Daniell, Princii>le8 of Physics, p. 327.
* Experiments with the Toluca iron ; Analysis No. 108.
60 THE OASES IN ROCKS.
proportions on account of the reducing action of iron on carbon dioxide
and water- vapor. Meteorites of the stony type, unless absolutely fresh,
are more open to the suspicion of terrestrial hydration and carbonation.
But the Allegan meteorite gathered up, still hot, within five minutes of
its fall, has not been subjected to outdoor exposure, though it may have
absorbed a small amount of moisture and carbonic acid from the atmo-
sphere since being placed in the National Museum. It yielded somewhat
more than half of its own volume of gas.^ Fresh material from the interior
of the E^tacado, Texas, meteorite, heated in a vacuum in the presence of
phosphorus pentoxide for five hours at 150°, and then allowed to remain
untouched for several days to enable the drying agent to take up the last
traces of moisture in the tubes, still yielded at red heat 0.86 volume of gas,
of which 36.25 per cent was hydrogen.*
These gases from stony meteorites resemble those from some igneous
rocks. That this correspondence should exist, is entirely in accordance
with the view that meteorites have been derived from the disruption of
small planetary bodies of the nature of the asteroids. As in the meteorites,
so in the rocks, that portion of the gases which can not have been produced
by chemical reactions at elevated temperatures, nor from the bursting
of rock-bound cavities, may fairly be assigned to occlusion. The computa-
tions indicating the excess of hydrogen obtained from quartz and beryl '
over that which might have arisen from the interaction of iron and water
under the most generous assumptions show that, in some cases, more gas
may arise from a state of occlusion than from ordinary chemical action.
The amoimt of occluded gases may be actually greater than that indicated
by demonstrating the inadequacy of other modes of holding gas. But in
basic rocks containing hydrated minerals and an abundance of ferrous
salts, the resulting volumes of hydrogen must doubtless come largely from
the decomposition of the water of constitution, and the amount of occluded
gases, if any, is beyond determination by these methods.
The gases argon and helium, which, according to current chemical
views, do not form compounds, must exist within rocks either mechanically
entrapped or in a state of occlusion. There are those, notably Ramsay
and Travers, who believe in the combining properties of argon and helium;
but the balance of opinion seems to be on the other side, so far as ordinary
terrestrial conditions are concerned. Lord Rayleigh concludes his paper
on the inactivity of these two gases, with this sentence: ''There is, there-
fore, every reason to believe that the elements, helium and argon, are non-
valent, that is, are incapable of forming compoimds."^
As the chemists' supply of helium comes from certain minerals, chiefly
those containing compounds of uranium, its occurrence in rocks is a well-
known fact. Recent studies have revealed the existence of helium in beryl.*
Argon is perhaps more widely distributed than helium, Gautier having
detected this element in ordinary granite. The waters of many springs
^ AnalyBis No. 106.
' Analysis No. 107.
» Ante, pp. 46-48.
* Lord Kayleigh, Proc. Roy. Soc., vol. 60, p. 56.
» R. J. Strutt, Nature, Feb. 21, 1907, p. 390.
BIGNIFICANCB OF THE THREEFOLD STATE. 61
bring up both helium and argon, proving the presence of considerable
quantities of these elements within the earth. This rather wide distri-
bution, when taken in connection with their supposed chemical inertness,
strengthens the presumption that occlusion, or some form of gas diffusion,
is prevalent in rocks. Though of much interest to chemists and physicists,
these gases, on account of their comparative scarcity, do not play a very
important rAle in general geological problems. Their presence in small
quantities within the rocks of the earth's crust being established, quan-
titative determinations become of less value. In the analyses made for
this paper, whose prime purpose was to determine the range and distribu-
tion of the common gases, the separation of helium and argon from nitrogen
was not usually attempted. These gases when present are included in
the figures given for nitrogen. In the case of pitchblende and carnotite,
however, helium was so important a constituent of the gas that its pro-
portions were determined. Carnotite produced 1.28 per cent of helium,
amounting to 0.04 volume, while pitchblende gave 38.48 per cent, or 0.37
volume.^ In both of these cases nitrogen also was abnormally high.
In general, therefore, helium and argon, together with at least as much
of the other gases as can be shown not to have been produced by chemical
reactions or the biursting of inclosing walls, are to be attributed to occlusion
or some form of diffusion not distinguishable from occlusion. In many,
and perhaps most, rocks this will not be the major part, for, of the three
gas-liberating processes, that by chemical interaction imder the influence
of heat appears to be the dominating one.
SIGNIFICANCE OF THE THREEFOLD STATE.
GAS IN CAVITIES.
While chemical reactions and the phenomena of occlusion imply that
gis exists in the interior of the earth, the presence of gas inclosed in ca\d-
ties under great pressure adds the further implication that the gas often
ezeeeded the point of saturation of the magma, at least at the stage of
solidification. Cavity gases are most abundant in minerals of poorly
developed cleavage, pointing perhaps towards a strong tendency to escape
along cleavage planes during, or after, crystallization. The gas inclusions
in quartz may, however, owe their abimdance not so much to the absence
of cleavage as to the fact that quartz is generally the last mineral to crystal-
lise out of a magma, and hence such absorbed gases as did not enter into
the other crystals would become concentrated in the siliceous residue and
might supersaturate it.
It is possibly this freely-moving gas above the point of saturation which
contributes most to the mobility of lavas. Dissolved gases and vapors,
while favoring fluidity, would seem to be relatively less effective. But
the foregoing investigations imply that gases mechanically entrapped in
crystalline rocks are not very abundant, and suggest that perhaps the
theory of liquidity due to gas is overworked. On the other hand, it is
true that as the lava cooled down to the point where the last mineral crys-
* Analyses 93 and 94.
62 THE OASES IN ROCKS.
tallized, its gas-solvent powers would be increasing, allowing some of the
gas to pass into solution. At the same time free gas might be occluded by
the growing crystals. The experiments upon the reabsorption of gas by ex-
hausted rock powder indicate that a portion of the gas unites chemically as
the heat diminishes. Because of these processes, liquid lavas may be sup-
plied with free gas, even when the solidified rocks retain but little free gas.
As the imprisoned carbon dioxide frequently remains in the liquid
form up to the critical point (30.9^ C), it must be subjected to a pressure
of at least 73 atmospheres, which is the critical pressure of this gas. Since
a pressure of 73 atmospheres corresponds to a column of water 2,470 feet
in height, quartz cnrstals formed from aqueous solution, under hydrostatic
pressure simply, can not contain liquid carbon dioxide up to 30.9^ unless
developed at depths exceeding 2,470 feet. It is to be recognized, however,
that such crystals might be formed at lesser depths if mechanical pressure
operated with hydrostatic pressure or replaced it.
If the quartz crystallized from a lava, say at 1100^ C, the effect of
cooling down to ordinary temperatures upon both the size of the cavity
and the pressure of the inclosed carbon dioxide must be taken into account.
If we take the case of a cavity found to be entirely filled with carbon dioxide
at the critical point (30.9^ C. and 73 atmospheres), it is possible, by the
use of Van der WaaPs equation, to calculate the pressure to which the gas
would be subjected if the quartz were heated to 1100^. This pressure is
found to be 756 atmospheres,^ provided the size of the cavity remains
constant. But as most minerals contract on cooling, the volume of the
cavity diminishes at the same rate as though it were filled with the material
of the inclosing walls.' The coefficient of expansion of quartz is given as
0.00003618. Assuming for the sake of simplicity that the rate of expansion
does not vary greatly with changing temperatures,' quartz, cooling from
1100® to 31®, would contract to an extent of about 3.87 per cent of its
original volume. Since the contraction of the quartz diminishes the size
of the cavity and increases the pressure by 3.87 per cent, the original
pressure need be only 727 atmospheres, which corresponds to the pressure
beneath 9,100 feet of average rock. To fill cavities forming in crystals at
1100® with carbon dioxide which is so condensed that it will pass into the
liquid state just at the critical temperature when the rock cools down,
a preaaure corresponding to a depth of at least 9,100 feet, or its mechanical
equivalent, would seem to be required. If, when warmed under the micro-
scope, the liquid carbon dioxide is found to pass into the gaseous state at
temperatures below 30.9®, and the cavity contains only carbon dioxide,
or carbon dioxide and water, these must have been entrapped under a
pressure less than 727 atmospheres, or else the crystal was formed at a
temperature above 1100®.
* By starting with the equation p « ~Z/h ~ ^ ** ^^® critical point where the values
of the constants are taken as i?»0.003684: a«0.00874; &-0.0029; v»36->0.00S7,
and substituting for the critical temperature, r« 1100^+273^, the theoretical value of 756
atmospheres for the pressure at 1100^ is obtained.
^Daniell. Principles of Phvsics, p. 370.
' It would, however, slowly increase with the increase of temperature.
SIGNIFICANCE OF THB THREEFOLD STATE. 63
The estimate of 9,100 feet for the minimum depth (where weight alone
acts) at which igneous quartz crystals now containing carbon dioxide,
liquid up to 30.9^, could have been formed/ applies best to those cases in
which only carbon dioxide exists in the crystal cavities. If there are other
gases and liquids present in appreciable quantities, this figure becomes less
applicable, since the constants a (denoting an internal force or attraction)
and b (representing the sum of the spheres of influence of all the molecules
in the space i') used in Van der Waal's equation are not the same for
all gases. At how much greater depths than this the crystallization of cer-
tain specimens of quartz actually did take place, if rock weight alone was
involved, may, perhaps, be estimated by a painstaking determination of
the pressure under which the imprisoned carbon dioxide exists in these
minute cavities. This might be accomplished by piercing one of the larger
cavities while submerged in mercury or other liquid, and noting the expan-
sion of the freed bubble, as first suggested by Sir Humphry Davy.
Though naturally subject to limitations, it is nevertheless possible to
throw considerable light upon the nature of cavity inclusions by the use
of the microscope. Some of the conditions may be stated:
(1) If, at slightly under 30.9°, the cavity is entirely filled with a liquid
which completely vaporizes at 30.9°, it contains only carbon dioxide.
(2) If, at slightly under 30.9°, the cavity is filled with two immiscible
liquids, one of which passes into the gaseous state at 30.9°, the liquids are
probably water and carbon dioxide.
(3) If the cavity, when just below 30.9°, contains a liquid and an appre-
ciable gas-bubble, and the liquid does not disappear when the slide is
warmed above 30.9°, the liquid is probably water, and the bubble water-
vapor with perhaps some of the difficultly liquefiable gases, such as hydrogen,
nitrogen, or methane.
(4) If , as is often the case, the temperature at which the liquid in a
cavity disappears is found to be several degrees below the critical tempera-
ture of carbon dioxide, two interpretations are possible: either the carbon
dioxide is subject to a pressure less than 73 atmospheres, or else there is a
small proportion of another less liquefiable gas present. If the cavity be
opened and only carbon dioxide be found, the pressure under which the
gas existed, and from that something as to the conditions under which
the crystal was formed, can be computed from the temperature at which
the liquid disappeared. If another gas, such as hydrogen or nitrogen, be
found and identified, it is possible, by using an equation,' to calculate the
relative proportions of the two gases from the critical temperature of the
mixture. Thus a cavity containing a mixture of carbon dioxide and
nitrogen which had a critical temperature of 29° would hold 98.7 per cent
* This figure is based on the assumption that the quarts crystallized at 1100^; if it
Is desired to use other temperatures, they can be substituted in Van der Waal's equation
and the corresponding pressures computed.
1 ^ « W| + (l(X)--n;<, ^here t is the observed critical temperature of the mixture, t,
100
and tt are the theoretical critical temperatures of the two liquefied gases. Then n equals
the proportion by weight of the first liquid and 100 ~n equals the proportion by weight
of toe second liquid.
64 THE OASES IN ROCKS.
of the former and 1.3 per cent of the latter. A critical temperature of 28°
would indicate 98.1 per cent of liquid carbon dioxide and 1.9 per cent of
liquid nitrogen, while 27** would mean 97.5 per cent of the dioxide and 2.5
per cent nitrogen. The figures for carbon dioxide and hydrogen are of the
same general order. In the estimates of the depths at which cavity-bearing
crystals were formed, made by different methods,* it has been usual to
assume that only the pressure arising from the weight of the overlying rock
was involved.
Sorby examined those cavities which contained only water, or a saline
solution, and a vacuole left by the contraction of the liquid, as a result of
the lowering of the temperature- By noting the relative size of the bubble
and the volume of the liquid, he estimated the temperature to which the
mineral would have to be heated for the liquid to completely fill the cavity,
and from this, together with the elastic force of the water-vapor, he com-
puted the necessary existing pressure in feet of rock. The highest tempera-
ture found by this method was only 356® C, at which point Sorby belie ved
that the trachyte of Ponza solidified, while the lowest temperature was
89® C, obtained from a study of the main mass of granite at Aberdeen.
But Sorby considered it more probable that the granite crystallized at
about the same temperature as the trachyte and, assuming that the solidifi-
cation took place at 360®, he computed that the granite of Aberdeen was
formed under a pressure of 78,000 feet of rock.' These estimates are based
upon the unwarranted supposition that when the crystals were formed
the volume of liquid water was such as to just fill the cavities, and that in
each case a meniscus at once appeared with a loss of heat. He overlooked
the fact that the meniscus could not appear until the water reached the
liqmd condition, no matter at what temperature the growing crystal sur-
rounded the vesicle of highly compressed water-gas.
The highest temperature at which a vacuole of this sort can appear
must, therefore, be the critical temperature for water, or 365® C. In order
to study this problem, we may, perhaps, best take the special case in which
the inclosed water passed through its critical state (at 365® and 200.5
atmospheres pressure) during the cooling of the crystal. The vesicle formed
in this case may be termed the critical vacuole. It may be assumed that
the growing crystal inclosed the water at some temperature in the neigh-
borhood of 1100®, which is an average temperature for the solidification of
lavas. Starting thus with a cavity formed at 1100®, in order to allow the
water on cooling to pass through the critical state, an original pressure
of 1,070 atmospheres is necessary according to Van der Waal's equation,'
provided the size of the cavity remained constant. But if 2.66 per cent is
allowed for the shrinkage of the cavity while cooling down^ from 1100®
^ See Geikie's Textbook of Geology, vol. 1, pp. 144-145.
' Sorby, Quart. Jour. Geol. Soc. London, vol. 14, p. 494.
■ p— — r — =-. In this case the values of the constants for the critical point
were taken to be ««.003607: a-.01173; 6-.00151; V-36-.00453. By substituting
for the critical temperature, r« 1100 +273, the equation gives the theoretical value of
1,070 atmospheres.
♦ Ante, p. 62.
SIONIFICANCE OF THE THREEFOLD STATE. 65
to 305^, the original pressure need be only approximately 1,040 atmospheres,
a preBBure which corresponds to 13,000 feet of rock approximately.
In this critical case the meniscus appears as soon as the temperature
falls below 365^. Since pressure exerts but little influence on the volume
of liquids, the shrinkage of the water in the cavity, and hence the growth of
the gas-bubble, is largely a function of the fall in the temperature, and, with
a knowledge of the varying coefficient of expansion, the relation between the
sise of the bubble and the volume of the liquid could be computed for any
temperature. The correction for the constriction of the cavity between
365^ and 20^ amounts to a little more than one per cent, which is to be
added to the size of both the cavity and the vacuole in computation.
If these principles be true, a vapor-bubble relatively smaller than the
critical vacuole may be interpreted to mean that the meniscus did not appear
in the cavity until the crystal had cooled below the critical temperature,
t. e., that at this temperature the water was more than normally condensed,
owing to a pressure exceeding the critical pressure. On the other hand,
a vapor-bubble relatively larger than the critical vacuole means that,
although it did not appear until below 365^, it began as a sizable vesicle
when it did start, owing to tbe lower pressure and more rarefied condition
of the water-gas.
On the basis of his experiments, Sorby estimated that a vesicle amoimt-
ing to 28 per cent of the volume of the liquid in the cavity would vanish
when the water was heated to 340® C. According to this figure, a vacuole
occupj^g in the neighborhood of 30 or 35 per cent of the volume of the
liquid should correspond to a shrinkage of the water from the critical
point to ordinary temperatures. But this figure has not been confirmed
by other investigators. Unfortimately the figiures obtainable for the ex-
pansion of water up to the critical point vary within such wide limits that
it does not seem advisable at the present time to attempt to calculate the
rdative dzes of the critical vacuole and the inclosing liquid.
The difficulties involved in applying these principles are considerable.
SSrkel has pointed out that, even in cavities within the same crystal, there
is much variation in the relative volume of the vapor-bubble and the
liquid, from which the inference is drawn that the vapor-bubbles are due
to causes other than contraction on cooling.^ Before this conclusion can
be accepted with confidence, due consideration must be given to the loca-
tion of the cavities within the crystal, and also to the evidence that they
are all primary inclusions. In an ascending lava subject to a steadily
diminishing pressure, those cavities formed during the early stages of
crystallization may be developed under conditions quite different from the
cavities later inclosed in the outer parts of the crystals. If systematic
differences in the cavities can be foimd to correspond with variations in
thdr location, something might be learned of the history of the lava during
the period of crystallization. Secondary fluid inclusions, formed subse-
quent to the solidification of the magma, must obviously be recognized
and avoided, whenever possible, in attempting to estimate the conditions
under which crystallization took place.
^ Zirkel, cited by Geikie, Textbook of Geology, 1, p. 145.
66 THE OASES IK ROCKS.
A diflSculty of a more serious nature, apparently, suggested by Professor
Iddings, lies in the change of volume of the magma in the passage from the
liquid to the crystalline form. Some magmas, such as those of granitic
rocks, contract so appreciably upon crystallization that it is conceivable
that the last crystals to form, those of quartz (which also contain the most
liquid and gas inclusions) might crystallize under reduced pressures in
spaces inclosed by crystals of the minerals already formed. The relative
size of the bubble of vapor in the cavity and the accompanying liquid
would, in such cases, not correspond directly to the depth beneath the
surface at which crystallization took place, even when nothing but hydro-
static pressure affected the lava column.
In the present defective state of knowledge as to the modes and condi-
tions which obtain in lavas penetrating the shell of the earth, it is by no
means safe to assume that the pressures to which an igneous intrusion is
subject are merely those represented by the overl3ring rock or a lava
column reaching to the surface. An ascending tongue of lava may extend
to great depths and be affected by pressures brought to bear upon it in its
lower part, which might be in excess of those represented by the depth of
the head of the column, to an unknown degree. So also it is possible that
lavas may become involved in mechanical deformations and thus be subject
to special pressures in no close correspondence to their depth.
WATER AND HYDROGEN.
The reversible reactions involving hydrogen, water, and iron com-
pounds, which cause imcertainties in the extraction of gases by heat,
are also operative within the earth. In the laboratory, when either ferrous
salts and water, or ferric compounds and hydrogen, are heated in tubes
without the removal of the products, reversible reactions set in until a
condition of equilibrium is established. Hydrogen and water, ferrous and
ferric salts are all present in a state of balance. In the interior of the earth
the heated, though solid, rocks should, it would seem, behave similarly,
though hindered by the slowness of diffusion. Nor should liquid magmas
constitute any exception to the law. Both hydrogen and water-gas, theo-
retically, should be present in liquid magmas and heated solid rocks. The
chief uncertain factors are high temperatures, and pressures.
The effect of pressure on chemical equilibrium is to favor the formation
of that system which occupies the smaller volume, but if there is no change
in volume, in passing from one system to the other, the increase of pressin-e
presumably has no influence on equilibrium.^ In the reaction
3FeO+HaO :!l^ Fe,04+H,
considered as a thermochemical equation, the number of gaseous mole-
cules, and hence the volume of gas, always remains the same, so that it
is not likely that this action will be influenced by change of pressure. A
rise of temperature favors the formation of that system which absorbs
heat when it is formed.' A comparison of the amoimt of heat liberated by
oxidizing three molecules of FeO to Fe,04 and one molecule of H, to HjO
^ Jones, Physical ChemistiT. p. 514.
' Van't Hoff, Lectures on Tneoretical and Phys. Chem., Pt. 1, pp. 161-164.
SIGNIFICANCE OF THB THBEEFOLD STATE. 67
shows that, in the former case, 73;700 calories are evolved, and in the
latter 68,300; that is, 3FeO+H,0 — ► Fe,O4+H, + 15,400 calories. As
heat is evolved in this process, a rise of temperature would accelerate the
reaction in this direction less than the reverse. In other words, the higher
the temperature, the more would the formation of ferrous oxide and water
be favored as compared with the conditions at lower temperatures.
Be<^use of this, there is much reason to suppose that, at the depths
where lavas originate, hydrogen and oxygen exist combined as water,
since up to temperatures of 2000^ C, the dissociation of water takes place
only to a limited extent. If a state of equilibrium between hydrogen,
water, and the iron compoimds were established in the heated interior
where a magma originated, as soon as it commenced its way upward and
began to lose heat the condition of equilibrium would be destroyed. With
the falling temperature, the tendency to re^tablish equilibrium would
favor the formation of that system which was produced with the libera-
tion of heat, i. e., magnetic oxide and free hydrogen. In ascending lavas
which are losing heat, the tendency, therefore, is to produce hydrogen and
magnetite, or ferroso-ferric compoimds. This is doubtless an important
source for the hydrogen which is so copiously exhaled during a volcanic
eruption. At the same time, this process accoimts for the widespread
occurrence of magnetite in igneous rocks. The considerable deposits of
magnetite, formed apparently from magmatic segregation, which are com-
mon in various regions, may, perhaps, owe their origin to a combination of
causes, in which this equilibrium reaction is an important factor.
In general, these reversible reactions tend to show that it is but a short
step from hydrogen to water, and from carbon dioxide to monoxide, and
mee versa, and that all of these must occur within the earth owing to the
processes tending toward equilibrium. Whether hydrogen, in a particular
ease, occurs in the magmas in the free state, or in the form of water-gas,
therefore becomes relatively unimportant. Because of this variation of
state, the problem becomes more complex and broader in scope. For the
most part, these water-gases are to be regarded as truly magmatic, and
not derived from surface-waters penetrating to the liquid lavas, as will
be brought out later. They are here put forward as essential factors in
the evolution of the magmas from the original planetary matter.
The reactions working towards equilibrium are able to supply hydro-
gen and carbon monoxide under conditions favorable to their absorption
and retention, even if they were not originally present as occluded gases.
The sources of the gases obtained from rocks are so complex that it is
diflBcult to determine how much is to be assigned to each. Because of the
penetration of surface-waters containing carbonic acid in solution, through-
out the accessible rocks of the earth's exterior, it is likely that, in many
eases, the bulk of the gas obtained by heating powders in vacuo has been
<terived from acquired water and carbonated compounds. But in fresh
meteorites, which presumably have not been subjected to action of this
sort, occlusion is relatively more important.
From the constitution of meteorites, some of the principles of early
terrestrial evolution may, perhaps, be inferred, though the growth of the
68 THE QA8E8 IN ROCKS.
earth was probably not quite analogous, in all respects, to the formation
of the meteorites. Whether we take the meteoritic material to repre-
sent the heavier part of the original matter of the solar system, or the
stellar system, as a whole, matters little in the geologic problem. If, in
truth, the unoxidized, heterogeneously aggregated material of meteorites
be typical of the original heavy material of the earth, it becomes evident
that, in the case of our planet, other factors have been at work which are
not operative in the bodies of which the meteorites are supposed to be
fragments. These visitors from space are characterized by such minerals
as cohenite, (Fe,Ni,Co),C, lawrencite, FeCl,, oldhamite, CaS,, and schreib-
ersite, (Fe,Ni,Co)3P, which, next to nickel-iron, is the most widely distributed
constituent of iron meteorites,^ though of less importance in the stony
specimens. Such compounds imply an absence of both free oxygen and
water in notable quantities. Of like import is the absence of hydrated
minerals, such as micas and amphiboles. Water and an oxygenated atmo-
sphere appear to be the agents which are lacking in the bodies from which
the meteorites were derived, but which have been the operative factors in
working over the outer portion of the earth.
But the original source of the earth's atmosphere and hydrosphere
is taken to be gas occluded, or absorbed, in the primitive meteoritic material.
These original gases, escaping, furnished both atmosphere and hydrosphere
when the earth became of sufficient size to retain them. A self-regulating
system was inaugurated. In the early stages of the hydrosphere, when
growth by infalling planetesimals was rapid, much water was buried
within the fragmental crust. This material, worked over by volcanic
activity, brought to the surface and subjected to weathering and erosion,
and buried beneath more material, has undergone assortment and altera-
tion until the accessible rocks at the present time are very different from
the meteoritic matter. Since the earth attained its growth and the infall
of planetesimals slackened, much less water has penetrated to great depths
below the surface. Post-Archean sedimentaries have not yet reached
thicknesses sufficient to carry inclosed water down to the depths from
which the lavas arise. Deep mines indicate that fractures and fissures
do not convey water down to very great depths at the present time. If
water does not penetrate so rapidly now, and hydration and carbonation
are less effective, it is also probably true that subsiding vulcanism brings
less gas to the surface.
It is essentially a system of balance. At the same time that water is
being buried with sediment, its elements, hydrogen and oxygen, the latter
in the form of the oxides of carbon, are exhaled from the earth's interior
through volcanic outlets. But the system here suggested is very different
from the postulated limited cycle of underground water which, following
Daubr^'s famous experiment,' has crept into geologic literature as the
origin of volcanic vapors and the modus operandi of vulcanism. Instead
of surface-waters following cracks and fissures down to the hot lavas there
to be absorbed, the water already is present, and is a part of the rocks and
> Farrington, Jour, of GeoL, vol. 9, pp. 405-407 and 525-526.
' Daubrte, ^Itudes Synth^tiques de G^ologie Exp^rimentale, 1. 1, pp. 236-246.
VXTLCANISM. 69
magmas in the interior, whether actually combined as water, or as its
elements held in solution, or chemically united in other compounds. These
gaBeoua elements form an integral part in the magmas, having been vital
dusters in their development from the primitive planetary matter. That
this process of reworking has gone on to considerable depths, if we are to
start with typical meteoritic material, is evidenced by the fact that the
deep-seated plutonic rocks are characterized by micas and other hydrous
minerals, while mineral species of the meteoritic type are absent.^
The more restrictive phase of the problem of water will be discussed
under the head of vulcanism.
VULCANISM.
In the actual dynamics of vulcanism, provided the gases are original
in the magmas, the state in which they occur is not of vital importance,
except in so far as it determines the conditions under which the gases be-
come free, from occluded or chemical bonds, to perform their part in the
mobility of lavas, in the explosions which sometimes accompany erup-
tions, and in the phenomena of fumaroles and volcanic vents. The dis-
tinction between cavity, occluded, and chemically united gas, which is
made in the case of solid igneous rocks, can not be extended to the liquid
lavas. In the liquid lava the gas may be supposed to be imprisoned
mechanically, or else to form a part of the magmatic solution. On the
solidification of the mass, the gas, formerly existing in the free state, may
enter chemical cembinations at the lower temperature, may be occluded
by the solid rock, or may become entrapped within the minerals last to
crystallize. So, too, it is possible that some of the gas dissolved in the
magma may, because of cooling and crystallization of adjacent portions
of the solution, reach a supersaturated condition and appear in the solid
rock also as gas inclusions. Otherwise, it would pass into the solid rock
occluded or chemically combined. The condition of the gases examined
in the laboratory need not, necessarily, correspond to a particular state
of occurrence in the lava before crystallization.
Gases mechanically distributed throughout the lava would always be
an operative factor in vulcanism, while such gases as were chemically
combined in the solution would, presumably, only become free, and hence
fully operative, upon the lowering of the temperature and the relief of
pressure,' and probably but partially then. Since vapors and gases in the
free state are the cause of volcanic explosions, they can be traced as far.
down in the conduits as explosions occur. From the nature of these explo-
sions, which appear to be due to the accumulation of vapor gradually work-
ing upward until suddenly able to relieve itself, it is fair to suppose that
^This statement should perhaps be qualified. The basalt at Ovifak, Greenland,
contains iron stronglv resembling the meteoric metal, and in which the minerals cohenite,
lawrencite, and doubtfully schreibersite have been recognized. The occurrence of this
terrestrial iron would indicate that material of this sort still occurs at points within the
outer port of the earth.
'A falling temperature favors the liberation of hvdrogen from water by ferrous
compounds (see p. o7), while carbonates are most easily decomposed at low pressures
(we p. 49).
70 THE GASES IN ROCKS.
aqueoTis vapor and the auxiliary gases are present in the free state at
still greater depths.
It has been the observation of those who have studied volcanic erup-
tions that water- vapor is by far the most abundant of the gaseous products
of volcanoes. Water is also the principal compound of the element hydro-
gen, which is quantitatively the most important gas obtained by heating
igneous rocks in vacuo. According to one of the common theories of
vulcanism, it is water, circulating underground and necessarily dissolving
and absorbing mineral and gaseous material, which penetrates to the
lavas and gives to them their supply of vapor and gases. Water, then, is
a critical element in the theories of vulcanism, and likely to be a decisive
factor, upon the basis of which many of these theories may stand or fall.
It is, therefore, of great importance to know whether the aqueous vapor,
which is so copiously exhaled from volcanic vents and plays such a rdle in
vulcanism, is derived originally from the magmas, or is merely underground
water which has been incorporated by the lava in its journey upward. A
decision of this question will carry with it the solution of the allied question
concerning the ultimate source of the other gases, and also throw much
light upon some of the more comprehensive theories of vulcanism.
Appealing to the fact that chlorine, in the form of hydrochloric acid
and volatilized chlorides, is one of the products of volcanoes, one of the
standard hypotheses attributes the cause of vulcanism to the penetration
of sea-water to the heated interior. If this were so, isolated volcanoes
far out at sea would be expected to yield much more chlorine than those
on the continents. But the Hawaiian volcanoes exhale comparatively
little chlorine or sublimed chlorides. It has been claimed that rain-water,
sinking into the cone, would have sufficient head to exclude the sea-water
from the neighborhood of the hot lava. Rain, however, falls upon but a
small part of the whole cone, whose greater portion is under the sea. It
would seem that if rain-water, falling upon a cone built up from the ocean
bottom, is able, by means of its head, to keep out the sea-water which
covers the lower slopes, the same amount of water precipitated upon a
continental volcano would be even more efficient in preventing the general
underground water from coming in contact with the lava in the conduit.
Whatever may be the reason for the small amoimt of chlorine given off
by the volcanoes of Hawaii, sea-water does not reach the heated lavas in
sufficient quantities to affect them appreciably.
On accoimt of the pressure exceeding the crushing strength of the
rock, pores and crevices can not exist at depths greater than 30,000 feet
according to the most generous estimate,^ and it is probable that continu-
ous cracks cease much short of this. Beyond this extreme figure, meteoric
waters can not be regarded as of any quantitative importance, on accoimt
of the extreme slowness of diffusion through solid bodies not containing
minute fractures. Liquid carbon dioxide still existing under great pres-
sure in sand grains of Pre-Cambrian age is a concrete example of this
slowness. While, theoretically, water may extend downward to the limit
of the zone of fracture, the testimony of deep mining appears to show that
^ HoBkios, 16th Ann. Kept. U. S. Geol. Surv., p. 853.
VULCANI8M. 71
meteorio waters grow relatively scant, as a rule, below the uppermost
1,500 to 1,800 feet of the earth's crust.^ This shallowness of meteoric water
increases the difficulties encountered by the hypothesis that the lava beds
are supplied from this source, since they rise from far greater depths and
only the upper portions of their conduits would be exposed to these waters.
It is in this portion of the zone of fracture that Daubr^'s much quoted
experiment upon the Strasbourg sandstone ' finds its application, if any-
where, since numerous capillary pores with plenty of water are requisites
for the operation of this principle. This famous experiment demonstrated
that, owing to its force of capillarity, boiling water will pass through a
disk of sandstone, 2 centimeters in thickness, against a slight steam-pres-
sure on the other side. But it was only necessary for the steam-pressure
to reach 686 millimeters, or nine-tenths of an atmosphere, in order to
prevent any more water from passing through the sandstone. It is a long
jump from this trivial capillary force, equal to less than one atmosphere
of steam-pressure, to the great pressures which would have to be overcome
in the depths of the earth's crust in order to reach the hot lavas, even though
it be allowed that the water-vapor, if it came in contact with the lava,
would be absorbed. Capillary force seems quantitatively inadequate.
To reach the critical pressure of water due to the hydrostatic column,
it is necessary to penetrate the earth to a depth of about 6,900 feet. At
depths less than this, water passing into the vaporous condition, in the
neighborhood of hot volcanic conduits, at temperatures below the critical
point, should leave behind more or less of the matter held by it in solution,
since the condensation, and hence molecular attraction of the vapor for
solutes, is less than that of the water. Thus even if vapor from underground
waters should enter the lavas, as Daubr^ has suggested, in the outer
6,900 feet of the earth's crust, much of the chlorides, sulphates, carbonates,
and silicates, dissolved in the water, would have been left behind. At
depths between 6,900 feet and 25,000 feet, beyond which water can not
penetrate, owing to the closure of all pores by the pressure of superin-
cumbent rock, mineral matter dissolved in the water would probably still
remain in solution when the liquid passed into the gaseous state at the
eritieal temperature, since the density of the gas is equal to, or greater
than, that of the liquid.
The lava, being under considerable pressure, may be supposed to occupy
aQ the cracks and crevices in the adjacent rocks, except those of capillary
dimensions. If, therefore, in the passage of underground water into vapor,
preparatory to entering lavas in the outer 6,900 feet of the earth's crust,
much of the dissolved mineral matter be deposited in the minute pores
leading to the lava, they should quickly become sealed, preventing any
farther access, even of water, to the lava. To test this principle experi-
mentally, a cylinder of medium-grained Potsdam sandstone from Wis-
conon, 40 millimeters in diameter and 28 millimeters in thickness, was
soldered into a short piece of iron piping, fitted at one end with an elbow
* Kemp, Economic GeoL, vol. 2 (1907), p. 3; Finch, Proc. Col. Sci. Soc, vol. 7 (1904),
pp. 193-252.
' Daubrfe, Etudes Synth^tiques, t. 1, pp. 236-246.
72 THE OASES IN ROCKS.
to serve as a receptacle for water, and at the other with a cork and a con-
denser. When ready, the receptacle was filled with Lake Michigan water
and a Bunsen burner was placed so as to heat the sandstone cylinder within
the iron tube. One side of the sandstone was thus kept at a temperature
slightly above 100°, while the other face, in contact with the water, remained
just at the boiling-point. Water was found to penetrate the porous cylinder
readily, evaporating and leaving its dissolved material within the mass
of the sandstone, and escaping as steam on the farther side. The rate at
which the water passed through the sandstone at the outset was not deter-
mined, but after 5 liters of lake water had been used, it was found that
129 cubic centimeters traversed the rock and were condensed in one hour.
The rate slowly fell as the experiment progressed. While the thirteenth
liter was being used, only 73 cubic centimeters passed through the sand-
stone per hour. It was evident that the pores were becoming clogged,
but to complete the experiment with Lake Michigan water, which contains
only 150 parts of solid matter, per million, would have required too much
time. To hasten the process, a saturated solution of calcium sulphate
was substituted. This soon caused a marked slackening of the passage of
water through the rock, and doubtless would have sealed the pores com-
pletely, if allowed sufficient time.
From thb experiment, it appears certain that water, evaporating in
the pore spaces of a rock and escaping as steam, will leave behind what-
ever material is in solution, until the crevices become clogged and the
penetration of water ceases. This principle may be applied to the outer
6,900 feet of the earth's crust; in the superficial portion of this zone it
should be very effective, since the conditions more nearly approach those
of the experiment; in the lower portion of this belt, as 6,900 feet and the
critical pressure (as well as temperature in the neighborhood of hot volcanic
pipes) is approached, the density, and hence the solvent powers, of the
water-vapor approach those of the liquid. The vapor, also, should escape
less readily from the liquid at these depths, since the expansive force of
the vapor drives the water back along its path with more difficulty. Toward
the critical point of water, therefore, the application of this principle
becomes more uncertain, but it would seem to be operative also at these
depths, though more and more slowly as the critical point is neared.
It might be objected that the passage of water into vapor, involving
the latent heat of steam, would keep the adjacent rocks cool and cause the
deposition to take place at the very contact where the hot lava could fuse,
and dissolve, the precipitated salts. But it is very doubtful whether the
vaporization of such a small quantity of water, taking place with the slow-
ness imposed upon it by the minuteness of the capillary pores, would keep
the contact rocks at a temperature below 365°. The gap between 365°
and 1100° is too great for there not to be a space, if of a few inches only,
at an intermediate temperature. It is also to be remembered that the
latent heat of steam diminishes with the pressure until, at the critical
point, it becomes zero. The testimony of the country rocks through
which a volcanic conduit has passed is that metamorphism has usually
progressed to some distance from the contact of igneous intrusion. In a
long-established volcano, where the rocks surrounding the conduit have
VTJLCANISM. 78
been heated to high temperatures, the deposition of the solutes from any
penetrating water should have sealed the capillary tubes and fissures at
a distanee from the lava such that the latter cannot absorb them and
keep the water-way open. Kemp has stated in a recent paper ^ that at the
eontaets with eruptives, limestone rocks, instead of being porous, are
prevailingly dense and compact, and often very hard to drill, as if due to
deposition within their interstices. However, the author assigned this
supposed deposition to magmatic waters from the intrusion. This brings
up a widely established view that magmas, instead of absorbing water from
the intruded rocks, give it off, depositing matter in solution to form veins in
the sone of fracture.
To quote Van Hise : '
In the bdt of cementation, in oonsequenoe of the porosity of that zone, the material
of the magma, both by direct injection and by tranamisBion through water, may pro*
foondly affect the aTerage chemical composition of the intruded rock for great distancee
from the Intrusive mass.
Geikie cites a case in Bohemia, where certain Senonian marls, invaded
by a mass of Tertiary dolerite, begin to get darker in color and harder
in texture at a distance of 800 meters from the contact, while, as the intru-
sive mass is approached, the interstratified beds of sandstone have been
indurated to the compactness of quartzite.*
But considering only meteoric waters at depths greater than 6,900
feet, where water remains liquid up to the critical temperature, it is less
probable that the pore spaces will be filled up in this manner. Nor does
it seem likely that Daubr^'s theory that water may penetrate rocks
against a steam-pressure can operate at these depths, since that principle
is dependent upon a marked difference between the capillarity of water
and of steam, while at the critical point, the density of water-gas being
the same as that of water, this force should be absent. The problem then
becomes a question of equilibrium between the hydrostatic column of
water and that of the lava, in which the pressure of the lava at a depth of
7,000 feet should be in the neighborhood of 2.7 times that of the water,
though this preponderance steadily diminishes as the water-gas becomes
eondensed, with increasing depth, at a rate higher than lava. Whether
under these conditions lava can absorb water-gas, is an open question.
Water can only penetrate from 25,000 to 30,000 feet below the surface
on account of the closure of all crevices by pressure. But on the assumption
thai the temperature gradient in the outer part of the earth's crust is 1^ C.
for each 100 feet of descent (which is probably too high) the critical tem-
perature will not be reached, except in the neighborhood of volcanic intru-
sions, until at a depth of about 36,000 feet. Hence, over the greater part
off the earth, water will remain in the liquid state as far down as fractures
and fissures will allow it to seep, and no appeal can be made to the more
rapid and potent gaseous diffusion to carry it beyond 30,000 feet. But
because of their heat, lavas must originate at much greater depths below
the surface, and hence far beyond the reach of surface-waters, which can
* Kemp. Economic GeoL, vol. 2, p. 11.
* Van Hise, Monograph 47, U. S. G. S.. p. 714.
'Hibsch, dted by Geikie, Textbook of Geology, vol 2, p. 774.
74 THE GASES IN ROCKS.
only come in contact with them, and only doubtfully then, in a very limited
portion of the throat of the volcano.
These considerations seem to indicate that, for the most part, the volcanic
gases and vapors have not been supplied to the lavas by ground waters,
but are original constituents of the magmas. Doubtless at the beginning
of an eruption, following a period of quiescence, much of the steam merely
comes from such rain-water as may have accumulated in the crater and
upper part of the cone, but this does not account for the gaseous emanations
from the lava itself, nor from those volcanoes, such as Stromboli, and the
well-known Solfatara near Naples, which maintain a mild form of eruption
for long periods. Such meteoric water could contribute to the volcanic gases
little except some dissolved air, together with a trace of carbon dioxide,
and perhaps hydrogen from chemical action. Such soluble salts as this
water might dissolve from the crater walls were brought up from the in-
terior in the first place (making some allowance, however, for weathering),
and so have little bearing on the case.
The hypothesis that the gases and vapors are originally from the mag-
mas, is greatly strengthened by the volcanic activity in the moon, if, as
is rather generally believed, the great pits on the surface of the moon are
craters produced by volcanic explosions; if not, of course the argument
does not hold. The gases and vapors which caused the tremendous out-
bursts can not be ascribed to the penetration of surface-waters and gases,
for the moon has neither appreciable atmosphere nor hydrosphere, and,
according to Stoney's doctrine, never could have held either, owing to its
feeble gravitative control. Such gases as are implied by these explosions
must be supposed to have arisen from within the interior of the moon.
The extent of this explosive lunar vulcanism, in the absence of any appre-
ciable atmosphere or hydrosphere, furnishes a strong argument against
the belief that surface-waters and atmospheric gases are essential factors
in terrestrial vulcanism.
Thus far evidence of a negative nature has been brought forward to
show the difficulties in the way of thinking that surface-waters play a
prominent rdle in volcanic phenomena. But more positive evidence can
be presented to support the view that the hydrogen and water in the deep-
seated rocks are truly magmatic. Micas are prominent constituents of the
plutonic rocks. The immense granitic bathyliths, which were probably
formed beyond the reach of ground-waters, are characterized by this group
of minerals. In fact, micas are more abundant in the deep-seated rocks
than in the surface lavas of similar composition. Yet all micas contain
hydrogen (or hydroxyl) and yield water upon ignition. This varies with
the minerid species and locality, ranging up to 4 or 5 per cent. If these
micas in the missive intrusions are primary minerals, as they seem to be,
and were out of the reach of ground- waters until long after they were crys-
tallized, there appears no other alternative than to consider this hydrogen
as inherent in the magma itself. The general petrological principle that
plutonic rocks are micaceous and hornblendic, while their more superficial
equivalents are more frequently characterized by pyroxenes which are
less hydrous, may point toward the suggestion that the magmas originally
contain considerable water or the elements which can produce it, but as
VULCANISM. 75
they approach the stirface much of the hydrogen and water-vapor escapes
and pyroxene minerals crystallize instead of these hydrous micas.
All of these facts and deductions lead to the general conclusion that
our surface-waters have been derived from the interior of the earth, and
oppose the idea that to explain the presence of hydrogen, or water, in
magmas and rocks, we have merely to appeal to the penetration of surface-
waters. The meteoric waters are limited to their superficial place and
function, both in the evolution of magmas and in vulcanism; an ultimate
source is found for these waters; and a steady supply of water and gases
is furnished to the earth to offset the loss of vapor into space, and thus
contributes to the globe one of the factors necessary to a long period of
habitability for living organisms.
VOLCANIC GASES.
The gases which escape from fumarolic vents are in many respects
similar to those obtained by heating igneous rocks in vacuo, but with the
addition of oxygen and vapors of chlorides, fluorides, boric acid, and other
high-temperature volatilizations. Though nitrogen is much more con-
spicuous in the analyses of volcanic gases than in those from rocks, this is
doubtless due, in the main, to a mixture with atmospheric air. However,
the greater heat of the volcano would also favor a higher proportion of
nitrogen, as shown by my experiment. Much of the oxygen also is probably
from the air. But an analysis of gas escaping from a stream of lava
flowing on the sea bottom at Santorin gave Fouqu^: oxygen, 21.11 per
cent; nitrogen, 21.90 per cent; and hydrogen, 56.70 per cent.^ This would
suggest that the dissociation of water also contributes free oxygen.
Fouqu^'s studies at Santorin confirm the law of variation in composi-
tion of volcanic gases, first established by Sainte-Glaire Deville,' namely,
that the nature of the gas evolved depends upon the phase of volcanic
activity. Hydrochloric acid, with free chlorine and fluorine, is given off
only from the hottest fumaroles where the heat is sufficient to liberate
these gases from chlorides and fluorides. At less active vents, sulphur
dioxide is the most noticeable of the corrosive gases, while the cooler fuma-
roles exhale chiefly hydrogen sulphide, carbon dioxide, and nitrogen.
Carbon dioxide and nitrogen escape from all the fumaroles. Fouqu6
found that the relative importance of hydrogen increased with rise of
temperature, and that his marsh-gas (which, owing to an imperfection in
the method of analysis in 1867, may have been carbon monoxide, or a
mixture of carbon monoxide and marsh-gas) diminished as the activity
increased. These observations are entirely in accord with the results of
my differential temperature experiments with rock powders. Hydrogen
sulphide and carbon dioxide are the gases expelled from the rocks at the
lowest temperatures; carbon monoxide and marsh-gas appear at inter-
mediate temperatures, while hydrogen is most prominent when the heat
is carried to bright redness. Nitrogen is most abundantly liberated at red
heat; hence the presence of that gas at the cooler vents and fissures is
chiefly due to atmospheric air.
* Fouqu6, Santorin et see Irruptions, p. 230.
> Sainte-Oaire Deville, Ann. de Chim. et Phys., 52 (1858), p. 60.
76 THE GASES IN ROCKS.
While carbon dioxide escapes from all fumaroles in greater or less
degree, it is at those vents whose activity has subsided beyond the point
where hydrogen and the noxious gases are evolved that this gas is most
conspicuous. For this reason, carbon dioxide has come to be regarded as
marking the dying of the volcanic activity. A source for carbon dioxide
after the disappearance of the other gases has been sought in the neigh-
boring limestone formations, either from baking or from the chemical
action of halogen or sulphur acids. The obvious difficulty confronting
this conception is that limestone is not always present to furnish carbon
dioxide. Experiments show that below 400^ G. carbon dioxide is the
principal gas evolved from rock material, and as the lava solidifying in
the crater, or conduit, has net lost all its gas, it is only a part of the natural
sequence of events that the escape of carbonic anhydride from the cooling
lavas should continue for some time after the volcano has settled into
quiescence. Some of this carbon dioxide doubtless also comes from previ-
ous lavas which, warmed again by the fresh lava, give up some of the carbon
dioxide which my experiments show them to contain.
AMMONIUM CHLORIDE DEPOSITS.
Among the various substances which are deposited around fumaroles,
sal-ammoniac, or ammonium chloride, is, in some respects, one of the most
remarkable. Compounds of ammonium have not yet been recognized in
igneous rocks, although rock powders often give o£f small quantities of
ammonia gas when heated in vacuo. Chemical analyses of spring-waters
report ammonium salts only in traces, such as may have been derived
from the decay of organic matter. If ground-waters be, for the most part,
unable to reach the lavas, even this rather doubtful source of ammonium
compounds is not available. If the elements of the radical NH4 be supposed
to have come from the interior magma, there are two alternative hypotheses
still open. The first assumes that the radical NH4 existed intact in the
magmatic solution in the form of ammonium salts and, volatilized by the
heat upon the relief of pressure, gradually collected on the cooler portions
of the crater. This hypothesis must, however, explain the apparent absence
of these compounds in igneous rocks. The second believes that the am-
monium chloride was formed synthetically in the throat of the volcano,
from the nitrogen, hydrogen, and hydrochloric-acid gases. This would make
it a direct product of volcanic gases.
The presence of ammonia, or its vaporized salts, in volcanic emana-
tions leads to the formation of another interesting compound. Silvestri ^
has foimd iron nitride, as a lustrous metallic deposit, at a fumarole on
Etna. This compound is due either to a reaction between the sublimed
ferric chloride and free ammonia gas or to the ignition of the iron-bearing
lava in the presence of ammonium chloride vapor. The appearance of iron
nitride around fumaroles throws no direct light upon the question of its
existence in the magmas, though it indirectly leads to the hypothesis that
the nitrogen in the ammonia and its compounds came originally from iron
nitride within the magma.
» Savertri, Pogg. Ann., vol. 167 (1876), pp. 166-172.
8UBTBBRANBAN QASBS. 77
SUBTERRANEAN GASES.
The atmosphere id now being fed by gases which escape through out-
lets other than those of active volcanoes. Work in the shafts of many
deep mines in different parts of the world is often impeded by the exhala-
tion of gases from the rocks. This iS; of course, familiar in the case of
organic rocks, such as coal, in which the decomposition of organic substances
is in progress. Reference is here made especially to gases escaping from
crystalline or other inorganic rocks. An exhalation of this kind is a
notable phenomenon in several of the mines in the Cripple Creek region of
Colorado, where nitrogen and carbon dioxide are poured into the workings
in considerable quantities when the barometer is low.^ Two analyses of the
gas escaping into the Conundrum mine at Cripple Creek gave the following:
1st: Carbon dioxide, 10.2; oxygen, 6.7; nitrogen, 84.1; total, 100.
2d: Carbon dioxide, 8.3; oxygen, 10.2; nitrogen, 81.5; total, 100.
No carbon monoxide, marsh-gas, or hydrocarbons were detected.
The gas from the Elkton mine, which was analyzed by Dr. A. W. Browne,
of Cornell University, consisted of nearly the same gases as from the Con-
undrum mine: Water-vapor, 1.4; carbon dioxide, 14.7; oxygen, 5.6; nitro-
gen, 76.8; argon, 1.5; total, 100.0. Hydrocarbons, methane, and hydrogen
were absent.' The authors estimate that this gas may be considered to be
25 per cent of air, 59 per cent of nitrogen and argon, 15 per cent of carbon
dioxide, and 1 per cent of water-vapor. The gas apparently is derived
from greater depths than those at which it issues, since it is warmer than the
air of the mines, and since practically no gas was encountered in the oxidized
zone. They regard the outpouring as the last exhalation of the extinct
volcano, around whose neck the Cripple Creek mines are located.
In some of the potash mines in the vicinity of Strassf urt trouble is caused
by the escape of combustible gas into the workings. According to Precht,'
blowers of this gas once lighted have burned continuously for periods as long
as two months. An analysis of this gas by Precht shows it to be largely
hydrogen. His figures are: Hydrogen, 93.05; methane, 0.778; carbon dioxide,
0.180; carbon monoxide, trace; oxygen, 0.185; nitrogen, 5.804; total, 100.002.
This investigator believed that but little of the hydrogen could have come
from the decomposition of organic matter; instead, he sought a source for it
in the possible oxidation of ferrous chloride in the salt by water, according
to the reaction:
eFea, + 3H,0 = 2Feaae + Fe,0, + SH,
This source of hydrogen is somewhat analogous to the production of the
same gas by the action of water upon ferrous compounds at high tempera-
tures, which has already been discussed, except that in the salt beds the
supposed action has taken place at the ordinary underground temperature.
But these gases coming from the sedimentary salt beds of the Upper Per-
mian represent, of course, gas merely restored to the atmosphere, and not
an original contribution to it.
^ Lindgren and Ransome, Prof. Paper 54, U. S. G. S., pp. 252-270.
> H. Precht, Ber. Deutsch. Chem. GeeeU., vol. 12 (1879), pp. 557-561.
* Loe, cit.
78 THE QA8ES IN ROCKS.
Nitrogen with an abnormal amount of inert gas (probably both argon
and helium) occurs, under high pressure, in a gas-well at Dexter, Kansas.^
However, instead of being derived from igneous rocks, this comes from a
gas-bearing sand near the contact of the Permian with the Upper Carbonif-
erous. An analysis of this gas gave:' Oxygen, 0.20; methane, 15.02;
hydrogen, 0.80; nitrogen, 71.89; inert residue, 12.09; total, 100.
Neither carbon dioxide nor carbon monoxide was present in this gas.
The methane, and perhaps the hydrogen also, may be attributed to the
decomposition of organic matter, since natural gas-wells exist at no great
distance away. But the remarkable feature of this analysis is the large
amount of nitrogen with the very abnormal percentage of inert gas.
From this analysis, and the testimony of many spring-waters which give
off considerable quantities of argon and helium, it would appear that
gases often collect underground somewhat in proportion to their chemical
inertness. The chemically active gases apparently are more largely retained
within the rocks by combination, while nitrogen, having less power to
unite chemically, more largely escapes from the rocks and accumulates in
reservoirs. Argon, still more inert than nitrogen, thus may reach such a
high proportion as 12 per cent.
GENERAL RELATIONS.
RELATIVE TO THE HYPOTHESIS OF A MOLTEN EARTH.
These studies show that, within the range of temperature employed,
heat causes the expulsion of gases in whatever form they are held, and
that the greater the degree of heat the more quickly and completely the
gases are given off. There is reason to believe that this principle applies
to the molten state as well as to the solid condition. If it be applicable
to liquid lavas, it would favor the belief that a molten globe would have
boiled out most of its gaseous matter before solidifying. Gases near the
surface should escape rapidly. It might, perhaps, on first thought, be held
that, while much of the gas in the outer portion would be lost, that exist-
ing in the central part of the sphere would be retained and slowly recharge
the peripheral portion after a crust had formed and prevented further
escape; but the molten globe, by hypothesis, grew up gradually, and essen-
tially every part was once superficial. Even to-day, in an essentially solid
earth, there are movements of lava that bring up gases from unknown
depths, and it is reasonable to suppose that the molten sphere was stirred
up by still more effective convection currents which facilitated the expulsion
of gases and vapors, and that almost all of the gaseous material of the
globe would have been boiled out before solidification set in.
The complete validity of this view depends much upon the fate of the
gases after they have reached the surface. If they were retained in the
form of a dense atmosphere, a condition of pressure-equilibrium might
be established between the atmosphere and the gases in the liquid earth,
by means of which the latter would retain some appreciable amount of
gas. But if, as some believe, our atmosphere is about all that the earth
1 Haworth and McFarland, Science, vol. 21 (1905), pp. 191-103. > Loc. cit.
GENERAL RELATIONS. 79
ean control/ the gas expelled from the molten sphere in excess of the mass
of the present atmosphere would escape and be lost to the planet. Geo-
logical evidences — early Cambrian glaciation, Paleozoic periods of aridity,
and the general testimony of life — all point toward the conclusion that
eariy terrestrial atmospheric conditions were not radically different from
those of to-day. If the hypothesis of a heavy atmosphere be not permissi-
ble! it becomes very difficult to explain the presence of original gases and
gas-producing compounds in plutonic rocks on the basis of the Laplacian
or other hypotheses that postulate original fluidity.
RELATIVE TO THE PLANETESIMAL HYPOTHESIS.
After the gaseous matter of the ancestral sun was shot out from the
solar surface to form the two arms of the spiral nebula, as postulated by
the planetesimal hypothesis, the rock-producing portion is supposed either
to have aggregated into planetesimal bodies, or to have been gathered,
molecule by molecule, into the nucleus of the earth. The planetesimal
bodies gathered in gas molecules of the atmospheric class both by chemical
union and by surface adhesion or occlusion. As the earth grew by sweep-
ing in the planetesimals, whatever gases they contained became entrapped
in the body of the growing planet and well distributed throughout its
mass. At first, the gravity of the earth may possibly have been able to
hold only the gases brought in by planetesimal aggregates of rock material
and those that became impounded in it by impact, but at a later stage,
when increased mass enabled it to hold gaseous molecules, gases may have
been added to the atmosphere directly from the nebula, and these, by
chemical reactions, may have become united with the surface rocks. As
soon as vulcanism commenced, a system of exchange was set up. While
gases were being fed to the atmosphere by volcanic action, water, carbon
dioxide, oxygen, and nitrogen were being buried with the surface rock
material, partly by chemical union and partly by mechanical entrapment,
as the growth by infalling matter continued. It is thus quite easy to
understand how the earth came to be affected by these gases throughout
its mass, and how they came to exist there in all available forms of retention.
While the carbon monoxide and methane derived from rocks by heat-
ing in vacuo are doubtless chiefly produced from the carbon dioxide and
water present in the rock material, there seems good reason to suppose
that similar reactions took place within the earth, as the surface material
became buried and heated, and hence that carbon monoxide and methane
exist, as such, in the earth's body, and are to be reckoned among the natural
gases of the rocks.
RELATIVE TO ATMOSPHERIC SUPPLY.
The fact that many of the igneous rocks are able to yield hydrogen
firom reactions between water and ferrous compounds, at high tempera-
tures, indicates that the material of the earth's crust is in a condition of
partial oxidation only. Near the center of the earth there is probably
very little oxygen, and even up to the surface, barring the weathered
> R. H. McKee, Science, vol. 23 (1906), pp. 271-274.
80 THE GASES IN ROCKS.
mantle, the rocks are suboxidized. Yet the earth is surrounded by an
oxygenated atmosphere. Since oxygen is not developed in the combustion-
tube, and does not appear to exist as a free gas in igneous rocks, it is not
likely that this constituent of the atmosphere has come directly as an
exudation from the interior of the globe. It is to be sought, rather, in a
dissociation or decomposition of compound gases by physical or organic
agencies. Originally, enough oxygen was derived from water-vapor, by
physical means, to permit the beginning of plant life; after vegetation ap-
peared, an abundant source of oxygen was found in the carbon dioxide.
The average gas content of igneous rocks, as determined by the analyses
now made, may be used to test the competence of the rocks to yield the
present atmosphere. Taking the average volume of nitrogen per volume
of rock to be 0.05, which b probably nearer the truth than the figure 0.09
given in table 16 ^ (owing to leakage of air), it would require the liberation
of all the nitrogen in the outermost 70 miles of the earth's crust to produce
the nitrogen in the present atmosphere. For an estimate of the amoimt
of igneous rock necessary to yield the carbon dioxide which is now locked
up in limestone and coal deposits, we may take Dana's figure of 50 atmo-
spheres of this gas, and an average of 2.16 volumes of carbon dioxide per
volume of rock. To produce these 60 atmospheres of carbon dioxide, it
is found that a thickness of 66 miles of crust would have to be deprived
of its carbon dioxide' — a figure which corresponds fairly well with the
estimate for nitrogen. If the water of the rocks be placed at 2.3 per cent,
a depth of 70 miles would supply the hydrosphere.
On the planetesimal h3rpothesis, gas has been supplied from the interior
to the atmosphere ever since an early stage of the earth's growth, prob-
ably from the earliest stage at which an atmosphere could be held, which
' may be placed at the time when the earth's radius was about 2,000 miles.
From this it appears that only a small fraction of the full gas-producing
possibilities of the rocks of the earth was required to supply the atmo-
sphere. The fact that gases are still being given forth through volcanoes,
and that the ejected lavas still have gas-producing qualities, makes it clear
that all the resources of the interior are not yet exhausted. The working
qualities of the planetesimal hypothesis, therefore, do not seem to be found
wanting in either past possibilities of supply, present output, or prospective
reserve.
ACKNOWLEDGMENTS.
In conclusion, I wish to express my special thanks to Dr. Julius Stieglitz
for constant advice in the conduct of the chemical researches; to Dr. Oskar
Eckstein for much valuable assistance in the laboratory; to Dr. R. A.
IGllikan for helpful suggestions pertaining to physical principles and the
designing of new pieces of apparatus ; and to my father. Dr. T. C. Cham-
berlin, for proposing the investigation, and for constant sympathy and
criticism during the progress of the work.
' Ante, p. 28. ' The limestones, of oourae, are not here included.
WiniSIinONS 1» COeilOGOKT aid tee FUXDAIiraTAL PSOBLEm or OEOUIfiT
THE TIDAL AND OTHER PROBLEMS
T. C. OHAKBEBLIN, F. B. MOtJI/TON, 0. 8. SLIOHTBB,
W. D. HaoHILLA^, ABTHDB 0. LXTNlf,
AKD JULIUS ariBQLrrz.
WASHraOTON, D. C.
r TBI CAurKia ImrmmoM or Wuhukhov
1009
GABNEQIE INSTITXJTION OF WASHINGTON
PUBUOATION No. 107
PRE88 OF J. B. UPFINCOTT OOMPAKY
PHILADELPHIA
I. THE TIDAL PROBLEM.
The Fobmeb Bates of the Eabth's Rotation and theib
Bearings on its Deformation. By T. C. Chambesiin.
The Botation-Pebiod of a Hetebogeneous Sphebold.
By C. S. Sliohtbb.
On the Loss of Enebgy by Fbiotion of the Tides.
By W. D. MaoMillan.
On Cebtain Belations among the Possible Changes in
THE Motions of Mutually Attbacting Sphebes when
Distubbed by Tidal Intebactions. By F. R. Moulton.
Notes on the Possibility of Fission of a Contbacting
Rotating Fluid Mass. By F. R. Moulton.
The Beabing of Moleoulab Activity on Spontaneous
Fission in Gaseous Spheboids. By T. C. Chambeblin.
n. GEOPHYSICAL THEORY UNDER THE PLANETESfflAL HYPOTHESIS.
By Abthub C. Lunn.
m. RELATIONS OP EQUILIBRIUM BETWEEN THE CARBON DIOXIDE OP THE ATMOS-
PHERE AND THE CALCIUM SULPHATE, CALCIUM CARBONATE, AND CALCIUM
BICARBONATE OF WATER SOLUTIONS IN CONTACT WITH IT.
By Julius Stiegutz.
CONTENTS
PAOB8
Tbx FoBifBB Ratbb of thb Earth's Rotation and thbib Bbabingb on its
DSPORIIATION l-fi9
The Astronomical DeductioDS d-20
CoDsideratioDS Based on the Older Cosmogonies 6-8
Considerations Based on the Planetesimal Theory 8-14
Genesis of the Moon 14-20
Evidences of a Present Change of Rotation 20
Deductions from the Tides Theinielyes 21-46
Tidal Phenomena of the Atanosphere 21-23
Tides of the Uthosphere 23-25
Pulsations of the Uthosphere 25-31
Tides of the Hydrosphere 31-37
More Radical Mode of Treatment 37-46
Geological Eyidenoes 46-59
Evidences from the Lithosphere 46-54
Evidence from the Hydrosphere 54-57
Minor Evidences 58
Accelerative Agencies 58-59
Conclusion 59
Ths Rotation Pbriod of a Hstbroobnboxtb Sphbboid 63-67
Thb Lobs of Enbbqt bt Fbiction of thb Tidbs 71-75
Cbbtain Rblationb among thb Pobsiblb Chanobs in thb Motions of Mdtu-
ALLT Atthacttino Sphbbbb whbn Disturbbd bt Tidal Intbbactions. 79-133
Introduction 79-83
General Ek^uations 84-86
Lapladan Law of Density 87
Moment of Loertia for the Laplacian Law of Density 88
Special Case i-0 h-0 a,-0 <-0 5-0 89-94
Application of Section V to the Ek^h-moon System 95-99
Case i-0 h-0 i,-0 a,+0 «-0 5-0 100-104
Application of Section VII to the Ek^h-moon System 105
Application to Binary Star Systems 106-110
Casei-O t\-0 i,-0 a,-0 «=»=0 5-0 111-115
Application of Case X to the Earth-moon system 116
Caset+O h-0 a,-0 «-0 5-0 117-118
Api^cation of Case XII to the Earth-moon System 119
Case i-0 i,-0 «-0 a,-0 5=1=0 120-125
Secular Acceleration of the Moon's Mean Motion 125-126
Summary 127-133
NoTBS ON thb Pobbibiutt of Fission of a Contractino Rotating Fluid
Mass 137-160
Introduction 137-138
Ellipsoidal Figures of Equilibrium of Rotating Homogeneous Fluids 139-141
Poincar^'s Theorems Respecting Forms of Bifurcation and Exchange of Sta-
bilities 142-144
Figures of Equilibrium of Rotating Heterogeneous Fluids 145-147
Ekjuations of Equilibrium for Constant Moment of Momentum 148-149
Application to the Solar System 150-152
Ai^cation to the Binary Stars 153-157
Sunmuuy 158-160
m
IV CONTENTS.
Bbabino of Molecular Activiti on Spontaneous Fission in Gaseous Sphe-
roids 163-167
Qbophtsical Theory under the Planstebimal Hypothesis 171-231
SynopsiB 171
Introduction 172-179
Theory of Fisher .* 180-200
General Equations 180-184
History of the Compression 184-190
Thermal Problem 190-200
Critical and Supi^ementary 201-218
Thermodynamic Theory 219-231
The Relations of EQumnRixTM between the Carbon Dioxide of the Atmos-
phere AND the Calcium Sxtlphate, Calcium Carbonate, and Calcium
Bicarbonate of Water Solutions in Contact with it 235-264
Ek^uilibrium in Aqueous Solutions of Calcium Carbonate, Calcium Bicarbon-
ate, and Carbonic Add in Contact with an Atmosphere containing Carbon
Dioxide, and the Solubility of Calcium Carbonate in Water containing Free
Carbonic Acid 237-246
The Determination of the Solubility Constant for Calcium Carbonate 245-249
Equilibrium between Calcium Carbonate and Gjrpsum 250-255
Calcium Sulphate, Carbonate, and Bicarbonate in the Presence of Sulphates. 255-258
Calcium Sulphate, Carbonate, and Bicarbonate in the Presence of Sodium
Chloride 259-262
The Effect of Temperature Changes 262-263
Summary of Results 263-264
OONTSIBIinONS TO COSKOGONT AND THE FUNDAMENTAL PROBLEMS OF aSOLOGT
THE FORMER RATES OF THE EARTH'S ROTATION
AND THEIR BEARINGS ON ITS DEFORMATION
BY
THOMAS CHEOWDER CHAMBERLIN
Ptofes»OT of Otology University of Chicago
THE FORMER RATES OF THE EARTHS ROTATION
AND THEIR BEARINGS ON ITS DEFORMATION.
In the treatment of the earth's deformations, which is to be the subject
of a following paper, it is essential to know whether changes in the rate of
the earth's rotation must be regarded as one of the important factors or
not. If the rate of rotation has appreciably varied during geological history,
it is almost certain that the oblateness of the earth-spheroid has also
varied, for unless the rigidity of the earth greatly exceeds that of any
known substance, it must have been modified in form under changing
rotation so as to approach the shape it would assume if it were a perfect
fluid. It would be an error to assume, as is sometimes done, that the
earth would conform to the fluidal shape perfectly, but that it would
approach to this with a measurable degree of closeness seems to be beyond
question. If there was a change from a high rotational speed, and con-
sequent high degree of oblateness, to a slower speed with less oblateness,
the surface area of the earth must have been reduced, because the nearer
such a body approaches a sphere, the less the area of its surface, the greater
its average gravity, and hence the greater its degree of compression. This
is brought out numerically, with a high order of approximation, in the
accompanjring paper of Professor Slichter. There will be occasion in the
course of the present paper to consider in detail the application of this
supposed reduction.
Whatever therefore may be the difficulties attending a treatment of
past rates of rotation of the earth, it is imperative that this element of
the problem of deformation be recognized and evaluated so far as lies in
our power.
The problem may be approached on two rather distinct lines, one of
which is astronomic but rests back so radically on postulates derived
from theories of cosmogony that it may almost be called cosmogonic, and
the other of which is geologic and rests on the direct or implied teachings
of terrestrial e\adence.
The ulterior purpose of this paper is to set forth the latter, but the
cosmogonic considerations can not be passed without notice, for the cogency
which will be thought to attach to geological evidences is certain to be
measured in no small degree by the presumptions which are entertained
on cosmogonic grounds, or on astronomic grounds with an essential cos-
mogonic factor. The recent literature of the subject indicates that a
belief in a former high rate of rotation of the earth based on cosmogonic
and tidal grounds has a strong hold on astronomers and, to some large
5
6 THE TIDAL PROBLEM.
extent, upon geologists. The extent of this belief is due in large measure,
no doubt, to the masterly papers of Sir George Darwin upon the origin
and tidal influence of the moon. It is obvious that if the arguments in
favor of a former high rate of rotation are accepted as decisive in them-
selves, such geological data as seem to conflict with them are likely to be
received with skepticism, or to be given interpretations consistent with
the accepted conclusions. It is therefore appropriate, if not necessary,
to review at the outset the grounds for the conclusions that have been
drawn from cosmogonic postulates and from tidal and other considerations
based upon these, so far at least as these have been thought to be weighty.
There is the more reason for this in the present series of papers^ because
of the very different basal postulates which may be grounded on the mode
of planetary genesis set forth in them.
THE ASTRONOMICAL DEDUCTIONS.
CONSIDERATIONS BASED ON THE OLDER CJOSMOGONIES.
It scarcely needs to be recited that, during the past century, astrono-
mers and geologists almost universally accepted the hypothesis that the
earth was formed from the condensation of a spheroid of gas, and that
current doctrines as to the earth's early rates of rotation were founded
on premises derived from some form of this hypothesis. Under the original
Laplacian view it was affirmed that the rotations of the sun and the plan-
etary masses were progressively accelerated as they shrank from a more
expanded to a more dense condition. The rotation of the parent earth-
moon spheroid was supposed to have reached, at a certain stage, such a
velocity that a ring of matter was separated from its equatorial tract and
formed the moon by subsequent condensation. It was held that the speed
of rotation of the residual spheroid further increased, or tended to increase,
by reason of its continued contraction, and hence that the primitive rota-
tion of the earth was exceedingly rapid. As the present rotation of the
earth is relatively slow, it followed, as a necessary inference, that a very
marked decline in the earth's rotatory velocity took place in the course
of geological history.
In the modification of the Laplacian view introduced by Sir George
Darwin,^ the material of the moon is supposed to have been separated
^ G. H. Darwin:
On the bodiljr tides of vibcoiib and semi-elafltic spheroids, and on the ocean tides
upon a yielding nucleus. <Phil. Trans. Roy. soo. Lond., part 1, 1879, pp. 1-35.
On tlu9 procession of a viscous spheroid, and on Uie remote history of the earth.
<Phil. Trans. R07. Soc. Lond., pi^ 2, 1879, pp. 447-538.
Fkoblems connected with the tides of a visoous splieroid. <Pliil. Trans. Roy. Soc.
Lond., part 2, 1879, pp. 539-593.
The determination of the secular efifects of tidal friction by a graphical method.
<Proc. Roy. Soc. Lond., No. 197, 1879, pp. 168-181.
On the secular changes in the elements of the orbit of a satellite revolving about a
tidally-distorted planet. <Phil. Trans. Roy. Soc. Lond.. vol. 171, part 2, pp.
713-891. 1880; Proc. Roy. Soc., vol. 29, 1879, p. 168, and vol. 30, 1880. p. 255.
On the tidal friction of a planet attended by severad satellites and on Uie evolution of
the solar system. <Phil. Trans. Roy. Soc. Lond., part 2, 1881, pp. 491-535.
Enc. Brit., article on "tides"; ''The Tides." 1899.
Also Thomson and Tait's Natural Philosopny, 2, articles on tides.
THE ASTRONOMICAL DEDUCTIONS. 7
from that of the earth after the condensation of the common mass had
reached the liquid or perhaps even incipient solid state. In what precise
form the separation took place is not specifically affirmed and is not material
here, where the only essential point is the high rotatory velocity assigned
the earth at the time of the moon's separation.
Most or all of the meteoritic hypotheses of the earth's origin — using the
term meteoritic in the restricted sense defined in this series of papers —
agree essentially with the gaseous hypotheses in assigning to the earth,
at its earliest separate stage, a molten condition and a rate of rotation
either identical with or closely approximate to that of the Laplacian
hypothec and of its modifications. The presumption, therefore, that the
rotation of the primitive earth was of a high order of velocity had the
sanction of these two classes of cosmogonic theories, and, as they occupied
the field almost exclusively during the past century, this common inference
from them came to have, naturally enough, a strong hold upon the beliefs
of astronomers and geologists. If there shall finally be found reason to
set these conceptions aside, it should still be recognized that they have
been powerful instrumentdities in advancing knowledge and in stimu-
lating inquiry, and that the investigations founded upon them have been
scarcely less than necessary steps toward a final solution.
Besides being at one in postulating a rapid rate of primitive rotation,
these older hypotheses were essentially in agreement in assigning to the
earth a molten condition in its early stages, as already stated, and this
postulate has entered pervasively into the tidal and deformative theories
of the earth that have had currency. Until the later decades of the last
century, it was commonly believed that a molten condition was retained
by the interior of the earth, or by some notable part of it, throughout
the geological ages. In the latter part of the century, the conception of a
solid earth came to be more generally entertained, but there went with
this, almost universally, the postulate of such a degree of viscousness as to
profoundly influence conclusions relative to tidal deformation and earth-
movements generally. At the present time, when belief in an essentially
solid earth has gained a large, though not universal, adherence, the con-
ception that the spheroid is to be regarded as a viscous body in the treat-
ment of all the larger geological problems is still widely prevalent and
not only enters profoundly into the study of these problems but takes on
forms exceedingly difiicult to adjudicate. The embarrassment does not
arise so much from the theoretical recognition of a viscous property in
the substances of the lithosphere, as from the lack of firm grounds for
estimating its actual participation in the deformations and internal move-
ments of the earth. One of the most vital questions of earth-dynamics
relates to the respective values of viscousness and of elastic rigidity in
terrestrial diastrophism.
In this discussion the elastic rigidity of the earth will be regarded as
the dominant factor in its morphology, and the tidal deformations of the
lithosphere will be regarded merely as strains in an elastic body, involving
viscous or liquid flowage only as an incident affecting those portions of
the earth's body which are in a molten, gaseous, or temporarily unattached
state.
8 THE TIDAL PROBLEM.
Working upon the cosmogonic grounds prevalent in the past century,
and supported by the nearly universal consensus of opinion regarding the
early stages of the earth, Sir George Darwin, in a memorable series of
mathematical investigations/ developed the well-known doctrine of the
tidal retardation of the earth's rotation from a primitive period of less
than 6 hours 36 minutes to the present period of four times that length.
Besides being grounded in presumptions that were commonly accepted,
it had the merit of bringing these presumptions into historical consistency
with the existing state of things. Not only that, but the investigation
started with what then seemed to be a present rate of retardation deduced
from astronomical observations, and proceeded backward by logical steps
and current assumptions to the supposed original state, or at least to a
close approach to it. The confidence that has been reposed in the conclu-
sions so reached has not been placed without persuasive reasons, whatever
conclusions may ultimately be reached from radically different cosmogonic
postulates and from revised astronomical data.
Not a few inferences of vital geological importance were drawn from
this classic investigation, and specific data to support them were naturally
sought in the geologic record. For the greater part, this search met with
negative results, or with results which could be regarded as giving but
meager or equivocal confirmation. Notwithstanding this, the logical force
of the tidal argument as developed by Sir George Darwin, when its cos-
mogonic postulates were taken for granted, was such that inharmonious
geological phenomena were generally explained away, largely by assuming
that the internal solidification of the earth took place at a relatively late
date.
CONSIDERATIONS BASED ON THE PLANETESIMAL THEORY.
As to the grounds for postulating a radically different constitution of
the lithosphere, growing out of a new hypothesis of earth-genesis, I must
content myself here with references to what has already been written'
and to a fuller exposition elsewhere in this series of papers. It is appro-
priate, however, to bring again to mind those inferences which are drawn
from the rotational features now shown by the solar system, since these
bear specifically upon the question in hand.
The doctrine that a prevalent forward rotation of the planets could
only mean that they were formed through gaseous or quasi-gaseous con-
densation, was one of the bulwarks of the older hypotheses. It was only
^ at., anU.
> A group of hypotheses bearing on dimatic changes, T. C. Gbamberlin, Jour. QeoL,
yol. 6, No. 7, 1897, pp. 6^-083.
An attempt to test the nebular hypothesis by the relations of masses and momenta,
Certain recent attempts to test the nebidar hypothesis, T. C. Cnamberlin and F. R.
Moulton, Science, vol. 12. Aug. 10, 1900.
The origin of the eartn, Chamoerlin and Salisbury, chap. I, vol. 2, Qeology, pp. 1-81
Dec., 1906.
Evolution of the solar system, F. R. Moulton, chap. XV, Introduction to Astronomy,
pp. 440-487, liar. 24, 1906.
THE A8TBONOMICAL DEDUCTIONS. 9
when the inapplicability of this doctrine to natural cases was detected,
about a decade ago/ that there was a clear path opened and logical grounds
provided for developing a hypothesis of planetesimal accretion. It is
pwhaps not too much to assume that the previous papers of this series
have shown that this hypothesis is even better fitted than gaseous con-
densation to give rise to the various rates of rotation actually presented
by the solar system. Under the planetesimal hypothesis, the primitive
rotation of the earth was not necessarily rapid, nor was the body of the
earth necessarily molten. Thus two of the primitive conditions, that were
formerly taken for granted on the basis of a nearly universal consensus of
opinion, have been brought into question and may now be fairly regarded
as being at best no more than working competitors with the alternative
of a solid elastico-rigid earth, a view which is hampered by no compulsory
presumption as to any particular rate of primitive rotation, but is hospitable
to any rotational state which the direct evidences, astronomical, geological
and otherwise, may require.
The speculative freedom relative to primitive rotations, which the
planetesimal hypothesis thus affords, directs attention anew to the actual
facts and to their unembarrassed implications. The most fundamental
case is that of the controlling body of the solar S3rstem itself. The present
rotation of the sun is relatively slow and its axis is inclined appreciably
to the common plane of the planetary system. When it is considered that
the mass of the sun is more than 700 times that of all the planetary deriva-
tives combined, this rate afld this inclination assume radical importance.
This dow rotation and this inclination of axis are perfectly consistent with
the planetesimal hypothesis and have peculiar suggestiveness in that
relationship. On the other hand, they seem to me very difficult to reconcile
with any theory under which the outlying bodies are supposed to be derived
from a gaseous or quasi-gaseous spheroid by contraction, particularly
any theory which postulates that the derived bodies were discharged from
the central mass by the equatorial velocity of its rotation. Obviously
the planetary material thus separated should be accurately adjusted to
the sun's equatorial plane, and to the common plane of the system. Obvi-
ously also the great residual mass should have a rate of rotation appro-
priate to such a discharge. Having separated a succession of masses
from its equator to form the planets, and having further shortened its
radius some 36,000,000 miles after the last known planetary mass was
detached, the sun should have a rotatory velocity somewhat near that
requisite for another planetary separation. The velocity of rotation at
the equator of the solar nebula when it was supposed to have detached
the material for Mercury must have been, according to the Laplacian hy-
pothesis, about 28 miles per second. The equatorial velocity requisite to
bring the centrifugal and centripetal components of the sun's equatorial
motion into equality if the sun now had a radius of 1,000,000 miles is 176
miles per second; the velocity required to bring about this state at the
present surface of the sun is 270 miles per second. We should then expect,
^ Journal of Geology, vol. 5, 1897, pp. 009-609
10 THX TIDAL PROBLEM.
under any hypothesis that rests on centrifugal separation, that the present
speed of the sun's equator would certainly be much greater than 38 miles
per second, and should approach the higher figures given. As a matter of
fact, the Sim's equatorial velocity of rotation is only about 1.3 miles per
second. Such a rate seems, therefore, to be altogether inconsistent with
the doctrine of centrifugal separation. If, for a moment, the thought be
entertained that tidal retardation may have reduced the sun's rotation
from a high primitive rate consistent with the centrifugal hypothesis, to
the present rate, it will become obvious, on a study of the nature and value
of the tidal influence of the planets on the sun, that this is wholly unten-
able. The quantitative estimates of Sir George Darwin are decisive on this
point.^ So also is the remarkable fact that the equatorial portion of the
Sim has a higher rotational velocity than the portions in higher latitudes
instead of lagging, as it should if it were affected by tidal retardation.
The obliquity of the sun's axis is a further grave objection to all forms
of the doctrine of centrifugal separation. On the other hand, some such
obliquity is extremely probable under the hypothesis that the system was
developed by the influence of a passing star, for the axis of the ancestral
sun might obviously sustain any relationship to the orbital plane of the
disturbing body. The position of the present axis, under this hypothesis, is
the result of a composition of moments of momentum derived in part from
the ancestral rotation and in part from the passing star, and it could not
therefore be expected, except by a remote chance, to be exactly normal
to the common plane of the planetary system. Under this h3rpothesis the
obliquity of the sun's axis, together with its slow rotation, suggest, if they
do not distinctly imply, that the direction of rotation of the ancestral sun
was opposite to that of the present sun, and that its axis was more inclined
than now to the plane of the present system.
Of similar rotational import is the relationship between the time of
rotation of Mars and that of the revolution of its inner satellite, Phobos.
It is obvious that, under any hypothesis of centrifugal separation, if a
revolving spheroid acquires by contraction an equatorial velocity sufficient
to leave behind the material of a satellite, and afterwards continues to
contract until its radius is but a small fraction of its value at the time of
separation, the rate of rotation of the spheroid must be greatly increased
and its period must be much shorter than the revolutionary period of the
derived satellite, unless some very potent agency intervenes to reverse
the systematic process of the evolution. Now, the satellite Phobos revolves
around Mars about three times while the planet rotates once. In an anal-
ogous way the little bodies that make up the inner edge of the inner ring
of Saturn revolve about that planet twice while the planet rotates once.
These are, on their face at least, seriously out of accord with the doctrine
of centrifugal separation by planetary contraction. Darwin has suggested
that tidal retardation may be a possible solution in the special case of
Phobos, but Moulton has called attention to the insuperable difficulties
of applying this explanation consistently to the Satumian case and the
Martian case at the same time.'
» Trans. Phil. Soc. Lond., 1881. * Astrophya. Jour., vol. 11, Mar., 1900, p. 109.
THE ASTRONOMICAL DEDUCTIONS. 11
The retrograde revolution of the ninth satellite of Saturn still further
and seriously complicates the case from the centrifugal point of view.
The rotations of Uranus and Neptune are unknown, but, whether they
are concordant with the revolutions of their satellites or not, they present
difficulties under the centrifugal hypotheses because of the great obliquity
of the planes of revolution of their satellites to the common plane of the
sjrstem, and because of their retrograde motions. This has long been
recognized, but these difficulties gain not a little in force when they are
associated with other rotational difficulties which have been insufficiently
considered in connection with them.
There are still other rotational features of the existing planets which
seem to be inconsistent with all forms of a contractional-centrifugal hypoth-
esis of planetary origin, and, what is especially to the point in the matter
in hand, which seem inconsistent with the very high rotational velocities
which such a hypothesis necessarily postulates. Among these are the great
differences in the rotational features of the members of the system. If
the system started from a common spheroid and if its derivatives were
shed by systematic centrifugal action, it is very difficult to see how so great
variety of rotational velocities, so varied inclinations of the rotational
axes, and so diverse directions of rotation as the system actually presents,
could have arisen.
Under the planetesimal hypothesis, the rotation of each planet is held
to have arisen independently of every other planet. Its rate of rotation
depended on the special conditions that attended the expulsion of its
nucleus from the sun, and on the mode of accession of the rest of its material
from the planetesimal state — conditions that were quite certain to vary
with each planet. Rotations, rapid or slow, direct or retrograde, with
inclinations of any degree, are consistent with the hypothesis. There is,
however, a decided balance of presumption in favor of forward rotations,
of moderate inclinations of axis, and of moderate velocities of rotation.
As none of the planets rotate at speeds that even remotely approach that
requisite for equatorial discharge, as their rates of rotation differ widely
frbm one another^ as the inclinations of their axes vary greatly, and as the
majority of their rotations are direct and the minority retrograde, this
hypothesis seems to be concordant with the facts of the case.
If we are thus permitted to start with a genesis which leaves us free
to suppose that the rate of the earth's rotation at the outset may have
been essentially what it is to-day, or may have been faster or slower in any
degree, the preconceptions that have led to former rotational views do not
trammel us. The determination of the past history of the earth's rotation
rests unhampered upon the evidences presented by its own phenomena
and upon those deducible from the necessary influence of its neighbors.
The most important of the rotatory influences of neighboring bodies on
the earth is the friction of the tides, particularly of the lunar tides. The
assigned mode of this action is familiar and may be stated briefly as fol-
lows: If the tidal protuberance has a position in advance of the position
of the moon, as at A in fig. 1, a component of the moon's attraction tends
to antagonize the earth's rotation and to accelerate the moon's motion.
12
THE TIDAL PBOBLEM.
If a tidal protuberance has a position behind the moon's position, as at
jB, a component of the moon's attraction tends to accelerate the earth's
rotation and to retard the moon's motion. If a protuberance rises directly
beneath the moon's position, its forward and backward pulls are equal.
On the opposite side of the moon
there are complementary protu-
berances, as A'^ B\ fig. 1, whose
rotatory effects are the reverse
of those on the moonward side,
but whose greater distances give
them less efficiency. It is merely
this difference in effectiveness
growing out of difference of dis-
tance that is usually appealed to
as influencing rotation.
I shall endeavor to show later
that, while the foregoing reason-
ing seems to be unimpeachable in
itself, there are counterbalancing
factors which seem to have been
overlooked, and which nullify the
value of this mode of treatment.
They do not, however, nullify
the proposition that tidal friction
tends to retard the earth's rota-
tion. It seems best, however, to
review the subject first on the
accepted lines.
The tides represented in fig. 1
are such as are assigned to the
direct pull of the tide-producing
body and are known as ''direct.^'
The protuberanpe A, fig. 1, repi^
sents a tide which is int^^reted
as having lagged in its forma-
tion, and hence has been carried
forward by the rotation of the
earth to a position in front of
the moon's position; B represents
a tide which has been formed
behind the moon's position, but
both may be regarded as falling within the class of '^ direct" tides. This
class of tides are said to be built up when the natural free period of the
tidal wave is less than that of the tidal forces. If the natural free period
of the tidal wave is greater than that of the tidal forces, the tendency is to
produce '' inverted" tides. The law underlying this difference of result, to
which Newton first directed attention, is thus stated by Darwin:^
Fio. 1.
» The Tides, pp. 171-172.
THE ASTRONOMICAL DEDUCTIONS.
13
Now, this ample esse illustrfttos a general dynamioal principle, namely, tbat if a
qrttom capable of oeoiUating with a certain period ia acted on by a periodic force, when
the period of the force is greater than the natural free period of the system, the osdl-
laiions of the system agree with the oscillations of the force; but if the period of the
loroe is less than the natural free period of the system the oscillations are inverted
with reference to the force.
This principle may be ap-
plied to the case of the tides in
the canal. When the canal is
more than 13} miles deep, the
period of the sun's disturbing
force is 12 hours and is greater
than the natural free period of
the oscillation, because a free
wave would go more than half
round the earth in 12 houra.
We conclude, then, that when
the tide-generating forces are
trying to make it high water, it
wfll be high water. It has been
shown that these forces are tend-
ing to make high water immedi-
ately under the sun and at its
antipodes, and there acccmiin^y
win the Idgh water be. In this
case the tide is said to be direct.
But when the canal is less
than 13} miles deep, the sun's
disturbing force has, as before,
a period of 12 hours, but the
period of the free wave is more
than 12 hours, because a free
wave would take more than 12
hours to get half round the
earth. Thus the general prin-
ciple shows that wtoe the forces
are trying to make high water,
there will be low water, and vice A
versa. Here, then, thm wOl be
low water under the sun and at
its antipodes, and such a tide
is said to be inverted, because
the oscillation is the exact in-
version of what would be natu-
rally eiqiected.
All the oceans on the earth
are very much shallower than
fourteen miles, and so, at least
near the equator, the tides ought
to be inverted. The conclusion of
the equilibrium theory will therefore be the exact opposite of the truth, near the equator.
This argument as to the solar tide requires but little alteration to make it applicable
to the lunar tide.
The positions of a set of ''inverted" tides corresponding to the fore-
going set of "direct" tides are shown in fig. 2.
Now since the rotation of the earth gives its surface an angular motion
greater than that of the tide-producing body, its effect must be to carry
Fzo. 2.
14 THE TIDAL PROBLEM.
the protuberances forward towards positions which, according to the
above interpretations, are in some cases more favorable for influence on
the earth's rotation and in other cases less favorable, and if a wave is followed
through its whole course, it sustains various relations, favorable and unfa-
vorable, retardative and accelerative. The sum total of influences is thus
seen to be a product of much complexity. It has been held that the retar-
dative positions predominate in effectiveness. The case has usually been
treated on the assumption of a continuous ocean belting the earth and
permitting the tides to follow the tidal forces consecutively about the
earth. It will be shown later that this is not the actual case, and that
the tides are essentially limited to individual water-bodies. This further
and greatly complicates the case. The problem is still further complicated
by the past relations of the moon to the earth, and this claims attention
before further considering the eflSciency of the tides.
THE GENESIS OF THE MOON.
(1) The hypothesis of Laplace in its original form took no account of
tidal action. Under it the rotation of the earth, when it had become
condensed to a molten globe, was assumed to have had the velocity which
at .an earlier stage was necessary to separate the lunar ring, plus that
which was added by subsequent contraction. How this high rate of rota-
tion was reduced to the existing rate was not explained.
(2) The supplementary hypothesis of Sir George Darwin replaces this
defect of the Laplacian hypothesis by postulating a centrifugal separation
of the moon-mass from the earth-mass after the parent-body had been
condensed to a liquid or perhaps even incipient solid state, and a subse-
quent recession of the moon by tidal influence, accompanied by a reduction
of the earth's rotation as its dynamic reciprocal. The postulated method
of this tidal action has been stated above. The fundamental proposition
that tidal friction will tend either to separate the two interacting bodies
or to draw them together — according to the precise nature of their rela-
tions — is not questioned, as it seems to be solidly founded on the laws of
energy, but it is necessary to consider the precise relations of the bodies to
determine the character of the action under the preceding mode of inter-
pretation, and we shall find occasion to question the mode itself. Darwin's
method of starting with what was thought to be a fairly reliable astronom-
ical indication of the present value of the earth-moon interaction and of
working backward mathematically to the primitive state, or so far as the
mathematical process would carry, is beyond praise. But as the present
value of the earth-moon interaction is open to serious question and is not
now replaceable by an unquestionable value, and as the postulates for the
backward tracing are themselves in question, it is necessary to consider
the hypothesis on more general lines. The value assignable to the tides in
each of the earth's ages depends on the assumptions made regarding the
physical states of the earth's interior. If the body of the earth be assumed
to be molten, or viscous in such a degree that the body tides are important
and are of the liquid or viscous type, the results will be very different from
THE ASTRONOMICAL DEDUCTIONS. 15
those which will be reached on the assumption that the body tides are
merely strains in an elastico-rigid earth. So, too, if the ocean has been
growing in volume during the geologic ages, or has been changing in form
in any notable degree, the results would need to be modified accordingly.
EiVen when approached on the admirable lines of backward tracing by
computation, the results are therefore subject to wide variation accord-
ing as the postulates arising from one cosmogonic hypothesis are used or
those of another.
If we were to follow carefully the first stages of the moon's evolution
under Darwin's hypothesis, it would be seen how critically dependent
that hypothesis is on an underljring theory of cosmogony. The moon is
assumed to have been separated from the parent earth-moon mass by
some form of centrifugal action. While the precise form may have been
either one or another of two or more alternatives, the principle of action
is the same up to a certain point and is best illustrated by supposing that
the moon-mass separated as a unit, and that just after separation it was
a spheroid close beside the earth-spheroid and revolving in the period of
the latter's rotation. An objection to this supposition •will be considered
later.
Now at this critical stage the earth was subject to the tidal action of
the sun, which, according to the fundamental theory of the hypothesis,
should tend to retard the earth's rotation. The earth was also subject to
contraction from loss of heat, which should tend to accelerate its rotation.
If the former was the greater influence, the lunar tide, which would have
begun to be generated as soon as a difference arose between the moon's
revolution and the earth's rotation, must have fallen behind the moon's
position and, according to the hypothesis, must have tended to draw it
backward and bring it down to the earth. To permit the evolution to
proceed at all it m necessary to suppose that the contraction from loss of heat
was a greater influence on the earth's rotation than were the solar tides. Now,
the earth's contraction from loss of heat at the present time is exceedingly
small. If, therefore, the constitution of the earth has been much the
same as it is now ever since its growth practically ceased, as assumed by
one cosmogonic hypothesis, the supposition that the tidal evolution of the
earth-moon system was started in the right direction for lunar recession
by the superiority of the influence of contraction over that of the solar
tides is either untenable or else the solar tidal influence was extremely
small. The initiation of the tidal evolution postulated by this hjrpothesis
is thus seen to be tied up with a very high rate of loss of heat in the initial
earth-stages, and this is only assignable under certain cosmogonic assump-
tions which give to the earth a very hot surface. These are, however,
not necessarily confined to the gaseous or meteoritic hypotheses. They
may possibly be made under the planetesimal hjrpothesis, but in any case
they are as speculative as the hypotheses themselves, and in the latter
case somewhat less well grounded, because the alternative phases of the
hypotheses seem to be the more probable.
It is worth while to note further in this connection that the evolution
of the lunar tide, under the theory of Darwin, would be a very slow process,
16 THB TIDAL PROBLEM.
and might be reversed before it escaped the critical conditions named
above. The speed of the earth's rotation and the speed of the lunar tide in
this early stage must not be conf ounded, for the speed of the tide is depend-
ent on the difference between the angular rate of the earth's rotation and
the angular rate of the moon's revolution. At the instant of separation
the two rates were the same, and the tide, if the stationary protuberances
caused by the moon can be called a tide at all, would be infinitely slow,
and the period of the tide — ^the time required for the tide to make a circuit
of the earth — ^infinitely great. At the first stage of difference in angular
rate, if the earth's rotation was accelerated by contraction more than the
solar tides retarded it, the movement of the tide would be infinitesimally
slow in the present direction, and the tidal period sub-infinitely long. The
movement of the tide over the face of the earth would be accelerated only
as the contractional acceleration continued to be superior to the solar
tidal retardation reinforced by the lunar retardation. If the rate of loss
of heat, which must have declined rapidly as the supposed molten earth
crusted over and the crust became thicker, fell below that at which its
accelerating value on the rotation of the earth was superior to the retar-
dational value of the solar and lunar tides, and the latter then became supe-
rior, the time of rotation of the earth might be forced back to coincidence
with the revolution of the moon, and the lunar tides temporarily suspended
and, a little later, reversed, and the moon brought back to the earth accord-
ing to the fundamental postulate of the theory.^ To escape this contin*
gency it is necessary to suppose that the contractional influence of the loss
of heat continued to be superior to the retardational influence of the solar
and lunar tides until the lunar tide, though developed with extreme slow-
ness, had extended the moon's revolutionary period so much that when
the retardational influence became superior to the contractional influence
it was too late for it to reduce the rotation-period of the earth to the revo-
lution-period of the moon at that end of the evolutional series. These
considerations serve to indicate how delicately poised were the initial condi-
tions assumed by the hypothesis and how completely they were dependent
on the heat-emission of the earth at the critical stage, which in turn was
dependent on the cosmogony that preceded it.
The argument that the balance of influence must have lain on the side
of heat-loss or else the moon would not be where it is at present, would
be pertinent if the earth-moon evolution were absolutely shut up to one or
the other of the alternatives just considered, but it has no force against a
hypothesis which entirely avoids these critical alternatives.
It is to be noted further that the tidal reactions in the initial stages of
the hypothesis of Darwin must apparently have been those of the earth's
body, for if the heat had been so far dissipated that the earth was crusted
over and the oceans were permitted to mantle the earth, the loss of heat
would possibly have been too small to start the evolution in the postulated
direction. The atmosphere must then have been of that vast vaporous
* The correctness of this is dependent on the soundness of the theory that the vonUon
of the tides determines their acoelerative or retardative character, which will be con-
sidered later.
THB ASTRONOMICAL DEDUCTIONS. 17
kind made so familiar to us by the geologic rhetoric of the last century.
Until the outward reach of this atmosphere was escaped by the receding
moon, the atmospheric friction must have kept the moon in consonant
revolution with itself and tidal action could not have been inaugurated.
This must have prolonged the critical stages and made the triumph of
contraction over the solar tides all the more doubtful.
The separation of the moon from the earth after the common mass
became a liquid spheroid is subject to another serious contingency, based
upon the same principle ol differential attraction as the tides themselves.
Roche has shown that a satellite revolving within a given distance from its
primary will be torn to fragments. The fragments must revolve at velocities
strictly dependent on their distances from the center of the primary and
hence must disperse themselves into a ring of the Satumian type.^ The
fragments so produced would be subject to further reduction by collisions
with one another, by changes of temperature, and by internal reactions,
and would probably only reach an approximately stable condition as to
size when they were well comminuted. For the earth-moon combination
the Roche limit of disruption lies about 11,000 miles from the earth's
center.' The cogency of Roche's reasoning, supported by that of Clerk-
Maxwell and others, and the example of the rings of Saturn, seems to leave
no alternative but to suppose that a body of the mass of the moon could
not pass from the earth outward by tidal reaction without being torn to
fragments and converted into a ring, unless the fission of the earth-moon
mass and the initiation of the lunar tide took place outside the Roche
limit, which is difficult to believe under the Darwinian hypothesis, though
consistent enough with the Laplacian. The laws of revolution seem to
forbid the supposition that the fragments produced by tidal disruption
could have been aggregated for any appreciable length of time on one
side of the earth so as to act jointly in producing an effective tide. Even
if some tide could be so produced there would still remain the question
whether it would have carried the fragments outward by reaction suffi-
ciently far for them to have escaped the dangers of reversal by the solar
tides, as pointed out in a preceding paragraph. There seem to be no
cogent theoretical grounds upon which it can be affirmed that the frag-
ments of a disrupted body of this kind would evolve into any other condi-
tion than that of a ring of discrete particles during the time available for
starting the recessional movement of the moon. They might perhaps
move outside the Roche limit or be drawn down to the planet in a period
sufficiently long, but probably not in the available period. The fact that
the Satumian rings are present at this stage in the history of the solar
system suggests, if it does not definitely imply, that this form of organiza-
tion is one of much persistence.
If we pass by these peculiar difficulties that embarrass the supposed
separation of the moon from the earth, and if we set aside the special
consequences assigned to a molten or viscous earth-body, the remaining
^ On the stability of motion of Saturn's rings. < Scientific Papers of James Cleik-
:wdl, vol. 1, pp. &8-376.
* Darwin's ^' Tides," pp. 368-360.
18 THE TIDAL PROBLEM.
problems respecting the influence of the tides on the earth's rotation are
essentially the same, whatever the genesis of the moon, and so these further
problems may well be deferred until it is seen where and how the various
genetic theories come onto common ground and may be treated in a com-
mon manner.
(3) There is nothing in the planetesimal hypothesis that is, in itself,
necessarily prohibitive of an origin of the moon by centrifugal separation
from the earth-mass, for under it the planetary bodies may have had very
high rates of rotation. So also the mass of the planetary nucleus may
have been so large and the ingathering of the planetesimals may have
been so rapid, by hypothesis, that a molten or even a gaseous condition
could have arisen. In the case of the larger planets such a primitive state
is quite within the limits of the probabilities. The case of the earth is
debatable, but it will be of no service in this discussion to follow the
gaseous or molten alternative, as it would be essentially identical with the
preceding.
There are two other possible modes of origin of the moon, in neither of
which was the moon-mass ever a part of the earth-mass. In both of these
it is supposed that the nebular nuclei of the earth and the moon were
separate knots of the parent spiral nebula. In the first case, they are
supposed to have been companions in projection from the ancestral sun,
and to have revolved about their common center of inertia from the out-
set. In the second case, the nuclei are supposed to have been at the outset
independent knots having separate orbits about the sun but near one
another. The two are supposed to have come into their present relations
in the course of the segregation of the parent nebula. Rather grave
dynamic difficulties attend this latter view, and it need not be pursued
further here, as the rotatory problems under it are not essentially different
from those of the first and much more probable alternative.
In this preferred alternative, the nuclei of the earth and moon, at the
instant they left the ancestral sun, are supposed to have been a single mass
which was given a forward rotation by the unequal resistances on its oppo-
site sides to the expelling impulse, for which there are assignable reasons.
Just after leaving the sun, the mass is supposed to have separated as an
incident of the expulsion, but the two parts are supposed to have continued
to revolve about their common center of inertia essentially as before, i.e., as
a rigid body. After separation, however, each was subject to the rotational
effects of the accession of planetesimals, and when their rotations came to
differ from their revolution about the mutual center of inertia they were
subject to tidal reaction. The extent to which such differences of rotation
arose is an essential part of the problem under this hypothesis.
There were many possible alternatives, theoretically speaking, as to
the relative sizes of these nuclei and the distances to which they separated
under the initial impulse, but there were limitations to these. If the com-
bined masses of the two nuclei were one-eighth of the combined mass of
the present earth and moon, the moon could not have been more than
460,000 miles from the earth, but as this is farther than it is at present the
THE ASTRONOMICAL DEDUCTIONS. 19
hypothesis is not hampered by this limitation. If the joint mass was
larger, their initial distances may have been greater; if smaller, they
must have been less. Within the limits thus imposed by the mechanics
of the case, the nuclei may have been separated by any distance, abstractly
considered, from the maximum permitted down to siu^ace contact.
In fact, however, the degree of nearness consistent with the present
state of things, was limited by the consequences of growth, for the increase
of the masses of the nuclei by the ingathering of the planetesimals may
have drawn the nuclei toward one another, or even together. This was
conditioned by the moment of momentum which the accessions carried
into the nuclei, which varied widely. Separation later by centrifugal
action would be theoretically possible, under assignable conditions, but not
at all inevitable, perhaps not at all probable. If the initial distance of the
nuclei were sufficient, however, the nuclei might approach one another so
long as growth was a ruling influence. Tidal action would nm concur-
rently with this and would oppose approach, under most conditions, but
during the more rapid stages of growth, the tidal effect may possibly have
been less than the effect of increasing mass. But the tidal effect would
increase as the bodies were drawn toward one another, while in the later
stages of growth the increase of the mass would decline in rate. At a certain
stage the two effects may be presumed to have balanced one another, after
which recession would begin through the preponderance of the frictional
effect of the tides. From that stage, the history would proceed along the
lines determined by the mutual interaction of the matured bodies.^
It will be seen that the range of specific assignments under this phase
of the planetesimal hypothesis has a wide amplitude, embracing the per-
missible assignments as to the original distances between the nuclei, as
to the original masses of the nuclei, and hence as to the amount of their
growth, as to their planes of revolution, their eccentricities of orbit, etc.
Under this amplitude, it is possible to suppose that the two bodies at the
climax of their approach reached precisely the relations which were indi-
cated by Darwin in his backward tracing of their history. On the other
hand, so far as the h3rpothesis itself is concerned, it is equally possible
that the approach of the bodies was much less close, and hence that their
recession under tidal influence was correspondingly less. It will be seen,
therefore, that this h3rpothesis has very much greater adaptability than
the hypothesis of centrifugal separation, and does not equally hamper us
respecting subordinate hypotheses, such as a molten state, a viscous
interior, or a particular amount or a particular distribution of the hydro-
sphere. We are quite free to follow backward from the present observa-
tional data, when it shall be possible to do this on firm ground, with the
utmost complacency as to the results, and to accept these as indicating the
original relations, whether they imply a former state of coalescence, or of
close approach, or of more distant approach. It is possible that the earth
and moon were drawn together by their growth into just those relations
which Darwin assigned to them when, in his backward tracings of their
^ The effects of contraction are here neglected.
20 THE TIDAL PROBLXIC.
history, his mathematics ceased to tell what lay beyond. At the same time,
the hyi>othesis is hospitable to any smaller numerical values for the fric-
tional effect of the tides which revised data may be found to imply.
We are now prepared to inquire with equanimity what is the degree
of trustworthiness of the astronomic data relative to the recent time-
relations of the earth and moon.
THE EVIDENCES OF A PRESENT CHANGE OF ROTATION.
Near the middle of the last century Adams, from a study of certain
data relative to the secular acceleration of the moon's mean motion, reached
the conclusion that the earth was then losing time at the rate of 22 seconds
per century. It is proper to add, however, that Adams laid but little stress
on the actual numerical values which he used in computation, and that he
was of the opinion that the amount of tidal retardation of the earth's
rotation is quite uncertain.^ At a later date, Newcomb made a computa-
tion based on the data then available, with the result that the rate was
reduced to 8 seconds per century.' Darwin verified the computative part
of Adams' results and added a neglected factor for the obliquity of the
ecliptic and the diurnal tide which raised the estimate to 23.4 seconds per
century. Newcomb's estimate similarly revised is 8.3 seconds.'
A reliable answer to the question whether the earth's rate of rotation
is or is not now departing from constancy, and at what rate, depends not
only upon extremely refined astronomical observations, but upon the
interpretation of these observations by means of a perfect theory of the
lunar motions. This latter has not yet been attained. In a case where the
suspected variation from constancy is so slight, and where the logical
structure to be built upon it in tracing it back through tens of millions of
years involves so great a multiplication of any error it may contain, it is
obvious that extreme accuracy and complete soundness are necessary to
trustworthy results. In the judgment of cautious astronomers, these
prerequisites are not yet attainable. It is not, therefore, too much to say
that the deductions thus far made have not a sufficiently secure obser-
vational basis to give them authoritative value. This is not to say, by any
means, that these results, based on the best data heretofore available,
do not fully justify the elaborate mathematical investigations based upon
them, for these have proved extremely illuminating and stimulative,
and were almost necessary as precursors to the more critical work on both
observational and theoretical lines which is necessary to give the firm
foundation so eminently to be desired.
> Thomson and Tait's Natural Philosophy, n, p. 419; also pp. 416-^620 and 503-^6,
edition of 1890, and the papers of Darwin previously referred to.
' Researches on the motion of the Moon, Washington, 1873. See also Thomson and
Tait's Natural Philosophy, n, p. 418.
* Thomson and Talt's Natural Philosophy, II, p. 505.
DEDUCTIONS FROM THE TIDES. 21
DEDUCTIONS PROM THE TIDES THEMSELVES.
As astronomical observations thus leave it uncertain at what precise
rate rotation is changing at the present time, it is necessary to fall back
upon such other evidences as the tides themselves present, and after that
upon the geological evidences. Each of the three fundamental divisions
of the earth, the atmosphere, the hydrosphere, and the lithosphere, is
affected differentially by the attraction of the moon and sun, and hence
they are all, theoretically at least, affected by the tides. They furnish a
suggestive combination for study in that the first is a highly fluent elastic
body, susceptible of great and easy changes of form and volume; the second
is extremely mobile, but sensibly incompressible; while the third is solid,
at least externally, and probably rigid as a whole and possessed of effective
elasticity of form. Because of the markedly different properties of these
three components of the earth, it would seem that comparisons of their
individual responses to the differential attractions of the moon and sun
might throw special light on tidal phenomena.
THE TIDAL PHENOMENA OF THE ATMOSPHERE.
Because the atmosphere is a highly symmetrical envelope, because its
continuity is broken by no barriers, because it is extremely mobile, because
it has great elasticity of volume, and because it presents greater differences
of distance from the tide-producing bodies than the hydrosphere or the
lithosphere, it would seem that it should give a tide of declared charac-
teristics. We are, however, almost wholly without evidences of such a tide,
notwithstanding the large mass of barometrical data at command. These
data stretch over a long term of years and are refined enough to show
several small periodic oscillations, but none of these, at least none of those
commonly recognized, are timed with the moon. Atmospheric tides play
no part in the science of modem meteorology. Laplace discussed the
tides of the atmosphere briefly and theoretically and found that if the sun
and moon were in the plane of the earth's equator and if the two bodies
were in the same line and at their mean distances, the variation of the
barometer would be 0.63 mm.^ Darwin, without entering upon their
discussion, expresses the opinion that they are undoubtedly very minute.'
Other methods of estimating the atmospheric tides support Laplace in
showing that the amount of the forced tides should be just within the
limits of observation, from which it is inferred that they should become
quite appreciable if they were much reinforced by the codperation of free
waves. The chief light which their scantiness seems capable of throwing
on the general problem in hand is that which bears on the dependence of
the actual tides upon the reinforcement of the forced waves by the com-
mensiurable action of the free waves that spring from them.
The best observational data relative to the rate of propagation of a
free atmospheric wave arising from a forced oscillation are those furnished
1 Mtouuque COeste, Pi. I, Bk. IV, and Bk. XIII, vol. 5, p. 337.
> Eno. Bnt., "Tides," p. 353.
22 THE TIDAL PROBLEM.
by the great explosion of Erakatoa on August 27, 1883, as set forth by
lieutenant-General Stracbey in the monograph of the Royal Society on the
'' Eruption of Erakatoa and subsequent phenomena." He says:
The observed facts dearly establish that the successive repetitions of the disturbance
at the numerous stations, after var3ring intervals of time, were caused by the passage
over them of an atmosphOTic wave or oscillation, propagated over the surface, of the globe
from Krakatoa as a center, and thenoe expanding in a circular form, till it became a great
circle at a distance of 90^ from its origin, after which it advanced, gradually contracting
again, to a node at the antipodes of Krakatoa; whence it was reflected or reproduced,
traveling backwards again to Krakatoa, from which it once more returned in its original
direction; and in this manner its repetition was observed not fewer than seven times at
many of the stations, four passages having been those of the wave traveling from Kra-
katoa, and three those of the wave traveling from its antipodes, subsequently to which
its traces were lost (p. 63).
The velocities of propagation of these waves were found to vary from
674 to 726 miles per hour — somewhat below the normal rate of sound at the
surface of the earth, which is 757 miles per hour at 10^ C. and 780 miles at
22^ C. The average temperature of the air at its base is 15^ C. to 17^ C,
from which the temperature declines with ascent, as does also the density.
The mean time occupied by the Erakatoan waves in making a first
circuit of the earth, for the computation of which 27 stations were avail-
able, was 36 hours and 24 minutes, the angular rate being 9.89^ per hour;
the mean of the second circuit, for which 18 stations were available, was
36 hours and 30 minutes, the angular rate being 9.86^ per hour; the mean
of the last observed circuit, for which 10 stations were available, was 37
hours and 50 minutes, the angular rate being 9.77^ per hour.
Now if the forced tidal wave be analyzed into instantaneous impulses
and these be regarded as discontinuous, they may each be treated as though
they gave rise to free waves similar to those derived from the volcanic
impulses of Erakatoa. If we compare the intermediate rate determined
for the free Erakatoan waves with the angular rate of the forced lunar tide,
it will appear that the latter would outrun the former at the rate of about
4.6^ per hour. The free wave would therefore soon begin to flatten the
surface configuration of the forced tide by extending its amplitude, and in
less than ten hours its influence would begin to be antagonistic to the
forced tide. This antagonistic influence would reach its maximum about
ten hours later, but would continue with declining force for nearly another
ten hours, beyond which, because of the relatively high viscosity of the air,
it may be regarded as negligible. It appears therefore that the periods of
the free atmospheric waves are not such as to effectively reinforce the
forced waves and hence they do not rise to appreciable value.
In addition to this there seems reason to suspect that the compressi-
bility and the relatively high viscosity of the air may combine to cause a
portion of the atmospheric tide to take the form of an elastic wave rather
than of a fluidal movement; that is, the tidal force may produce alternate
expansion and compression of the air such as would not be possible in
water because of its incompressibility. Such expansional and compressional
states of the atmosphere would be relieved by a prompt return to the un-
strained condition as fast as the tidal forces were in any measure withdrawn
DEDUCTIONS FROM THE TIDES. 23
and this would reduce the amount of fluidal movement on which a mass-
tide depends. It is not unlikely, therefore, that some part of the scantiness
of the atmospheric tide is due to the elastic constitution of the atmosphere.
There is a semi-diurnal wave of atmospheric pressure which has its
maximum about 10 o'clock a.m. and p.m. Lord Kelvin, interpreting
this as an increase of mass corresponding to the increase of pressure,
has computed that it would accelerate the rotation of the earth about
27 seconds per century.^ If however this oscillation is merely a transient
increase of elastic pressure at the base of the atmosphere due to basal
heat, the ezpansional effects of which are resisted for the time by the inertia
of the air above, as seems not impossible, the wave would have no direct
accelerative effects on the earth's rotation.
THE TIDES OF THE LITHOSPHERE.
There is reason to suspect that the water-tides are in part derived from
the pulsations of the lithosphere.' It will therefore be best to discuss
these first. Since no body is absolutely rigid, and since abundant evidence
shows that the lithosphere is appreciably yielding, there can be no theo-
retical doubt that there are tides of the lithosphere of some kind and of
some magnitude. The only vital questions therefore relate to their magni-
tudes and their specific forms.
The experimental efforts of Sir George and Horace Darwin,* of Von
Rebuer-Paschwitz,^ and of Ehlert,* resulted in detecting only slight indica-
tions of body tides, and even these indications were of somewhat doubtful
interpretation. It appears, however, that the effort of these investigators
was directed toward the detection of the general deformations of the
spheroid directly assignable to the tidal forces, and it is not clear that the
observed results are to be interpreted as equally adverse to the existence
of shorter pulsations assignable to the normal vibrations of the spheroid,
induced by the tidal strains. The nature and likelihood of such shorter
pulsations will be considered later.
So far as opinion as to the value of the lithospheric tides is entitled to
weight we can not do better than to quote the conclusions of Sir George
^ Natural Philosophy, Thomson and Tait, ed. 1890, p. 418.
' It should be understood that this is merely an individual view unsupported by the
expressed opinion of any special student of the tides, so to as I know, and without r&oog-
niuon in the literature of the subject. It is based on the conviction that while the direct
rise and fall of the surface of the lithosphere in response to attraction similarly affecting
the water tends to reduce the amount of the water-tides, the tiltin^^ of the hthoepherio
bed in which the oceans lie first on one side and then on the other m the course of the
procress of the lithospheric wave must develop an inertia tide very similar to the waves
poroduoed by the rockmg of artificial basins. It is also my view that the various free pul-
sations that may arise fiom the forced deformations of the lithosphere may give impulses
to the waters rating in basins on its surface and that water-waves may spring from these
<iuite independently of the direct attraction of the tide-inxxiucing body, though of course
indirectly dependent on it.
* Reports to the Brit. A. A. 8. on Measurement of the Lunar Disturbance of Gravity,
York meeting, 1880, pp. 93-126, and Southampton meeting, 1882, pp. 95-119; also
"Tides," G. H. Darwin, 1893, pp. 108-148.
« Das Horisontaljpendel, Nova Acta Leop. Garol. Akad., 1892, vol. 60, No. 1. p. 213; also
Brit. Assoc. Rents., 1893; also Ueber Horizontalpendel-Beobachtungen in Wilnelmshaben,
Potsdam, und Puerto Orotava auf Teneiifa, Astron. Nadhrichten, vol. 103, pp. 194r-216.
*Horisontalpendel-Beobachtungen, Beitrage zur Geophysic, vol. 3, Pt. I, 1896.
24 THE TIDAL PROBLEIC.
Darwin relative to their present magnitude, remarking by way of pre-
caution that the quotations given, separated as they are from their context
and the qualifications it carries, are liable to convey misconception of the
author's views on points other than that for which alone they are quoted
here, viz, the magnitude of the tides of the lithosphere. He says:
The chief result of this paper [on Bodily tides of viscous and semi-elastic spheroids,
and on the Ocean tides upon a yielding nucleus] may be summed up by saying that it is
strongly confirmatory of the view that the earth has a very effective rigidity. But its
chief "^ue is that it forms a necessary first chapter to the investigation of the precession
of imperfectly elastic spheroids, which will be considered in a future paper. I shall then,
as I believe, be able to show, by an entirely different argument, that the bodily tides in the
earth are probably exceedingly small at the present time.'
And again, at the end of the later paper referred to:
The conclusion to be drawn from all these calculations is that at the present time
the bodily tides in the earth, except perhaps the fortnightly tide, must be exceedingly
■mall in amount; that it is utteriy uncertain how much of the observed 4' of acceleration
of the moon's motion must be referred to the moon itself, and how much to the tidal fric-
tion, and aocordin^y that it is equally uncertain at what rate the day is at present being
lengthened.'
It has already been made clear that Darwin's inquiry involved the
assumption that in an earlier state, when the earth was more largely
molten or viscous, the body tides were much greater and more effective
than now. But if we substitute the view that the rigidity of the litho-
sphere has been nearly what it is at present through the whole history of
the earth, as is permitted by the planetesimal hjrpothesis, the conclusions
quoted will apply to the whole period, with such modifications as may be
required for differences of distance between the earth and moon.
The substitution of an elastico-rigid earth for a viscous one affects the
rotational influences of the tides qualitatively also. If tidal deformation
causes a movement of the molecules of the lithosphere over one another
in fluidal fashion, friction is the result, and the tide, under present condi-
tions, must have a retardational influence. If, on the other hand, the mole-
cules are merely strained elastically in their relations to one another, but
do not shift these relations as they do in fluidal motion, the strain and
the resilience from it act almost coincidently with the straining force, the
original form and relations are almost perfectly restored on relaxation,
the friction is slight, and the rotational effect will be essentially negligible.
Now, when it is considered that a tidal protuberance, at the very most, can
warp a line of molecules only in some such measure as 5,00^,0^0 ^ 15.00^.000 1
it seems clear that the deformation lies far within the strain-limits of crys-
talline rock, and probably within the strain limits of all rigid substances
in the lithosphere. The only known substances within the outer half of
the lithosphere that probably move as fluids under tidal stress, are the
relatively trivial threads, tongues, or pools of lava within it, and the iso-
lated molecules or groups of molecules here and there in the free form in
the rigid rock. If we postulate an earth of such a degree of elastic rigidity
^ Phil. Trans. Roy. Soc. Lond., 1879, p. 31.
' On the precession of a viscous spheroid, etc. <Phil. Trans. Roy. Soc. Lond., Ft. II,
1879 (1880), pp. 483-484.
DEDUCTIONS FROM THE TIDES. 25
as seems to be required by the concurrent evidences of astronomical,
geophysical, and seismic phenomena, it seems quite inconsistent to suppose
that a brief deformation of the tidal sort can be other than a minute, highly
distributive strain, which involves no flowage motion of the molecules
upon one another, with the exceptions noted, and hence no friction of the
fluidal t3rpe. There is a large body of geological evidence which seems to
indicate that the lithosphere is able to accumulate stresses for long periods,
which are then relieved by permanent deformations. It is difficult to
understand how an earth could be possessed of this ability, if it yielded
fluidally to such transient and moderate stresses as those of the tides of
the outer part of the lithosphere. We therefore assume with confidence
that, whatever the amount of the lithospheric tide, it is only an elastic
strain which relieves itself almost instantly on the removal of the force
which caused it and involves little friction.
It does not appear probable, therefore, that the body tides of the earth,
under this view of the earth's constitution, are an efficient agency in reduc-
ing its rotation.
This conclusion, however, even if fully accepted, does not appear to
cover the entire possibilities of the case; for, even if the primary tidal
deformation of the lithosphere has little or no rotational effect, it may
possibly give rise to pulsations in the spheroid itself which will be com-
municated to the water upon its surface and give rise to water-tides. If
the periods of these pulsations are commensurate with those of the water-
bodies arising from the direct attraction of the moon and sun, they may add
something to these by sympathetic action, even though their independent
value might be inconsiderable. This leads to an inquiry as to the natural
oscillations of the spheriod and their relations to the oscillations of the
lunar and solar tides.
THE PULSATIONS OF THE LITHOSPHERE.
It appears to be possible to reach an approximate determination of
the fundamental susceptibilities of the lithosphere to oscillations of differ-
ent classes by combining the good offices of theoretical computations and
observational inductions. The types of oscillation which need to be consid-
ered here embrace those which traverse the interior as well as the surficial
parts of the earth as distinct waves of propagation, and those oscillations
of shape which affect the form of the earth as a whole. The latter are
treated as harmonic pulsations and may spring either from the transmitted
oscillations or from differential stresses arising from variations of attraction.
The data relative to transmitted oscillations have been furnished chiefly by
seismologists; the treatment of harmonic pulsations and fundamental sus-
ceptibility to such oscillation has thus far been chiefly mathematical.
Lamb, following earlier work by Kelvin, has shown that several different
species of harmonic oscillations may arise from both the longitudinal and
transverse waves transmitted through the earth.^ For a steel body of the
size of the earth, he found the period of the slowest fundamental mode of
1 On the vibratioDB of an elastic sphere, by Horace Lamb. kFtoc. Lond. ICath. Soe.,
vol. 13, 1882, pp. 189-212.
26 THE TIDAL PROBLEIC.
oscillation which assumed the form of a harmonic spheroid of the second
order to be 78 minutes. A series of other oscillations of lesser lengths
would be developed. He found that the compressibility of the matter is
not a vital factor, for if t be the time required by a wave of distortion to
traverse the earth's diameter^ and if P be the period of oscillation of shape,
then P ^ t 0.848 if the material is incompressible, and P — t 0.840 if
the material preserves uniconstancy. Bromwich,^ bringing into the com-
putation the effect of gravity, found that the gravest free period of a sphe-
roid of the size, mass, and gravity of the earth, with a rigidity about that
of steel, is 55 minute. The corresponding period, if the effect of gravity
be neglected, is 66 minutes. If the rigidity be about that of glass, the
period is 78.5 minutes if the effect of gravity be included, and 120 minutes
if gravity be neglected.
Nagaoka ' has made a study of the pulsations connected with the Kra-
katoan eruptions of August 26 and 27, 1883, as recorded by the gasometer
at Batavia, 94 miles from Erakatoa, on the supposition that these pulsa-
tions were derived directly from the volcanic explosions and thus registered
their relative times. He reached the conclusion that the series of eruptions
were rhythmical with a unit-period of 67 minutes and a tendency toward
the grouping of these shorter periods into larger ones of about 200 minutes.
The former he interprets as an expression of the fundamental period of
oscillation of the earth as a spheroid. Referring to the results of Brom-
wich, he cites the coincidence of the Erakatoan periods so deduced with
the computed periods when the assumption is made that the rigidity of
the earth lies between that of steel and that of glass. Nagaoka also cites
the apparent relationship of this period to seismic phenomena, and the
apparent connection of certain of these phenomena with the Chandlerian
nutation of the pole.
The correspondences may be carried appreciably further. While exact
determinations of the velocities of seismic tremors recorded at a distance
from an earthquake are not yet available, the time required by the fore-
most waves to traverse the earth's diameter may be taken provisionally
at 22.6 minutes. These vibrations are generally interpreted by European
and American seismologists as compressional waves and as passing through
the earth along chords, or along curves of adaptation departing slightly
from chords. The second set of tremors, generally interpreted as distor-
tional, require about 50 per cent longer for chords up to 140^, and perhaps
up to 180^, which would make their diametrical period about 33.75 min-
utes. For the chords between 140® and 160% and perhaps up to 180®,
Oldham inferred a longer period from the available observations, which
are, however, thus far not sufficiently numerous for positive conclusions.
These problematical vibrations may be directly transmitted or may be
reflections. The period deduced for them is approximately double that of
the compressional waves. The foremost large seismic waves, which have
approximately a uniform velocity and which are interpreted as following
^ On the influence of gravity on elastic waves, and in partictilar on the vibrations of
an elastic globe, by T. J. A. Bromwich. <Proo. Ix>nd. Math. Soc., XXX, 1899.
* Nature, May 25, 1907, pp. 89-91.
DEDUCTIONS FROM THE TIDES. 27
the surface of the earth, pass from the point of origin around to the anti-
podal point in about 112 to 115 minutes. The largest and strongest group
of these large waves takes about 135 minutes. Comparing these with one
another, it appears that the period of the maximum group of the large waves
is six times that of the compressional waves, the former, however, travers-
ing a semicircumference and the latter a diameter. The period of the second
set of short vibrations for most chords is one-half more than that of the
first set, while that of the problematic set is approximately twice that of
the first set. Three times the period of the compressional waves, twice
that of the best recorded distortional waves, one and a half times that of
the problematic waves, and half that of the maximum long waves are
each approximately 67.5 minutes, or essentially the same as Nagaoka's
unit-period for the Krakatoan pulsations. The gasometer record of the
Krakatoan eruptions is rather coarse and can not be read with exactness,
but, taking Nagaoka's readings, the discrepancies between the recorded
times of the twelve eruptions of August 27 and the periodic times on the
67.5 minute basis are as follows in minutes: (starting-point, first erup-
tion on August 27); -29 (—33.5, the half period, + 4.6); +14; -1.5;
+2; +16; -3.5 (strong); + 3 (strong); +23.5 ( = 22.6, one-third period,
+ 1); +33.5 (half period); -1.5 (the great eruption); -9 (the final erup-
tion, strong). In the interpretation it is assumed that the eruption of
August 26 at 6^ 20™ p.m. started a series of oscillations in the lithosphere
which, at the end of the sixth period of 67.5 minutes, with a lag of 4 min-
utes, had developed sympathetic relations with the volcanic forces and
stimulated the first of the twelve eruptions that followed. These have the
degree of correspondence to the assumed period just shown. Each erup-
tion falling at or near the critical stage of the pulsation previously developed
may be supposed to have strengthened the succeeding oscillations until
the series reached a first double maximum at the seventh and eighth erup-
tions, and a second and greatest maximum at the eleventh and twelfth.
If this interpretation be justified, it may mean that the vibrations which
arose from the earthquake developed into the form and periodicity of the
fundamental vibrations of the earth-spheroid. The inadequacy of the
data, quantitatively and qualitatively, to establish this positively is obvi-
ous, and it may not be safe to rest much upon it; but the following are
curiously related to it.
The moon's synodical period, 1,490.5 minutes, is 22 times 67.75. The
solar period, 1,440 minutes, is about 21.25 times 67.75. If 67.75 minutes
be taken as the normal period of spheroidal oscillation, 22 of these con-
stitute a lunar day, 314 approximately the average fortnightly excursion
of the moon north and south of the equator, and 628 the lunar month.
If a represent the northerly fortnightly excursion, and a' the southerly,
each of these equaling 314 earth-pulsation periods, they will obviously
have close commensurate relations at the periods represented by aa\
a (a'\'a'), and (a'\'ay, whose numerical values are 218, 436, and 870 days,
respectively. Now, 436 days is the recent estimate of Kimura * for the
* Fhymco-Math. Soc. Tokyo, Pt. II, 24, pp. 357-364, 1905; Sci. Obs., July 26, 1906.
Pop. Astr., Oct. 1906, p. 469.
28 THB TIDAL PSOBLElf.
larger circular element of the Chandlerian nutation of the pole, and this
period of Eimura is perhaps to be regarded as a closer approximation
than the earlier estimates of 427 to 430 days. As the sums of the tides
formed when the moon is on the equator, is north of the equator, and is
south of the equator, respectively, are different from one another, partly
because of the differences in the moon's position and partly because of
differences in the configurations of the lands and seas on the two sides of
the equator, there seems to be a fair presumption that there would be a
periodic difference in the tidal influences on the rotation of the earth
about its axis corresponding to the fortnightly excursions, which would
express itself in a nutation. Now if this period of forced nutation happens
to be commensurate with the free period of the earth as a rotating body,
the effect would be cumulative. Euler long ago computed that the period
of free nutation of the axis of the earth, if it were an absolutely rigid body,
would be 305 days. Newcomb, on the assumption that the earth has the
rigidity of steel, found that the period would be increased to 447 days.
This seems to imply that the earth is somewhat more rigid than steel
and has a free nutation period somewhere about 427 to 436 days. As the
fortnightly group of tides have a cumulative period commensurate with
the latter, the nutation of 436 days may perhaps be due to the agency
of this tidal group.
In addition to this larger circular nutation, whose radius is about 15
feet, there is a smaller elliptical nutation, of about 4 feet by 14 feet, with
an annual period. This is assignable to the annual migration of the sun
north and south of the equator, which gives rise to a variety of dynamic
effects in the form of changes in the circulation of the atmosphere and of
the ocean, in the accumulation and melting of snow and of ice, etc. This
is in line with the common explanation of this minor nutation.
It is an established principle that when the normal period of oscilla-
tion of a body is less than the period of the periodic force acting on it,
the oscillations of the body will agree in phase with those of the force.
On this principle the oscillations of the lithosphere should agree in phase
with the period of the tidal forces. There should therefore be direct co-
operation between the waves of the lithosphere and the forced water-waves.
On account of this close coincidence there is an obvious difficulty in dis-
tinguishing the contributions of the lithosphere to the water-tides from
those tides which spring directly from the attraction of the tide-producing
bodies. The two should merge into a common tide, but, if the view here
entertained relative to the development of water-tides through oscillations
of the lithosphere be valid, the actual tides are to be regarded as composite.
If the tides of the lithosphere were of the fluidal type and acted in strict
coincidence with the water-tides, they would reduce the latter to the extent
of their own magnitude, as urged by Kelvin and Darwin;^ but in so far as
the pulsations of the lithosphere have the effect of a series of tiltings of the
basins on the lithospheric surface, they must impart oscillatory movements
to the water held in the basins. It is safe, on observational groimds, to
' Thomson and Tait, Natural Philosophy, Pt. II, p. 439.
DEDUCTIONS FBOlf THE TIDES. 29
aflBrm that all the oceans behave, in the main, as if they were isolated bodies
held in basins on the surface of the lithosphere. There are no effective tidal
belts stretching around the earth parallel to the equator and furnishing an
opportunity for the development of a continuous tide of the canal type.
The Southern Ocean, once regarded as such, does not prove to act in this
way, nor do the Pacific and Indian Oceans act as a common body, as
represented on the old tidal charts.^ For the purposes of this discussion it
may be assumed with practical safety that the seas occupy a chain of irregu-
lar basins linked to one another in various unsystematic manners, and that
each of these bodies is subject, in its own way, to such oscillations as the
rocking of its basin may impart to it. If the tides of the lithosphere are
as small as present evidence seems to indicate, this may not be important in
its own first effects, but as a periodic action it may become, by commen-
surate accumulation, a not unimportant factor. Some of the peculiar fea-
tures of the tides seem to be much more intelligible on the supposition that
they arise from the oscillations of the lithosphere than from the direct action
of the lunar and solar attractions. The rocking action of the basins would
generate tides as freely on the eastern as on the western sides of the oceans,
whereas the attraction of the moon and sun should be accumulative toward
the western side. The tides are, however, rather higher on the eastern than
on the western sides of the oceans. We shall have occasion to return to this
significant feature.
When a strain, or a deformation, or a movement of any kind is being
impressed with increasing or declining intensity upon an elastic body which
is already in a state of constant pulsation, as is the lithosphere, the super-
imposed action becomes itself pulsatory, however continuous and uniform
the increment or decrement of the superimposed action may be in itself,
for the existing pulsation of the body alternately opposes and coincides
with the superimposed action and gives it a corresponding pulsation. The
water-tides assigned to the rocking of the containing basins may therefore
be treated as composite pulsations, each advancing and each declining
phase consisting of an undetermined number of pulsations, each of which
gives rise to its own partial free wave. These, as do all waves of what-
ever source, react on the lithosphere. Each such reacting pulsation, so
far as it takes the form of a compressional wave, passes through the litho-
sphere to the antipodes in about 22.5 minutes. It there constitutes an
impulse acting at an angular distance of about 5.6^ in the rear of the corre-
sponding part of the antipodal wave, tending, in its minute degree, to
strengthen it, but with a slight increase of amplitude. The return of this
wave requires an equal period which brings it into action at about 11.2°
in the rear of the crest of the wave from which it sprang. This wave will
therefore act several times in an approximately commensurate way before
any appreciable incommensurate effects will be developed, and by that
time its force will largely be spent. In so far as the reaction of the original
wave develops an undulatory wave on the surface of the lithosphere,
this wave will reach the antipodes in about 2.25 hours and will act in a
1 See Tidal charts of the U. S. Coast and Geodetic Survey, Rept. Sept. 1900, App. 7,
Outlines of Tidal Theory, Pt. IV a, Rollin A. Harris. Also Pt. IV b, 1904.
30 THE TIDAL PROBLEM.
similar way on the antipodal wavci tending to increase its amplitude and
to reduce its surface gradient. On its return it will fall so far behind the
original wave as to have appreciable incommensurate effects. The waves
of intermediate period will have corresponding intermediate effects. In
so far, therefore, as water-waves react upon the lithosphere and develop
waves in it, these, while cooperating with the original waves commensur-
ably for a time, will tend to distribute the oscillatory action into broader
amplitudes. This may be looked upon as a tendency to develop a distribu-
tive series of small pulsations in lieu of the original more concentrated one.
It is observed that small pulsations attend the incoming and outgoing of
the tides, but they have not, so far as I know, been made the subject of
sufficient study to determine whether they are systematic or irregular,
and whether their periods are at all in accord with the natural periods of
the lithosphere or not.
It is assumed in the foregoing that tidal pulsations will move through
and over the lithosphere at the same rate as seismic pulsations, which
probably does not involve any essential error, though pulsations vary some-
what in their speed, even when of analogous classes. But only the general
order of velocity is of special moment here.
According to the mathematical investigations of Lamb, there should
be a double series of modes of oscillation in the spheroid derivable from
the initial impulse, of which one set should spring from the compressional
waves and another set from the distortional waves, and these should differ
in period. Only the gravest periods have been cited above. Without
attempting to determine what these shorter periods are in the case of the
lithosphere, it is probably safe to say that such as are commensurate with
the tides of any body of water would cooperate to build these tides up and
such as are incommensurate would have the opposite influence. Now if
the tides of each body of water are essentially individual and are radically
influenced by the breadth, depth, and configiuration of the water-body,
it is not improbable that different species of both series of natural pulsa-
tions may cooperate with the tides of different oceans and assist in their
perpetuation and development.
If the distribution of the strains developed by increasing or diminishing
attraction takes place at a velocity similar to seismic vibrations, even the
larger tidal movements of the lithosphere will act almost simultaneously
with the tide-generating forces, for no strains of the same phase will extend
more than 4,000 miles from the center of development of that phase. The
extreme movement from the center to the circumference of the strained
area would occupy, at the observed rate of compressional waves, less than
12 minutes. In so far, therefore, as retardation of the earth's rotation is
dependent on lag of the tide, it will be inconsequential for this class.
Relative to the tides of an elastic earth, Darwin says:
The other hypothesis considered is that the earth is very nearly7perfectly elastic.
In this case the semi-diurnal and diurnal tides do not lag perceptibly, and the whole of
the reaction is thrown on the fortnightly tide, and moreover there is no* perceptible tidal
frictional couple about the earth's axis of rotation. From this foUows^the remarkable
conclusion that the moon may be undergoing a true secular acceleration of motion of
DEDUCTIONS FROM THE TIDES. 31
toiTwthing Imb than S.G* per century, whilst the length of the day may remain almost
unaffected.
The leeuits of theoe two hypotheses (a viscous spheroid and a nearly perfectly elastic
spheroid) show what fundamentally different interpretations may be put to the phenom-
enon of the secular acceleration of the moon.
Under these circumstances, I can not think that any estimate having any pretension
to aoeuraoy can be made as to the present rate of tidal friction.^
THE TIDES OF THE HYDROSPHERE.
If the preceding views are tenable, practically the whole tidal effect
on rotation at present is concentrated in the water-tides. A part of these
are assigned to the immediate action of lunar and solar attraction and a
part to the mediate action of the lithosphere. While the lithosphere is
thus supposed to contribute to the formation of the water-tides, this sup-
plementary action is supposed to be qualified by its distributive action
as previously explained. The water-tides are thus interpreted as more
complex in origin than they have usually been thought to be. This must
doubtless be regarded as an unwelcome infliction, for even under the simpler
conception of their origin from direct attraction only, they are, in many
of their phases, beyond complete mathematical treatment. These added
complexities put them still further beyond the reach of such treatment.
But this added complexity may, after all, only help to force us on toward
the adoption of naturalistic methods. It has been becoming increasingly
clear for some time that, to secure reliable results, the tides must be studied
on a direct observational basis. The more hopeless the purely theoretical
method becomes, the more assiduously is the observational method likely to
be pursued. If theoretical methods are given precedence, they should be such
as are based on the fundamental laws of energy, which hold good irrespective
of special forms of action, however multitudinous and irresolvable.
As already remarked, the tidal water-bodies have no systematic, much
less have they any symmetric, distribution. Innumerable idealizations
as to the forms and relations of the oceans have been framed, but beyond
a few of a very general sort, they are notable principally for their imdue
emphasis of amenable concurrences and their neglect of refractory non-
concurrences. The north-south extensions of both the eastern and western
continents are particularly unfavorable for the development of a con-
tinuous forward movement of the tides. The southern ocean furnishes
the only continuous east-west belt of ocean encircling the earth's axis of
rotation, but, according to the cotidal charts of the U. S. Coast and Geo-
detic Survey, this is not affected by a continuous westerly tide. The
Pacific tides move easterly from New Zealand and, by interpretation of
the scant data available, easterly all the way to the straits between South
America and Antarctica, through which they move eastward and then
northward along the Patagonian coast. The tides of the northeastern part
of the Indian Ocean move easterly into the straits between Australia and
Asia, while the Pacific tides enter on the opposite side and the two sets
1 On the procession of a viscous spheroid, etc. <Phil. Trans. Roy. Soc. Lond., Ft. II,
1879 (1880), p. 629.
32 THB TIDAL PROBLEM.
meet one another within the inter-idand water-bodies. There seems to
be no definite perpetuation of the Pacific tides into the Indian Ocean,
these bodies, though connected both north and south of Australia, acting
in essential independence. While the long-prevalent view that the tides
of the Atlantic and Arctic Oceans are derivatives from the Southern Ocean
still has the apparent support of observational data, there are many facts
that seem to indicate that this is only a part of the truth. A derivative
wave should gradually die down as it progresses; but, notwithstanding
the distance of the North Atlantic from the Southern Ocean, the tides are
higher there than those of the South Atlantic. A derivative wave should
be intensified in passing a constriction and should be lowered in an expanded
water-body beyond; but, notwithstanding the reduction of the Atlantic
breadth between Brazil and Sierra Leone, the tides are particularly high
in the lee of the great nose of Africa north of this constriction. The north-
easterly-trending coast of New England and the Provinces stands directly
athwart an unobstructed stretch of sea reaching back to the Southern
Ocean along the line of assumed propagation, and yet the average tide on
this coast is notably less than that on the European coast of the same lati-
tudes, though this lies behind the African projection. Comparing the tides
on the Atlantic islands — whose isolation should render them measurably
free from local influences, save those of their own basal slopes and their
harbors — it is notable that the tides on the islands of the South Atlantic
average less than half as much as those of the islands of the North Atlantic.
The tides on the islands in the far North Atlantic, and even some of those
in the borders of the Arctic Ocean, are singularly high, such as those of the
Faroes, Shetlands, Orkneys, Hebrides, Iceland, Greenland, Jan Mayen, and
some Arctic Islands of North America. On the purely derivative theory,
these tides must be supposed to have been traveling 24 hours or more
since they left the place of their origin, and those in the high north have
been subject to the damping e£fects of polar ice.
There are not a few anomalies that are very puzzling on the supposition
of a westward drag of the waters by the moon and sim. The northeast
coast of South America trends northwesterly and the Central American
states continue the trend in fair alignment. Over against this, the North
American coast has a southwesterly trend, meeting the projection of the
northwesterly trend of South America on the coast of Guatemala, thus
forming a wide eastward-facing angle. From this point to Cape Race,
the angular distance is 40° and from it to Cape St. Roque, 50°, while the
open eastward-facing mouth between Cape Race and Cape St. Roque is
about 50°. On the hypothesis of a westward-moving tide, cumulative
toward the west, we should expect high tides in the Antilles, the Caribbean
Sea, and the Gulf of Mexico. If it is thought that the last two bodies are
protected from this high tide by the Antilles, the tides on the eastern side
of the Antilles should be markedly high. The record does not show this.
The tides on the African and European coasts opposite are notably higher
than those which might be supposed to be unusually concentrated within
this angle. As bearing on any supposed protection of the Gulf of Mexico
and the Caribbean Sea by the Antilles, a comparison may be made with
DBDUCnONB FBOM THB TIDES. 33
Hudson Bay, which is far more land-locked and is in a much higher latitude
and much farther from the assigned source of derivation. Singularly
enough, the tides in Hudson Bay are several times as high as those of the
Mezioan Gulf and the Caribbean Sea.
If it be objected that the North Atlantic is too narrow and too peculiar
in its relations to give these singular features much weight, a similar line
of inspection may be applied to the Pacific, whose breadth and equatorial
position make it preeminently favorable for a westward accumulation of
the tide. From the mass of data now made available by Harris's compila-
tions and harmonic reductions, it appears that the tides on the eastern
side of the Pacific as recorded on the American coasts are notably stronger
than the average tides on the Asiatic coasts. A comparison of the tidal
heights on the Pacific islands, though the data are inadequate, also fails to
show a concentration on the western side.
On the old cotidal charts, and more definitely on the new ones of Harris
(fig. 3), it appears that the dominant tide of the Pacific originates in a
singular loop near the Galapagos Islands, off the coast of South America,
from which, on one side, a wave moves easterly and southeasterly to the
South American coast in strong force, then down it to the extremity of
the continent, where, according to Harris, it rounds Cape Horn and moves
up the eastern coast of Patagonia to about the mouth of the Rio de la
Plata. On the other side of the Galapagos loop, a wave moves north-
westerly along the North American coast, and then westerly toward the
Asiatic coast. In the heart of the Pacific, Harris locates three amphi-
dromic centers of practically no tide (fig. 3). While these are not directly
based on observations, they are believed by this industrious and original
student of the tides to be in accord with the data derivable from the observed
tides of the central Pacific Islands. This singular dispersion of the Pacific
tides from the vicinity of the Galapagos Islands near the eastern border
of the great ocean, as shown on the old charts, suggested to me, perhaps
a decade ago, that the water-tides might be largely derived from the litho-
sphere rather than directly from the attraction of the moon and sun on the
water itself. The rocking of the basins, first by a lift on the east side and
later by a lift on the west, under the progressive influence of the tide-
producing body, seemed to me more compatible with this behavior of the
tides than direct attraction on the water itself, which I supposed should be
less effective on the east side than on the west. The greater strength of
the tides on the east side of the Atlantic also strengthens the impression
that, whether this suggestion of derivation from the pulsations of the
lithosphere be of any value or not, the actual evolution of the tides involves
much more than the simple upward pull and westward drag of the waters
by the moon and sun.
The recent theory advanced by Harris,^ that the tides are largely due
to the cumulative agency of stationary oscillations in such segments of
the oceans as may act commensurately with the tidal forces, goes far to
relieve the foregoing and similar features of the tides of their seeming
t "Outlines of tidal theory/' Rollin A. Harris, Rep. U. S. Coast and Geodetic Sunr.
1900, app. 7, pp. 636-e09.
34 THE TIDAL PROBLEM.
incompatibility with the theory of direct attractional action on the water
itself; indeed, the concrete features of Harris's theory seem to have been
developed largely by a study of these remarkable features, and to be an
attempt to give them a dynamical expression in terms of the direct-action
hypothesis. Whatever shall be the final judgment regarding particular
aspects of this theory, whose author claims for it only a partial explana-
tion of the tides, it seems eminently probable that commensurate oscilla-
tion is a vital factor in building up the waves of the actual tides. The
attempt to work out a concrete theory of the tides from their specific phe-
nomena is greatly to be commended, for by concrete application alone are
the proximate sources of the tides likely to be determined. This may be
said without derogation of the value of the more general theories.
Concerning the insufficiency of simple attraction without sympathetic
intensification to explain the actual tides, Harris says:
In approaching the question of the actual causes of the tides, upon which so much
labor has been expended and oonceming which so much has been written, one may well
surmise that the subject does not admit of accurate or complete treatment. It is there-
fore natural to consider, in the first place, only those sources which would seem to account
for the dominant tides in any given region under consideration, and to postpone, perhaps
indefinitely, the consideration of those sources whose importance in the production of the
tides must be relatively small. Considering the actual distribution of land and water a
few computations upon hypothetical cases will suffice to convince one that as a rule the
ocean tides, as we know them, are so great that they can be produced only by successive
actions of ibe tidal forces upon oscillating systems, each having as free a period, approxi-
mately the period of the forces, and each perfect enough to preserve the general character
of its motion during several such periods were the forces to cease their action.*
In another place he says:
Unless the free period of a body of water, or of some portion of this body, approxi-
mately agrees with the period of the tidal forces, the tide in the body proper must be small,
and generally smaller than the theoretical equilibrium tide for the body in question. But
in many parts of the oceans, the tide is several times greater than that which could be
raised by the forces, even if we could suppose sufficient depths and sufficiently complete
boundaries for enabling equilibrium tides to occur. Hence regions the dimensions of
which approach critical values must exist in the oceans and account for the principal tides.
That stationary oscillations of unexpectedly large amplitude exist in the oceans there
is abundant evidence. In fact, a glance at the charts will show regions of large ranges
over each of which the time of the tide varies but little. As a nodal line is approached the
range diminishes, and the time of the tide changes rapidly in a comparatively short distance.
Moreover the dimensions of the oceans are such that areas having nearly critical lengths
can be readily discovered; these respond well to the forces, and their tides must be the
ruling semi-diurnal tides of the ocean.'
Harris has attempted to detect those portions of the oceans whose
lengths, depths, and relations make them susceptible to the development
of free oscillations whose periods are sufficiently near to the periods of the
tidal forces, or to some simple fraction of them, to permit cooperation in
building up effective stationary systems of oscillations. Of the major
order, he finds a northern and a southern system in the Indian Ocean, a
South Atlantic system, a North Atlantic system, and two systems in the
Pacific, as well as a large number of systems of the minor order. Diagrams
and details of the main systems are given in the original paper.
^ Loc. cit., p. 624. ' Nature, Feb. 22, 1906, p. 388.
DEDUCTIONS FROM THE TIDES. 35
Harris does not claim that the demonstration of these is complete
or finali and he recognizes that much additional data will be required for
a full verification of the postulated systems and for completing the full
cat^ory of systems, but he believes that those announced correspond
fairly well with existing knowledge. Question has been raised as to whether
the phases of oscillation which he assigns to his systems are such as would
naturally arise from the forced waves. Question has also been raised as
to the adaptability of the oceanic segments to oscillate as postulated.^
Harris contributes a new map of cotidal lines for the world (fig. 3) in which
interpretation is conveniently combined with the observed data which
are also given separately in tables and in sectional charts. By comparison
of this cotidal map with previous charts, it will be seen that Harris's inter-
pretation departs rather markedly from that implied by the previous
cotidal charts.
A significant feature of Harris's cotidal map is its amphidromic nodes,
centers of little or no tide about which the tidal wave swings in the course
of the twelve-lunar-hour period. Three of these amphidromic points are
located in the Pacific, one in the North Atlantic, and one in the Indian
Ocean (fig. 3). These points are associated with nodal lines that separate
the oscillating sections of the systems to which they are assigned. That
there should be such nodes of little or no tide in the heart of the great
oceans, where under the familiar mode of interpretation the tidal waves
should have their freest sweep and greatest strength, well expresses the
extent of Harris's interpretation^ departure.
It appears then that, under the broad mantle of the postulate that the
tides are due to the attractions of the sun and moon, there are three special
or proximate views as to the immediate origin of the actual tides: (1) the
direct attraction of the tide-producing bodies on the water; (2) the effect
of stationary oscillations promoted by such direct attraction; and (3) the
to-and-fro tilting of the rock basins in which the water-bodies rest by
the tides of the lithosphere. The older view has always recognized the
supplementary effect of the natural oscillations of the water-bodies, but it
has never given them a prominent place nor quite that distinctive form
which has been assigned them by Harris. Since the cooperation of oscilla-
tions is independent of their source, any waves that may come from the
lithosphere are as available for building up systems of stationary oscilla-
tions as are those springing from direct attraction, since they are likely to
be timed quite as well. Harris's theory, or any theory of its kind, may
therefore find as good a working basis in tides derived from the lithosphere,
so far as these go, as in those formed by direct attraction.
The vital question here is the bearing of the deductions from these three
points of view — assuming that each of them represents some truth — on
the rotational problem. Let us first consider this on the familiar assumption
that retardation is dependent on the position of the waves, assuming that
to produce retardation the wave on the moonward side must be in front
of that body pulling it forward and being itself in turn pulled backward
^ Nature, Sept. 4, 1902, p. 444; Apr. 23, 1903, p. 583; Jan. 11, 1906, p. 248; Feb. 22
1906, p. 388.
3
36 THE TIDAL PBOBLElf.
over the earth's surface (figures 1 and 2). Afterwards let us consider
the phenomena from a more radical point of view founded on the laws of
energy and the configuration of the interacting bodies.
From what has already been said, it is clear that no continuous tides
are being dragged around the earth acting as a frictional band. No single
tide moves westerly so much as one-half of the earth's circumference, and
most of the tides have a much less movement in that direction. On the
other hand, many tides move easterly and still others move northerly and
southerly. The position of these relative to the moon is various, and the
attraction of the moon upon them may be accelerative or neutral as well as
retardative, so far as instantaneous attraction while in the given positions
is concerned. A wave starting in the Southern Ocean and moving north
through the Atlantic for more than a day will rim the whole gamut of
positional relations to the moon and sun, and will, considered simply as an
attached protuberance, be retardative, neutral, and accelerative in turn.
In the case of waves that move to and fro across the water-bodies in seiche-
like fashion, it is obvious that the positional relations may be various.
The most interesting cases are those of water-bodies whose periods of
oscillation are nearly commensurate with the periods of the tidal forces.
The breadth and depth of a water-body may be such that a wave started
under the moon when it passes over the eastern margin will cross to the
western side and return to the eastern just in time to fall imder the moon's
next crossing of the eastern margin, and so be reinforced by every return.
In a body a little wider or a little shallower, the return of the wave would
fall behind the moon's arrival and at its turn tend to retard the moon's
motion, while in a body a little less wide or a little deeper the turn will
come before the moon's arrival and the wave, at its turn, will tend to
accelerate the moon's motion. But if either of these waves were to be
followed through its whole course and its relations to the moon observed,
it would be found to be accelerative, retardative, and neutral at different
points.
Pursuing this linQ of inspection, it may be seen that the waves developed
in the basins of the lithosphere must have a wide range of periods, some
longer, many shorter, than the period of the tidal forces. Their rotatory
influence on this basis of treatment is thus extremely difficult to analyse
and evaluate, and the algebraic sum of all such influences is quite beyond
mathematical determination.
The case is even more complicated when we consider amphidromic
systems and those whose oscillations lie in lines oblique to the axis of
rotation and to the moon's course. In the case of a stationary oscillation
neither forward nor backward drag seems to be predicable as a total result,
on this basis of treatment.
When all of the multitudinous phases are considered, it is clear that
the case becomes so extremely complex that' it can not be solved with any
assurance of a reliable conclusion by analyzing the rotational effects of
individual cases and summing the results. Some more basal method, so
chosen as to escape these complications and the uncertainties of their
interpretation, is required.
DEDUCTIONS FROM THB TIDES. 37
Poincar^y after a mathematical treatment of the influence of the water-
tides on the earth's rotation in the endeavor to simplify the case, reached
the following conclusion:
L'infliienoe des mardes oo^aniennes sur la durde du jour est done tout k fait minime
et n'sit nuUement comparable k I'effet dee mar^ dues k la viscosity et k T^lasticitd de la
pariie aolide du g^be, efifet sur lequel M. Darwin a insiBt^ dans une s6rie de M^moires du
plus haut int6r^.*
A MORE RADICAL MODE OF TREATMENT.
To the foregoing method of treating the rotational effects of the tides
on the basis of the poaitiona of the tidal protuberances and depressions,
as such, there seem to be, as previously intimated, graver infelicities than
those of mere complexity. The method appears to be defective in neglect-
ng the cooperating effects of the changes of kinetic and potential energy
that are associated with these differences in the distribution of matter.
These protuberances are not fixed masses of matter, but rather aggregates
of variations in the paths of the molecules of water in their revolutions
about the earth's axis. In the production of these protuberances and
depressions there are reciprocal increases and diminutions of the potential
and kinetic energies of the water particles involved. In analyzing the
influences of these on rotation, it will be serviceable to separate the factors
of inertia and friction — ^including under the generic term friction all obstruc-
tive effects growing out of the relations of one particle to another — because
the fimctions of these factors are contrasted, since the inertia tends to
perpetuate any given state of motion, while the friction tends to reduce the
amount of motion. There are also certain advantages in considering each
particle separately as a body in revolution about the axis of rotation.
Let therefore the lithosphere be regarded as a perfect spheroid sur-
roimded completely by an ocean of uniform depth, and let the matter of
each particle be regarded as concentrated into a point and separated from
its fellow particles by a complete vacuum, but let the collapse of the particles
be prevented by a hypothetical force taking the place of the resistance to
condensation which affects the water in nature. We shall then have an
ocean made up of mass-points which move in perfect freedom from frio-
tional and other obstructive relationships; in other words, these points
will constitute satellites of the lithosphere which may here be regarded as
a rigid body acting as a massive point at its center. The behavior of the
mass-points may then be treated, qualifiedly, according to the principles
of celestial dynamics. In fig. 4, let E represent the earth, the circle L the
surface of the lithosphere, and the circle A BCD the ideal surface of the
hydrospheric satellites when revolving without perturbation by the moon.
Then, according to the principles of celestial mechanics, first applied to
this dass of cases by Newton,' the orbit of a particle, p, will be a closed
curve, abed, closely resembling an ellipse, whose major axis is transverse
to a line joining the centers of E and M. The general configuration, it
will be noticed, is that of the ''inverted tides." The particle p in passing
> Bulletin Aatronomigue, vol. 20 (1903) " Sur un Thter^me Q^n^ral Relatif auz Mar6M/'
par M. H. Poincar^, p. 223.
' Mouhon'B OdestiA] Mechanics, art. 156, p. 243.
88
THB TIDAL PROBLEM.
from a to b will lose velocity — and hence kinetic energy — and gain potential
energy. At the point b it will have the minimum of motion and the maxi-
mum of potentid energy. From b to c it will fall back toward the center,
a portion of its potential energy being converted into kinetic, and its veloc-
ity being increased and reaching a second maximum at c. In the second
half of its orbit, cda, similar
exchanges of kinetic and poten-
tial energy will take place.
If p is affected by no friction
or obstruction in its course,
these exchanges of kinetic and
potential energy will be com-
pensatory and maybe continued
indefinitely without affecting
the rotation of E. The case is
that of an inner satellite or an
inner planet when all the bodies
involved are considered as rigid
bodies or massive points. But
if now friction be introduced
at any paint in the orbit of p,
heat wiXL be developed and
dissipated, and energy lost to
the system. Looked at in
detail, it would seem that the
retardation of p by friction on
E in some phases of its orbit
would be accelerative to E'a
rotation, and in other portions,
retardative, for in some por-
tions p'a angular motion is
greater than E'a, and in others
less, but traced out in its full
history it appears that what is
seemingly accelerative in one
phase is retardative in another
and that the ulterior effect is
precisely measured by the loss
of mechanical energy by con-
version into radiant energy and
dissipation. For example, fric-
tion between p and the normal
periphery of J? as represented
by A BCD, at or in the vicinity of e, will be accelerative in its immediate
phase because p in this part of its orbit is moving faster than the contact
portion of E, but the retardation of p in this portion will reduce the rise
of p in the section at and near d which is retardative in its immediate
phase because in this position p is moving more slowly than the normal
DEDUCTIONS FROM THE TIDES. 39
periphery ABCD. These would be precisely compensatory — outside action
n^eeted — ^if there were no loss of energy by dissipation. This loss reduces
the mechanical action of the system by precisely the mechanical equivalent
of the heat lost.
In the actual system the distribution of the loss of energy is necessary
to a complete solution. When the earth-moon tides alone are considered,
it is clear that the lost energy must come either from the rotation of the
earth or the revolution of the moon, or from both partitively. Now the
moment of momentum of the system must remain constant and the distri-
bution of the loss of energy must be such as to meet this requirement.
Just what this distribution must be in a given case depends on the con-
figuration of the system which then obtains, and is a question of celestial
mechanics rather than of tidal theory. For the purposes of the present
discussion it is of decisive moment to know whether the configuration of
the earth-moon system at present is such that the earth and moon may
recede with reduced rotation of the earth if the dominant phase of the tides
is what has been called retardative, or may approach with accelerated
rotation of the earth if the phase is what has been regarded as accelerative,
or whether the configuration of the earth-moon system is such that it can
move in one direction only when loss of energy by tidal friction takes place,
bearing in mind that all tides give rise to friction and dissipation of energy.
For a solution of this I appealed to Dr. Moulton, who found that, under
the present astronomic relations of the earth and moon, any loss of energy
by their interaction requires that the bodies recede from each other and
that the rotation of the earth be diminished. This he finds to be rigorous
under the laws of energy, and it seems to follow as a necessary inference
that the special phase of the tidal action which caused the loss of energy
is immaterial. While this determination at the time was wholly indepen-
dent, it was soon recalled that Sir George Darwin had made a similar deter-
mination so far as the dynamic relations of the earth and moon under loss
of energy are concerned. It does not appear, however, that he drew the
inference we have just drawn, for this seems to exclude any differential
e£Fect dependent upon the positions or phases of the tides, other than such
as is expressed in the amount of friction involved, which is dependent
solely on the amount of the water-movement and not on its phase.
There are assignable configurations of the system in which a movement
in the opposite direction would take place if energy were lost by interaction.
Such a configiuration would obtain if the centers of the earth and moon
were within 9,000 miles of each other or their surfaces about 4,000 miles
apart. The precise figure reached by Dr. Moulton, all computable influ-
ences being taken into account, is 9,241 miles. Dr. Lunn computed this
independently, using slightly different values for some of the factors — essen-
tially those of Darwin — and neglecting some inconsequential ones, and
foimd the distance 9,113 miles. From this the conclusion seems inevitable
that if, for any reason, the moon were separated from the earth within
this distance, the tidal interaction of the earth and moon would tend to
bring them together, an adverse tendency which the fission theory must
face in addition to those cited before.
40
THl TIDAL PBOBUBIC.
Relative to the question whether the lost energjr, in the present con-
figuration of the earth-moon system, must oome wholly from the rotation
of the earth or in part from the revolution of the moon. Dr. Moulton finds
Fio.6.
that the loss would be derived partitively in the ratio of 27.3 from the
rotation of the earth to 1 from the revolution of the moon.^
*The general relations here involved are shown graphically by Sir Georse Darwin
in Thomson and Tait's Nat. Phil.. II, p. 511. These are developea in much deti^ and with
many api^cations in the paper ot Dr. Bfoulton in this series, some of the details and appli-
DlDUCnONS FROM THB TIDES. 41
W Now, it being thus rigorously shown that the energy dissipated by the
reaction between the tide-producing bodies must be taken from the mechan-
ical energies possessed by the bodies, respectively, in the proportions
given, and it being further shown that the evolution must take this direc-
tion and no other, it is clear that it will be possible to determine the effect
of the water-tides on the earth's rotation if we can estimate the total
energy dissipated by the tides, for the proper proportion of this can then
be subtracted from the kinetic energy of the earth's rotation. We can of
course use only imperfect data at present, but the uncertainties of these
can be covered by allowances so as to approximate the true order of mag-
nitude, and if the value does not prove to be a critical one, the order of
magnitude may be as decisive for all geological purposes as a precise deter-
mination. If the value proves critical, the serial method may be applied
to make the results cover the whole range of uncertainty. By thus dealing
with the whole of the friction, and by applying it by means of the rigorous
laws of energy, we not only avail ourselves of a radical mode of treatment
but avoid the tangle of special interpretations and the discrepancies of tidal
theories.
Preparatory to an attempt to compute the total friction, it is worth
while to note that an altogether exaggerated impression of the friction
of the tides is inevitably conveyed by their association with wind-waves,
river-currents, sea-currents, and other water-movements. As all of these,
or nearly all of these, are the products of energy communicated to the earth
by the sun's radiation, they may be assumed to be neutral in their rota-
tional effects. If all these adventitious elements be removed in imagination,
and the hydrosphere be made to take on a perfect calm, save as affected
by the tidal forces, the picture of the frictional effects will be radically
transformed, as may be seen by simple inspection. In the mid-ocean a
water-particle will merely describe a circuit of a few feet in twelve hours,
its movement on its fellow particles which are pursuing a somewhat similar
circuit being almost imperceptibly slow; the movement on the bottom will
be very slight. On a shore shelving at the rate of 1 foot in 50 feet, and
with a tide of 6 feet, the edge of the tide would advance 250 feet in about
6 hours, or a little over 40 feet per hour. On the exceedingly low slope
of 1 foot in 800 feet, the advance of a 5-foot tide would be only 660 feet
per hour. There are of course concentrations of motion in bajrs, straits,
cations of which were woiked out after this was written. In the acoompanyinc graphic
fllustration by Dr. A. C. Lunn, the relations of the factors are somewhat (ufferently
arranged. In the upper diagram, fig. 5, the line " O = Af | " represents the angular velocity
of the rotation of the earth; the une, «, the angular velocity of the moon in its orbit; r, the
Hi'g^ni^ between the centers of the earth ana nu)on; N, the relative days in the month;
X = Mm, the orbital moment of momentum of the system. In the lower diagram, E repre-
sents the total energy of the earth-moon system; Ki, the kinetic energy of the earth's
rotation.
The crossing of O and «. where the angular velocity of the earth's rotation and the
itngiiUr velocity of the moon's revolution are eqiial and they move as though a rigid body,
is somewhat over 9,000 miles from the line of reference at the left. From this point of
orosnnff to the left, where the distance of the centers is declining as shown by r, the total
energyls declining as shown bv J^ in the lower diaipam and loss of energy promotes move-
ment to the left. To the right of the crossing of O and u the centers move apart, the
total energy declines, and loss of energy promotes movement to the right.
42 THE TIDAL PBOBLEM.
and other special situations, that give quite notable movements, bat the
proportion of these to the whole is eadly exaggerated. From a simple
inspection of this kind it may be seen that the actual movement of the
water, as distinguished from the wave-form, is of a low order and hence the
friction is relatively small.
If one were to entertain the thought that the energy of movement of
the tides arrested by the continents is a factor of much value in rotation,
it should be recalled that the arrest of the easterly moving tides is to be
set over against that of the westerly moving tides, and in the case of tides
moving obliquely to the earth's axis there are easterly components to
offset the westerly components. We have previously noted that the tides
on the eastern sides of the Atlantic and Pacific average higher than those
on the western side, so that, on the face of things, the balance of influence
would, if the reasoning were fundamentally sound, favor rotational accel-
eration. But an analysis of the action shows that it results chiefly in a
return wave, and this throws the problem back upon the dissipation of energy
through friction. Some part of the energy of impact is not recovered in
the return wave either through elastic resilience or the increased head which
actuates the tidal ebb, but if the return wave is computed as though it were
equal to the advance wave the loss will be covered.
Hough has made an important contribution to the frictional phase of
the tidal problem by an investigation of the influence of viscosity on tidal
waves and currents,^ in which he included stationary oscillations as well
as progressive waves. He considered separately the tides whose wave-
lengths are large in comparison with the depth of the water-body and
those in which the wave-lengths are small. If attention were confined to
the main tidal phenomena, it might be thought that the wave-lengths
small in comparison with the depth of water would have no representatives
in tidal action, since the greatest depths of the sea are small compared with
the amplitude of typical tidal waves, but, on consideration, it is apparent
that these waves of great amplitude, by their various interactions upon
one another, by the modifications which they suffer in their approaches
to the shores, their arrests and their retreats, as also by their multitudinous
interactions with wind-waves, sea-currents, etc., give rise to an indefinite
number of waves of lesser amplitude, and that the length of these second-
aries may be short compared with the oceanic depths. Hough did not
consider the special effects of the irregularities of coasts and some other
modifying conditions and these must be recalled in interpreting his extra-
ordinary results. When these various agencies of modification are duly
considered, it seems probable that after a very few days the original tidal
oscillations largely lose their primitive amplitudes and take on shorter
ones, and that in this way they pass beyond observation long before they
actually die out.
Hough did not attempt to determine the complete destruction of the
tidal waves by friction, but merely the time required for their reduction
to the value represented by lie, or 0.368 of their original value. He found
> Proc. Lond. Math. Soc., vol. 28, pp. 264-288.
DEDUCTIONS FROM THE TIDES. 43
that for waves of the class corresponding in his terminology to n^^O, in
waters of abysmal depths, the modulus of decay (to 0.368 value) is 42.6
years; for the class corresponding to n~l, 31.7 years; while for that
corresponding to n = 100, it is 4.8 years. The t3rpes, n«=0, n = l, represent
the longest and strongest waves; while those in which n has a high value
represent waves of much shorter and feebler type and hence those first
to be reduced to the limit named, as indicated by the moduli. In a case
where the depth is 200 meters — the prevailing depth on the edge of the
continental shelf — the moduli of decay for the types n«>0 and n« 1 are 4.5
and 3.3 years, respectively. For waves of 100 meters amplitude in water
with a depth of 1 meter, the modulus of decay is about 80 minutes, bottom
friction included. If there were no friction at the bottom, the modulus of
decay would be about 2.25 years. In summation Hough says:
These results indicate how little can be the effects of viscosity upon the motion of
the sea, except possibly in usually confined waters. It seems that wherever the depth
exceeds a very moderate amount, say 100 fathoms, the rise and fall of the waters due to
the sun and moon will not be appreciably affected by friction.^
These determinations have a significant bearing upon the question how
much of a given tidal wave is due to the force that has just been acting
upon it dming the current tidal period, and how much to the residual
motion inherited from previous tides. If the motion of the waters when
once generated requires these long periods for subsidence, it is obvious
that each tidal wave may be perpetuated so as to codperate with a long
series of forcing actions in succeeding periods, if its period is commensurate
with these. This supports the view, previously discussed, that the waves
observed in those portions of the ocean most favorable for sympathetic
accumulation are the products of a considerable series of forcing actions.
This means that, at least in such cases, the element added with each tidal
period is not measured by the actual waves observed, but by some minor
fraction of it, and hence that the tidal friction which is daily exerted on
the earth is by no means the amount necessary to reduce the observed
tidal movement to zero, but merely that which is necessary to offset the
daily increment, or its equivalent, the daily factor of decay. A wave of
the type n»0 in water 200 meters deep, if commensurate action were per-
fect and dissipation, by giving rise to derivative waves, were wholly absent,
would need to have less than ^-^ of its value added to it daily to maintain
its value. This must not be taken as representing an actuid case, but it
appears from, considerations of this kind that the total energy of motion
expressed by the tides daily is by no means a safe basis for estimating
the energy lost through them. This must be computed directly from the
water^movements under the conditions that actually affect them.
There is a check on carrying considerations of this kind too far, in the
fact that the spring and neap tides and other special tides that depend on
variations in the relations of the tide-producing bodies pass through their
climacteric phases within one or two days of the astronomical configura-
tions that give origin to them. They increase and die away with a relative
» Loc. cit., p. 287.
44 THB TIDAL PBOBUBIC.
promptness which shows that no tidal agency perpetuates its speeial eflfeots
in a special phase for a very long period.
There is a minor qualification of Hough's results that is worthy of
passing notice. He took for his coefficient of viscosity 0.0178, which is
one of the determinations for 0^ C. The viscosity of water is much in-
fluenced by temperature. The coefficient for 17^ C. is 0.0109. As this is
about the average temperature of the surface of the earth, it may be taken
as roughly, though not accurately, the average temperature of the ocean
water. Salinity increases the viscosity. While I do not know of any direct
determinations on sea-water, the determinations for normal solutions of
sodium chloride imply that the viscosity of sea-water at 17^ C. would be
about 0.012, which makes Hough's results very conservative, so far as aflfeoted
by the coefficient of viscosity used. This, however, is not so much the point
as is the difference in the viscosity of the bottom and the surface water,
respectively, in the low latitudes, that of the deep water being somewhere
about 0.0195, while that of the surface in the tropics is about 0.0099,
or but little more than half as great. This difference must increase rela-
tively the motion of the water on itself and reduce that upon the bottom.
It will thus substitute a distributive movement within the water, in which
the friction is very low, for a more concentrated movement between the
film of water attached to the bottom and that immediately above it, in
which the friction is relatively high. The total effect is to reduce the frio-
tional value.
Radical as are the suggestions of Hough's inquiry, a way to apply them
directly, so as to secure a numerical expression of the total vaJue of the
friction of the tides in terms of work done, has not been found, and I have
therefore tried to shape the problem so that it could be treated by the
method of the engineer, as a given mass of water, with a given amount of
flow, in a given time, under assigned conditions. To do this it seems neces-
sary to substitute for the actual ocean an equal body of water in a more
tractable form, but subject to equivalent friction. In doing this I have
endeavored to give to the movements of the substitute ocean at least as
great friction as that of the actual ocean. For the purposes of frictional
treatment the ocean may be regarded as consisting of three portions, (a)
the shallow water between the coast and the edge of the continental shelf,
(b) the water on the slope between the edge of this shelf and deep water,
and (c) the deep portion.
(a) The coast line is taken at 120,000 miles, which is nearly double the
simple outlines of the ocean and about 5 times the earth's circumference.
The depth of water on the outer edge of the continental shelf is taken at
600 feet, the accepted depth. The water is made to deepen uniformly
from the coast to the edge of the shelf, which very greatly exaggerates its
shallowness near the shore, where the friction is relatively greatest. The
area is taken at 12,000,000 square miles, or 20 per cent more than Murray's
estimate for this part of the actual ocean.
(b) The portion b is given a width of 50 miles, a descent from 600 to
9,000 feet, and an aggregate length of 120,000 miles. This length is much
greater than the actual length of the continental margin, the excess being
DlDUCnONS FBOM THB TIDB8. 45
intended to include a liberal allowance for the slope-tracts of the oceanic
islands, which, however., are not subtracted from area c.
(c) The abysmal section is given a depth of only 9,000 feet and an
area 20 per cent greater than that of the deep ocean, this reduction of
depth and increase of area being intended to offset the frictional effects
of the inequalities of the actual bottom.
The mean height of the tide in the substitute ocean is taken as 4.09
feet, which is equivalent to 4.9 feet for the actual ocean. The mean range
of the tide for the 280 available stations given in Harris's table of tides
harmonically analysed is 4.548 feet.* As these stations are chiefly in harbors
where local concentration is felt, 4.548 feet is probably rather high for the
average range of the tide, even on the coasts, and it is certainly much too
high for the mean range over the whole ocean. In using the equivalent of
4.9 feet for the substitute tide in addition to the large allowances made
above, it would appear that the computation is amply guarded against
underestimation.
In using the foregoing guards against underestimation, which seem to
me excessive, I have been somewhat influenced by the thought that there
are derivatives from the observed tides which are not recognised and
measured as such, but whose dissipation of energy should be covered by
the computation. But, however well guarded, it is not presumed that
any results now attainable will have much value beyond indicating the
order of magnitude of the total friction. With the foregoing precautions
the results should not be seriously less than the actual fact. But, if they
are thought to be so by any one, the results can easily be multiplied accord-
ingly.
With these data, a computation was made by Dr. W. D. MacMillan
in the manner set forth by him in a following paper of this series, p. 71.
This computation, it will be observed, was made for continuous motion,
but in estimating the rate, 12.5 minutes between each lunar tide were
allowed for the turn of the tide. He finds the yearly loss of energy to be
38,918 X 10*' foot-pounds. The rotational energy of the earth, reckoned
on the assumption that the Laplacian law of density obtains, is 157 X 10^
foot-pounds. At the computed rate of loss, this amount of energy would
last 40,440,000,000 years. The length of the day would be increased one
second in about 460,000 years. In 100,000,000 years the total lengthening
of the day would be about 3.6 minutes.
If this result does not wholly misrepresent the order of value of the
friction of the water-tides, it follows that, even if the allowances for the
irregularities of the tidal water-bodies be greatly increased, and if the
formulas of the engineers for the effects of friction be multiplied several
times, and other allowances be made in the most generous manner, the
effects of the water-tides on the rate of rotation of the earth during the
known geological period are negligible. If the friction of the body-tides
and the air-tides is also very small, there is no reason to expect to find in
the geological evidences any appreciable deformations of the earth's body
the distinctive characteristics of tidal effects. On the other hand,
' Rept. Coast and Geodetic Surv., 1900, pp. 664-677.
46 THE TIDAL PROBLBIC.
if the deductions which have heretofore been drawn from the older eos-
mogonic and geophysical conceptions are true, there should be geological
testimony to support them. We turn therefore to the geological evidences
with heightened interest.
THE GEOLOGICAL EVIDENCES.
Perhaps the most important of the geological lines of approach to the
rotational problem, is found in the evidences of an appropriate change
or lack of change of the earth's form. At least it is this problematic change
of form that gives the subject its obvious importance in diastrophism, to
which this discussion is a preface. If the rotation of the earth were once
appreciably faster than now, either the form of the lithosphere would have
been more oblate than it is at present, or the surface-waters would have
been accumulated at the equator by the increased centrifugal force, or
both actions would have taken place conjointly; and a change from this
configuration to the present one must have followed. If the lithosphere
has changed its form appreciably within known geological times owing to
reduction of rotation, such a change should be manifest in its structural
deformations, especially in the deformations of the early ages. If the
lithosphere has not essentially changed its form because of reduced rota-
tion, but the waters served as the accommodating factor, this, if it were
of sensible amount, should have been manifested by deposits of the kinds
that imply prevalent and deep submergence in the equatorial regions
and by erosions signifying prevalent and pronounced emergence in high
latitudes in the former ages of higher rotation, and by the reverse in the
later ages, both of which would be shown by the geological records of those
regions.
THE EVIDENCES FROM THE LITHOSPHERE.
The bearing of a possible change of form, assignable to a change of
rotation, on terrestrial diastrophism has long been recognised in some
measure by geologists, but the first attempt to reduce it to numerical terms
seems to have been that made by President Van Hise several years ago.^
He inspired Prof. C. S. Slichter to make the computations necessary to
show in numerical terms what would be the reduction in surface area if
the rotation were changed to the degree postulated in Darwin's interpre-
tation of the past history of the earth and moon. It was thought by
him sufficient to base the computations on the convenient hypothesis of a
homogeneous density. The change of surface area was shown to be large
and this made it clear that, if such a change of rotation has taken place, it
is an important factor in deformation. Even if the chief deformation took
place early in the history of the earth, the effects should be apparent still
in the inheritances of the regions most affected, and the record should show
them. For the purpose of a more critical study of the subject. Professor
Slichter has been kind enough to recompute for me the requisite data on
the basis of a distribution of internal density as near that of the actual
earth as our present knowledge permits. For this purpose Laplace's law
.» Van Hise, Jour. GeoL, vol. 6, pp. 10-64, 1898; Slichter, ibid., vol. 6, pp. 66-68, 1898.
THE GEOLOGICAL EVIDENCES. 47
of increase of internal density was taken as perhaps the best expression
of this factor and as being in fair accord with astronomical data. Pro-
fessor Slichter extended the computation to other constants of the earth
than those requisite for this inquiry and these give to his paper a value
quite independent of its application to the present problem. His paper
and table will be found on pages 61--67 of this volume.
Column 7 of this table (page 67) shows that ten periods of rotation
have been selected, ranging from 3.82 hours to the present period. Darwin's
hypothesis * leaves unassigned the precise period of rotation when separa-
tion took place, but from an inspection of the configuration of the spheroid
at the rotation-period, 3.82 hours, and of the gravity in different parts of
the spheroid at that stage, it seems safest to assume that a rotational
period less than 3.82 hours would be necessary to cause fission. It seems
best also to assume that at the 3.82-hour stage the earth was solid on the
exterior, whatever may have been its internal condition. If this shall not
seem so to any one, the arguments based upon the data of this rotational
period can easily be shifted to the numerical values of the next period of
4.03 hours, or to any of the later periods given in the table.
From column 11 it will be seen that the equatorial circumference at
the rotation-period, 3.82 hours, was 1,131 miles greater than it is at present,
while the meridional circumference was 495 miles less. In changing to
the present form, the tract immediately under the equator must have
become shorter by 1,131 miles. The tracts under the parallels adjacent to
the equator north and south would have become shorter by less amounts,
those still farther away by still less amounts, until a little beyond 30^
latitude, north and south, parallels are reached under which the crust would
have theoretically remained unchanged so far as this immediate factor is
concerned. These are the latitudes of mean radius for each stage of rota-
tion and are shown in column 9. It will be noted that these shift from
lat. 33^ 20' to 35^ 13' in the course of the series, but it is sufficient for our
purpose to speak of the neutral zone as lying at 35^ latitude, north and south
respectively. The equatorial belt between these parallels, 70^ in width
roundly speaking, would therefore, by the postulated change, have become
shorter along its central line by 1,131 miles, since the rotation-period of
3.82 hours. On its borders it would have suffered no change, and between
the borders and the central line it would have suffered a graded series of
shortenings.
That portion of the meridional circumference which lay within the
equatorial belt should have been shortened in the course of the change
from the rotation-period of 3.82 hours to that of the present, but the whole
meridional circumference should have been lengthened 495 miles. It
is obvious, therefore, that the areas north and south of the neutral zones
must have become extended meridionally 495 miles plus the amount of the
contraction in the equatorial zone, the precise value of which is unimpor-
tant here. It will be convenient to call these areas of expansion polar caps,
though they reach down to about 35^ latitude. In the course of the change
named, the surface at the poles should have been raised and the curvature
« The Tides, p. 360.
TBI TIDAL FBOBLIH.
of the c&ps iDcreased at all points. The extension should have been great-
est at the poles and should have died away to aero at the parallel of no
change. There was therefore a olimacterio stretching at the poles and a
climacteno compression at the equator.
Fig. 6 is intended to illustrate the nature of the ohange as seen from
a point of view above one of the poles. To keep the view as true to pa-
spective as practicable, the equatorial belt is foreehorteaed. The excess
of area in the equatorial belt is represented by the black triangles, whioh
are too small on account of this foreshortening. The deficiency of area in
the polar repon is repreeented by the white ground, which ia more neariy
in true proportion.
Changes in the oruat of the earth of this magnitude, or of such leoso-
magnitude as would have followed a change from any of the other eariy
periods of rotation to the present one, could scarcely have taken place
without leaving a record of them-
selves in the form of compreesional
and tensional phenomena. We
may, to be sure, suppose that the
interior of the earth has always
been sufficiently mobile, in one
form or another, to permit intei^
nal shift of material from areas of
compression to areas of tension,
and so to accommodate itself to the
progressive change of form, but
this can not reasonably be sup-
posed to have taken place in the
outer shell without having left evi-
dences of itself, for this shell must
be assumed to have been solid from
an early state and, being at the
surface, it was not under such prea-
sure as to Sow and hence must
have been deformed in the familiar
modes that characterise surface
thrust and tension respectively. It
is known from abundant geological
observation how the shell of the earth deports itself under conditions of
compression and tension resulting from forces of the kind that would arise
from the changes asogned. The data of Sliohter's computations may there-
fore be interpreted by the usual methods.
The equatorial belt of the earth of the 3.82-hour rotation-period would
differ from that of the present earth to the extent of a broad swell 180
miles high. In settling down this might doubtless relieve its excess of
length in cross section by thrusting northward and southward into the
areas of tension, but aa its equatorial length was 1,131 miles greater than
the present equator, it would seem that in an east-west direction the tract
must fold, crumple, and overthrust on itself after the familiar fashion of
■t33°20'Utilu[|e.
TBI GEOLOGICAL EVIDBNCB8. 40
folded mountains. To estimate the result comparison may be made with
estimates of the amount of crustal shortening involved in the formation
of folded mountains. It is obvious that the estimates which assign the
greatest amount of shortening to given amounts of folding are those which
would give the least mountain production to the sinking of the equatorial
belt in question and are hence the most conservative. One of the highest
estimates of the crustal shortening involved in the formation of a familiar
range of mountains, made by a competent geologist on the basis of much
personal field work, is that of Professor Albrecht Heim for the formation
of the Alps, which is 74 miles. Somewhat comparable estimates are those
of Dr. Peter Lesley for the folds of the Appalachians west of Harrisburg,
which is 40 miles, and that of Dr. G. M. Dawson for the Laramide Range
in British Columbia, which is 26 miles. In the opinion of some other geol-
ogists these estimates are too high. If therefore we apply these to the
equatorial belt the results will be relatively conservative. If we use Heim's
figure, the sinking of the equatorial belt to the assigned amount should
give 15 mountain ranges of the magnitude of the Alps standing across the
equator. They should be short ranges d3dng away within 35^ of latitude
on either side. If we apply Lesley's estimate there should be 28 ranges of
the order of the Pennsylvanian Appalachians standing across the equator;
if the estimate of Dawson be used, there should be 45 ranges of the magni-
tude of the Laramides of British Columbia.
If we start with the 4.03-hour rotation-period instead of the 3.82-hour
period, these figures become, 13, 26, and 40, respectively; and they may
be easily reduced for later periods.
If, as an alternative, we choose to assign mcnre mashing of the shell and
less corrugation, it will merely give us a massive equatorial ridge with less
cross-folding. If, as another alternative, we choose to assign more com-
pression into denser rock, we shall have greater resistance to subsequent
erosion and higher specific gravities to account for.
Under no tenable hypothesis, so far as I can see, can an equatorial
protuberance of 180 miles comparable to the 3.82'hour period, or of 180
miles comparable to the 4.03 period, or of 87 miles comparable to the 6.35-
hour period be assumed to have subsided to the present equatorial dimen-
sions without having left a distinct record of itself in the form of transverse
ranges of mountains, or of irregular protrusions, or of indurated terranes,
or of some combination of these or of the other modes in which exceptional
tangential stress is accustomed to express itself in the shell of the earth.
It is to be noted that, by the terms of the retardational hypothesis,
the tangential stress must have been applied constantly from the beginning
to the present time. It was indeed more rapidly applied in the earlier
stages, but some stress has been added constantly ever since. If compres-
sion to a more compact form is to be assigned at all, in any important
d^ree, it must be assigned to the first stages of stress, and the later pro-
trusions would be all the more enduring on account of this early induration.
Now it is for every one to examine for himself the equatorial tract to
see if it presents the character which the hypothesis requires. For myself ,
I am quite unable to find it. There is not even an equatorial belt of land.
50 THB TIDAL PBOBLEM.
much^less an elevated girdle accidented by cross-folds, or knotSi or con-
torted protuberances; nor do I find evidences of the truncated remains of
these. Since the rotation-period of 15.63 hours, 40 miles of shortening
should have been added to all that preceded, and 15 miles of this should
have been added since the 19.77-hour period. Even if these were remote
in years, they should have served to perpetuate a phenomenon that in its
nature must have been dominant from the beginning, for it is difficult to
assign any other agency of deformation that should have overmastered
this, if it had this degree of efficiency. On the contrary, other agencies of
deformation should, according to an accepted generalization derived from
observation, have reenforced the deformation assigned to this cause, for
old lines of 3delding usually determine new ones.
As a matter of fact the depressions below sea level on the line of the
equator are fully as great as the amount normal to a great circle; about
three-fourths of the equatorial zone is submerged and one-fourth emergent.
The oceans crossed are normally deep; the mountains of the tract are
scarcely normal in height or massiveness, the Andes of Ecuador being the
only conspicuous range within the equatorial tract. The mountains which
cross the equatorial tract show no special signs of limitation to it, as they
should if they were essentially dependent on the agencies involved in the
retardational hypothesis.
If we take into consideration the whole compressional belt from 35®
north to 35® south, it is found to embrace but little more than the average
amount of land; indeed, the emergent surface within it is less, in proportion
to the submerged area, than in the region north of it, though it is more
than in the region south of it.
If we turn to the tensional areas that should, under the hypothesis of
reduced rotation, lie between 35® north and south and the poles, the inspec-
tion is unembarrassed by any doubt about the effect of the stress upon
the density of the rock, for appreciable stretching can not be assigned to
rocks, except as it expresses itself by Assuring and equivalent modes,
which leave an appropriate record. It is to be observed here again that,
while the larger part of this tension was brought to bear in the early stages,
it was, according to the hypothesis, continuous throughout the whole
history. The results naturally assignable to this progressive tension would
be a persistent fissuring and gaping radial from the poles, somewhat as
implied by fig. 6. This must have run through all geological time, except as
counteracted by some other agency. The cooling of the earth, or its shrink-
age from internal molecular change or from any similar pervasive agency,
would antagonize this, and if equal to it might prevent the actual opening
of the fissures. But, to be consistent, this shrinkage must be applied gen-
erally and such application would intensify the difficulties in the equatorial
belt in proportion as it relieved those of the polar caps. Simply to counter-
act the 495 miles of stretching required by the hypothesis in the rotational
reduction from a period of 3.82 hours to the present, leaving out of con-
sideration its special distribution, would require about 78 miles of vertical
shrinkage in the polar regions and of 1,600 miles in the equatorial belt.
But there is a special difficulty of distribution. The stretching required by
THE GEOLOGICAL EVIDBNCSS. 51
the hypothesis of rotational reduction is concentrated toward the poles,
and hence, if tension is to be avoided in high latitudes, a very much larger
radial contraction than the amount named must be postulated.
It must also be considered whether cooling, or any other similar con-
tractional agency that can be postulated consistently with the early states
of the earth assumed by this hypothesis, would be competent to offset the
tensional effects imposed by the change of rotation in the polar regions.
If, to escape the difficulties arising from exceptional tension in high
latitudes, it be assumed that the whole shell of the lower latitudes crowded
toward the poles, this would involve meridional crowding and the forma-
tion of a system of folded ranges pointing to the poles, while east-and-west
ranges should be absent proportionately, and thus the effects should be
expressed in a distinctive manner. So it seems safe to conclude that, in
one way or another, the high-latitude tension should have expressed itself in
a characteristic way and, on account of its magnitude, its expression should
be declared.
In comparing the facts with the theoretical requirements it must again
be noted that the earlier formations should show the most evidence of
tension, the Archean most of all. As a matter of fact, the Archean of high
latitudes, as of low latitudes, shows abounding evidences of compression.
It was my privilege in 1894, as geologist to the Peary Auxiliary Expe-
dition, to see something of the ancient crystalline rocloi of Greenland at
latitudes as high as 77^. They bore the same evidences of crumpling,
contortion, foliation, and thrust-stress generally as are commonly shown
by the Archean rocks in lower latitudes. All descriptions of high-latitude
formations of this age are identical in dynamic characters with those of
lower latitudes, so far as my knowledge extends. The Archean terranes
of Scandinavia and Finland lie far within the area of hypothetical tension,
as do also those of Scotland and Canada, and even those of central Europe
and the northern United States. The Archean and Proterozoic rocks of
these regions bear evidences of tangential thrust of a most declared type, and
no distinction between the most ancient rocks of the high-latitude and the
low-latitude regions, in the matter of compressional characters, has, I believe,
ever been detected. The literature of the subject does not show any special
distribution of veins, dikes, normal faulting, and other evidences of tensional
stresses correspondent to latitude. Apparently these]f eatures are essentially
as prevalent in the equatorial belt as in the polar circles.
If the equatorial belt has been subjected from the beginning to constant
increments of tangential stress and of gravity (column 4 of Slichter's table,
page 67) while the polar regions have been concurrently subjected to incre-
ments oif tension and decrements of gravity (column 5), it would seem that
volcanic action would always have found adverse conditions in the former
region and favorable ones in the latter, certainly so if pressure is adverse
to liquefaction and if tensional faulting facilitates eruption. It does not
appear, however, that volcanoes are in any appreciable degree infrequent
in the tropical zone or that they are specially frequent in high latitudes.
The prevailing impression is that they are somewhat more abundant in
the tropics than in high latitudes, but there is little, if any, warrant for any
latitudinal discrimination.
62 THE TIDAL PROBLEM.
The distribution of folded mountains appears to be quite indifferent
to the latitudinal distinction which the hypothesis of rotational reduction
involves. The great Cordilleran belt of the Americas begins far within
the southern tensional area, is strong where it crosses the southern neutral
belt, is ako strong in the southern ha^ of the equatorial belt, becomes weak,
scattered, and tortuous in the northern half of this belt, attains strength
and broadens as it crosses the northern neutral aone, and reaches great
breadth and aggregate mass in the lower part of the north tensional area.
About 30^ within that zone, still strong, it swings about toward the Asian
continent. The great tangled mass of mountains of central Asia lies chiefly
in the northern tensional area. According to Suess, the thrust movement
was generally from the northwest; that is, from the more highly tensional
to the less highly tensional area. The great east-westerly range of southern
Europe and Asia lies chiefly in the lower tensional and neutral zones and
only at the east passes obliquely into the equatorial belt. A thrust from the
compressional zone toward the tensional zone is indicated in the western
portion and the opposite in the eastern portion. If, neglecting the latter,
we fasten upon the former as dynamically probable under the hypothesis,
it is to be noted that, with tension increasing in the direction of the thrust,
it is not apparent whence came the resistance that was necessary to the
intricate folding and distorting of the east-west ranges. Rather should
we expect 3delding in the direction of the tensional area and lateral crowd-
ing of the shell as it was pushed from the periphery toward the center of
the tensional cap, with short meridisnal ranges as the result. Without
reviewing the multitude of minor mountains, it may be sufficient to note
that the Urals, the ranges of Scandinavia and of the British Isles, the
Appalachians, and the mountains of Greenland testify to the dominance
of thrust phenomena in the northern zone of tension. Statistically con-
sidered, the facts now known give this northern zone precedence over all
others in thrust phenomena. The great Archean tracts of Canada, Green-
land, Scandinavia, and Finland carry the dominance of this thrust phenom-
ena back to the earliest known ages. Taking the facts as we now know
them, there seems to be no observational support for the compressional-
tensional distribution which the hypothesis of great tidal retardation
involves.
In the discussion thus far, agencies of compression and tension, other
than rotational, have largely been ignored for the sake of following out,
consecutively and iminterruptedly, the consequences of the hypothesis
of rotational reduction and comparing them with observed facts. It is
proper now to consider whether the intercurrence of other agencies of
deformation would mask the results of tidal retardation, if these were of
the order of magnitude implied by the fission theory of the origin of the
moon, or even the close approximation of the moon to the earth in its
early history under the planetesimal hypothesis. The existence of other
causes of crustal deformation is of course fully recognized. To bring these
under consideration in connection with the hypothetical tidal effects, it
is necessary to note first their qualitative relations and second their relative
values.
THE GEOLOGICAL EVIDENCES. 53
Essentially all of the other assignable causes of deformation of the
major class seem to be general in their application and to affect all latitudes
practically alike. This is true of cooling, of internal redistribution of heat,
of molecular rearrangement, whether chemical, crystalline, or diffusive, of
atomic transformations and decompositions, of radioactivity and presum-
ably of igneous extravasations. Grant to these agencies whatever sep-
arate or combined effect may be their due, that effect, if it be general
and essentially indifferent to latitude, as it seems that it must be, should
be distinguishable from the effect of a superimposed agency that is pro-
nouncedly correlated with latitude, because of this peculiarity. Granted
a given amount of uniform earth shrinkage as the result of the general
agencies named, or any of them, the crustal stress arising from this in the
equatorial belt would be intensified by the addition of the stress of the same
kind arising from the retardation of the earth's rotation, while the crustal
stress which arises from these agencies in the polar regions would be propor-
tionately relieved by the tension arising there from rotational retardation.
A difference of result equal to the algebraic sum of the retardational and
general stresses should be manifest in the resulting deformations. The
conspicuousness of this difference must depend largely on the relative values
of the two classes of agencies, which is our second point of consideration.
If, on the tidal side, we take the higher deformative values given in
Slichter's table (page 67), and if, on the other side, we take estimates of
shrinkage made from a study of the foldings and faults of the earth, a
comparison may be made. Quite without thought of this application, I have
recently reviewed the data of the latter class in the endeavor to form a
reasonable estimate of the amount of shrinking which the earth has prob-
ably undergone; and, while this estimate has little claim to value in itself,
it may perhaps be taken to fairly represent the import of the present
imperfect data. It is as follows:
If one is disposed to mimmize the amount of folding, the estimate may perhaps be
put roundly at 50 miles, on an entire circumference, for each of the great mountain-making
periods. If, on the other hand, one is disposed to give the estimates a generous figure so
as to put explanations to the severest test, he may perhaps fairly place the shortening at
100 n^es, or even more. For the whole diortening since Oambrian times, perhaps twice
these amounts might suffice, for while there have been several mountain-making periods,
only three are perhaps entitled to be put in the first order, that at the dose of Uie Faleo-
soic, that at the dose of the Blesosoic, and that in the late Oenosoic. The shortening in
the Fhyterosoic period was considerable, but is imperfectly known. The Archean rocks
suffered great compression in their own times, and probably shared in that of all later
periods, and if their shortening could be estimated dosely, it might be taken as covering
the whole. Assuming the circumferential shortening to have been 50 miles diuing a given
great mountain-folding period, the appropriate radial shrinkage is 8 miles. For the more
generous estimate of 100 miles, it is Id miles. If these estimates be douUed for tJie whde
of the F^eosoic and later eras, the radial shortening becomes 16 and 32 miles, respectiydy.'
If we assign to the Proterosoic era a shrinkage equal to the Paleozoic,
Mesozoic, and Cenozoic eras combined, and to the known Archean twice as
much, the minimum and maximum estimates are 64 miles and 128 miles
of radial shrinkage, respectively, or roundly 400 and 800 miles circumfer-
ential shortening, respectively.
' Geology, vol. 1, Chamberlin and Salisbury, 1904, p. 551.
54 THl TIDAL PBOBLEM.
The assigned equatorial shrinkage from reduction of rotation since the
3.82-hour rotation-period is 180 miles, which is to be compared with the
minimum 64 miles or the maximum 128 miles of the above estimate. Of
course, it must be recognised that the 180 miles covers a period preceding
the known Archean, which is not embraced in the latter figures; but if an
allowance of two-thirds be made for this, the remaining 60 miles vertical
shrinkage still bears a sufficiently large ratio to the stratigraphicai estimates
to make its effects certainly discernible, when the contrasted influences
in polar and equatorial regions are brought into comparison.
The computation for maximum rotational change gives a meridional
elongation of 495 miles; the stratigraphic estimate gives a meridional
contraction of 400 and 800 miles minimum and maximum respectively.
Allowing two-thirds of the 495 miles for the period preceding the known
Archean, there remain 165 miles of elongation to reduce the effects of the
400 or 800 miles of contraction.
Combining equatorial and polar effects, the case stands 777 (400+377)
vs. 235 (400-165), on the minimum basis, and 1,177 (800+377) vs. 635
(800- 165), on the maximum basis, when two-thirds of the retardation is
assigned to pre-Archean times. It would seem that differences of this order
of magnitude should be clearly manifest in the phenomena.
THE EVIDENCE FROM THE HYDROSPHERE.
If there be any doubt about the practicability of detecting the influ-
ence of any great change in the rotation of the earth by the distinctive
features of the deformation of its shell, we certainly have a very delicate
means of detecting deformations in the position of the sea-level relative
to the land. The position of the sea-level has been recorded by a series of
shallow-water and shore deposits extending from the Cambrian period to
the present, and this record was made with sufficient frequency and fidelity
to answer every purpose of an inquiry of this kind. To a much greater
extent than has usually been recognized, the known stratigraphic series
is the product of shallow water, as shown by shallow-water life and appro-
priate physical evidences. In many cases some latitude must be allowed
in the interpretation of these criteria of depth, but this can be the source
of no essential error in a problem of deformation whose units are miles
rather than feet; but, if required, a sufficient number of cases of irreproach-
able accuracy can be given, for at not a few geologic epochs there were
emergences and submergences between which some stage of the transition
marks the relations of the water surface to the land with positiveness and
exactness. If, for instance, we know that in the critical regions, whether
poleward or equatorward, a given horizon has been above the water-
level and below the water-level respectively at two successive stages, we
know that between these stages it was absolutely at the water-level. By
means therefore of the successive emergences and submergences of given
horizons, the relations of the sea to the land can be determined very accu-
rately for a sufficient number of geological stages to be wholly decisive in
such a problem as that in hand, and approximately for most of the other
periods.
THE GEOLOGICAL EVIDENCES. 55
If the litbospbere could be supposed to have acted under tbe forces of
gravity and rotation so nearly as tbough it were a perfect fluid tbat its form
would be at all times perfectly adapted in all its parts to its rate of rota-
tion, bowever mucb tbat may bave cbanged, the argument here introduced
would bave little or no force. If this assumption is made here, it must
of course be carried consistently throughout the whole range of deforma-
tive interpretation. If this is done faithfully, very grave difficulties will
be encountered, so grave that, for myself, I bave found them insuperable.
It is indeed commonly thought consistent with experiments and geological
observations, to regard tbe litbospbere as a solid which acts rigidly toward
stresses of short period, and quasi-fluidly towards those of long period.
Under this proposition it is possible to assume tbat the accommodation of
the earth to a steady change of rotation might be so nearly perfect tbat
▼ariatioiis would escape detection by even so delicate a registration as
that of tbe sea-surface. But if this is done, it should be with tbe full con-
fldousness tbat this is not a deduction from the proposition, but merely
an assumption under it; for the general proposition that tbe litbospbere
will yield under stress applied for a sufficient time does not in itself carry
tbe conclusion tbat it will yield under the given stress in tbe given time.
A quartz crystal is under self-gravitative stress and may bave also been
under terrestrial gravitative stress for eons, and yet it shows no signs of
becoming a gravitative spheroid. Mountains and continents are under
gravitative stresses and they probably jidd to these, but at wtiat rate is a
practical question of much geological importance. Tbe postulate of quasi-
fiuidal accommodation is not a solution; it is only a broad generalization
under which a solution may be sought by specific evidence.
The shell of tbe earth is chiefly an aggregate of interlocking crystals
which are possessed of specific elasticities of form, and the whole aggregate
dearly has elasticity of form. If the great mass of the earth or even the
deep outer portion be similarly an elastico-rigid solid, deformations will
only take place when the stress-differences rise to equality with the elastic
resistances, except in the limited form of strain, and to tbe limited degree
permitted by tbe individual transfer of molecules from one rigid attach-
ment to another. Deformations in this case await a certain accumula-
tion of stress-difference. As the crux of the whole deformative problem
lies largely in these basal conceptions, we may do well to turn to geological
phenomena to ascertain, if possible, whether the earth does habitually
yield concurrently with the accumulating stress-differences and thus con-
stantly accommodate itself to stress-demands, or whether stress-differences
do actually accumulate until the elastic limit is reached when deformation
proceeds with relative rapidity until an approximate equilibrium is reached.
This is but stating in dynamic terms tbe question of periodicity in geo-
logical deformation. On this question, a consensus of geological opinion
can not now be cited without qualification. Apparently views differ and
reserve predominates among cautious geologists. It appears to me, how-
ever, that strong evidence is steadily accumulating, from various quarters
of the globe, tbat there were great periods of base-leveling of essentially
world-wide prevalence, with concurrent sea-transgression, separated by
56 THE TIDAL PROBLEM.
briefer periods of deformation of similar prevalence. For these it does not
appear that there will be found a consistent explanation except in the
ability of the great body of the earth to accumulate stresses to a notable
degree during the long periods of relative quiescence necessary for the base-
leveling and the sea-transgression.
It would be going beyond the proper limit of this paper to try to estab-
lish this thesis by the citation of evidence, for this would involve a review
of some large part of the great mass of stratigraphic, paleontologic, orograic,
and physiographic data possessed by geology. Suffice it therefore to note
here that this is the one of the alternative views of the earth's deformative
methods that seems at present best supported by geologic evidence. It
is not wholly necessary to the following considerations, though it lends much
strength to them.
Let it be assumed merely that the earth-body ofifers some appreciable
resistance to deformation, an assumption which can scarcely be questioned,
since the irregidarities in the form of the geoid imply this, even when allow-
ances are made for differences in the distribution of density. Let a limited
slackening of the earth's rotation take place. This will disturb the preced-
ing equilibrium between the centripetal and centrifugal forces and both the
body of the earth and the water on its surface will experience stress-differ-
ences which give a tendency toward a new equilibrium. This equilibrium
may be established by the subtraction of matter from the equatorial regions
and its transfer to the polar regions internally or externally. The earth-
body certainly offers some resistance to this transfer while the water on its
surface offers practically no resistance at all because it is in circulation as
the result of solar influence, and to effect the new distribution it is only
necessary that it stop where the new demands of gravity require, and in
this friction will lend its aid. The water surface may therefore be supposed
to fall in the equatorial regions and rise in the polar regions until the new
water surface of the globe conforms to the new equilibrium required.
This must relieve, in some large part at least, the stress upon the body of
the earth, for if the newly developed equilibrium required more matter
in the polar regions the water would supply it, unless it were previously
exhausted. Local stresses might remain where the land was left pro-
tuberant, but geological evidence shows that such protuberances can be
maintained for long periods by the effective rigidity of the earth, if they do
not exceed a certain measure. Such a protuberance of the equatorial land
may be treated as any other local protrusion of the earth's body. When,
therefore, in the case in hand, an equatorial mass became protuberant
above the surface of the geoid sufficiently to overcome the effective rigidity
of the part of the earth affected, the appropriate deformation would follow.
The determination of what mass is sufficient for such deformation is
qualified by the available time. Given infinite time and the requisite mass
would doubtless be relatively small in a body like the earth, even on the
hypothesis of elastic rigidity; for, even within the limit of elasticity,
deformations may take place by the transfer of molecules from one rigid
attachment to another individudly. But whatever might be the results if
indefinite time were available, the practical case is one of limited time and,
THE GEOLOGICAL EVIDENCES. 57
as implied above, geological evidences seem to show that stresses do accu-
mulate to certain large magnitudes before sensible deformations take place.
Meanwhile surface transfers by wind and water action are in progress.
The protuberant equatorial belt postulated must ever have been shedding
material northward and southward, mechanically and by solution, thus
building up sedimentary series in the flanking sea-borders. As the pro-
tuberant tendency was ever renewed by slackening rotation, this should
have become a perpetual process and, as we have seen, should have been
a pronounced factor, if not the dominant one, in the earth's deformation,
if the reduction in rotation was as great as the hypothesis of earth-moon
fission requires. It appears, therefore, that an annular latitudinal dis-
tribution of the sediments and of the lands derived from the sediments
should have arisen, and this should have codperated with the tendency of
the waters to polar accumulation in giving a distinctive configuration to the
distribution of land and water. Yet, as a matter of fact, the surface con-
figuration is singularly free from latitudinal sones. There is a very rough
tendency toward a meridional arrangement, but the essential fact is that
the arrangement is irregular. The protuberances and depressions consist
of an unsymmetrical interspersion of independent triangidar, quadrangidar,
oval, and scarcely definable areas.
Going more into detail, and in this insisting only on the obvious general
proposition that the water-level in the equatorial sone should have tended
to a low position relative to the land and to a high position in the polar
regions, we may note that the Greenland Archean embossment not only
stands high above the water-level to-day but is singidarly free from evi-
dences of submergence in the past. At various periods from the Cambrian
onwards, the water-level has stood low about its base and has risen above
and fallen below the present shore-line. Much the same may be said of
the great Archean tract of Labrador and of the region west of Hudson's
Bay, as also of that of Scandinavia and Finland. It is a remarkable fact,
in the light of the matter in hand, that the old lands which are now best
exposed, the lands that seem to have been longest out of water, and that
have been most persistently above sea-level, are more largely the lands of
high latitude than of low latitude.
A candid and critical survey of the relations of land and water in high
and low latitudes alike, and in all longitudes, especially in the northern
hemisphere where best known, and where the protuberant lands furnish
the best record, seems to me to reveal a singidar constancy of relations,
subject only to oscillations measured by a few thousands of feet at most,
an order of magnitude quite out of harmony with any hjrpothesis which,
to dte a very conservative example, requires that the equatorial tract
should have been 8 miles higher than at present when the rotation-period
of 14 hours prevailed.
If the moon were once much nearer the earth than now the tides should
have been much stronger and the littoral deposits of the early ages should
show not only greater coarseness but greater vertical range. Geologists
have not been generally convinced that the earlier sediments are different
in any such systematic way from those of later times.
58 THE TIDAL PROBLEM.
MINOR EVIDENCES.
If the earth's rotation were much more rapid than now in early times,
the gyratory component affecting the courses of the winds would have been
strengthened and probably trees would have required a corresponding
strengthening of the trunks, branches, and roots to meet this successfully.
Such provisions are not certainly detectable. In the coal-accumulating
eras trees grew to great heights without tap roots, and in some cases they
appear to have grown on accumulations of vegetal debris which could not
have furnished a very secure hold, and yet there is no evidence that they
were especially subject to overthrow. In no way is it clear that the life of
the early ages, either vegetal or animal, was adapted to atmospheric move-
ments essentially different from those of to-day.
A more rapid rotation should have caused a stronger deflection of the
streams to the right hand in the northern hemisphere and to the left in the
southern. This should have resulted in tilted aggradation planes. How-
ever, these might not now be capable of detection, even if present.
It is probable that some changes would arise from the shortness of the
day and night, but it is not clear just what these would be nor what would
be the criteria for their detection.
It seems safe to say, in summation, that no geological evidence of any
unquestionable kind, or even probable kind, is found that supports the
theoretical postulate of a former high rate of rotation of the earth.
The geological criteria are not delicate enough, however, to forbid the
belief that the rotation of the earth has changed in some minor degree
during the time over which the record extends. If the deformative effects
of such changes were small compared with those of the other diastrophic
agencies, they might be so far^masked as to escape ready detection.
AOCELERATIVE AGENCIES.
There are some agencies, apparently not very potent ones, which tend
to accelerate the earth's rotation and to offset the influences of the tides.
Of these the most familiar is the shrinking of the earth. It was noted in
the review of the hypothesis of Darwin that in the initial stage the shrink-
age of the earth was made more effective rotationally than the tides of the
sun. It was of course assumed as a basis for this that the loss of heat at
that stage was quite exceptionally great. The computations of Wood-
ward ^ and others have shown that the present rotational effects of loss of
heat, assuming the correctness of current estimates, is exceedingly small.
Even if the estimates of loss of heat need to be increased, as seems probable,
such loss can not be a very efficient agency. Shrinkage from other sources,
as molecular rearrangement, atomic reconstruction, or other agencies, may
have a more considerable effect. The rotational results of the contraction
of the body of the earth from a radius of 4,160 miles to 3,960 miles, with
intervening stages, as computed by Dr. MacMillan on the assumption that
the Laplacian law of density is maintained, are as follows:
^ " The effects of secular cooling and meteoric dust on the length of the terrestrial
day." < Astro. Joiir., No. 502, 1901 : ^' From this it appears safe to oondude that the length
of the day will not chanse, or has not changed, as tne case may be, by so much as a half
second in the first ten mulion years after the initial epoch. ** p. 174.
CONCLUSION.
5»
LmngOi of day for variouB lengths of earih's raditu — Laplacian law of dennty.
Radiu of earth
(mUes).
iDcreaae
(miles).
Length of day.
3,960
24h 0^ 0*
3,970
10
24 7 17
8,980
20
24 14 36
3,990
30
24 21 64
4,000
40
24 29 14
4,010
60
24 36 36
4,020
60
24 43 68
4,040
80
24 68 46
4,060
100
26 13 39
4,110
160
26 61 10
4,160
200
26 29 08
If current views of the shrinkage of the lithosphere founded on the folds
of the shell and on overthrust faulting are valid, its accelerative effects on
the earth's rotation are greater than the retardative effects of its water-tides
as hereinbefore computed, and perhaps greater than all the tides combined.
It is possible that the earth may respond to the radiant energy of the
sun as a thermal engine and that its rotation may be influenced by this,
but the subject is obscure and elusive, if not delusive, and no attempt will
be made to develop it here.
If, as seems probable, the evening sky, because of clouds, dust, etc.»
offers more resistance than the morning sky to the passage of solar radiation
tangential to the earth, there is a slight preponderance of light-pressure in
favor of acceleration of rotation, but it must be very small.
CONCLUSION.
The application of the most radical and the most rigorous method of
estimating the frictional value of the present water-tides, a method which
brings to bear practically all the friction of these tides as a retardative
agency, irrespective of their positions or directions of motion, seems to
show that they have only a negligible effect on the earth's rotation.
From the best available evidence I conclude that the tides of the litho-
sphere are chiefly elastic strains and have little retardative value, while
Uie tides of the atmosphere are too small to be measured.
The accelerative influences seem to be also negligible, so far as geological
applications are concerned.
In close accord with these deductions, the geological evidences indicate
that there has been no such change in the rate of the earth's rotation
during its known history as to require it to be seriously considered in the
study of the earth's deformations.
I desire to acknowledge my great obligations to my colleagues, C. S.
Slichter, F. R. Moulton, A. G. Lunn, and W. D. MacMillan, for the indispen-
sable aid which their several contributions have rendered to these studies
and for criticisms and suggestions relative to my own paper.
OONTKIBUnONS TO COSMOGONY AKD THE FUNDAMENTAL PROBLEMS OF OEOLOOT
THE
ROTATION-PERIOD OF A HETEROGENEOUS SPHEROID
BT
GHABLES S. SUOHTEB
Profe$9or of Applied MathemaHa, UnivertUy of Wiicontin
61
THE ROTATION-PERIOD OF A HETEROGENEOUS
SPHEROID.
It is a simple problem to determine the rotation-period of an ellipsoid
of revolution^ if it be postulated that the density of the body is uniform,
and that the form is that assumed by a perfect liquid under like conditions
of rotation. A table of the rotation-periods of such a body having the same
volume and mean density as that of the earth, computed for various values
of the eccentricity of an elliptic meridian section, will be found on p. 327 of
Part II of Thomson and Tait's Natural Philosophy (edition of 1890). It
is the purpose of the present investigation to obtain analogous results for
an ellipsoid of variable density, assuming a law of increasing density from
surface to center approximate to that actually possessed by the present
earth. The law of density assumed in the computation is the well-known
law of Laplace: ^^ sing a ^^^
in which the symbols have the meaning given on page 64. According
to this law the internal layers or shells of equal density gradually change
from the shape of the surface to forms more and more nearly spherical
as the center of the spheroid is approached. The forms of these layers
are best expressed in the case of ellipsoids of revolution by the ellipticity
of a meridional section. This number is computed by subtracting the
length of the polar or short axis from the length of the equatorial or long
axis and dividing the result by the length of the equatorial axis.^ The
variation in the value of the ellipticities is shown by the dotted line in
fig. 7. In this diagram the polar axis is represented as divided into ten
equal parts. The ellipticities of the shells of equal density are expressed
as percentages of the ellipticity of the surface. Thus the ellipticity of the
shell that cuts the polar axis at 0.5 of the distance from the center to the
surface is equal to 85 per cent of the surface ellipticity, while the ellipticity
of the central shell is about 80 per cent of the surface ellipticity.
If we assume a surface density of 2.75 and a mean density of 5.50, the
above expression takes the form:
4.365 ao . 2.4605 a ,^,
/o = ^sm (2)
The variation in density according to this law is shown graphically by
the continuous curve of fig. 7.
' The mean axis or the polar axis is often used as the divisor. There is little differenoe
in the^results for the small values of the ellipticity usually involved.
63
64
THa TIDAL PBOBLSM.
An ioBpeotion of the diagram sbowB that the dennty inoreuM quite
uniformly tor a considerablfl distance as we pan from the Barfaee tovard
the center. We finally come to a central nueleua oS naariy uniform deooi^.
The density at the crater, required by the Lapladan law, is 10.74. ^s
value would be modified if values different from 2.75 and S.SO be aanuned
for the surface dendty and mean density respectively.
1 1 1 1 1 I 1 1 1 X li
0/Z3456''fl9iO
Polar Axis of Spuaoio
U8T OF BYMBOIB USED, WITH HEANINO.
a« ■■ mean radius of surface of spheroid.
a -= mean radius of any homogeneous shell in interior of spheroid.
6 -= equatorial radius of any homogeneous internal shell of sphwoid
of revolution.
&s " equatorial radius of surface of spheroid.
c^ polar radius of any internal homogeneous shell of spheroid.
Co» polar radius of surface of spheroid.
0-=a constant '=4.366 for earth-epheroid.
q = & constant B' 2.4606 for earth-spheroid.
/> = density of any homogeneous shell of mean radius a.
/)q — surface density of spheroid - 2.76.
a^ — equatorial attraction or value of gravity at equator of any spheroid.
Qp— polar attraction or value of gravity at pole of any spheroid,
m — ratio of centrifugal force at equator to gravity at equator.
e —eccentricity of any homogeneous internal shell of spheroid.
fg ~ eccentricity of surface of spheroid.
THE BOTATION-PBBIOD OF A HSTEROGENEOUS SPHEBOID. 65
6-C'
-r— — ellipticity of any homogeneous internal shell of spheroid.
c, " ellipticity of surface of spheroid.
M » centripetal acceleration at equator of spheroid.
While the Lapladan law of density originated in an assumption of
Laplace which had little to recommend it beyond mere plausibility, the
law is now believed to be fairly close to the truth. The computed values
of the earth's precession based upon this law of density agree well with
the observed values. The law is probably quite as near to the truth as is
the measured value of the earth's mean density, which must enter as a
basal number into any formula of density we may adopt.
The plan of the investigation is substantially as follows: The attrac-
tion of the heterogeneous spheroid of given ellipticity is first found for
points on the equator and at the poles of the spheroid. These results are
substituted in Clairaut's well-known equation connecting gravity at the
pole and at the equator with the equatorial centripetal acceleration, and
hence with the rotation-period of the earth. In thii9 manner the rotation-
period for any given ellipticity of meridional section becomes known.
Clairaut himself gave expression to formulas which give the attraction
at external points of any rotating liquid ellipsoid.^ For polar and equa-
torial points these may be written as follows (referring to the preceding list
for the meaning of the symbols) :
Equatorial ( f<^o f^t \
attraction = a, —|.-^,KJ / pda^+1..^ j pd(flh)l (3)
Polar
attraction =
In both formulas £ is a constant whose value depends upon the units
of measure in which the various magnitudes are expressed.
These expressions assume that each stratum of density p has the
ellipticity t that would exist for the given rotation-period if all strata
were perfectly liquid. In other words, the formula is built upon the hypoth-
esis of the perfect fluidity of the spheroid. If we apply these expressions
to the present earth we assume that the rigidity of the earth' is not sufiScient
to withstand for geological intervals of time the stresses that would exist
if the form of its surface differed materially from that of a free liquid.
The above expressions (3) and (4) can not be integrated until we sub-
stitute for p the appropriate law of density from (1) above. Using the
notation:
<p(a)=-^ (sin qa^—qa^ cos qa^) (5)
> See '' History of the theones of attraction and the figure of the earth/' l^ I. Todhi^
London, 1873, vol. 1, p. 220.
66 THB TIDAL PBOBLBM.
we may obtain
'pd{ah)^5a.\i.^^)<p{a.) / pda*^df{a.) (6), (7)
from which we obtain
..-^K.W(l+f[^-f]) (8)
..-ft.W(i-^[.-f]) (.)
The relation between m and c^ depends upon the ''degree of hetero-
geneity" of the earth: that is, it depends on the departure of the surface
and central density from the mean density. For a homogeneous earth we
may write, as is well known:
2c.-|m (10)
but for a body possessing the law of heterogeneity given by equation (2)
above, we can deduce the expression ^
The change of the denominator of the fraction on the right-hand side
of the equation from 2 to 2.536 is brought about by the change of hy-
pothesis from
{Central density™ 5.50
Mean density » 5.50
Surface density — 5.50
to the hypothesis
{Central density » 10.74
Mean density «- 5.50
Surface density — 2.75
If we further assume that the mean attraction at the surface of the
earth is 982 dynes per gram of attracted matter, we may write the equa-
tions of equatorial and polar attractions in the simple form
a. = 982(l-0.1739eo) (12)
ap = 982(l+0.3477eo) (13)
M = 982(1.0144)eo (14)
It is upon the numerical values here written that the results of the
following table have been obtained. The results can be checked by sub-
stituting for e in equations (3) and (4) the expression
£ = [0.09645 (a»+a») +0.8071]€o (15)
which is an algebraic function approximately equivalent to the transcen-
dental relation between e and a, within the interval with which we are
* See a "Treatise on attractions, Laplace's functions, and the figure of the earth/' by
John H. Pratt, London, 1871, p. 116.
THE ROTATION-PERIOD OF A HETEROGENEOUS SPHEROID.
67
concerned. This expression was found in 1898 by Mr. H. C. Wolff, then
a graduate student at the University of Wisconsin.
Investigations concerning the properties of the spheroid have usually
hjrpothecated an ellipsoid so nearly spherical that small error would be
introduced by neglecting the square of the ellipticity in comparison with
its first power. Such has been done in the present instance. For that
reason it is hardly possible to extend the computations to spheroids of
greater ellipticity than those given in the table. As a matter of fact, the
writer believes that he has extended the computations as far backward as
is practicable without straining the approximate formulas beyond their
limit of significance.
The Rotation-period of a Heterogeneous Spheroid,
(1)
Polar radius
(mean radius
= unity).
c
(2)
Equatorial
radius
(mean radius
= unity).
b
(3)
EllipUcity.
(4)
Equatorial
attraction
(dynes).
a*
(5)
Polar
attraction
(dynes).
Op
(6)
Centripetal
force
at equator
(dynes) .
M
(7)
Rotation-
period of
hetero-
geneous
spheroids
(hours).
T
(8)
Rotation-
period of
homo-
geneous
spheroids
(hours).
T
0.997726
.9967
.9947
.9933
.9867
.9767
.9667
.9600
.9533
.9428
.9333
.9267
.9167
.9065
1.001137
1.0017
1.0027
1.0033
1.0067
1.0117
1.0167
1.0200
1.0233
1.0286
1.0333
1.0367
1.0417
1.0467
0.003411
.005
.008
.010
.020
.035
.05
.06
.07
.0859
.10
.11
.125
.1402
981.4
981.0
980.6
980.2
978.6
976.1
973.6
971.6
970.1
967.2
964.9
963.1
960.5
958.1
983.1
983.5
984.7
985.4
988.8
993.8
999.6
1002.0
1006.0
1011.0
1016.0
1020.0
1025.0
1030.0
3.398
4.98
7.97
7.96
19.92
34.86
49.8
59.8
69.7
85.5
99.6
109.6
124.5
139.7
23.934
19.77
15.63
14.00
9.91
7.51
6.30
5.76
5.35
4.84
4.49
4.29
4.03
3.82
23.934
22.21
• • • •
• • • •
10.94
....
....
....
• • • .
5.49
• • • •
....
• . . .
4.37
(9)
Latitude of mean
radius.
(10)
Equatorial contrac-
tion
(per cent).
(11)
Equatorial contrac-
tion
(mUes).
(12)
Meridional elonga-
tion
(per cent).
(13)
Meridional elonga-
tion
(mUei).
35* 13"
35 12
35 10
35 8
35
34 47
34 35
34 27
34 15
34 5
33 53
33 45
33 33
33 20
0.0000
0.06
0.16
0.21
0.55
1.03
1.55
1.88
2.21
2.73
3.21
3.54
4.04
4.54
00
15
40
52
137
256
386
468
550
680
799
881
1006
1131
0.00
0.05
0.12
0.16
0.32
0.53
0.76
0.90
1.06
1.30
1.48
1.60
1.80
1.98
00
12
90
40
80
182
190
225
265
825
370
400
450
495
(mTRntunoNS to cosHoeomr and the fundamental problems op geology
ON THE LOSS OF ENERGY BY FRICTION
OF THE TIDES
BY
WILLIAM D. MacMILLAN
ON THE LOSS OF ENERGY BY FRICTION
OF THE TIDES.
In this paper the waters of the ocean will be conceived as concentrated
in a basin, rectangular in shape, the width of which will be taken as 2,860
miles and the length as 60,000 miles. The bottom will have a uniform
slope from the surface to a depth of 600 feet at 100 miles from the shore,
dropping then to a depth of 9,000 feet at a distance of 150 miles from the
shore, and then parallel to the surface out to the middle of the basin, the
opposite side having the same shape. The tide will be supposed to rise
4iV f^t in 6 hours — falling at the same rate.
Sec. a I Sec.b
I so mi.
Sec. c !
1280 jni. \
i
Fio. 8. CroM-teoiion of barin (ahowing oim-IiaII).
The rigorous determination of the motion of the water in such a basin
on the principles of hydrodynamics seems to be unattainable at the pres-
ent time. It is true that^ to start with, we have the equations of motion of
a viscous incompressible liquid, but I have not succeeded in finding a solu*
tion for them with the assigned boundary conditions, and therefore am not
able to give an exact statement of the rate at which energy is dissipated.
We may, however, approach the problem through some of the formuka
of hydraulics and obtain an approximation, which, even though it be
rough, will permit us to form some idea of the order of magnitude of dissi-
pation. If we liken the ebb and flow of the tide to the flow of water in a
canal we can use the formula of engineers for the loss of head due to fric*
tion and viscosity, and consequently the loss of energy.
Weisbach * gives us the following formula:
A-^vl irih V ^^^^^ vel.y wetted perimeter
"" ^^ 2g area of cross-section
where h is the total fall of water in the canal necessary to maintain the flow,
( is the coefficient of friction, and g is the acceleration of gravity. We will
* ThecMretical Mechanics. Translated by E. B. Coxe. 8th Amer. ed., sec. 475.
71
72 THE TIDAL PROBLEM.
take (jf » 32 feet per second per seoond. The coeflBcient $ itself is dependent
upon the velocity. As a result of many experiments the following value is
assigned:
f-0.007409(
.1920 \
V )
where v is the mean velocity. From this formula it will be observed that
$ increases as the velocity decreases.
This formula is applicable to a canal in which the cross-section is uni-
form throughout its length. In order to adapt it to a canal of variable
cross-section and velocity an integration is necessary. Consider an element
of the canal between parallel cross-sections at distances I and Z+cK from the
upper end, and put
V » velocity a «> area of crossHsection p » wetted perimeter
We have then, from the above formula,
dl "^2»a
or, since 2^«64 and
f = .007409(1+^)
this may be written
This expression represents the slope at the point I necessary to main-
tain the flow. The rate of fall of the water in this element of the canal is
obtained by multipljring the slope by the velocity, that is
r-3f--«rate of fall of the water
dl
The distance through which the water falls multiplied by its weight gives
the amount of work done expressed in foot-pounds. Consequently
E
where E is the amount of work done per unit time and w is the weight of
water in the element considered. The volume of water is equal to adl and
the weight of a cubic foot of water is approximately 62.3 pounds. Therefore
11; -62.3 adl
These values substituted in the expression for E give
/*»
g ^ 62.3X^7409 ^ (v* + A92W)pdl
ON THE LOSS OF ENERGY BT FRICTION OF THE TIDES. 73
Consider now a section of the basin 1 foot wide, and by means of this
formula compute the loss of energy in this strip in one second of time with
such a tide as has been supposed.
SECTION "a"
Let us begin with section '' a. '' We will suppose the surface of the water
is level throughout the section. A tide of 4^^ f^t in 6 hours means a
constant flow throughout the section of 600 feet per hour or one-sixth
foot per second, i.e., v^l. The wetted perimeter is the bottom only,
that is p»l, and thiis also is constant throughout the section. We have,
therefore,
»s2noo
„ 62.3X0.007409 / F/l V^n mon/l Vlwi
where E^ is the work done in section ''a'', the length of the strip being
528,000 feet. Evaluating the above expression there results
E„ a 37.939 foot-pounds per second
SECTION "b"
The flow in section ''a" shows that the surface sinks at the rate of
j^ foot per second. We will suppose that the surface of section '^b*'
sinks at the same rate, remaining always level. If Z be measured from the
beginning of the section the velocity at the distance I is given hj v^^,
when q is the volume of water flowing by the point I and a is the area of
the cross-section. Since the surface sinks j^ foot per second
?-100+^ a-600+ "
5280 220
Therefore
1 528000+^
*" 6 ^528000 +2«
The wetted perimeter is constant throughout the strip, so that p»l.
The length of the strip is 50 mileB, or 264,000 feet. The dissipation of
energy in this strip is then
„ 62.3X0.007409
E, p
ni r 528000+f y 0.1920r 528000+f V \„
6»L628000+2aJ "*■ 6* L528000+2«J /
By putting l~2MOO0x this expression becomes
„ 62.3 X 0.007409 X 264000
*" 64X216
/{t^r-"»i^i'}''
74 THB TIDAL PBOBUBM.
which may abo be written
» 62.3X0.007409X264000 , >it ■ *' i . oooeai, . ^' i i^
^*" 512X216X14* * ^ll + ^-T-r7-l+32.256|l + ^^v^| V(fa
CW-rM-^H^-r^n
From the reduction of this expression is obtained
£5 »i 1 . 1 24 foot-pounds per second.
SECTION "c"
Making the same assumptions with respect to this section that were
made with respect to section "b", we find in the formula v^^
a
'-^^o+sAo °-»*^
80 that
792,000+1
9000 X 5280
As before, we take p »i 1 . The length of the section is 1 ,280 miles, or 1,280 X
5,280 feet. If now we put {«5280xl280x, the expression for the lost
energy is
^ _ 62.3X0.00m9X^280X 1280 f |(i5 + i28»)«+172.8 (16+128.)*}d«
which reduces to
E^ » 142.484 foot-pounds per second
Combining these results, we have
£;«-37.939 S^-1.124 5,-142.484 Total, 181.547
The total work done in the entire strip is therefore 181.547 foot-pounds
per second. If the work done on the opposite shore of the basin be the
same, the total work done per linear foot of basin is 363.094 foot-pounds
per second. In a basin 60,000 miles long this would amount to 11,503 X 10^
foot-pounds per second, or 36,300 X 10^^ foot-pounds per year of 365} days.
The kinetic energy of the earth due to its rotation is given by the
formula ^^^j^
where T is the kinetic energy, / is the moment of inertia of the earth, and
a> is its angular velocity. The moment of inertia depends upon the law of
density of the earth's interior which is not known. We will probably be
not far astray in using the Laplacian law of density, t.e.,
r*
sm m—
5 = ^
r
a
* See Tisserand, M4c. C^., 2, p. 234.
ON THE LOSS OF ENERGY BT FRICTION OF THE TIDES. 75
where S is the density at the distance r from the center, a is the earth's
radius, and
m - 141^ 40' 28'^ = 2.4727 - 4.426
With the Laplacian law of density, then
-|'^«'
/ iih^'^i^iii)
8
—•=-;ra'— i[(3m'— 6) sin m— (m*— 6m) cos ml
o m
With the same law of density the mass of the earth (Af ) is
'btOa!
'jf 7'^''"'i^(i)
— 4;ro*— iFsin m^m cos ml
Consequently, by the division of these two expressions,
.^ 2 r(3m'— 6) sin m— (m'— 6m) cos ^ Iw 2
""am^L sin m—m cos m J
Substituting in this the numerical value of m, there results
/-.335Jlfa'
and therefore
r-.168MaV
Taking the radius of the earth at 3,968 miles, its mean density at 6.5, and
the sidereal day as 86,164 seconds, it becomes
T - 159 X 10" foot-pounds
At the rate of loss due to tidal friction as calculated above, this amount
of energy would last 43,900X10* years. The day would be lengthened
by 1 second in about 500,000 years.
CONTSIBIinONS TO COSMOGOITr ASD THE FUNDAMENTAL PROBLEMS OF GEOLOftT
ON CERTAIN REUTIONS
AMONG THE POSSIBLE CHANGES IN THE MOTIONS OF
MUTUALLY ATTRACTING SPHERES WHEN DIS-
TURBED BY TIDAL HfTERACTIONS
BY
FOREST BAY MOULTON
Aiiociate Profenor of A$tronomy, UnivenUy of Chiooffo
77
ON CERTAIN BEUTIONS AMONG THE POSSIBLE CHANGES IN THE
MOTIONS OP MUTUALLY AHRACTING SPHERES WHEN
DISTURBED BY TIDAL INTERACTIONS.
I. INTRODUCTION.
Sir George Darwin has written a number of classical memoirs on the
subject of tidal friction which are remarkable not only for the profundity
and the thoroughness of the mathematical anal3rsis, but also for the charm
and lucidity of the exposition of the nature of the problems treated, the
hypotheses upon which the investigations were based, and the conclusions
which were reached. Frequent references will be made to these memoirs
in this paper, and for simplicity they will be designated by numbers as
follows:
1. On the bodily tides of viscous and sezni-elAstio spheroids, and on the ocean tides upon a
jrielding nucleus. <Phil. Trans, of the Royal Soo., Part 1, 1879, pp. 1-^.
2. On the precession of a viscous spheroid, and on the remote history of the earth. <Phil.
Trans., Pfeirt H, 1879, pp. 447-538.
3. On the secular changes in the' elements of the orbit of a satellite revolving about a
tidally distorted planet. '<Ph]l. Trans., Plirt U, 1880, pp. 713-891.
4. On the tidiJ fdotion of a planet attended by several satellites, and on the evolution of
the solar system. <Phil. Trans., Part U, 1881, pp. 491-535.
5. The determination of the secular effects of tidal friction by a graphical method. < Pro-
ceedings of the Royal Society of London, vol. 29 (1879), pp. 168-181.
Darwin's method of treatment is to express the tide-generating poten-
tial as a sum of terms, each of which is the product of a second-order
solid harmonic and a simple time harmonic, and then to derive the corre-
sponding surface harmonics which define the tidal deformations when the
system has assumed a condition of steady movement. The results are
adapted to viscous or elastico-viscous spheroids, the heights and lags of
the several tides being expressed in terms of the speeds of the tides and
the viscosity, or the rate of decay of elasticity of the tidally distorted body.
The effects of these tides upon the motions of the disturbed and disturbing
bodies are then derived with rare skill.
Apparently Darwin's work can be questioned, if at all, only where he
applies his analysis to the earth-moon system. Here he reaches the con-
clusion that very probably the moon once separated from the earth by
fission, and that it has been driven to its present distance by tidal friction.
In reading these conclusions we should heed his warning:^
TIm lesolt at which I now arrive affords a warning that every conclusion must always
be read along with the postulates on which it is based.
^ 2, p. 532, footnote.
79
80 THE TIDAL PROBLEM.
It is quite evident from Darwin's discussions that he acceptSi as a general
basis for reasoning on the problems of cosmogony, the Laplacian nebular
hypothesis with its implication of a one-time fluid earth; indeed, in 2,
p. 536, line 3, and in 4, p. 530, first paragraph, he explicitly states that
he adopts this hypothesis in its main outlines. It is obvious that this
point of view might tend to give one a confidence, perhaps without his
realizing fully the postulates upon which it was based, that, even if the
spherical harmonic analysis should not be strictly applicable to a hetero-
geneous earth whose liquid parts are broken up by continental masses,
it still would be sensibly correct when applied to a fluid body such as the
earth was supposed, according to this theory, to have been in the past.
It is now known that there are very grave, and I believe fatal, objections
to the Laplacian ring theory. At any rate, one would not now make it a
postulate in a discussion involving so many and such serious complexities
as arise in the theory of tidal evolution, or allow it seriously to influence
his conclusion as to what is the most probable of the various possible
hypotheses. Darwin examined with great thoroughness the character of
the results for various conditions of viscosity and semi-elasticity, and only
where he undertook to say what seemed to him the most probable of vari-
ous possible series of events was he influenced, possibly, by his preconcep-
tions as to the early condition of the earth. To illustrate the delicacy of
the discussion we shall enumerate a few of his conclusions together with the
hypotheses upon which they were based.
In 3, Part IV, and in its summary, pp. 871-876, Darwin discussed
the inclination of the moon's orbit and the obliquity of the ecliptic Con-
sidering first the hypothesis of imaU viscosity and tracing back the qrstem
until the day and month were equal, he found that, if this hypothesis is
true, the lunar orbit and the earth's equator must initially have had con-
siderable mutual inclination. " If this were necessarily the case, it would
be difficult to believe that the moon is a portion of the primeval planet
detached by rapid rotation, or by other causes. " (3, p. 873.) Then taking
up the hypothesis of large viscosity and supposing that it was 'Marge
enough," he found, tracing the system back, that when the day and month
were of equal length, then the lunar orbit was sensibly in the plane of the
earth's equator, which was inclined 11^ or 12^ to the plane of the ecb'ptic.
His final conclusion from this discussion (pp. 875-876) was that it wfll be
most nearly correct to suppose that the earth in the earliest times, though
plastic, possessed a high degree of stiffneaa, and that now the greater part if
not the whole of tidal friction is due to oceanic tides, and not to bodily tides,
for in this way the theory of the fission of the parent mass into two bodies
and the present inclination can be best reconciled.
In 3, Parts V and VI, the effects of tidal friction upon the eccentricity
of the lunar orbit were considered. The equations were integrated on the
hypothesis of small viscosity, and it was found that in past times the
eccentricity was much smaller than at present, nearly vanishing when the
day and month were equal. If it had been assumed that the viscosity was
very large, the eccentricity of the lunar orbit would have been the greater
the farther back the system was traced. Since a large original eccentricity
INTRODUCTION. 81
is incompatible with the fission hypothesis, it is found necessary to
conclude that, at least for a part of the earth-moon history, the viscosity
of the part of the earth distorted by tides has been amaU. Hence that
viscosity which best explains the inclination of the lunar orbit causes
trouble when considering its eccentricity. Since numerical details were
not worked out in the discussion of the eccentricity for large viscosities, it
is not known to what quantitative extent the two things are antagonistic,
and it may very well be that a viscosity could be assumed which would
explain largely the inclination and not be particularly unfavorable to the
eccentricity. There were so many partial contradictions and so much
uncertainty that Darwin attempted to draw no final conclusion from this
discussion (3, p. 879).
Another question, still more critical, is the distance from the earth to
the moon when the day and month were of equal length. In 2, section 18,
neglecting part of the action of the sun, Darwin found that the day and
month were equal at 5** SO"", corresponding to a distance between the centers
of the earth and moon of 10,000 miles. In 3, section 22, including all the
action of the sun, he found that the initial period and distance were less.
The numerical results were not obtained, but he stated (3, section 22, p. 836) :
It 18 probaUe that an accurate solution of our problem wouki differ oonsiderably from
that found in "Precession" (5^ 3$™), and the oonunon angular velocity of the two bodies
might be very great.
In the summary of this same paper, p. 877, he said:
In section 22 it [the sun's action] is only so far considered as to show that when there
is identity of periods of revohition of tiie moon and earth, the angular velocity must be much
greater than that given by the sdution in section 18 of "IVecesBioa."
Computations given at the end of the present paper, section 14, show
that the difference is actually unimportant.
Apparently, in order to make this fission hypothesis workable it must
be shown that if the day and month ever were equal they had such a period
that the distance from the earth to the moon was much less than 10,000 miles.
There are many places where it would be easy for a careless reader
of Darwin's work to lose the connection between the conclusions and the
hjrpotheses upon which they were based. For example, taking approxi-
mately that viscosity which would produce the most rapid tidal evolution,
he found ^ that 57,000,000 years ago the day was 6^ 45" long, and that
the length of the month was 1.58 of our present days. Notwithstanding
the fact that this is quite a different thing from having proved that
the earth-moon system has actually gone through this series of changes,
undoubtedly many first-hand and more second-hand readers of Darwin's
work have suppoiBed that this computation gives a fairly certain and
definite account of the evolution of these bodies. But it is interesting to
find in the same memoir, section 14, under the hypothesis that the observed
secular acceleration of the moon's mean motion is due entirely to tidal
friction, and also that the earth is purdy viscous^ the conclusion that the
length of the month is now being increased at the rate of only ^ 20" in
> 2, Section 15, Table IV, and pp. 529-531.
82 THE TIDAL PBOBI.EM.
100,000,000 years. Very different results were obtained by assuming that
the earth is dastiahviscous. Under this assumption in 700,000|000 years
the day will be about as long as at present, the month nearly a day shorter
than at present, and the obliquity of the ecliptic about 6^ less than it is
now. The discussion of the secular acceleration of the moon was closed
with these remarks (2, p. 483, last paragraph) :
The ooncluflion to be drawn from all these calculations is that, at the praent time,
the bodily tides in the earth, except perhaps the fortni^tly tide, must be ezoeeding^y
small in amount; that it is utteily uncertain how much of the obs^ved 4' of aocelnation
of the moon's motion must be referred to the moon itself, and how much to the tidal frictkm,
and accordingly that it is equally uncertain at what rate the day is at present being
lengthened.
Notwithstanding these uncertainties, in the general discussion at the
end of the final paper of the series (4, pp. 632-533) Darwin states:
The previous papers were prindpaUy directed to the case of the earth and moon, and
it was there found that the primitive condition of those bodies was as follows: The earth
was rotating, with a period from two to four hours, about an axis inclined at 11^ or 12^ to
the normal to the ecliptic, and the moon was revolving, nearly in contact with the earth,
in a circular orbit coincident with the earth's equator, and with a periodic time only slightly
exceeding that of the earth's rotation.
Then it was proved that lunar and solar tidal friction would reduce the system from
this primitive condition down to the state which now exists by causing a retardation of
terrestrial rotation, an increase of Ixmar period, an increase of obliquity of ecliptic, an
increase of eccentricity of lunar orbit
It was also found that the friction of the tides raised by the earth in the mocm would
explain the present motion of the moon about her axis, both as regards the identity of the
axial and orbital revolutions, and as regards the direction of her polar axis.
Thus the theory that tidal friction has been the ruling power in the evolution of the
earth and moon completely coordinates the present motions of the two bodies, and leads
us back to an initial state when the moon first had a separate existence as a satellite.
This initial configuration of the two bodies is such that we are almost compelled to
believe that the moon is a portion of the primitive earth detached by rapid rotation or
other causes.
The problem of tidal evolution is an extremely complicated one and
the uncertain factors which enter into it are very many. Darwin's treat-
ment of it as a mathematical problem was masterly and worthy in every
respect of the highest admiration. He was generally very cautious in
drawing conclusions with respect to the actual earth-moon system. The
danger lies in the 'formidable and protracted analyses coming in between
the hypotheses and the conclusions, which might lead one to suppose that
results drawn from a particular set of postulates necessarily belong to the
earth and moon, particularly if they, in a general way, coincided with his
preconceptions of cosmogony. Even though Darwin may have been with-
out fault in this respect, it is not certain that less critical minds, especially
if they were without the illuminating experience of finding by actual com-
putation how great changes in the results would be produced by admissible
changes in the hypotheses, would not attach imdue importance to some
particular computation. There is nothing deduced from observations so
far made or from Darwin's investigations that would prevent one, if it
suited his fancy, from drawing the conclusion that the motions of the earth
and moon have been for 100,000,000 years about as they are at present.
INTRODUCTION. 83
In queBtioQS of cosmogony, where immense intervals of time are in-
volved; the problem of tidal evolution is obviously one of great importance,
unless it shall some time be shown that it is not a sensibly efficient factor.
The two most obvious methods of determining its efficiency are by direct
attacks from the mathematical standpoint, or by comparing its certain im-
plications with as many facts given by observation as possible. The first
is mainly the method of Darwin, and he has written what will certainly
always be an extremely important chapter in the question when considered
in the broadest possible way. His results can be improved, apparently,
only by a determination of the physical properties of the earth as a whole,
and by an estimate of the loss of energy in the ocean tides. While there
is hope for the former from seismic vibrations, certain astronomical phe-
nomena, and the character of the crustal deformations as revealed by
geological studies, the results are not now so well established that they do
not need support from other sources. The second method, that of com-
paring the positive implications of the tidal theory with observed facts in
as extended a way as possible, is broadly speaking that adopted in this
paper. Since there can be no test of time-results except on the basis of
other doubtful hypotheses, and since it is impossible to draw any certain
conclusions in the questions involving the time, this variable has been
entirely eliminated from the discussion except in section 14. In a general
way it may be said that the energy of the system has been taken as the
independent variable, for it is known that under any sort of friction it
must degenerate into heat. The results are characterised by certainty
so far as they go, but as compared with Darwin's they are in most oases
much less explicit as to particulars. The discussion is mostly attached
to the fundamental equations of moment of momentum and energy. After
the work was well advanced it was found that Darwin had applied fun-
damentally the same methods to illustrate his results in a paper supple-
mentary to his main series and published in a diiBTerent serial (No. 5 in the
list previously given). The variables he used were different from those
employed here, but, though for certain purposes they may be more conve-
nient, nevertheless, for the sake of complete independence, those originally
selected have been retained.
When the discussion is based simply upon the moment of momentum
and energy equations, the number of quantities to be determined is greater
than the number of the determining equations. To attain the greatest
simplicity the general problem has been divided up into a number of
special cases covering altogether the entire field. In this way each special
problem is very easily understood and the question as a whole is much
illuminated.
6
84 THE TIDAL PROBLEM.
II. GENERAL EQUATIONS.
The problem treated will be that of the tidal interactions of m^ and m^,
which, in certain cases, are supposed to be disturbed by a large distant
mass is. The masses m^ and m, will be assumed to be spherical. Then let
a^^ radius of m^; a, » radius of m,
r«- distance from the center of m^ to the center of m,
/>.«« — r» distance from m^ to center of gravity of m, and m,
/>,— ^ — fB distance from m, to center of gravity of m^ and m,
X, y, g^B&i of fixed axes with origin at center of gravity of m^ and m,
i ""inclination of orbits of m^ and m, to xy-plane
ii"- inclination of plane of equator of nii to xy-plane
{,»i inclination of plane of equator of m, to xy-plane
e^i eccentricity of relative orbit of m^ and m,
a » major semi-axis of relative orbit of m^ and m,
B angular velocity of revolution of m^ and m,
P» ^ -B period of revolution of m^ and m,
a>] B angular velocity of rotation of m^
a>s-" angular velocity of rotation of m,
Di— -= — -"period of rotation of m.
* a>i
!),—-= — —period of rotation of m,
iS "- a distant disturbing mass
r^«" distance of S from center of gravity of nti and m,
jP» period of revolution of S
M, M\ M^— whole moment of momentum of system, exclusive of S,
about the z, x, and y-axes respectively
£■> whole energy, both kinetic and potential, of system exclu-
sive of S
The moment of momentum of m^ about the z-axis is the moment of
momentum about its axis of rotation multiplied by cos t\ plus its moment
of momentum of revolution around the z-axis. The rotational moment
of momentum of tn^ around an axis through its center and parallel to the
f-axis is .
Cifnfi^Wi cos t|
where C| is a constant depending upon the law of density. The corre-
sponding quantity for m, is
c^mTfl^a)^ cos t.
The moment of momentum of revolution of m^ about the 2-axis is
m, pi' cos i — 7 — \~^r,r^ cos i
(1)
GBNSBAL EQUATIONS. 86
and the oorresponding quantity for m, is
m, pJ cos i — 7 — \ ^ .^ T^O cos %
Therefore the total moment of momentum about the 2-axis is
ilf = — ^^-r^O cos i-^cMjiit^a). cos it+e.mMJo). cos u
The whole energy of the system is the kinetic energies of rotation, plus
the kinetic energies of revolution, plus the potential energy.
The kinetic energy of rotation of m^ is ic^mia^^oji^, and there is a similar
expression for the kinetic energy of rotation of m,.
The kinetic energy of revolution of m^ is
and there is a similar expression for the kinetic energy of revolution m^
The potential energy of the system is ^-^, where k^ is the gravita-
tion constant. ^
Therefore the total energy is
From the two-body problem we have
Hd=±*^(m,+m,)a(l-e») (3)
the determination of the ambiguous sign depending upon the direction
of revolution, and
2wo«
r*
By means of (3), (4), and (5) equations (1) and (2) reduce to
„_ +m,OT,fc«P* r; — :; .• . 2m:^m^a^* cos t, . 27rc,wt /i,' cos t, .
^-(2;r)»(t»,+»g»V*-' «'»*+ D, + D, <®^
the sign of J^ depending upon the direction of revolution, and the signs of
D| and D, upon the directions of rotation of m^ and m, respectively.
86
THB TIDAL PROBLBH.
The units are so far arbitrary. We shall choose them so that
unit of time = mean solar day
(2«)Hin,+m,)»~^
Then equations (6) and (7) become
X Di Cj \o,/ Vf
(8)
(9)
(10)
If we represent the ascending nodes of the phuies of the orbit and of the
equators of m| and m, by Q, Sii, and Q, respectively, the moment of
momentum equations for the x and y-azes are similarly
M--p».Ji:::^sin*sina+ '"'°^!;°^^' +g?fty "^°^^°^^'
>^« ni n 7 • • /-> , TO. sin t, cos Q, , c,/a,\* m, sin*. COB ft,
M'-P»^l-c»smtcoBft+-* ^^ »+-|(^) -=i ^
THE LAPLACIAN LAW OF DENSITY. 87
III. THE LAPLACIAN LAW OP DENSITY.
In treating the rotations of such bodies as the earth it is not permissible
to regard them as homogeneous, for in the case of the earth the density
of the snrface rock averages about 2.75, while the average density of the
whole earth is 5.63. If we let a represent the density of the sphere m,
the well-known expression for it suggested by Laplace is
— ; — . (11)
a
m
where r is the distance from the center of m, and where and /i are con-
stants depending upon the constitution of the body. According to this
law the density of the body increases from the surface to the center, and is
finite at both the surface and the center. We shall determine O and /i by
making both the surface density and the mean density agree with the
results furnished by observation.
The mass is found from the equation
— 4jr / m^dr — 4to"G / — sin (;*— ) d — — — 3~Csin /i— /i cos /i] (12)
Let a^^ represent the surface density and a the mean density. Then
and fi are determined by the equations
<7»^ -<? sin /I a;£> - 3G [sin ju— /i cos /i] (13)
The density at the center is
a'^'^Qfi (14)
In the case of the earth a^^ « 2.76, a^ — 5.63, whence it is f oimd from (13)
and (14) that
<?- 4.39633 /i- 2.46679 (7/^^-10.840 (16)
In our ignorance as to the density of the surface material of the moon,
m^ we shall assume that for it /i,—/! and determine (7, from (12) so that
81.7m, -m,. We find, taking Oj- 3,958.2 miles and a, -> 1,081.5 miles, that
whence
a, -3.32
0,-2.6393
(16)
ff,»>-1.66
«T,<*-.6.61
(17)
88 THB TIDAL PROBLEM.
IV. MOMENT OP INERTIA FOR THE LAPLACIAN
LAW OF DENSITY.
Letting / represent the moment of inertia of the sphere m, we have
/= / r*cos*^dm= j j j ar^coa^fpdrdfpdO (18)
""2
Substituting the expression for <r from (11) and integrating, we find,
/-^^[3 (/i»-2) sin /I-/I (/i»-6) cos /i] (19)
or, making use*of (12),
J ^ 2aHn [3 (/i^— 2) sin /i-/i (/i»~6) cos p] .^
3p? sin fi—fi cos fi
Hence the values of c^ and c,, occurring in (1), are determined by equations
of the form
^ ^ 2 [3 (/i»~2) sin pt^fi (/i»-6) cos ft] .^n
3/1* sin fi—pt cos pt
When m^i represents the earth and nt, the moon, we find from this equation
and the value of /i given in (16) that
c,-c- 0.33594 (22)
instead of 0.4, the value for homogeneous spheres^
SPECIAL CABB.
89
V. SPECIAL CASE i«0 i\"0 a,-0 e»0 ^^O.'^
This means that two bodies undisturbed by any exterior force revolve
in circles, that the radius, mass, and angular velocity of rotation of one of
them are so small that its rotational momentum and energy may be
neglected, and that the axis of rotation of the other is perpendicular to the
plane of their orbit. In this case equations (9) and (10) become, writing D
in place of D^,
ilf=pi+:^ (23)
^ D
(24)
We may choose the direction of revolution of the bodies as the positive
direction. Then only a positive P can have a meaning in the problem,
since a revolution in one direction can not be reversed without a collision
of the bodies. D is positive or negative according as the rotation is in the
same direction as the revolution or the opposite. Under the hypotheses
adopted M is rigorously constant. When Z) =■ + oo then P « AP; when
D-^then P«0; when D=Ztm(0+0 then P«-oo; when D«Zim(0— t)
then P = + 00 ; when D — — oo then P = HP. Consequently the curve
defined by (23) is as given in fig. 9. The part of the figure to the right of
the P-axis belongs to the case where the rotation of m^ and revolution of
m, are in the same direction, and the part to the left where they are in
opposite directions.
P
Fio. 9.
The slope of the curve, or the ratio of the rate of change of the period
of revolution to that of rotation, is found from (23) to be
dP / dP
dt / dt
(26)
* For a nmilar treatment of this problem see No. 5 of Darwin's papers.
90
THE TIDAL PROBLEM.
The values of P and D must also satisfy equation (24). Starting from
any epoch, P and D must change, if at all, so that E shall decrease, for if
there is (tidal) friction in the system, some of its energy will degenerate
into heat and be dissipated.
The relations between P and E and D and E are found from (23) and
(24) to be respectively
AP (26)
p,_2MPi-^=m,^
m,
D'
E
D^ {MD-m^y It
(27)
In these equations P may vary only from to + oo while D may vary
from— ooto + oo.
The curve whose equation is (26) has two forms according as E^ con-
sidered as a function of P, has a finite maximum and minimum or not.
In case there are a maximum and a minimum it has the form /, fig. 10.
Since E can only decrease, it follows that if at < «fo ^^ period is on the part
Fio. 10.
of the curve ob it will decrease to the abscissa of the point h; if it is at h
it will permanently remain at that value; if it is between 6 and c it will
increase toward the value at h; if it is at c it will remain there unless the
system suffers some exterior disturbance, when it will increase toward h
or decrease toward d according to the nature of the disturbance; if it is
between c and d it will continually decrease toward zero.
When the curve has no finite maximum and minimum it has the form
//, and then whatever may be the value of P at <«^o ^t will continually
decrease toward zero with decreasing E,
It is unnecessary to draw the curve whose equation is (27), for the
relation between the change in P and the change in D is given in (25).
SPSaAL CABB.
91
It is seen from this equation that when the rotation and revolution are
both in the same direction P and D either both increase or both decrease,
if they change at all.
The necessary condition for a maximum or a minimum of B is, from (26),
^Plg^Pl-AfP+m.-0 (28)
The corresponding condition from (27) is
The only D having a physical meaning is real. Since no real negative D
satisfies this equation, it follows that when D is negative E has no finite
maximum or minimum. In this case by fig. 10, dotted curve, the period
of revolution must always decrease and the two bodies ultimately fall
together. Taking the last four terms to the right and extracting the cube
root, we have,
Z)t=JlfD-m,
Since the roots of this equation are the same as those of (28), it follows that
whtnEis a maximum or minimum D '=> P and the system moves (U a rigid body.
The real roots of (28) are the abscissas of the intersections of the curves
y-PI
y="ilfP— nij
(29)
It is evident from fig. 11 that there are two, or no, intersections of
these curves.
. 11.
For a given ntj the value of M may always be taken so great that there
wiU be two real roots; or, so small that there will be no real roots. The
limiting value of M, as it decreases, for which the real roots exist is that
value for which they are equal. The condition that (28) shall have equal
roots is
|P.-M
(30)
02 THB TIDAL PROBLEM.
Since this value of P must also satisfy (28) we have
(iy^*-(iy^*+»»i-o
whence
M^^m,k (31)
as the value of M for which there are two equal roots of (28). For greater
values of M (28) has two distinct real roots. The roots are the abscissas
of the points 6 and c, curve I, fig. 10. The smaller root corresponds to a
maximum of £, and the larger to a minimum. When the roots are equal
the curve has a point of inflection with tangent parallel to the P-axis.
We may express the rates of change of P and D in terms of the rate of
change of E by differentiating (23) and (24) and solving. We find
dP 3P»Z) dE dD PD" dE /jg)
di "" 2;r(P-Z)) di dt ^ 2m,7t{P-D) dt
Consider the case first where the direction of rotation and revolution
dE
of nil are the same, i.e., when D > 0. Since --g- can be different from sero
only when P%D, then when P>D both P and D must increase whatever
may be the character of the tides as determined by the physical condition
of mi,* and when P<D both P and D must decrease. When m^ rotates
in the negative direction, i.e., when D<0, P must always decrease and
D always numerically increase. When P'^D equation (23) becomes
-^(Pt-MP-hmi)-O
for which, by (28), £ is a maximum or minimum.
We are supposing the orbit a circle and the axis of tn^ perpendicular
to this orbit. Consequently when P'^D there can be no change in the
motion of the system due to the tides. Therefore the right members of
(32) must carry (P—Dy as a factor, where 7 > 1. The exponent ; can not
be fractional for then the rate of change of P would be imaginary for P<D.
Consequently / is 2 or some greater positive integer. The velocity of the
p D
tide with respect to the surface of m^ is , and the tidal force is pro-
1 "^
portional to -,. If we assume that the friction is proportional to the height
of the tide and the first power of its velocity over the surface of m^, and
that the loss of energy, i.e., the work done against friction, is proportional
to the square of the friction, equations (32) become
(33)
dP e^P-D
dD c, D(P-D)
dt~r* PW
dt m^r* P
where c^ is a constant depending upon the physical condition of m^.
SPECIAL CASE. 93
Equations (32) are very instructive, for they prove rigorously, under
the hypotheses we have adopted in this section, that the rates of change
of both the day and the month are proportional to the rate of the loss of
energy, however it may be lost. That is, if tidal friction is now almost
exclusively in the ocean tides, as Darwin supposed,^ in so far as the earth-
moon system satisfies the hypotheses of this section it is quite immaterial
whether the energy is lost in the manner described by the spherical har-
monic analysis when applied to the viscous theory, or whether it degen-
erates after the waves have been time after time reflected from the con-
tinents and have run into narrow bays or into the high latitudes. The
relation of the tidal wave to the moon is not directly involved as it is in the
elementary geometrical discussions of tidal friction, though of course the
rate, and therefore the phase, of the friction depends upon the viscosity
of the water. This would increase one's faith in the spherical harmonic
analysis for such an earth and ocean as we have if it were not for the fact
that the irregularities in the depth of the ocean and in the outlines of the
continents undoubtedly greatly change the whole amount of friction.
The moon sets up motions in the waters of the ocean, but not all of the
energy possessed by this water is lost. At the succeeding disturbance of
the same region by the moon the phase of the tidal deformation still per-
sisting may be such that the moon's attraction will tend to destroy it rather
than generate a new wave; or, the phase may be such that the moon will
augment the tide. On a world covered with oceans of many dimensions and
depths we should expect to find places where the natural periods of oscilla-
tions in water basins are such that the moon's disturbance builds up consid-
erable tides, and others where they are kept low. In the former case the
friction of the water prevents their becoming excessively large; if the water
were entirely frictionless they would increase until the resulting alteration in
their period would lead to their destruction by the moon's disturbing forces.'
One method of finding the present rate of tidal friction, at least so
far as it is due to ocean tides, is to compute from tidal observations in all
parts of the earth, and from the frictional properties of water, the actual
waste of energy.' If one were to observe the energy manifested when the
tide rims through a strait on our coasts, he would be apt to overestimate
the work the moon is doing upon the earth. In the first place such condi-
tions are quite exceptional, and in the second place only a very small part
of that energy degenerates into heat. When the run of the tide ceases the
kinetic energy has very largely become potential, and it becomes kinetic
again when the tide runs out. If the outgoing tide has the same energy
as the inflowing tide there has, of course, been no loss, and, according to
equations (32) and (33), there is no tidal evolution in such a system as
we are considering in this section.
In the units employed the kinetic energy of rotation of m^, the kinetic
energy of revolution of mj and m, about their common center of mass,
' 2. pp. 483-484.
' For a discussion of the observational evidence see Harris, U. S. Coast and Qeod.
Survey (1900), app. 7, pp. 53^600. Also Chamberlin's paper, ante, pp. 5-60.
' See paper by Macwillan, ante, pp. 71-75.
94 THE TIDAL PROBLEM.
and the potential energy of ntj and m, are respectively
(rotation) ^^^ (revolution) ^ (potential) _23r ,,-.
Then from equations (32), (33), and (34) we find
(rot.) (rcT.) (pot)
dEmi P dE dEm^-k-mt D dE dEm^+mt -"2P dE r^g)
di "P-D di di ^ P-D dt di " P-D di
whence
(rot) / (rev.) (rot) / (pot)
dEm^ / dE(Mi-\-mt) ^ , P dEmi / dEmi^mt JP (36)
di / dt ^D dt / dt '^ 2D
When the directions of the revolution and of the rotation are the same,
the loss of energy of rotation is to that of revolution of both bodies as
the period of revolution is to that of rotation, and the potential energy
gains twice the loss of the revolutional energy.
The number of periods of rotation in one of revolution is, from (23),
When (28) is satisfied iV -> 1. The maximum value of iV is defined by
ni,^-(M-|pi)-0 (38)
whence, at maximum N,
'-g^ "-m <»>
APPUGATION OF SECTION V TO THB EABTH-HOON 8Y8TEU. 95
VI. APPLICATION OF SECTION V TO THE EARTH-
MOON SYSTEM.
This discussion neglects the rotational momentum and energy of the
moon, the eccentricity of the moon's orbit, the inclination of the equator
of the earth to the moon's orbit, the slight oblateness of the earth, and the
disturbing action of the sun. These factors have probably been of slight
importance in the series of changes which P and D may have undergone.
The observations show that at present, taking for the radius of the earth
that which would give it a volume equal to that of the actual oblate earth,
a^ » 3,968.2 miles 8 « 332,000 m^
a,» 1,081.6 miles D =0.997270 mean solar day
r' - 92,897,000 miles P « 27.32166 mean solar days
m| «81.7 m, P' « 366.26636 mean solar days
(40)
From the two-body problem we have ^
P'^S+m^+m^ ^*^^
Consequently the second equation of (8) becomes
With the data of (40) the first equations of (8) and (42) give
imit of time "mean solar day a^ » 0.688303 1
unit of length « 1 .46286 a^ m^ - 0.62631 1 (43)
unit of mass « 1.69666 m^
Using (40) and (43), equations (23) and (24) become
M - 3.01187+0.62803-3.639901 .^.
E - -0.34632 + 1.96689 - 1.61067 J ^ ^
It is found from (26) that at the present time the ratio of the rate of
the change of the month to that of the day is
f/f-.7.U (45,
With the value of m, given above we find
|j«,*-1.56
^ There m, of coune, a correspondmg equation for the motion of the moon about the
earth, but nnoe the direct perturbing action of the Bun increaaes the period for a given
distance of the moon, the k found from this equation would be too smalL
96 THE TIDAL PROBLEU.
Since M is greater than this number it foDows that (28) has two real roots.
It is easily found by approximation processes that the roots of (28) are ^
Pj -0.20535 days =4.9284 hours P, -47.705 days (46)
Since the present value of P lies between these limits it must alwajrs remain
between them and continually approach P,. From the formula
27ta^
P-
w -v Wl| + t?l J
where k is the Gaussian constant, and the units are the mean solar day,
r', and S, we find that the distances fi| and R^ corresponding to P^ and P,
*~ fii =9,194.35 mUes fi, =345,355 miles (47)
The maximum possible number of days in a month is at once found
£«»m(39)tobe iV-29.659 (48)
The corresponding length of the month expressed in terms of present
mean solar days is, from the first equation of (39),
P= 20.345 (49)
Since the month is increasing and now greater than 20.3 days, the system
is already beyond the condition of maximum number of days in a month.
Let us apply these results to Darwin's hypothesis of the separation of
the moon from the earth by fission, remembering, of course, that a number
of factors involved in the actual case have been omitted. At the time of
separation their periods of rotation and of revolution about their center of
gravity must have been equal. But the solution shows that they moved as
a rigid body when the surface of the moon was 9,194.4— (3,958.2 + 1,081.5)
-> 4,154.7 miles from the surface of the earth, which contradicts the hypoth-
esis that they had just separated by fission. But this is neglecting the
earth's oblateness, which must have been great. To get an idea of the
possibilities let us examine a number of modif3ring hypotheses. First let
us suppose that the earth was then so oblate that its equator reached to
the moon, and that the law of density was such as to keep its moment of
momentum and volume unchanged. Then we find that
equatorial radius = 8, 112.9 miles polar radius = 942.2 miles
Obviously the spheroid would have broken up long before it attained this
degree of oblateness, and under the hypothesis that the moment of momen-
tum was as it would have been in a sphere the equatorial zone would have
been so rare that one could not account for the matter in the moon.
In order to avoid the difficulty of the rare periphery, forced by the
condition on the moment of momentum, we may waive this condition.
Assuming simply that the earth was oblate, let us find the qualitative
effects on the initial distance of the moon. The moment of momentum for
* Darwin in 2, p. 508, taking the earth as a homogeneous spheroid and other data
somewhat difiPerent, found P, =^ 5.6 h., P, = 65.5 d.
APPUCAHON of BBCTION V to THB BABTH-UOON BTBTBIC. 97
a given volume will have been greater than that which we have used, and
we may adjust our formulas for it by increasing the c^ which occurs in the
original equations (6) and (7). It foUows from (8) that the unit of length is
now greater than before. Then by (42) the numerical value of m^ is greater
than with the original e^, from which it follows by (29) and fig. 1 1 that the
distance of the moon from the earth when the month and day were equal
was greater than that computed above. This would necessitate an increase
in the oblateness in order that the earth's equator should have extended
out to the moon, and the difficulty of having an earth already improbably
oblate is increased.
Another hypothesis is that the earth was initially larger and, since the
separation of the moon, has shrunk to its present dimension. This is
quite in accord with the general ideas prevailing in the fission theory. We
ean not apply directly the formulas which have been written down because
a change in volume would change the distance at which the system moved
as a rigid body. Consider an instantaneous change of any extent in the
radius of the earth. This does not change its rotational moment of mo-
mentum. Then if we employ the same units -=^ is not changed; that is, D
is changed so that when the new coefficient of m^ as defined in (8) and (42)
is used the quotient is constant. But an increase in the size of the earth
would result in an increase in D. Therefore m^ is increased to km^ and the
condition for equality of the day and month is
From this equation we find
dP m^
dk "M-4P*
which is positive for the smaller root of P^. Hence, if the earth has shrunk
from larger dimensions, the earth and moon moved as a rigid system at
a greater initial distance than that found above. That is, the hypothesis
that the earth has shrunk only adds to the embarrassment because of the
initial great distance of the moon.
We may try the hypothesis that the moon separated from the earth at
a distance of 9,194.4 miles, that the earth's law of density was such that
at that time its radius was equal to this number, and that the moment
of momentum of the earth's rotation was the same as if its density were
as it is at present. The latter condition is necessary, for the whole moment
of momentum is unchanged by contraction. This hypothesis amounts to
simply attempting to change the law of density as well as the volume so
that the implications of the hypothesis shall be reasonably satisfied.
We shall suppose the density is expressible by the Laplacian law, only
with different values of G and /i from those which are used above. Letting
/{|-* 9, 194.4, the moment of momentum of the whole system was
M-2«c'(«.+m,)^!-?^'(l+2)(|4?l-lVa,. (60)
98
THE TIDAI. PBOBUaf.
where </ depends upon the new law of density. We have in the units defined
in (8)
P,Af /3,958.2 V nnon,., /e,^
the Ci, -^, m^, M, P^, and R^ being given in equations (22), (40), (43),
(44), (46), and (47) respectively.
With this value of c' we must determine a new value of /c, say fi', from
(21). To facilitate the solution (21) may be written in the form
^(^0-|-c'-4+-^5^-o
(52)
3 ' ju'» ' 3 (tan /— /)
We may draw the graph of this function. It is the sum of two functions
and
J'' "3 -''-/'
4 tan ft'
''*"3(tan/-/)
(53)
(64)
Fio. 12.
For /i'—O we see that yi= — ooand y,« +oo, but that Vi+y,— 2— c'«1.98.
The curve for y^ is very simple. The value of y^ is positive while j/ varies
from to ;r, vanishing at /i^«;r. Then y, becomes negative and remains
negative until y,— — oo by the vanishing of tan /£'— /i' at j/^257^ 30'.
4 Stt
Then y, changes sign and descends from 4- <» to - at ^m' « — and to zero
3 2
APPLICATION OF SECTION V TO THE EARTH-MOON SYSTEM. 99
for fi'^2n. Then it decreases to — oo in the fifth quadrant, changes sign
to +00 and decreases to zero at p!^Zn and to — oo again in the seventh
quadrant. In a general way this cycle of changes is repeated indefinitely,
the points where y^ becomes infinite approaching nearer and nearer to
)u^»(2n + l) ^, n being an integer. The curves y^, y, are given in fig. 12.
Both yx and y, being even functions of pi the curves are entirely to the
right of the vertical axis.
From the diagram it is seen that the only places where Vi+t/, may
vanish are at the left of 9r as at a, to the right of ;r, as at b, to the right of 27r,
as at c, and in general to the right of n;r, n any integer. There are, in short,
an infinite number of determinations of p!. But when we consider that
the law of density is
G'sin/^^
fit
we see that if n<ik' <2n the density will be negative near the surface and
elsewhere positive; if 2n<p!<Zn there will be a single spherical layer
between the center and surface where the density will be negative; and if
nn<ii'< {n + \)n there will be n layers of negative density if n is odd, one
of them being at the surface, and n— 1 layers of negative density if n is even,
the surface density being positive. Consequently in considering such a prac-
tical question as the separation of the moon from the earth, in which nega-
tive densities would have no meaning, we need consider only the possibility
of a solution to the left of n. The value of fi', for which y^ vanishes, is
2^^i
^2-3c'
From fig. 12 it is seen that Vi+y, can not vanish between this point and n.
It is easily verified that no value of /i'< 142.5^ will satisfy (52). For
example, we find the following corresponding sets of values
(
/i'- 0^ 46^ 90^ 135° 140°
/(/!') -0.58 0.40 0.38 0.33 0.32
Consequently the smallest fi' satisfying the conditions is greater than n, and
the hypothesis of the separation of the moon from the earth requires, so far
as the factors and the law of density here considered are concerned, that we
assume that the surface density of the united mass was negative just previous
to the separation. If we had used the oblateness of the figure of the mass,
a still larger /c' would have been found. However, it is not impossible that
neglected factors may somewhat relieve the theory of these embarrassments.
Thus we see that when we add any of the hypotheses of an original
oblateness, shrinking, or different law of density singly, the difficulties of
the hypothesis are not relieved.
7
100 THE TIDAL PBOBLXlf.
VII. CASE t=0 t\-0 t,-0 a,+0 e-0 5-0.*
This case differs from V only in that the rotational momentum and
energy of m, are not supposed to be zero. In this case equations (9) and
(10) become, supposing that c^^Ci,
In these equations only P, D^, D^, and E can vary, the last decreasing
through loss of energy by friction. If by means of (55) we eliminate one
of the variables from (56) we have a relation among the other three. This
equation may be considered as defining a surface. Let the £-axis be pointed
upward. Then starting from any point on the surface the variables other
than E may change in any way, so far as these considerations show, so
that the point descends.
We shall now find the maximum and minimum values of E. In order
to simplify the algebra let us put
Using this substitution and eliminating P between (55) and (56) we obtain
g. -1 xJ^+J!L (58)
t: [M—u—vf m, icm|
The necessary conditions for a maximum or minimum of E are
mJM— w-vr dE
27t du
=— m^+u [M—U'—vf^O
— ^2^ --^=-tcmi+v[M-'U-vf^O
(59)
Multiplying the first equation by v, the second by u, and taking their
difference, we have ku=^v; whence, by (57),
D, = D, (60)
Then the first equation of (59) becomes
-— *+[M-(l+ic)u]»=0 (61)
♦ See 5, pp. 178-181.
CA8B vn. 101
which, after extracting the cube root, is
Af-(^')* + (l+/c)tt
Let the common value of D^ and D^ be D. Then this equation becomes
M -Z)» -f (1 -f ic)^» -D* +[m,+(^^Jm,}^ (62)
But when D| — Z), «* D, (55) becomes
M=P»+K + (^|)m,>i (63)
where P^ must be taken with the positive sign. Since (62) and (63) must
simultaneously be satisfied for a maximum or minimum, we must have,
when £ is a maximum or minimum,
P-D,-Z), (64)
That is, the energy is at a nutximum or minimum only if P ^ -h D^'^ -h D^,
i.e., if the whole system moves m a rigid body.
There can be a maximum or minimum only if (62) has real roots. The
treatment of this question is the same as that given in IV, except that we
must replace m^ of that section by m^ + f — ] m,. Hence the condition for
real roots, and therefore for a maximum and a minimum of E, is, by (31),
M^^[m, + (^|)' m,l* - A(i ^^)krn,^ (66)
Let us suppose the inequality (65) is satisfied and then consider the
surface defined by (58). It will be most convenient to give E a series of
constant values and to draw the corresponding equi-energy curves. To
simplify the treatment let
w^u-^v (66)
Eliminating u from (58) by means of this equation and solving for v we find
'Lti,.y,±jz^:;^^K^M^K (67)
K M K K{M--Wy KK
Consider first the function
^(^)._!^+a+jOg^+ (i+*)'».g (68)
K K{M—Wy KK
whence
''-±lv^w±^Jf(^ (69)
THE TIDAL PROBLEH.
The derivative of (68) is
dw
2io 2(i[+1)bi ,
" K "^KiM—XD)*
(70)
It depends only upon u> and is negative for all w > M. For w~M it ia
infinite. For u> leas than, but near to, Af it is positive; when M is large it
vanishes and becomes negative and again vanishes and beoomea positive,
as to decreases from JIf to 0. It is positive for all u)<0. From these facts
and equation (08) fig. 13 is drawn, the curves £„ . . .,Bf belonging to four
values of E such that ^,>B, >£,>£,.
From equation (69) and fig. 13 the equi-energj curves in fig. 14 are
drawn, the correeponding curves being similarly lettered. There are two
points of particular interest, A and B. As the energy decreases, the energy
curves descend to a point at A. That is, on a section through A approxi-
mately parallel to the v£-plane the point A is a minimum. With decreas-
ing E the energy curves separate at A and recede in opposite directions
nearly parallel to the to-azis. Hence a section by a plane through A approxi-
mately parallel to the to£-plane has the point A &s a maximum. Therefore
A is a minimaz point of the surface.
The minimax point A, fig. 14, corresponds to the point A, fig. 13, at
which /(io)-0. Therefore, by (69), for this point
(71)
CA8B vn.
103
At the point A, fig. 13, we have also
ic df(w)
2 dw
,(ic+l)mj ^
which becomes by virtue of (71) and (66)
^^*+[M-(ic+l)uP-0
agreeing with (61), the condition for a maximum or minimum (or minimax).
Fio. 14.
The point B of fig. 14 corresponds to the point B of fig. 13. At the
point B| fig. 13, we have
/(ti^)-O
which lead again to equation (61). Therefore the smaller real root of (61)
belongs to a minimum and the larger to a minimax.
However the system may change under loss of energy the w and v of
fig. 14 must always go from a curve of higher to one of lower energy. When
the energy is greater than that for which the point A appears in fig. 14,
the curves give us no positive knowledge regarding the series of changes
the system may undergo, except that if at any time t0>-ti+i'<Af it has
always been, and will always be, less than M, and the opposite. But if
the energy is less than that for which A appears in fig. 14, then, if at any
104 THB TIDAL PBOBLElf.
time the w belonging to the system is less than the abscissa of A, that is,
if the point is on the curve E^, it will always remain less than the abscissa
of A and will approach the point B as a limit; but if the w is greater than
the abscissa of A and less than Af it will always remain between these two
values. The abscissa of every point on the oval part of £, is in all possible
cases less than the abscissa of A, for the points on this branch of the curve
correspond to the points of the curve E^ of fig. 13 which are above the
t0-axis and to the left of the line w=*M. This part of the curve is always
to the left of A, for the curves of fig. 13 are all derivable from any one of
them simply by vertical displacements.
A simple case is that in which the two bodies are precisely alike in every
respect and have at any time similar motions. Then from the symmetry
of the problem the motions of the two bodies will always be the same.
Equations (55) and (56) become in this case
The treatment of these equations is precisely like that of (23) and (24)
of section V if we replace m^ of that section by 2fn^.
APPLICATION OP SECTION VII TO THE EARTH-MOON SYSTEM. . 105
VIII. APPLICATION OF SECTION VII TO THE EARTH-
MOON SYSTEM.
With the data given in (40) we find
(?)'^ -icmi -0.000914 m^ (73)
Consequently the moment of momentum and energy of the earth-moon
system are respectively
Af - 3.01 187 + 0.62803 + 0.00002 - 3.63992 1
f (74)
E - -0.34632 + 1.96689+0.00001 - 1.61058 J
The limiting value of M for which a maximum or minimum may occur is,
by (65), 1.5614. Since the actual M is greater than this quantity there
are a minimax and a minimum.
We shall now solve (61) and find the value of u corresponding to the
minimax. With the values of k and M given in (73) and (74) we find by
methods of approximation that the value of u satisfying (61) and corre-
sponding to the minimax is
u -3.04719 ii;-2u -6.09438 (75)
The corresponding common period of the system and distance of the moon
are respectively
Pi -0.20554 day -4.9329 hours R^ -9,200 mUes (76)
or more than 42 miles greater than when the moon's rotational momentum
and energy were neglected. The whole energy of the system for this value
^'^"^^ ^-10.666 (77)
At the present time in the earth-moon system we find from the data
of (40) and (73) and the condition Z),-P that
u -0.6280257 v -0.0000005 ti;-u+v -0.6280262 (78)
Since the present energy, given in the second equation of (74), is less than
that corresponding to il of fig. 13, which is given in (77), and since the
present value of w given in (78) is less than that corresponding to A, given
in (75), the energy will continually approach the value corresponding to
the minimum B, when the earth-moon system will move as a rigid body.
Although it is thus possible, under the hypotheses, to draw positive con-
clusions as to the conditions toward which the system is tending, it is, of
course, not possible to afibm, even aside from all factors neglected in this
section, that the system ever descended from the condition corresponding
to A. This discussion simply gives the numerical values belonging to the
condition corresponding to A, which may or may not have been verified.
Thus the real question of interest gets no conclusive answer by supposing
the moon of finite size; the introduction of this factor simply embarrasses
the fission theory a little more.
106 THE TIDAL PBOBLElf.
IX. APPLICATION TO BINARY STAR SYSTEMS.
If the earth and moon were derived by the fission of a parent mass,
the process has presumably been exemplified elsewhere. We shall conclude
that the earth has had an exceptional origin only as a last resort. It has
been many times suggested that the binary stars may have originated by
the breaking up of larger masses, and See especially has urged this view
and applied Darwin's formulas in an attempt to explain the dimensions
and eccentricities of their orbits.^
We shall apply the methods developed here to the problem. In order
to simplify it as much as possible we shall suppose first that the parent
mass divided into two similar and equal masses. We know that this rela-
tion of the masses of the two members of binary stars is, in a number of
cases, nearly fulfilled, and nothing is known to make this assumption seem
improbable. We shall assume that immediately after the fission the system
moved as a rigid mass. Then the equations for the moment of momentum
and energy are p i o^
differing from those in section V only in that m^ has been replaced by 2m, .
The condition for a maximum or minimum of ^ is [cf. eq. (28)]
P*-MP-h2mi=0 (80)
As has been shown, when M is sufliciently large this equation has two
real roots. The smallest Af for which there are real roots is that for which
the real roots are equal. The condition for equal roots of (80) is
4 i
M^:^P^=^^6fn, (81)
Consider the system moving as a rigid body with the two stars in contact .
Then their orbital moment of momentum will be — times their rotational
moment of momentum, or ^^
P*-^,(andP-i)) (82)
Consequently the whole moment of momentum of the system is, calling
this special value of the common period P^^,
M = (l+cOPo* (83)
In order that this may be at least as great as the value defined in (81)
we must have
c,5i (84)
The value of c^ depends only upon the law of variation of density of the
bodies. When they are homogeneous c^^OA. When the density varies
' Inaugural Dissertation, Berlin, 1892.
APFUCATION TO BINABT 8TAB SYSTEMS. 107
according to the Laplacian law and vanishes at the surface {fflSO^)^
we find from (21) that C| « 0.26. The more the matter is condensed toward
the centers of the bodies, the smaller in general c^ will be. It is apparent
from these figures that e^ can not be much greater than 0.33, and therefore
that M can not greatly exceed the value for which there are equal roots.
Consequently the two roots of (80) can differ but little, which means that
tidal friction is not competent to drive two equal stars originating in this
way far from each other.
Let us suppose Ci >0.33 so that E has both a maximum and a minimum.
By (82) we have 2m|>-C|Po*. Using the value of M given in (83), equation
(80) becomes
P* - (1 +c,)P.^P +c,P.^ -0 (85)
Since one of the roots of this equation is P«^ we may factor it into P^^P^^
and
P -CjPo*^' -e^P.^Pi - c,Po " (86)
whose real root gives the minimum value of E. To solve this cubic in
P*let
(87)
Then by the theory of cubic equations '
Pt -|P.* + ^—i^ -2^(Ci)P.* (88)
^jp
Let us apply these equations to the case of two stars each equal in mass
and dimensions to the sun. Taking the radius of the sun at 433,000 miles,
we find from
2TOi
that in this case
Po- 0.2324
We shall take for e^ the largest possible value, that is 0.4, which belongs
to a homogeneous sphere. We find from (88) the corresponding largest
value of P, to be
P,- 0.307
which is'the period for a separation of the centers'of the bodies of 1,042,400
miles. The original separation was 866,000 miles. That is, imder the
assumption that each of the two components of a binary system is equal
to the Sim in mass and volume, and that they remain of constant size and
1 Burnnde and Panton, Thtory of BquaHim$, p. 108.
108 THE TIDAL PROBLEM.
shapCi we have proved rigorously that after fission, tidal friction can not
have increased their initial distance more than 200,000 miles. Such an
inconsiderable increase as this in the distance between two stars can have
had no important effects on binary systems. If the system started at the
configuration corresponding to maximum energy of the curve of fig. 10,
either it must have gone toward the condition of minimum energy just
computed, or the two bodies must have fallen together.
Let us suppose the bodies to have shrunk as a consequence of loss of
heat so that their period of rotation would have become, except for tidal
friction, kD, Since the rotational moment of momentum is not changed
by shrinking, the m^ becomes tcm^, the mass changing its numerical value
because the definition of units depends upon the dimensions of the bodies.
Then equations (79) become
^ ^^ kD n P^^ K'D^ ^^^^
Eliminating D we have
1-^-^^^
The condition for a maximum or a minimum of £ is
Pt-MP+2/cmi=0 (91)
From the derivative -r-^-rrn — h- it follows that, for given values of M
and nil, the smaller k is the farther apart are the two roots of P. Let the
common initial value of P and D be P^. Then equation (91) becomes, by
(82) and (83),
P*-{1 +c,)P,iP+fx,P,^ =0 (92)
Since this equation is homogeneous in P and P^, its solution is of the form
i'*=/(C|,/c)Po* (93)
For a given amount of shrinkage of a body the constant k is determined,
because the moment of momentum is not changed by a decrease or increase
of volume. Hence, if we assume an initial Po and the #c, we may determine
the final (and greatest) P from (92), and compare the results with the data
given by observations of double stars. Or more simply, we may assume a
final P in accord with the data furnished by observations and compute the
K from
for various assumed values of Pq. Since the two stars are supposed to have
been initially in contact, the initial density may be expressed in terms of
Po, and the final density is determined by the initial density and the amount
APPUCATION TO BINABT STAR BTSTBlfS.
109
of shrinkmg, which is measured by k. Therefore the final density may
be determined, by the use of (94), in terms of the initial density and the
final period.
We shall assume that P»100 years, or approximately the period of a
Centauri and f Scorpii. There are many binaries known with much longer
periods than this, as well as many with shorter periods. Since they have
presumably all originated in a similar manner, any correct theory must
explain the long periods as well as the shorter ones. We shall assume that
the bodies have always remained homogeneous, whence C|»0.4, which
has been shown to be most favorable to the theory of a large increase of
period through tidal friction. From the formula for the period in the
two-body problem we find, using the volume times the density for the
masses, and assuming that the bodies were originally in contact, that P«
varies inversely as the square root of a^, or
p.
constant
(96)
The constant of this equation is determined by the fact that the density
of the sun is 1.41, while in the case of two such stars as the sun we have
found P^ to be equal to 0.2324. Therefore the constant is 0.2760.
From the fact that the moment of momentum of a body simply shrink-
ing must remain constant, we find that
'.'•'V*
where a^^ is the initial radius and a^ the final,
inversely as the cube of the radius we have
(96)
Since the density varies
-^ — I ^« " W^ -= final density
(97)
By formulas (94), (96), and (97), with P-100 years, Ci-0.4, the fol-
lowing table has been computed:
^0
Pq
K
^
1.4
4
10«»
3.8
10^
3.6
10"»
3.0
10"
1
10»»
0.2324 day
36.4 years
38.8 "
39.9 "
43.7 "
72.8 "
— 20,660,000
0.19
0.25
0.46
1.03
18X10*
00
46
io>»
28.6
10"»
9.7
10»»
3.8
10^
110 THE TIDAL PBOBLElf.
With an initial density of 1.4 the final density comes out imaginary, indi-
cating that incompatible conditions have been imposed. That is, it is impos-
sible for a star having the density and twice the mass of the sun to divide
and be driven by tides into a binary system having a period of 100 years.
When cTo— 4X10r^ the final density is infinite, another impossible result.
When the initial density is 10~^® we find ic«1.03, indicating an expansion
instead of a contraction. The only initial densities compatible with the
assumptions lie between 4 X 10~^^ and 10"^^ Cionsequently a double star
having two equal components with combined mass twice that of our sim and
a period of 100 years could not originate by fission except when it was in
Q
the nebulous state with a mean density not exceeding j^ of that of our
atmosphere at sea-level. On the other hand, if the close binaries revealed
by the spectroscope have originated by the fission of a single star, their
period can never become great through the effects of their mutual tides.
It follows from (92) that this relation between the initial period and final
period depends only upon the law of density and the amount of the con-
traction, and is entirely independent of the mass of the system.
The question may be raised whether the results would not be less
unfavorable to the theory if the mass of the system were unequally divided
between the two stars. The discussion of section VII shows that the problem
remains in a general way the same, 2m| being replaced simply by m^ + f ^j m,.
We must conclude from this discussion that approximately equal
binary stars with long periods can have originated by the fission process
only when the parent mass was yet in the nebulous state. In fact, it
removes the chief support of the belief that there is any such thing as fis-
sion among the stars simply because of rapid rotations. From other con-
siderations Jeans ^ has arrived at the same conclusion.
>The Afltrophysical Journal, XXII (1905), p. 101.
CASE X. Ill
X. CASE t-0 t\«=0 t,=0 a,»0 e+O S-0.*
In^thifl case equations (9) and (10) become
M
m, E 1 . m
'"V^+^ T- ^+P w
Eliminating D and P in turn between these equations, we get
IT pi
Sohringtorjt we obtain
(99)
Vp.-[«±v¥^ «-V>-D-f)^^^%^ ow
p»
The'conditions for a mazimum or a minimum of the first of (99) are
§Pi|^-Pi(l-O-Vl-^'^^+^i-0
(101)
The first equation may be satisfied by the vanishing of any of its three
factors. P^— is a phsrsical impossibility and makes E infinite. Setting
the second factor equal to zero and substituting in the second equation
we have m|— 0, which is impossible. The remaining possibility is e— 0,
and then equations (101) reduce to
u-0 Fn-MP+m^^O (102)
the second equation being precisely the same as (28).
Similarly the second equation of (99) gives as the conditions for a
maximum or a minimum
e -0 --Z)*(AfZ)-mi) + MD»-mi(AfZ)-.mO'-0 (103)
The latter of these equations gives
Dt-MD+mi-O (104)
which, together with the second of (102), shows that for either a maximum
or a minimum
c-0 P-D
* See 3, parts V and VI, and the appendix, pp. 88^-801.
112 THE TIDAL PROBLEM.
We shall now study the surface defined by the second equation of (100)
by considering the curves for various values of E. Cionsider first the curves
defined by
y.-l-5t(lf-^y (105)
From
^.^ („_».) („_^) <,„„
we find the positions of the maxima and minima of yi and we can then
easily construct its graph.
We must now consider the curves defined by equation (106) for various
values of E, both positive and negative. They all pass through the point
y, » 0, Z> « -jr^, and are tangent to the Z>-axis at this point. All the branches
of the curves are asymptotic to the lines D"0 and y^^EM^Iitt except in
the special case when £»0.
The sum vyi+y^ must then be considered. Whatever the value of
Ef all of these curves are asymptotic to D«0 and y» — oo . All of the
curves pass through the point 2)«Wy y°"ly c^d their slope is sero at this
point. Their slope vanishes at the points defined by
The solutions of this equation are Z)-» ± oo , D=»r^^, and the two roots of
M
the last factor. When E is large the roots are small numerically, one being
positive and the other negative. With decreasing E the negative root
recedes to — oo which it attains at E=0. It then becomes positive from
+ 00 and unites with the other positive root when
8mi
and for smaller values of E the slope is always negative for values of D
greater than -^^
The curve has a point of inflection for this value of E, for which D =
Those ourves which are t&ngeat to the axis are found by imposing the
condition (108) and
y.+y.-^'-O {109)
The first factors of (108) give no results of interest. Eliminating E between
the third factor and equation (109) we get
«-(«-?)■-»
agreeing with (103), the coaditioos for a maximum or a minimum. If
equation (103) has no real roots then there is but one intersection of each
curve with the £>-axis to the right of D~^. When the equation has real
roots then for certain values of E there are three intersections to the right
of D'—j-^. As E approaches — oo the curves approach the line ^— W
between y— and y=- +1.
We get the final equi-energy curves from
(110)
They are given in fig. 15. The point A is one of the solutions of (103)
and IB a mlmmax of E considered as a function of D and e. When this
point belongs to a positive value of E, as it does in the earth-moon system,
the curves for this and larger values of E are open on the right to infinity,
because the curves are asymptotic to the lines «=±*/n —
When
E decreases until EAf ' — —a the curves close bX +oo, and for smaller values
of E they are closed ovals until they vanish at a point B on the Z>-axis to the
right of A. This point corresponds to a true minimum of E considered as
a function of D and e. On the left of the «-axis the ourves are shaped
114 THE TIDAL PROBLEM.
somewhat like parabolas and vanish at ^oo for EM^^^n. They are the
analytic continuations of the curves on the right, the union being at infinity.
J|f2
When K recedes to— QoasJ? = — it reappears at +qo as iS continues
to decrease. ^
It follows from (105) and (106) that y^+y^Kl for all negative values of E,
Consequently for all points on the closed ovals we have \e\<l, and the
system can not be wrecked by a collision of the bodies. Therefore, if at
any time the configuration of the system corresponds to any point on one
of the closed ovals to the right of A, it will always tend with decreasing E
through tidal friction toward the configuration corresponding to the point B,
and the evolution will end with this configuration. Because of the sym-
metry of fig. 15 with respect to the line €»0, it follows that if at any
time eaiO it will always remain zero.
While under certain conditions the system will inevitably progress
toward a definite configuration, in the general case it is not possible, with*
out hypotheses as to the physical condition of the bodies, to determine the
character of the evolution. The question of greatest interest in the present
connection arises in the case where the conditions lead to doubtful results.
If the moon separated from the earth by fission and if its orbit were
originally circular, it would not become elliptic through tidal friction. Since
the orbit is now considerably eccentric, we must assume that it was some-
what eccentric at the time of separation. Cionsequently, let us suppose
the moon has just separated from the earth so that P^D, and, supposing
that e^O, let us find whether the moon will fall again to the earth or recede
from it. Since the orbital velocity will have been such that the moon's
motion will have fulfilled the law of areas, while the rotational velocity
will have been uniform, there will have been relative motion of the various
parts of the system, and consequently tidal friction. We are to find the
effects of this loss of energy on the distance of the moon.
Under what seem reasonable assumptions we have seen in section V,
equations (33), that when the orbits are circular the rate of change of the
month is given by
dP c P-D
dt ""r* PW
We shall now assume that when the orbit is elliptic the rate of loss of
energy at any instant depends upon the square of the product of the tide-
raising force and the angular velocity of the tide over the earth. This
assumption is equivalent to taking the circular case as applying instan-
taneously to the elliptic case, and omits the lag in tidal conditions due to
inertia. With this assumption the equation above becomes
dt r* 01 ^ ^
where c' is a positive constant.
* This result agrees with that found by Darwin in 2, p. 497, eq. (79) after change of
variables, notation, and proper specialisation of his problem.
CASE X. 115
By hjrpothesis the mean value of is equal to at, and we have therefore
from the two-body problem
5
r'^all— e COB cut +^ {1 — 0032(14)+ . . . .]
(112)
Consequently equation (111) becomes
^--c^e[2coBa;e+ye+^eco82arf+. . . .] (113)
where e" is a positive constant. The first and third terms produce no
secular results. The second shows that P secularly decreases. That is,
under the hypaiheeie thai the lose of energy ie proportional to the equare of the
product of the vdodty of the tidal vxivt and Ae magnUude of the tide^'aieing
force, U foUowB that if the moon had separated from the earth and originally
tiad been moving around it in a dighUy eccentric orbit in a period equal to
that of the rotation of the earth, then the friction of the tides generated by the
moon in the earth would have brought the moon back to the earths Since these
hjrpotheses certainly approximate the truth, we are led to the very probable
conclusion that the moon can not have separated from the earth in an elliptic
orbit and have been driven out to its present position by tidal friction.
Precisely similar reasoning applies to the hypothesis of the fission of a
star into a pair of equal stars, and is an additional strong argument against
the soundness of this theory regarding the origin of binary stars, which
generally have large eccentricities.
^ Under the amiinpticn that the planet k Tieoous and with diff«rait araroxiiiiatiQDS,
Darwin's equatiooa led to the fame refult. See 3, p. 864, eq. (202), also 3, p. 878 and p. 891
8
116 THE TIDAL PBOBLEM.
XI. APPLICATION OP CASE X TO THE EARTH-
MOON SYSTEM.
Observations show that in the case of the moon's orbit e^ 0.0549.
Then we find from (98) that in this case
M- 3.63082 £« 1.61057
Cionsequently for the moon at present the curves of fig. 15 are yet open at
infinity, and so far as this discussion goes, the eccentricity may increase
to unity, and the system be wrecked by a collision of the earth and moon.
The most interesting question relates to the least possible distance of
the earth and moon from each other. Suppose at the time of the assumed
separation the eccentricity of the orbit was very small, as apparently it
must have been if the bodies separated by fission. Then the configuration
at the time of separation corresponds to the point A of fig. 15. The abscissa
of this point is the smaller real root of equation (104), which for the present
value of M we find to be
I>«" 0.206008 days -4.944 hours
From the relation between the period of revolution and the distance we
find that the distance corresponding to this period is Rq^9,214.0 miles.
Neglecting the eccentricity of the moon's orbit we found for the initial
distance 9,194.4 miles.
But, as was explained in the preceding section, the initial eccentricity
could not have been sero for the present eccentricity is different from lero.
The larger it was the shorter the initial period and the smaller the initial
distance. It will be a liberal assumption to suppose it was «o»0.1, for then
the initial perigee and apogee distances differed by 1,800 miles. With this
value and putting D—P we find as the smaller root of the first equation
of (98) Do — 0.205797d « 4.939 hours. This corresponds to an initial mean
distance of 9,207.7 miles, not differing materially from that found when the
initial eccentricity was neglected.
CASS xn.
117
XII. CASE t4:0 t\4:0 a,»0 6-0 5-0.
If t is to be taken different from lero and subject to change it is neces-
sary to suppose that t\ may vary also, for all the interactions are mutual.
In this case equations (9^) become
Af'-P«BintsinQ +
nii sin ij sin Q^
M'' — F* sm t cos &2 H — * 1^
(114)
We may choose the xy-plane so that it shall coincide with the invariable
plane of the system. Then Af' — Af''— 0, and equations (114) give
Qj- Q+n;r (n— Oorl)
With this relation equations (114) become
[pirint±???i^]Bina-0 [P»8int±^jii^]co8a-0
whence
P*sm»±— ^-g — *—
(115)
In this discussion we shall take D as essentially positive and let % and <i
vary from to ;r. Then we must take the lower of the two signs in (116),
or Si^^Si+n. This equation and equations (9) and (10), under the
hypotheses of this section, give us
M
n
ni • • tn. sm i.
F* sm * i-yT — '
Di • • ^1 cos *i
•P* cost+ yv — -
1 ^mj
pi • Z)»
(116)
Eliminating D and t| from the last of these equations by means of the
first two, we have
^ ^+Pi-2McostP» + M«
n P«
The conditions for a maximum or minimum of E considered as a func-
tion of f and P are
^^-2AfP»sinf-0
in^bB^ 2mt2 1 2 Jfcosi
TT dP"3P«'*"3P» 3 Pl ""
118 THE TIDAL PROBLEM.
The first equation can be aatifified only by t »0 (the case t — « makes only
a change in direction of the axes in the final solution). Then the second
one gives, since P can not equal infinity, as the conditions for a mi^Tirrium
or a minimum
i=0 Pt-AfP+nii-O (117)
Then the first equation of (116) shows that either i|— or I>« oo . In the
latter case t| is indeterminate.
To investigate whether the roots of (117) correspond to a maximum
or minimum we form (at t « 0)
-^-2MP» irdidp"° Irak's piVs^^^; ^^^^>
The right member of the first equation is positive for both roots of (117).
We must consider the function
For the smaller root of (117) the right member of this equation is negative,
and therefore B is neither a maximum nor a minimum for this set of values.
For the larger root of (117) it is positive and the corresponding value of E
is a true minimum. Whenever the sjrstem arrives so near the condition
corresponding to this point that the equi-energy curves are dosed they
remain closed for all smaller values of E down to the minimum, and under
the influence of tidal friction the system will inevitably approach this
condition of minimum E, and having attained it will remain there.
It follows from the second of (116) that for either value of P satisfjring
(117) we have D^^P.
APPUOATION OF CASS XU TO THE SABTH-MOON SYSTEM. 119
XIII. APPLICATION OP CASE XII TO THE EARTH-
MOON SYSTEM.
There is a difficulty in attempting to determine the position of the
invariable plane of the earth-moon system, for the plane of the moon's
orbit is continually changed by the perturbative action of the sun. In
the course of about 9} years the line of nodes of the moon's orbit makes a
complete revolution, and the inclination of the plane of the earth's equator
varies all the way from 23.6^+7^ to 23.6^— 7^ We shall suppose here
that the sun is not disturbing the orbit of the moon, and that the inclina-
tion of its orbit to the plane of the ecliptic is sero. Let us take the plane
of the ecliptic as the original xy-plane. Then the angle between the plane
of the ecliptic and the invariable plane is given by
cos to
j^ P» + J» COS (23.6»)
(120)
VM»+M'«+Af- ^p,^2^»^^^(23.6-)+5l
Since SJj— Si +n we have
ti-23.6^-t (121)
In the case of the earth-moon system we find
»-4*» t,-19.6** (122)
Then M of (116) becomes Jf- 8.69664. With this value of M the smallest
root of (117) is P—0.20862. This period corresponds to a distance of
£—9,364 mfles. However, the initial inclinations could not have been
* precisely sero, and oonsequentiy the initial distance must have been some-
what less tlum this amount. But the chief point of interest is that the
factors neglected in the discussion of section V so far all make the initial
distance greater than that found in that place.
120 THE TIDAL PROBLEM.
XIV. CASE t-0 t'l-O e = o,-0 S+0.
The sun affects the evolution of the earth-moon system in two ways.
First, its direct perturbing influence on the moon makes the month longer
than it would be if the moon were revolving at its present mean distance
in an undisturbed orbit. In the second place the tides which the sun
raises in the earth retard its rotation and reduce the moment of momentum
of the earth-moon system. We shall consider these two influences separately.
The relation among the sum of the masses of the earth and moon, the
mean distance from the earth to the moon, and the moon's period is found
in the theory of the moon's motion to be
^a«[l+|(|7)' ]=-*Mtn,+m,) (123)
where P' is the length of the year. With this relation instead of having
equations (9) and (10) become, when 6»i»ii— a,— 0,
The disturbing forces which have made these changes in the equations
are mostly radial, and so far as the radial components are concerned can not
change the moment of momentum. The tangential components are periodic
with equal and symmetrical positive and negative values in a period.
Consequently the M will not have changed under these influences.
At the time of the supposed separation of the earth and moon P^D,
and the first of (124) gives for the determination of P
P*[l-|(py. . . .]-MP+mj-0 (125)
We find from the first of (124) that the value of ilf is in this case M —
3.63428. The smaller root of (125) for this value of M is Po"" 0-205760,
and this period corresponds to an initial distance of R^^ 9,206.2 miles, a
little greater than that foimd when the sun's action was neglected.
Consider the direct tidal friction of the sun upon the earth-moon system.
The sun's tides lengthen D without producing a corresponding change in
the motion of the moon. Consequently in this case M is not constant.
Let us assume, as before, that the amount of tidal friction is directly
proportional to the product of the square of the tide-raising force and the
square of the velocity of the tidal wave along the earth. It also depends
upon the physical condition of the earth. Then we have, including the
tides produced by both the moon and the sun,
dE rm,» (P-Z))» g»(P^-P)n ,.2ftx
CASS XIV. 121
where the first term comes from the tides raised by the moon and the second
from the tides raised by the sun. Since E can only decrease, C must be
positive. The factor of proportionality, C, is the same for both since it
depends only upon the physical condition of the earth.
Substituting (126) in the second of (32) we have the rate of change
of the day defined by
dD C m^ D
(ft " 2mi;r r* P
<''-"){'-(^)*(a'(?^)'©'}<"')
The length of the month is changed by the moon's tides alone, and
bears a definite relation to the rate of change of the day due to the moon's
tides. From (32) this relation is found to be
D" dt "Pt (ft
When the sun's action on the rotation of the earth is included we have
therefore
From the formulas for the month and year we have
fry /_mi+n^_y fPy
Since nii is large compared to m^, and S is large compared to m^, this rela-
tion becomes with sufGicient approximation
Then equation (128) becomes
If we integrate this equation and determine the constant of integration
by the present values of P and D, and if we then put P — D and solve, we
shall have as one of the roots the value of P at the time of the supposed
separation of the earth and moon. But P and D enter this equation in
such a complicated manner that it is not possible to express its solution
in finite terms. The critical values of the variables are
P.0 i,.o P-.>-o :.0(?rg)-@'.„
Only the third of these critical values will arise in the applications which
will be made here.
122 THE TIDAL PBOBLEM.
In the present condition of the earth-moon system we find that
It will appear in the computations which follow that as P and D
decrease in such a way as continually to satisfy (129), this function de-
creases rapidly until P — 0.25 and D » 0.25 approximately. For 1 ^ P > 0.25
it is convenient for the purpose of solving (129) to let
Q-^ + Pi-Af (130)
With this substitution equation (129) becomes
di 3 W P^KP-DJ VP7 di
If we should eUminate D from (131) by means of (130) the result would
have the form
^^F(P,Q) (132)
If we let Pq represent the present value of P, and M the total present
moment of momentum of the earth-moon system, then at P»Pt we have
Q»0, and an approximate solution of (132) is
F (Pfl) dP (133)
Successive approximations may be found by the series of operations
This series of approximations approaches the true value of the integral
provided the upper limit does not pass beyond a point for which F(P, Q)
has a singularity.^ In the application of these formulas we are explicitly
limiting ourselves to a region in which F(P,Q) is everywhere regular.
The integrations can be very easily carried out by mechanical quadratures
to the desired degree of accuracy.
Since we wish to trace the system back and find for what values of P
and D the two variables were equal, we must take P < Po in the integration.
» Picard, Traiti d'Avudj^, vol. 2, pp. 301-304.
CA8B XIY. 123
When the integration has been carried so far back that P— D is small
enough to make the second term in the right member of (129) the dominant
one, it is convenient to change from P to D as the independent variable
of integration. We may now write (129) in the form
'«'•"■' Z..[(P_D,. + (a)V_D,.(^)*] "'*)
For P— D=0 this equation does not have a singular point when D is the
independent variable, for at this point/ (P, D) vanishes.
Suppose corresponding values of P and D have been found by (132)
and (130) until P^ and D^ are obtained, and that their difference is small.
Then P may be expressed in terms of D by an expression of the form
P~P.-|il<(D-Do)< (136)
provided the modulus of D-^D^ is sufficiently small. Substituting (135)
in (134), expanding the right member, and equating coefficients of corre*
sponding powers of D—D^, we find
where
3m,(P,-D,)'P,l
(136)
A.
a/(P,, PJ 2m.(P.-D^(4P.-D,)
^»D.»[(P.-D.)» + g)'(P'-D,)»(J«)]
dP,
D^[(P.-D,y+[^\p'-D.y(^)*\
d/(Po,D^ 6m.(P,-I),)P.i
^^* D* [(P.-D.)»+(^)* iP'-D,y (Jj)]
6m. (P.- D,)' P.I [p,-D.+(^y (P'-D;) (^)*\
, _ .TO,
i5.'[(P.-D.)' + gy(P'-D,)'(fj)Y
124
THE TIDAL PBOBLBM.
Therefore
A,
,(P.-D,)^' + 3P,(P,-I),)'*'
[(P.-i>.)« + 0'(P'-D.)'(Jj)'] '
[P.-D.+3(^)*(P'-D.)»^*]
A^
[(P.-i>.V + (^y(P'-D.)«(^)T
Since these equations are applied only when Pq—Dq is small and D*
small compared to P^ we have approximately
A,
m,Do»PoV mj»Do*PoV
__Po__. (5Po-8Do) . ,
D.CP.-Do)'^^ 3P«(Po-Do)^
(137)
By successive application of (135) and (137) the corresponding values
of P and D can be followed until P—D passes through sero. The value
of P, for which P^^D, can then be determined by interpolation. The corre-
sponding values of P and D in the following table have been computed
from equations (130), (131), (135) and (137). The third column gives
p^ the ratio of the rate of the change in the rotation of the earth due to the
sun's tides to that due to the moon's tides.
p
D
p
27^166
0.99726
0.224
25.32166
0.86685
0.165
23.32166
0.76672
0.119
21.32166
0.68644
0.083
19.32166
0.61939
0.056
17.32166
0.56278
o.as6
16.32166
0.51291
0.022
13.32166
0.46847
0.013
11.32166
0.42786
0.007
9.32166
0.39012
0.003
7.32166
0.35410
0.001
5.32166
0.31875
0.000
3.32166
0.28229
0.000
1.32166
0.23966
0.000
0.20037
0.20037
00
The initial common period of rotation and revolution is found by this
computation to be 0.20037 day, corresponding to a distance of 9,045 miles.
THS SECULAR ACCSLSBATION OF THE MOON'S MEAN MOTION. 125
This is 149 miles less than that found when the action of the sun's tides
upon the rotation of the earth was neglected. That is, when the efifects
of the sun's tides upon the rotation of the earth are included, it is found
that the possible initial distance of the moon is not materially diminished,
and that the theory of the fission of the earth and moon is still subject to
the serious embarrassment of wide separation immediately after the sup-
posed division into distinct masses.
XV. THE SECULAR ACCELERATION OP THE MOON'S
MEAN MOTION.
As is well known, there is a secular acceleration of the moon's mean
motion of about M' per century which has not been explained by the ordi-
nary perturbation theory. It was long ago suggested by Delaunay that
it may be due to tidal friction, and Darwin has made an investigation of
the subject in 2, section 14.
If we accept the tidal explanation, the apparent acceleration of V is
due to an actual retardation of the moon, the only result possible accord-
ing to (32), and a greater retardation of the rotation of the earth. Since
the rotation of the earth is used to measure time, the period of revolution
of the moon on this basis apparently is accelerated. In making the discus-
sion we shall neglect the effects of a,, e, 1 1, i, and S.
Let — AV| be the gain in longitude of the moon in a century, and — Av,
the corresponding gain in the angular distance of rotation of a meridian
of the earth. Then we have
Ar,— AV|-4^
dP 2ndB 2n AVj P* AVj
dt 0* da 0* (lOOPO* 2;r (lOOP')*
dP _27c^do) 2n Ay, D* Ay,
(ft " CO* (ft " a>» (100P0*"2;r (lOOP')'
From these equations and (32) we find
dD P^D^ V dP 3mjP«
(138)
(ft 27r(Pi-3mi) (lOOP')' * 2;r(Pi-3m,) (lOOP')*
(139)
Representing the value of P at <— (; by P«, integrating the second equa-
tion, and determining the constant of integration, we have
P. -p.. +«. (A-^j »g ^^ (,-,.) (140)
The present rate of tidal evolution of the earth-moon system depends
upon the forces acting and upon the physical condition of these bodies.
If we regard the 4^' per century of apparent gain in longitude of the moon
as due to tidal evolution, we have a measure of the ^ of equations (32).
126 THE TIDAL PROBLEM.
We may use equation (140) to compute the time {t—Q corresponding to
any value of P provided the physical condition of the system, and partic-
ularly that of the earth, has not changed sensibly in the mean time. The
data furnished by geology are showing more and more that the earth has
been sensibly in its present state, except for approximately periodic oscil-
lations in its climate, for many millions of years. For the purposes of
computation we shall assume that it has been indefinitely so. While this
assumption is not strictly true, the actual observational data show that it
IS almost certainly much less in error than the assumption, stimulated by
the Laplacian theory of the origin of the earth, that our planet was fluid
in the not very remote past. Remembering the fact that we are assuming
simply that the apparent secular acceleration of the moon's mean motion
is due to tidal friction and that it is a measure of the rate of tidal evolution,
and that we are assuming further that the physical condition of the earth
and moon has not changed in the time covered by our calculations, we find
from (23) that when D=*20 hours the value of P was 24.096 days, and from
(140) that P had this value 220,700,000,000 years ago. If the action of
the sun had been included the interval would have been decreased by about
20 per cent. It is impossible to believe that the neglected factors, such as
the eccentricity and inclination of the moon's orbit, could reduce the time
enough to change the order of these results. This computation, which has
the merit of being based quantitatively on actual observations, points very
strongly to the conclusion that tidal evolution is so slow a process that it
can not have played an important rdle in the earth-moon system, even when
we consider an interval of a billion years.
There is, however, another possibility that may be considered. It is at
least conceivable that there may be unknown forces acting upon the earth-
moon system in such a way that they largely mask the relative secular
tidal acceleration of the moon. Any thing increasing the moon's distance
and period without otherwise disturbing its motion, or any thing acceler-
ating the rotation of the earth, would tend to ofifset the seciUar acceleration
produced by tidal friction. The possible secular contraction of the earth
is a factor working in the right direction. But from the numbers obtained
above it follows that, if we are to escape from the conclusion that tidal fric-
tion is now a negligible factor, we must assume that the actual relative tidal
acceleration of the moon is several hundreds of times V per century. Sup-
posing the reduction to V is due to the acceleration of the rotation of the
earth because of shrinking, it follows that at every epoch in the past the
day and the month were more nearly equal than they would have been
except for this factor. Finally, at the limit at which they were equal,
their common period was many times that computed above, and their
great initial distance fatal to the fission theory. If there are unknown
forces retarding the moon's revolution, the conclusions are the same.
SUMMARY. 127
XVI. SUMMARY.
The object of this investigation has been to examine the theory of tidal
evolution in order to find out, if possible, not what might take place under
certain assumed conditions, but how important this process has been in the
actual development of our sjrstem. The aim has been to avoid, as far as
possible, assumptions regarding the uncertain factors depending upon the
phjrsical conditions of the bodies involved. In order to compare the theory
with the actual facts the various methods of testing it have been carried to
quantitative results.
A large part of the discussion has been made to depend upon the com-
ponents of the moment of momentum and upon the energy of the system.
In section II the moment of momentum equations and the energy equation
are devdoped, and they are perfectly rigorous so long as the two bodies
are subject to no forces except their mutual attraction. Under this con-
dition the three components of moment of momentum are rigorously
constant, and the three equations which express these conditions are fixed
relations among the various quantities which define the djmamical state
of the system. The energy equation is a relation among the same quan-
tities, but unlike the components of the moment of momentum the energy
diminishes by friction. These relations are too few to determine the changes
which will actually take place, but they give important information about
them. They are particularly valuable, for they are true whatever the
phjrsical conditions of the bodies involved.
One of the conclusions reached by Darwin was that it is probable that
the earth and moon have developed from an original mass by fission. One
critical test of this hypothesis is the determination of the smallest distance
at which the bodies could have revolved around each other consistently
with the present moment of momentum and energy. This test has been
worked out quantitatively, first with the problem simplified so that the
conclusions are absolutely certain under the hypotheses; then the effects
of various modifying conditions, which seem more or lees probable, have
been examined, one after another, and their influence upon the final result
determined. The results reached are so near the border line separating what
is favorable to the theory from that which is unfavorable, that it is impor-
tant in applying this test to determine accurately the constants upon which
the system depends. One of these is the rotational moment of momentum
of the earth, which depends only upon the law of density of the earth as an
uncertain factor. In section III the constants of the density according to
the Laplacian law are worked out. It is found that according to this law
the density varies from 2.76 at the surface to 10.84 at the center. While this
is probably not an exact expression for the earth's density, the inherent
probabilities as well as the actually observed precessional phenomena lead
us to conclude that it is not sensibly in error for the purposes of this dis-
cussion. By the same law the surface and central densities of the moon are
respectively 1.65 and 6.51.
In section IV the moment of inertia for the Laplacian law of density
is found, and it comes out 0.336 times the mass instead of 0.4 times the
mass, as in the case of a homogeneous body.
128 THB TIDAL PBOBLBM.
Section V is devoted to a consideration of the problem in which the
two bodies revolve in circular orbits undisturbed by exterior forces, and
in which the rotational moment of momentum and energy of one of the
bodies are so small that they may be neglected, while the axis of rotation of
the other is perpendicular to the plane of the orbit. Although the condi-
tions assumed in this section are not exactly fulfilled in any physical prob-
lem, still they are near enough those prevailing in the earth-moon system to
throw much light on what may possibly have taken place. But the chief
value of this investigation is that the variables are so few that the results
are precise, except as to the time rate at which the possible changes will
take place. The thing of greatest interest is that the rates of change of
revolution and rotation are proportional to the rate of the loss of energy
through friction, and are not directly dependent upon the phases and
lags of the tides or the surface peculiarities of the tidally distorted body.
These results show that, so far as the hypotheses upon which they are
founded apply to the earth-moon system, if we could from the direct tidal
observations calculate the rate of loss of energy in tides raised by the moon
upon the earth, then we could compute the rate of tidal evolution at the
present time. While this problem undoubtedly presents serious difiiculties,
they are not more formidable than those of assigning to the earth a physical
constitution which shall agree reasonably with the truth.
Assuming that friction is proportional to the height of the tide and its
vdocity relative to the surface of the tidally distorted body, and that the
loss of energy is proportional to the square of the friction, equations are
developed, (33), giving the rates of change of the periods of revolution and
rotation. They involve only one unknown constant depending upon the
phjrsical constitution of the distorted body.
In section VI the equations of section V are applied to the earth-
moon system. The influence of the sun is neglected, later computation
showing that its effects upon the rotation of the earth are now about one-
fifth as great as the moon's. The rotational moment of momentum and
energy of the moon are small because of the small mass of the moon, its
small dimensions, and its slow rotation. Using the numerical data, it is
found that the moon's rotational moment of momentum is less thui one
thirty-thousandth that of the earth. Since the eccentricity of the moon's
orbit is small and the cosine of obliquity of the ecliptic not much less than
unity, it is seen that the conditions of this investigation really approxi-
mate rather closely to the actual earth-moon S3rstem. It is found that the
month has always been increasing and that it can not pass beyond 47.7
of our present days, at which period the month and day will be equal and
the system move as a rigid body. There is no way of telling by this investi-
gation how long a time will be required for the system to reach that state.
But it is a more interesting fact that the month can never have been less
than 4.93 of our present hours, this being the period of revolution when
the distance from the center of the earth to the center of the moon was
0,104 miles. Consequently we must suppose that when the moon broke
off from the earth it was at this distance from it, or 5,236 miles from its
present surface. Or, including the radius of the moon and supposing that
BUMMABT. 129
both the earth and moon were of the same density and shape as at present,
the distance from the surface of one body to the surface of the other was
immediately after fission 4,155 miles. Since this result is altogether incom-
patible with the obvious implications of the fission theory, we must either
abandon the theory or show that this number would be very largely reduced
by including the effects of the neglected factors. Consequently we examine
the effects of various neglected conditions and influences.
If the earth were rotating in 4.93 of our present hours it must obviously
have been very oblate instead of spherical as was assumed in the computa-
tion. In the absence of certain Imowledge we may assume that its equa-
torial radius reached out to the surface of the moon when the distance of
its center was 9,194 miles, that the oblateness was such that the volume
was the same as at present, and that the law of density was such that its
rotational moment of momentum was the same as it would have been if
it were spherical and the Laplacian law of density prevailed. We find
that under these hjrpotheses the polar radius would have been only 942
miles. A scale drawing shows that this oblateness is out of the question,
and a little consideration shows that the equatorial zone must have been
so rare as to make it impossible to account for the mass of the moon.
If we waive the condition that for a given period of rotation the law
of density was such as to keep the rotational moment of momentum the
same as when the body was supposed to be a sphere, we shall have to sup-
pose the moment of momentum was greater than this in order to get sufGi-
cient matter in the periphery to account for the origin of the moon. But
this supposition leads to the conclusion that the nearest possible distance
of the moon was greater than the 9,194 miles found before.
Another hypothesis is that the earth was initially larger than at present,
and has shrunk to its present dimensions as it cooled. It was found in this
case also that the initial distance of the moon must have been greater than
the 9,194 miles found on the original hypothesis.
An examination was made of the hjrpothesis that the earth originally
had a radius of 9,194 miles and a density such as to keep its moment of
momentum the same as if it were of its present size. It was supposed the
density varied according to the Laplacian law and the constants of the law
were worked out by the conditions of the hypotheses. It turned out that
the density of the surface must have been negative, a result having no
phjrsical interpretation and proving the falsity of at leaist one of the hypoth-
eses upon which the computations were made.
In sections VII and VIII the problem was treated without neglecting
the rotational moment of momentum and energy of the moon, but keeping
the earth and moon spherical. It turned out that the initial distance of the
earth and moon could not have been less than 9,200 miles. That is, when
this factor is included the result becomes less favorable to the fission theory
than when it was omitted.
Then in sections X and XI the hypothesis was made that the moon's
rotational moment of momentum and energy may be neglected, but the
eccentricity of the moon's orbit was given the value assigned by observa-
tions. It was found under these hypotheses that the initial distance of the
130 THE TIDAL PROBLEM.
moon could not have been lees than 9,214 miles, a result more unfavorable
to the fission theory than that obtained when the eccentricity of the moon's
orbit was neglected.
In sections XII and XIII it was assumed that the rotational moment
of momentum and energy of the moon and eccentricity of the moon's
orbit may be neglected, but the inclination of the plane of the earth's
equator to the plane of the moon's orbit was taken into account. Under
these h3rpotheses it was found that the initial distance of the moon could
not have been less than 9,364 miles, a result more unfavorable to the fisrion
theory than any of those heretofore derived.
All of the factors initially neglected and later taken up one by one
have made the initial distance greater than the originally computed 9,194
miles. Obviously all of them combined would operate in the same direction.
Since they only increase a difficulty which was in the first place serious, it
is not necessary to go to numerical results for all of them combined.
The factors which remain to be considered in attempting to test the
fission theory by computing the initial distance of the moon are the sun's
perturbations of the moon's orbit and its effect upon the rotation of the
earth. The first part of section XIII is devoted to a discussion of the direct
action of the sun upon the moon's orbit, and it is shown there that includ-
ing this influence alone the initial distance of the moon could not have been
less than 9,206 miles, which is somewhat greater than that found when the
sun's action was neglected.
The second part of section XIII treats the relative retardative effects
of the Sim upon the rotation of the earth. The magnitude of the tide-
raising force of the sun compared to that of the moon can easily be com-
puted. Other things being equal there are good grounds for assuming that
the rate of tidal evolution is proportional to the square of the tide-raising
force. The friction depends also upon the speeds of the tidal waves with
respect to the earth's surface. At the present time the speeds of the moon's
and the sim's tides are about equal, but if we trace the system back until
the month and day were approximately equal this relation is no longer
approximately verified. We must resort, therefore, to some specific assump-
tion as to the way in which tidal friction depends upon the speeds of the
tides over the surface of the earth. The assumption was made that friction
is proportional to the first power of the velocity, and therefore that the
loss of energy is proportional to the second power of the velocity. It is
practically certain that this assumption will give results which are sensibly
true. Using this hypothesis and supposing that the rotational moment
of momentum and energy of the moon, the eccentricity of the moon's
orbit, the inclination of the plane of the earth's equator to the plane of
the moon's orbit, and the direct action of the sun on the moon's orbit may
all be neglected, it was found that the initial distance of the moon was
reduced from 9,194 miles to 9,045 miles. Thus it is seen that the one
factor which makes the moon's initial distance less than that found in the
first computation is not only of no particular consequence, but also that
it is less than some of the factors which increase it. Using all those factors
whose^effects have been computed when they have been supposed to act
•epsrateljTr u>d suppCNBUig that they would be eBtentially tbe same when
•eting jointlr, we find ^t tbe ■mallest poaiible distance of the moon
eomimtible with present conditions is 9,241 mites. Fig. 16 sbows the earth
and moon and their initial distance to scale, and obviously one would need
azttemaly strong confirmatory evidence to convince him that the moon had
just broken off from the earth. The distance from the surface of the earth
to the surface of the moon is 4,201 miles, or 243 miles greater than the
radius of the earth.
As a conoesrion to the theory, we may assume that the earth and moon
have separated by fission so that their periods of rotation and revolution
are precisely equal, and then inquire whether the present system could
develop from it. If the original orbit were exactly circular the orbit would
always remain circular. Since the moon's orl^t now has considerable
eccentricity it follows that we must assume that the orbit immediatdy
after separation waa somewhat eccentric. But since the rotations would
be sensibly uniform while the revolution would be such as to fulfil the law
of areas, there would be relative motion of the various parts and therefore
tidal evcdution. Tbe question whether this friction would drive the moon
farther from the earth or bring it back and precipitate it again upon the
earth is treated in section X, and it is found there, under the assumption
that the loss of energy is proportional to the square of the tide-raising
force and the square of the velocity of the tide along the surface of the
earth, that the tide* vmdd bring tlie moon again to lAs earth. Thus, unless
some of the neglected factors can offset this result, the direct implications
of the theory destroy it, and it may be noted here that these remarks apply
with equal force to the hypothesis that the binary stars have originated
by fission and that their present distances from each other and the eooan-
tridties of their orbits are a residt of tidal friction.
If we neglect the rotational moment of momentum and energy of the
moon, the eccentricity of tbe moon's orbit, and the inclination of the plane of
the earth's equator to the plane of the moon's orbit, then it is certain, as was
shown in sections V and VI, that tidal friction will at the present time
lengthen both the day and the month, but at such relative rates that the
number of days in a month will decrease. Consequently if time be measured
by the rotation of the earth the moon will continually get ahead of its place.
132 THB TIDAL PROBLEM.
as predicted by the gravitational theory neglecting tidal evolution. This
relative gain in longitude, if established by observations^ will give us the
measure of the rate of tidal evolution at the present time, and we ean safely
apply it, by properly varying the factors depending upon the moon's dis-
tance, over any interval during which the physical condition of the earth
has been essentially as it is at present. Since the geological data go to show
that the physical state of the earth has been about as it is now for many
tens of millions of years, and do not give certain evidences of any radically
different general physical conditions, we are perhaps justified in boldly
applying results based on the present rate of gain of the moon for a very
long interval of time.
It is well known that a comparison of ancient and modem eclipses shows
that the moon has an acceleration in longitude of about V per century
which is not explained by perturbations. Let us assume that this is due
to tidal friction and is the measure of it at the present time. At this rate
it will take over 30,000,000 years for the moon to gain one revolution.
Consequently we see without any computation that it must have been an
extremely long time in the past when its period was a small fraction of its
present period.
The problem was treated in section XV, and it was found there that, if
the physical condition of the earth has been essentially constant, the length
of the day was 20 of our present hours, and of the month, 24 of our present
days not less than 220,000,000,000 years ago. It is extremely improbable
that the neglected factors, such as the eccentricity of the moon's orbit,
could change these figures enough to be of any consequence. This remark-
able result has the great merit of resting upon but few assumptions and
in depending for its quantitative character upon the actual observations.
If it is accepted as being correct as to its general order, it shows that tidal
evolution has not affected the rotation of the earth much in the period
during which the earth has heretofore been supposed to have existed even
by those who have been most extravagant in their demands for time.
And if one does not accept these results as to their general quantitative
order, he faces the embarrassing problem of bringing his ideas into harmony
with the observations.
If tidal friction has been an important factor in the evolution of the
earth-moon system, then presumably it has also been an important factor
somewhere else in the universe. Certainly one would expect to find the
theory encountering no difficulties in the case of any other planet. But
the place where it has been applied most is in the orbits of the binary stars,
which have been supposed to have become binaries through fission and
to have become widely separated as a consequence of tidal evolution. The
first difficulty is, as has been pointed out, that there must have been an
initial eccentricity of the relative orbit, and this eccentricity would cause
the bodies to reunite as a consequence of tidal friction. But there is another
important difficulty, as was explained in section IX. If we suppose the
binary to be composed of two equal suns moving just after separation
as a rigid body, and if we waive the effects of the eccentricity for the sake
of the argument, we find that there is a maximum distance to which the
sumiABT. 133
bodies ean be driyen from eaeh other as a consequence of tidal friction,
this maadmum distance corresponding to a minimum of energy, and that
this maadmum distance is not many times greater than the distance im-
mediatdy after sq[>aration. In particular, when the equations were applied
to two stars each equal in mass and dimensions to the sun at an initial
distance between centers of 866,000 mfles, it was foimd, on the basis of
condusiye reasoning, that the greatest distance possible as a result of tidal
evolution would be only 1,042,400 miles. These results, which were ob-
tained under the hypothesis that the stars suffered no shrinkage with loss
of heat, were not radically modified when they were supposed to shrink to
any extent whatever. The conclusion is that the widely separated binaries
which our tdeseopes reveal to us can not have originated by fission, at least
from masses condensed beyond the nebulous stage.^
In a word, the quantitative results obtained in this paper are on the whole
stronj^y adverse to the theory that the earth and moon have developed
by fission from an original mass, and that tidal friction has been an impor-
tant factor in their evolution. Indeed, they are so uniformly contradictory
to its implications as to bring it into serious question, if not to compel us
to cease to consider it as even a possibility.
^Tbm of coune rafen only to ipontaDeoiis finion without the aecenion of moment of
momentum from aome outvie body.
I
II
coirrjuBmoNS to cosmogony and the fundamental psoblems of geology
NOTES ON THE POSSIBILITY OF
FISSION OF A CONTRACTING ROTATING FLUID MASS
BY
FOREST RAY MOULTON
AsBOwUe Professor of Astronomy^ UniversUy of Chicago
135
NOTES ON THE POSSIBILITY OF FISSION OF A
CONTRACTING ROTATING FLUID MASS.
I. INTRODUCTION.
In the speculations on cosmogony there are two fairly definite hypoth-
as to the manner in which a single body may give rise to two or more
distinct masses without the intervention of external agencies. The first, as
outlined by Laplace, is that possibly a rotating fluid may abandon an equa-
torial ring, which will subsequently be brought by its self-gravitation into an
approximately spherical mass. The second, the fission theory, had its rise
in Darwin's researches on tidal evolution, and in his speculations on the
origin of the moon. It has found extensive application in attempts at
explaining the great abundance of binary stars.
The hypothesis of Laplace has the support of no observational evidence,
unless we regard the rings of Saturn as such, and rests upon no well-elab-
orated theory. On the contrary, there are well-known considerations of the
moment of momentum of our system which compel us to reject it as being
an unsatisfactory hypothesis for the explanation of the development of the
planets. But the fission theory of Darwin, even if the origin of the moon is
left aside as being doubtful, has strong claims for attention because of its
immediate application to explaining the origin of spectroscopic and visual
binaries and certain classes of variable stars. Besides, it is in a general way
confirmed by the investigations of Maclaurin, Jacobi, Kelvin, Poincar£, and
Darwin on the figures of equilibrium of rotating homogeneous fluids, and on
their stabilities. In particular, considering a series of homogeneous fluid
masses of the same density but of different rates of rotation it is shown that
there is a continuous series of figures of stable equilibrium beginning with
the sphere for sero rate of rotation; then, with increasing rotation, passing
along a line of oblate spheroids until* a certain rate of rotation is reached;
then, with decreasing rate of rotation but with increasing moment of
momentum, branching to a series of ellipsoids with three unequal axes, and
continuing imtil a certain elongation is reached; and finally, at this point,
branching to a series of so-called pear-shaped figures. It has been con-
jectured that if it were possible to follow the pear-shaped figures suflSciently
far, it would be found that they would eventually reach a point where they
would separate into two distinct masses. From this line of reasoning it has
been regarded as probable that celestial masses, through loss of heat and
consequent contraction, do break up in this way often enough to make the
process an important one in cosmogony.
Aside from the unanswered question as to what form the pear-shaped
figures finally lead, there are two reasons for being cautious in accepting the
conclusions. One is that the celestial masses are by no means homogene-
137
138 THE TIDAL PROBLEM.
OU8. When they have reached, or are in, that condition of steady motion of
slow rotation postulated in the investigation, they are undoubtedly always
strongly condensed toward their centers. The other, and probably more
important, one is that isolated celestial masses do not change their rates of
rotation except when they change their densities or distribution of densi-
ties. While the theoretical discussions to which reference has been made
regard the rate of rotation as the single variable parameter, in the actual
case there is a corresponding change in density. The importance of not
neglecting the latter is easily seen.
Consider a slowly rotating homogeneous fluid having the form of a nearly
spherical oblate spheroid. The eccentricity of a meridian section depends
CO*
upon the quantity ^ p t where w is the angular rate of rotation, X:* the
gravitational constant, and a the density of the mass. The eccentricity of
the axial section increases with the increase of this function, provided it does
not go beyond a certain maximum. Now suppose the mass contracts in such
a way as to remain homogeneous throughout, and so that it continues to
rotate as a solid spheroid of equilibrium. Because the moment of momentum
of an isolated mass is constant, the contraction implies an increase in o), and
therefore, as far as this factor alone is concerned, an increase in the oblate-
nees of the mass. But the contraction also implies an increase in a, and
therefore, so far as this factor alone is concerned, a decrease in the oblate-
ness of the mass. That is, keeping the moment of momentum constant, as
the dynamical situation requires, we find the eccentricity acted upon by
two opposing factors. If, under the influence of these factors, the figure
should become less oblate, the fission theory would get no support from the
discussion; if it should get more oblate, the question is at what rate the
mass must rotate and to what extent the contraction must proceed before
there is a possibility of fission. This paper will be devoted to a brief discus-
sion of these questions.
ELUP80IDAL FIGURES OP EQUIUBRIUM. 139
II. THE ELLIPSOIDAL FIGURES OP EQUILIBRIUM OP
ROTATING HOMOGENEOUS PLUIDS.
For the applications which follow it will be necessary to review briefly
the facts regarding the spheroidal and ellipsoidal figures of equilibrium and
0)
3
their conditions of stability.
Maclaurin^ has shown that for very small values of 'o~W~ ^^^^^ ^^^
two ellipsoids of revolution which are figures of equilibrium, one of them
being nearly spherical and the other very oblate, the limits for co =» being
respectively the sphere and infinite plane. For greater values of this quan-
tity, the figure corresponding to the former is more oblate and that corre-
sponding to the latter is lees oblate. For ^ j^ =■ 0.22467 . . . the two
figures are identical. For ^ hi > 0.22467 . . . there is no ellipsoid of
revolution which is a figure of equilibrium.
Jacobi has shown' that if » ^ < 0.18709 . . . there is an ellipsoid of
three unequal axes satisfying the conditions for equilibrium. When this
quantity is very small, the axis of rotation and one other are very short and
nearly equal to each other, while the third is relatively very long. With
greater values of this quantity the shorter axes are longer and the longest
a>'
axis is shorter. For » j^ =0.18709 . . . the figure becomes an ellipsoid
of revolution and is identical with the more nearly spherical Maclaurin
spheroid. For o 1.2 > 0.18709 . . . the Jacobian ellipsoids of revolution
do not exist. ^s
In the case of the Maclaurin spheroids the relation between ^ ^^ and
the eccentricity, e, of an axial section is given by the well-known equation^
where
e
^4{^--'-'}-*«
aF
«»
It follows from these equations that for i •- we have
and that for A « oo we have
0(i)-O e-1 -|^-co
> Treatise on Fluxions, Edinburgh, 1742.
' Letter to the French Academy, 1S34.
'Tisserand, M6canique Celeste, 2, Chap. VI.
140
THE TIDAL FROBLXIf.
mx)
It 18 well known that -^j^ passes through lero but onoe between X^O and
A— 00. Hence we may represent in figure 17 the faets so far enunciated.
As «u starts from sero and increases, there is a series of figures of equilibrium
starting from O and another from P, the two series coinciding and vanishing
at the point a.
Fio. 17.
We may also indicate the existence of the Jacobian ellipsoids of equi-
librium on this figure, without, however, being able to define completely
their shape by a single point on a curve. Let us represent the eccentricities
of the sections made by planes passing through the axis of rotation and each
of the other two axes of the ellipsoid by abscissas in figure 17. The Jacobian
ellipsoids branch from the Maclaurin spheroids at b. For a given value of
o JL2 the corresponding point on the curve 6c gives the eccentricity of the
section through the longest axis and the axis of rotation, while the cor-
responding point on bd gives the eccentricity of the section through the
remaining axis and the axis of rotation. These two points together com-
pletely define the shape of the ellipsoid.
However, we shall regard the two series of ellipsoids be and bd as dis-
tinct, the properties indicated by a point on either of them being sufficient,
when taken with certain equations of relation not represented on the dia-
gram, completely to define the figure. That is, each curve of the whole
diagram will be regarded as carrying with it a certain set of equations which
serve to complete the definition of the shape of the figure of equilibrium
corresponding to each of its points. Thus, OaP carries with it the equation
which sayB that the eccentricity of every plane section through the axis is e.
BLUP80IDAL ilGITBES OF SQUIUBBIUH.
141
The equations associated with db state that the figure is an ellipsoid, and
relate the eccentricity given by the point of this curve with the eccentricity
of the other principal section by means of elliptic integrals.^ From this point
of view the figures corresponding to points on the curve bd are quite dis-
tinct from those on be, and it is regarded simply as an interesting fact that
they are the same in shape and differ only by their orientation in space. For
sufiBdently small w there are two figures of this kind. When w increases
so that
2nk'a
—0.18709 • . . they become identical with each other and
with the Maclaurin spheroid and vanish at this point.
Fio. 18.
Figure 18 is a scale drawing of an axial section of the Madaiuin spheroid
corresponding to the point b of figure 17, and is therefore the figure from
which the Jacobian ellipsoids branch. The eccentricity of its axial section
is 0.813, or more than twice that of Saturn. In fact, there is no known
celestial mass condensed beyond the nebulous state which approaches the
oblateness of this theoreticflJ figure of equilibrium.
^ TiMennd, IMesnique Celeste, 2, Chap. VII.
142 THE TIDAL PROBLEM.
III. POINCARE'S THEOREMS RESPECTING FORMS OP
BIFURCATION AND EXCHANGE OP STABILITIES.
In a memoir ^ remarkable for its powerful methods and important results,
Poincar^ has proved the existence of an infinite number of other forms of
equilibrium. He considered the equations for equilibrium as functions of the
parameter co. For a definite value of to, as io^o)^, they have a certain
number of solutions. For example^ for ^ l2 < 0.22467 there is a solution
on Oa and one on Pa of figure 17. If -o~h~ < ^- 18709 there are also solutions
on he and hd. If for co=cO| two or more solutions unite and do not vanish
as (o passes through a>|, then the figure of equilibrium corresponding to w^
is a form of bifurcation. If after uniting they vanish, the figure is a limit
form. Thus, in figure 17, h belongs to a form of bifurcation, for at this point
the Maclaurin spheroids and Jacobian ellipsoids are identical. The point a
belongs to limit form, for at this point the two series of Maclaurin spheroids,
Oa and Pa, unite and vanish. Likewise the point h belongs to a limit form
for the series hd and he.
Poincar6 showed in the work cited that there are no forms of bifurcation
corresponding to points on the curve Oh, but that there is an infinite num-
ber of them on haP. He proved also that there is an infinite number of
them on he and hd between the point h and the axis a> =^0. That is, in addi-
tion to the spheroids and ellipsoids of equilibrium an infinite number of
other forms exist. The first one on he is at /, and its deviation from the
Jacobian ellipsoid to the first order of small quantities depends upon the
third zonal harmonic with respect to the greatest axis of the ellipsoid. It
is the pear-shaped figure referred to above. Since it is unsymmetrical with
respect to the axis of rotation, there are really two similar figures, differing
by 180^ in orientation, just as the two series of Jacobian ellipsoids differ by
90° in orientation. There is of course a precisely similar series on hd.
If two real series of figiures of equilibrium, A and B, cross, and if before
crossing A is stable and B unstable, then after crossing A is unstable and B
has at least one degree less of instability. Poincar£ has also proved an en-
tirely similar theorem in periodic solutions of the problem of three bodies.^
All the spheroids corresponding to points on the curve Oh of figure 17 are
completely and secularly stable. At h the spheroids lose their stability but
the branching Jacobian ellipsoids are stable. They remain stable until /is
reached. It is an interesting question whether the pear-shaped figures are
stable or unstable. Poincar^ threw the determination of the answer to the
question into a form capable of numerical treatment,* and Darwin has
made an elaborate and detailed discussion of it.* The rigorous answer turns
* Sur r^uilibre d'line masse fluide animde d'lin mouvement de rotation. < Acta Mathe-
matica, 7, 1885, 269-380.
' Les M^thodes Nouvelles de la M^canique Celeste, 3, pp. 347-349.
' Sur la stabilit^^ de r^uilibre des figures pyriformes affectdes par une masse fluide en
rotation. < Phil. Trans., A, 198 (1902), pp. 333-373.
*The stability of the pear-shaped figure of equilibrium of a rotating mass of liquid.
<Phil. Trans., A, 200 (1903), pp. 251-314.
THEOREMS ON BIFUBCATION AND STABILITY.
143
on the numerical value of the sum of an infinite series; and from a compu-
tation of its first terms Darwin regards it as certain, though not algebrai-
cally proved, that the pear-shaped figure is stable. However, liapounoff
has stated^ that these figures are unstable. If so, the line of completely
stable figures terminates at this point, and as soon as a body has passed be*
yond it a slight disturbance will cause it to undergo radical changes of form
and perhaps break into many fragments. Even if the pearH9haped figures
are at first stable, they may become unstable as well as all figures which
branch from them long before fission occurs. Indeed, this now seems prob-
able, for Darwin has found,' in a memoir on the figure and stability of a
liquid satellite, that a satellite loses its stability before it can be brought
near enough to its primary to coalesce with it.
Consequently if a slightly viscous fluid mass were originally turning
slowly and had the form of a stable Maclaurin spheroid of equilibrium, and
if in some way greater and greater rates of rotation were gradually impressed
upon it without violently disturbing its figure, then we should see the series
of changes in its shape described by a point moving along the curve Ob to
the point b, then branching on to the line be of stable forms, again branching
at/, if the pear-shaped figures are stable, and continuing along lines of stable
figures until they terminate or until fission takes place. At any rate there
is no possible chance of fission until the change of shape has' passed beyond
b, for up to this point there is secular stability and no branching. In fact,
we may feel assured that it can not occur until the shape of the mass has
passed at least to/. In order that we may see to what a remarkable extent
a rotating homogeneous fluid must depart from sphericity before there is a
possibility of fission starting, we give in figure 19 the most oblate section
Fio. 19.
of the ellipsoid belonging to the point/. Darwin has shown' that its eccen-
tricity is e» 0.9386. The eccentricity of the other principal section through
the axis of rotation is e' = 0.6021, and the eccentricity of the principal section
perpendicular to the axis of rotation is 6^=^0.9018.
* Aoad. Imp. dee Sd. de St. P^rsbourg, 17, No. 3 (1905).
* Phil. Trans., A, 206 (1906), pp. 161-248.
' On the pear^haped figure orequilibrium of a rotating mass of liquid. < Phil. Trans., A,
198 (1902), pp. 301-331.
144 THE TIDAL PBOBLSM.
The points of bifurcation and conditions of stability or instability depend
only upon the shape of the figures, while the shapes of the figures in the va-
rious series depend only upon the values of ^ ^ . Consequently if a rotat-
ing homogeneous fluid body contracts in such a way as to remain always
homogeneous, its shape and condition as to stability are determined by this
function throughout the whole series of changes.
BOU8 FLUIDS.
145
IV. FIGUKBS OP EQUIUBRIUli OP ROTATING
HKTEKCX&ENKOUS PLUIDS.
We nofw enter a fidd bent with fonnidaUe difficulties and in which
there aie but few p iM iti fe leeaUe. We shall consider masses which are
posaesBed off dow rotation, whieh grow continually denser toward their
eentersy and wlueh are a p projumatdbr qpherieal in form. To simplify the
pioblon we may mpp oee first that we have under consideration a body
composed off a number off tnesaiprBsnUe fluids of different densities, arranged
in order off increanig density from thesurfaoe to thecenter. This hypoth-
esis more neady agrees with the conditions found in nature than that of
homogeneity does» and the results obtained under it may be taken as throw-
ing h^t on the aetnal problems.
Oairant has diown* that such a body as we are considering will always
be less oblate than it would if its mass were uniformly distributed through-
out its Tdume. This is easy to see in conadering the limiting case of a
dense nudeus surrounded by a homogeneous atmosphere of vanishing mass.
That this result may be kgitimatdy applied to the cdestial bodies is proved
by the fact that the earth, Jupiter, and Saturn are all less oblate thim they
would be if they were homogeneous and rotating at their respective rates.
It is an interesting and important fact that the differences in oUateness
of these planets and the eoneqponding homogeneous figures of equilibrium
are greater the smaller the mean density. That is, if a low mean density
means the mass is largdy gaseous and compressible, we may condude that
the more a body b condensed toward its center the less oblate it will be for
a pvea rate of rotation. The f aets for the earth, Jupiter, and Saturn are
given in the following taUe, whoie e has been computed from equation (1).
OHmernd
eeecuiiicttjr of
avth
Jupiter.
use
.461
Mr
XKS
M7
.408
ZtX—
osnm
Condder still the ease of a heterogeneous incompresdble fluid mass with
greatest dendty at its center. Suppose the dendty is given by an equa-
tion of the form /o\
where a^ is the mean dendty of the whole mass, and where a^ is dways
fiidte and may be a continuous function of c. For t^O certain series of
figures of equilibrium are represented in figure 17. Let us suppose there is a
third axis in the figure perpendicular to the e and lo-axes. We shall mark
off vdues of c along this axis.
Let the conditions for a figure of equilibrium be
^«(«i.
x^,w,t)^0, 1-1,
(3)
See TfaMraad's MtauBiqoe Cfleste, 2, Chap. Xm.
146 THE TIDAL PBOBLEM.
a>»
For €»0 and ^ ^ < 0.22467 . . . certain solutions are represented by
points on the curves of figure 17. Suppose for w^w^ the solution x^^^x^^^^
is not a multiple solution. The F^ depend upon the gravitational potential
and the rotational energy, and are continuous functions of t. Consequently
the roots of (3) vary continuously with t, and we may represent their solu-
tion by a point in figure 17 in the plane e«£, where as before, each linear
series with respect to a parameter, as ^ u or e, carries with it a set of
relations which completely defines the shape of the figure of equilibrium.
If <o=a>o and e«0 belong to a multiple solution of (3) there is an a> near
io^ such that for e— e equations (3) have also a multiple solution. By this
process the curves of figure 17 become surfaces to ^every point of which
belongs a figure of equilibrium. If in this figure e is set equal to a small
constant a new set of curves will be obtained in a general way similar to
the old, and possessing maxima and points of bifurcation. In general the
greater i the greater will be their deviation from the forms of the curves
in figure 17.
Now consider the question of stability. The necessary and suflicient
condition for complete and secular stability of the figure of equilibrium
is that the total energy shall be a minimum for all variations preserving
constant moment of momentum.' All the quantities involved in these
conditions are continuous fimctions of c. Consequently, starting from an
ordinary point in figure 17 whose corresponding figure of equilibrium has
any properties of stability, it is found that the figure obtained by varjring e
through a sufficiently small range will have the same properties of stability.
Consider the curves obtained by giving £ a constant value. At certain places
the figures of equilibrium will change the character of their stability; but as
in the case of c = 0, treated by Poincar^, wherever the stability changes a
new series of figures branches out. Since the curves for e=e are in the
analytic sense the continuation of those for e = 0, the figures of equilibrium
for €=£ go through a series of changes of stability entirely analogous to the
changes in the figures for e = 0. Of course, it is possible that two curves,
Co' and Co^, might cross a curve Co at a single point for e-^O, and that the
corresponding curves, G/ and C/, might cross the curve C. in two distinct
points, or the opposite. For example, there might be such a definition of a^
that, for a certain value of e, the point corresponding to / of figure 17 would
fall on the point corresponding to 6. However, in the present connection
such exceptional cases are trivial. The point of interest is that for € = e>0
there is a line of stable figures of equilibrium corresponding to those for
which this parameter is zero.
In general, for e > 0, the point of bifurcation corresponding to 6, figure
17, will not appear for the same value of to as that belonging to b. We may
represent these two values of lo by a>. and u)^ respectively. The question
of interest in the present connection is which is the greater; or, in other
terms, whether with increasing rotation instability occurs first in the het-
erogeneous or in the homogeneous body. We shall not attempt a positive
> Thomson and Tait's Natural Phil., Part II, 778, (/) and ijk).
FIGUBES FOB HBTBROGBNEOUS FLUIDS. 147
answer to this question, but in view of the fact that for a given rotation
and mean density the homogeneous body is more oblate than the hetero-
geneous body, we are justified in concluding that probably instability first
appears in the figures of equilibrium for e « 0.
Now consider the case of a heterogeneous compressible rotating fluid.
The preceding remarks pertaining to the existence of the figures of equi-
librium still holdi though all the explicit defining equations are immensely
more complicated. But the question of stability has a new element in it.
As Jeans has shown in an important memoir/ gravitation itself becomes a
source of instability. It is easy to see that if for any reason there is a local
condensation in a compressible fluid, gravitation will tend to augment this
condensation. On the other hand, the elastic forces called into play tend to
destroy the condensation. The two kinds of forces are in conflict, and the
state of stability depends upon which will predominate when the equilibrium
is disturbed. Jeans showed that an infinitely extended nebulai of density
such that it possesses the properties of gases, is in unstable equilibrium
independently of its mean temperature and density. That is, in this case it
is possible to introduce such local variations in density that the decrease in
gravitational potential energy shall more than balance the increase in the
potential energy of the elastic forces.
There are as yet no quantitative results by means of which we may
measure the importance of this factor of instability. It is true that Jeans'
and Love * have proved the stability of the earth under certain assumptions,
but in view of its present existence, notwithstanding the many vicissitudes
which it has survived, this result may be taken as reflecting favorably upon
the method employed in obtaining it, rather than assuring us of the perma-
nency of our planet. In the absence of positive quantitative results we are
able to make only more or less probable hypotheses as to the importance
of this factor.
There are seen to be two opposing factors entering into the question of
stability of heterogeneous compressible masses. As compared with homo-
geneous incompressible masses, the central condensation tends strongly
toward sphericity, as is shown both by theory and by the observed shape of
the sun and planets, and therefore presumably toward stability. But the
compressibility tends toward instability for local deformations and altera-
tions in density. We shall assume as appearing reasonable that in bodies
having strong central condensations and continuous changes in density these
two opposing factors approximately balance. To the extent to which this
assumption is justified, we may draw conclusions in regard to the actual
celestial bodies from what is known regarding the forms and stabilities of
homogeneous fluids. For the purposes of additional safety in this procedure
we shall keep far from the conditions of possible disruption in the applica-
tions which follow.
* Tbe stability of a spherical nebula. < Phil. Trans., A, 190 (1902), pp. 1-53.
* On the vibrations and stability of a gravitating planet. <Phil. Trans., A, 201 (1903),
pp. 167-184.
* The gravitational stability of the earth. <PhiI. Trans., A, 207 (1908), pp. 171-241.
10
148 THE TIDAL PBOBLEM.
V. THE EQUATIONS OP EQUILIBRIUM FOR CONSTANT
MOMENT OP MOMENTUM.
Equation (1) is the relation connecting the shape of the Maclaurin
spheroid; its density, and its rate of rotation. Let m represent its mass, a
its polar radius, and M its moment of momentum. Then we have
m=|;ra«(l+;i')a (4)
J|f=|ma'(l+;i> (5)
Eliminating cj and a between equations (1), (4), and (5), we have
For a homogeneous body of given mass and moment of momentum, this is
a relation between the density and oblateness which must always be satis-
fied so long as the figure is a spheroid. It is easily verified that -r^ ifl
positive for all real values of X, from which it follows that when the left
member of the equation is given there is but a single real solution for X\
The Jacobian ellipsoids are defined by the equations^
/^ V(l-C^(l-AM^»COdC (7)
/ (H->l'C*)»(l+>l'*C*)»
to*
2ick?a
where the axes of the ellipBoid are a, b, and c, and
6»-o»(l+>l*) c»-a»(l+il'*)
Equation (7) defines the relation between X and X' which must be satisfied
by these figures of equilibrium, and equation (8) expresses — in terms of
X* and X'*. "
The equations corresponding to (4) and (5) are in this case
m-|jra»'y/T+^-y/T+A^ff Jlf =^(2+^>+^'»)<^ (9)
By means of these relations, equation (8) reduces to
25(|;r)«M» , (2+ A
SJfc'mV " " (l+A')
Now we may write (7) and (10) respectively
^'+^'') [ ^ c'a-C)dc (10)
*(1+>1")* / (l+>l'c')*(l+>l"C')*
0{X,X')=O ,piX,X')=Kai (11)
* Tisserand's M^anique Celeste, 2, Chap. VII.
EQUIUBBIUM FOB CONSTANT MOMENTUM.
149
where X is a constant depending upon m and M. From these equations we
have
dX
di!
Then we find
«--
\ K dd
Zd (tI dk
jda
aii'- +
1 K d«.
(12)
where
dX
fly
fli
flA'
flF
r, Since i and X' enter and f symmetrically, it follows that when the
ellipsoid branches from the Maclauiin spheroid, t.e., when X^H, we have
■TJ-— rw-. Hence as the ellipsoid branches from the spheroid because of
increasing denmty, the eccentricity of one principal axial section increases
while that of the other decreases. This continues indefinitely unless either
^ or Tw vanishesi which is extremely unlikely. This means the figure
tends to become cigarnshaped. At a certain elongation (see figure 19) the
so-called pearnshaped figure branches. Certainly in homogeneous masses
there can be no fission before this elongation, with its corresponding den-
sity, is attained.
150 THE TIDAL PBOBLEM.
VI. APPLICATIONS TO THE SOLAR SYSTEM.
In applying the formulas above to the solar syBtem we must remember
that they are strictly valid only when the masses are homogeneous. Now
the sun and planets are certainly not homogeneous, but we have seen reasons
for believing that nevertheless the formulas will give results which are not
remote from the truth. But, because of this uncertainty, in the applica-
tions which follow we shall not attempt to draw conclusions except where
the margin of safety is extremely great.
Let us consider first the sun. We shall find its density for various de-
grees of oblateness, and its oblateness for a certain very high density.* Re-
ferring to (6) we see that the greater the moment of momentum of a body
the less dense it will be for a given oblateness, and the more oblate it will be
for a given density. Consequently we shall be favoring the conclusion
that the sun will eventually suffer fission if we use too large a value of M.
The moment of momentum is most easily computed if we suppose the sun
is homogeneous, and the result obtained in this way will certainly be in
excess of the true value.
Using the mean solar day, the mean distance from the earth to the sun,
and the mass of the sun, as the units of time, distance, and mass respec-
tively, we find that the density of water is
<^wAter'= 1,567,500.
Taking the sun's density as 1.41 on the water standard, its period of rota-
tion as 25.3 days, and its radius as 433,000 miles, we find for its moment of
momentum oia
^"^^lo^
Now we may apply equation (6) to find how dense the sun will be before
the Jacobian ellipsoids branch off. It is hardly possible that the sun could
suffer fission before this point is reached, the shape of the spheroid being
given in figure 19. The computation shows that when the sun shall have
reached this degree of oblateness its density will be
<T=307 X 10" on the water standard.
This density corresponds to an equatorial radius of the sun of 11 miles.
Since this density is millions of times greater than it is supposed matter
ever attains under any circumstances, we must conclude that the oblate-
ness of the sun can never approach that for which the Jacobian figures of
equilibrium branch. Or, in brief, the sun can never contract so much that
its rotation will threaten it with disruption.
Notwithstanding the extreme character of these figures, one might still
be so ultra-skeptical as to doubt the conclusion, since it is based on the com-
putation of the point of bifurcation for homogeneous masses. However
that may be, we must admit that the sun will be stable until its oblateness
reaches that of Saturn at present. We have seen that the eccentricity of a
' Strictly speaking the computations are made for homogeneous bodies of the same
mass and having the same moment of momentum, but no confusion will result from this
mode of expression.
APPLICATIONS TO THE SOLAR 8TSTEM. 151
meridian section of a homogeneous fluid having the mass and the mean
density of Saturn, and rotating in its period, would be 0.607. Since the
constitution of the sun is in a general way like that of Saturn, it will have
the oblateness of Saturn about when the section of the corresponding
homogeneous mass has this eccentricity. Making the computation from
equation (6) we find that the sun will not become so oblate as Saturn is now
until its mean density becomes 148 X 10^ on the water standard. This cor-
responds to a radius of 37.3 miles. If we may regard this as an impossibly
large density we may conclude that the sun will never be so oblate as Saturn
is now, and that its stability will always be greater than that of Saturn at
present.
Apparently the chances that Saturn will separate into two parts because
of shrinking and rapid rotation are greater than that any other member of
the solar system will ever suffer fission. To examine the probabilities we
shall apply equations (4), (5), and (6) to Saturn. Taking the density, mass,
and period of rotation as 0.72, j-^f fti^d 10.25 hours respectively, and com-
puting the moment of momentum under the hypothesis that Saturn is now
a homogeneous sphere, in order to give the theory that fission is possible all
the benefits of the approximations, we find that when Saturn shall have an
oblateness equal to that of the spheroids from which the Jacobian ellipsoids
branch, its density will be 21 times that of water, its axial diameter 16,500
miles, its equatorial diameter 28,400 miles, and its period of rotation 1 hr.
24 m. The high mean density demanded seems to be fatal to the theory of
fission in this case.
In order to see how great changes in the density, dimensions, and period
the body will undergo by the time it reaches the state where the pear-
shaped figures branch, we may apply equations (7), (9), and (10) to Saturn.
Darwin has made the computations^ from equations equivalent to (7) and
(8), and has found that for this point
X = 0.7544 X' - 2.7206 k^^ — 0. 1420
Then equations (9) show that at this stage the mean density of Saturn
must be 93 on the water standard, its polar diameter 9,400 miles, its longest
*diameter 27,000 miles, and its period of rotation 46 minutes. That is, the
mass is about four and one-half times as dense when the pear-shaped figures
branch as when the Jacobian ellipsoids branch. While the computation
was applied to Saturn, it follows from equations (6) and (10) that this same
ratio for the densities at these critical forms is true, whatever the mass and
moment of momentum of the body under consideration.
The density which the earth will attain before it will reach one of the
critical forms is so great that the computation is without interest. But we
may examine the hypothesis that the earth and moon were originally joined
in one mass whose rapid rotation produced instability, and that resulting
fission gave rise to two bodies having great stability. It is to be observed
in the first place that the moment of momentum of the earth-moon system
has remained constant except for influences exterior to itself. There is none
' Phil. Trans., A, 198 (1902), p. 326.
162 THE TIDAL PROBLEM.
readily assignable which could have increased it. Among those which may
have decreased it apparently the solar tidal friction is the only one which
can have produced sensible results.
The effect of the moon's tides in the earth has been to transfer moment
of momentum from the earth to the moon. According to the results given
in the preceding paper on tidal evolution, section XIV, the maximum
moment of momentum the earth-moon system could have had, including all
that the sun could have taken from it, is that belonging to the system mov-
ing as a rigid mass with a period of 4.8 hrs. Let P represent this common
period. Then, neglecting the inclinations of the planes of the equators of
the earth and moon to the plane of their orbit and the possible eccentricity
of their orbit, factors which reduce the moment of momentum, we have by
equation (6), loc. ciL,
The quantities c^ and c, depend upon the distribution of mass through-
out m^ and m,, m^ representing the earth, and m^ the moon. If the masses
are homogeneous c^^c^^QA, If they obey the Laplacian law of density
Ci»C2« 0.336. If the distribution of mass is such that the densities increase
from the surfaces to the centers of these bodies, the values of c^ and c, are
less than 0.4. We shall certainly get too large a value for M by putting
C|=>C2»0.4. Adopting these values we find that for the earth-moon system,
in the units of tWs paper, Af =8X10~".
Since large moment of momentum tends to instability we shall favor the
theory of fission if we add 26 per cent to this number, supposing that per-
haps this amount may have been lost through meteoric or other friction.
Then, using Af =10~" and m = 3 X 10~*, equation (6) gives for this united
earth-moon mass at the time when the Jacobian ellipsoid branched a mean
density 215 times that of water. We find similarly that this hypothetical
earth-moon mass could not become even so oblate as Saturn is now imtU
its density had become 10.4 times that of water. Since the present density
of the earth is only 5.53, this means that if the hypotheses upon which this
computation was made are valid, the earth-moon system can not have arisen
from the fission of a parent mass under the influence of rapid rotation.
In the preceding paper, starting with the earth-moon system as it now
exists and following backward in time the effects of tidal friction, it was not
possible to get the earth and moon in close enough proximity to make the
fission theory seem possible. Now, starting with the supposed initial sys-
tem with the critical factor, the moment of momentum, determined from
observations, we do not find the figure approaching an unstable form until
the density is more than 40 times the present density of the earth.
APPLICATION TO BINABT STARS. 163
VII. APPLICATION TO BINARY STARS.
Direct telescopic observations prove the existence of very many binary
stars. According to Hussey and Aitken about one star in 18 of those
brighter than the ninth magnitude is a visual double. There are many
stars whose spectra periodically consist of double lines. This phenomenon
is taken as indicating that these stars are binary systems, made up of
two approximately equal components, whose orbital planes pass nearly
through the earth. There are many more stars whose spectral lines period-
ically shift. This is interpreted as meaning that these systems are binaries
in each of which one component is relatively non-luminous. There are many
variable stars whose light curves are explained by supposing they are eclips-
ing binary systems. Unless the interpretations of these phenomena are
very much in error a considerable fraction of all the stars are binaries.
Granting that the interpretations are correct, the evolution of binary sjrs-
tems is a standard celestial process. We are raising the question here
whether these binaries may have originated from parent masses by fission
without external disturbing factors, and if so at what stage of condensation.
Before considering the question of the fission of single masses into binary
systems, we shall write down some of the implications of the interpreta-
tions of the spectra of spectroscopic binaries, particularly as regards limits
on their masses and densities.
Consider first the case where both spectra are visible. Let i be the com-
plement of the inclination of the plane of the orbit to the line of sight.
Suppose the orbit is circular. Let v be the maximum observed radial
velocity, roi and m, the two masses, P the period of revolution, and a the
major semi-axis of the orbit.
Then we have
^^* k^[^^^, 2;rt^cos»t ^' )
Since i can not be determined in a star which is not also a visual binary we
have, reducing the units so that when v is expressed in kilometers per second
and P in mean solar days, the sum m^+m^ expressed in terms of the sun's
mass by the relation ...^
m^+nH^-^Pv^ (16)
Suppose the spectrum of only one star, m^, is visible, and let v^ be its
maximum observed radial velocity with respect to that of the center of
gravity of the system. Let r^ be the radius of its orbit, assumed to be cir-
cular, around the center of gravity of the system. Then we have
Pr, o . D 27rrii(mi+m,)
— V=27rr, wiir. — m/-, a^r^+r^ P— — ^ , — —
cost * II rj 11 j.^1
m*^^
^•("+5)'
whence '^^i V ^'^J (1«)
' 2wJk* cos* t
164 THE TIDAL PROBLEM.
In this case the lower limit of the mass of the system can not be computed
except an assumption be made regarding the relation between m^ and m,.
Suppose — -=fi> Then equation (16) gives
m,+m,= (l+/i)m,>^??J^±^* (17)
When P is expressed in mean solar days, and v^ in kilometers per second, we
have
m^ + nh^^Pv,\l+piy (18)
where m| + m, is expressed in terms of the sun's mass. If it were true that
the more massive star always gives the observed spectrum we should have
/«>land ..24
^i+^>-^^V (19)
Let Oi and a, represent the radii of m^ and m, respectively. Let the dis-
tance between their surfaces be represented by #ca. Then 0^#c< 1 and we
a.+a,-a(l-«) P= 2;r(a.+aOt ^^^
A;(l— ic)*-ym,4-m.
Suppose the two bodies have the same density a. Then we have from
m^^-K^noa^t m^^-^aa^ and equation (20), putting — ^=/i as before
Zn (1 +/!*)* .
The ratio /< may vary from zero to infinity, and x from to 1. From the
derivative a<; 3;r (i+;zi)^(l-;z»)
d;£ Jk2p2(l-K)» /iJ(l+;£)2
it follows that; for fixed values of P and k, a constantly increases while ii
varies from to 1, and then constantly decreases while /i varies from 1 to
00 . Therefore, since (21) is a reciprocal equation in /i, we have
^^'^ ^<^^ , .,./!., o. (22)
Or, changing the units so that a will be expressed in terms of the density of
water when P is expressed in mean solar days, we have
* ^^'^TT^TwA:^. (23)
100 (1 - kYP^ ^ - 100 (1 -i«)»F
The smaller k the smaller the limits for a. When the bodies are in contact,
and the period 4.57 hours, that of ^ Cephei,^ we find
2.2^(7^0.5
» The Period of f^ Cephci, by E. B. Frost, Astrophysical Jour., 24 (1906), pp. 269-262.
APPUCATION TO BINABT 8TAR8. 155
If they are separated so that their surfaces are at a distance from each
other equal to one-half the sum of their radii, i.e., it k^^, equation (23)
According to Darwin's results, loc. eit., p. 232, this is about the minimum
distance at which homogeneous masses could revolve in stable equilibrium.
If they were separated farther the already high limits on the density would
be still greater. It follows that we must conclude either that these very
short period binaries and eclipsing variables are very dense, or that hetero-
geneous masses are more stable than homogeneous ones.
Now let us return to the question of fission of binary stars. Denoting
the periods of rotation of m^ and m, by D^ and D, respectively, we have for
the total moment of momentum of the system ^
(2^)»(m,H-ng»"^ D, ^ D, ""^^^
The signs of the second and third terms in this equation are determined
under the hypothesis that both bodies rotate in the direction in which they
revolve. But this is a necessary consequence of the fission theory, and
therefore an allowable assumption in testing it.
If P > D| and if P > D, then the mutual tides of the two bodies tend to
bring P, D^ and D, eventually to the same value. In the case of widely
separated visual binaries the fission theory implies that the tidal evolution
has proceeded far, and that Dj and D, are closely approaching an equality
with P. If these conditions are satisfied and if m^ and m, are approximately
equal, we see from (24) that the inequality
^>(2^7*W,+m,)* ^^^^
is nearly an equality. For example, in the case of a Centauri, assuming that
each component has the dimensions of the sun and that D| » D, « P approx-
imately, the ratio of the first term to the sum of the other two is roughly
10,000 to 1. In general, the greater P and the more nearly equal m^ and
m,, the more nearly the inequality (25) approaches an equality, and the
opposite.
Let us suppose the two stars were originally in one spheroidal mass
m^m^-\-m^. Then equation (6) gives the relation between its density and
oblateness, which reduces by means of (25) to
25m,»m,»fctPI > ^ (1 H-^')« f (3 +k') \ .gO)
6(3^)*(m,H-m,)^^<-^— j-T-**^ ^"^j ^^^^
Now let ZT'^l^) then equation (26) gives
^<6(3^/fVa:^|(3+£)t,„_.,_3| (27)
Letting /(/i)-- ^^-^^ we find for /i-l that -1^-0, 4-t>0 and that ^'
d/£ ' dp? dft
* See equation (6), p. 85.
156 THE TIDAL PROBLEM.
vanishes for no other value of ft. Consequently the right member of (27)
has a single minimum at /i = 1. But for other values of fi the inequality of
(27) differs more from an equality, and hence we are not certain that /c= 1
gives the least value of a for a given P and A. But in many stars ft is
undoubtedly near enough unity to make equation (27) useful.
We shall suppose pL = l and compute o for the value of X for which ap-
parently there is first any danger of fission. If we suppose that this first
occurs for that value of X for which the Jacobian ellipsoids branch, that is
for X=' 1.395 . . . , we find, taking the units so that a will be expressed in
terms of the density of water when P is expressed in terms of mean solar
^*^®' 016
a<^ (28)
Or, if the fission occurred when the pearnshaped figures branched from the
Jacobian ellipsoid, we find similarly
^ 0.071 ™.
Consequently, this discussion leads to the conclusion that in all binary sys-
tems in which the two masses are approximaidy equal, and in which the periods
are at least several years, as they are in the vistud pairs, ti^e fission mtut have
occurred, if at all, while the parent mass was yet in the nebulous state. The
data regarding binary systems as a class are so meager that probably no
stronger conclusion than this can be drawn from this line of argument.
There is, of course, no a priori objection to the theory that binaries as a
class have originated by fission in the nebulous state. But there are at least
two rather distinct hypotheses as to how and why such fission may have
taken place. The first is that in the origin of a nebulous mass the factors
which have determined its initial condition may have brought it into exist-
ence with at least two nuclei of condensation whose magnitude and density
were sufficient to have led to a binary, even though the moment of momen-
tum may have been so low that, if the mass had been spheroidal with the
same mean density, it would have been a stable figure. Fission in this t3rpe
of masses is not under consideration here. The second is that the mass in
its earliest nebulous stage was in an approximately spheroidal form, densest
at its center with density decreasing outward through approximately sphe-
roidal layers, and that as a consequence of its high moment of momentum
it lost its stability and divided into two masses. This is the type of fission
under consideration here.
Suppose a nebula of this latter type suffers fission. At the time of fission
all parts are rotating at the same angular rate, and one of the two parts
must have a mean density less than, or at the most equal to, the mean den-
sity of the original mass. Consequently one of the two fragments because of
its lower density and equal rotation, must have at least as great a tendency
to fission as that which led to the division of the initial mass, unless cither
its form is one of greater stability, or the tidal forces of the other member of
the pair tend to keep it from breaking up. If, as seems probable, the ap-
proximate spheroid is the most stable figure of equilibrium, and if the mass
APPUCATION TO BINABT 8TABS. 157
under consideration has suffered fission by evolution along this line of figures,
as is assumed, then the former alternative is eliminated. It does not seem
that the tidal factor can tend toward stability. This intuition is strongly
supported by the results obtained by Darwin, loc, cit., attacking the problem
from the other end, viz., that it is not possible to bring two homogeneous
fitiid masses near enough to touch without their being certainly in unstable
equilibrium.
We observe next that the binary stars are now actual stars of consider-
able density. Consequently if they have originated from the fission of neb-
ulas they have undergone enormous contraction. The contraction implies
increased rotation which would increase the already dangerous tendency
for at least one part to suffer further fission. Tidal friction would offset
this tendency by decreasing the rotations, but considering all the factors
involved, it is seen that if a fitiid mass ever gets to the state where fission
occurs, there is at least great danger of its breaking into many pieces.
Consequently we are led to believe that if binaries and multiple stars
of several members have developed from nebulas, the nebulas must orig-
inally have had well-defined nuclei. The photographs of many nebulas
support this conclusion. But we observe that if we are forced to this
position we do not explain anything — we only push by an assumption
the problem of explaining the binary systems a little farther back into
the unknown.
158 THE TIDAL PROBLEM.
VIII. SUMMARY.
The problem under consideration is that of the fission of celestial bodies
because of rapid rotation when they are not disturbed by important exter-
nal forces. The attack is made through well-known results concerning the
figures of equilibrium and conditions as to stability of rotating homogeneous
incompressible fluids. It is recalled that for slow rotation a nearly spherical
oblate spheroid is a stable form of equilibrium; that for greater rates of
rotation the corresponding figure is more oblate; that when the eccentricity
of a meridian section becomes 0.813 the figure loses its stability and at this
point a stable line of three axis ellipsoids branches; that when the longest
axis of the ellipsoid becomes about three times the axis of rotation a new
series, known as the pear-shaped figures (or better, perhaps, the cucumber-
shaped figures) branches, and that before this point is reached there is no
possibility of fission. We are almost entirely ignorant as to what may happen
after this point is passed, and it must be remembered that it has not been
proved that in any case fission into two stable bodies is possible.
The celestial bodies differ from those just considered in two important
respects. In the first place their densities increase toward their centers.
For a given rate of rotation and mean density this central condensation
makes them more nearly spherical, as is shown both by theory and by com-
parison of the observed figures of the planets with the computed forms of
corresponding homogeneous masses. In the case of Saturn, for example,
the eccentricity computed on the hypothesis of homogeneity is 0.607 while
the observed value is only 0.409. It seems certain that this central conden-
sation tends toward stability. The second important difference between
the ideal homogeneous incompressible fluids and the celestial bodies is that
the latter are compressible. This latter factor, at least under certain cir-
cumstances, tends toward instability.
The opposing quantitative effects of central density and compressibility
undoubtedly differ greatly in different masses and can not be easily deter-
mined in any case. However, if we may assume that they approximately
offset each other, we may reach some conclusion respecting the possibility
of the fission of the actual celestial bodies by discussing the corresponding
homogeneous incompressible body. This is the assumption adopted here,
but, because of its uncertainty, in the applications to the solar system, where
it turns out fission is impossible, all approximations are made so as to favor
fission, and it is assumed that in the actual bodies fission may be immanent
long before it is possible in the homogeneous ones. These safeguards and
simplifications are possible and easy because it is a negative result which is
reached.
The actual problem is not one in which the rate of rotation changes
while the density remains constant, though this is the one heretofore treated
in the mathematical discussion. In the physical problem the rate of rota-
tion and the density change simultaneously with the shape in such a way
that the moment of momentum remains constant. Imposing this condition,
we arrive in the case of the spheroids and ellipsoids at relations between
the density and respective shapes, the coefficients depending upon the mass
SUMMABT. 159
and the moment of momentum. When the oblateness of the spheroid is
given there is but a single density satisfying the conditions, and when the
density is given there is but one spheroid satisfying the conditions.
For the applications we assume that an actual celestial body will not be
in danger of fission until the corresponding homogeneous incompressible
body arrives at the state where the Jacobian ellipsoids branch. The density
at this stage is less than one-fourth that at wUch the pear-shaped figures
branch, and actual fission in the homogeneous bodies is certainly beyond
this form, if indeed fission into only two bodies is ever possible. With this
very conservative assumption we proceed to some calculations.
(1) We find that the sun can not arrive at this critical stage until its
mean density shall have exceeded 307 X 10" on the water standard. This
corresponds to an equatorial diameter of the sim of about 22 miles.
(2) We find that the sun can not become so oblate as Saturn is now until
its mean density shall have exceeded 148 X 10^® on the water standard. This
corresponds to an equatorial diameter of the sim of about 75 miles.
Since even the latter density is impossibly great we conclude that the
sun will never become so oblate as Saturn is now, and that it will always be
more stable than Saturn is now.
(3) We find that Saturn can not arrive at the critical stage at which the
Jacobian ellipsoids branch until its mean density shall have become 21
times that of water. This corresponds to a polar diameter of 16,500 miles
and an equatorial diameter of 28,400 miles. We conclude because of the
great density demanded that Saturn will never su£fer fission.
(4) We assume that the earth and moon were once one mass and get
their original moment of momentum from its present value. In computing
it, however, we make certain approximations so as to get it too large and
thus favor the conclusion of fission, then we add[to it the maximum amount
the sun's tides can have taken from the earth, and finally we add 25 per cent
for fear there may be some unknown sensible factors omitted. Then we
find that this hjrpothetical earth-moon mass could not get even to the
critical point where the Jacobian ellipsoids branch until its mean density
became 215 times that of water, or about 40 times the present mean density
of the earth and moon. It would not become even so oblate as Saturn is
now until its density had become 10.4 times that of water. Therefore we
conclude that the hypothetical case was false, and that the moon has not
originated by fission from the earth in this way.
(5) In applications to the binary stars the results are less definite because
of the meager data regarding these systems. But assuming that fission in
stars will occur when the Jacobian ellipsoids branch in the corresponding
homogeneous masses, we find for the density a in terms of water at the time
of fission when the two stars are of equal mass
^ 0.016
where P must be expressed in mean solar days. Even though fission should
not occur until the density is ten times this amount (which, if true, makes
the evidence against fission in the solar system much stronger), all visual
160 THB TIDAL PBOBLKM.
binaries of two approximately equal masses must have separated, if they
have originated by fission, while they were yet in sr nebulous state. The
results are of the same order so long as the disparity in the two masses
of a binary is not very great, and this probably includes all of the visual
binaries.
(6) Certain formulas, not connected with the question of fission, were
developed for binary syistems. If P represents the period in mean solar
days, fi the ratio of the mass of the star whose spectrum is measured to the
mass of the other one, and v^ the maximum observed radial velocity ex-
pressed in kilometers per second, then the sum of the masses expressed in
terms of the sun's mass must satisfy the relation
When the spectra of both stars are measurable, and v represents the
maximum relative velocity of the stars, the corresponding formula is
If we let a represent the distance between the centers of a binary pair,
and ML the distance between their surfaces, and suppose they have the same
mean density a, then a must satisfy the inequalities
8
>a>
100(1 -/c)«P» = = 100(1 -/c)«P»
where a will be expressed in terms of water when P is expressed in mean
solar days.
The results obtained by the computations above are quite adverse to the
fission theory, in general, except if it is applied to masses in the nebulous
state, and seem practically conclusive against it so far as the solar system
is concerned, either in the future or past. Perhaps the hypothesis that
stars are simply condensed nebulas, which has been stimulated by a cen-
tury of belief in the Laplacian theory, should now be accepted with much
greater reserve than formerly. Up to the present we have made it the
basis not only for work in dynamical cosmogony but also in classifying the
stars. It may be the time is ripe for a serious attempt to see if the oppo-
site hypothesis of the disintegration of matter — because of enormous sub-
atomic energies, which perhaps are released in the extremes of temperature
and pressure existing in the interior of sims, and of its dispersion in space
along coronal streamers or otherwise — can not be made to satisfy equally
well all known phenomena. The existence of such a definitely formulated
hypothesis would have a very salutary effect in the interpretation of the
results of astronomical observations. We should then more readily reach
what is probably a more nearly correct conclusion, viz., that both aggre-
gation and dbpersion of matter under certain conditions are important
modes of evolution, and that possibly together they lead in some way to
approximate cycles of an extent in time and space so far not contemplated.
OOFTlUBFriONS TO OOSMOGONT AND THE FUNDAMEITTAL PB0BLEM8 OF GEOLOOT
THE BEARING OF MOLECUUR ACTIVITY
ON SPONTANEOUS FISSION IN GASEOUS SPHEROIDS
BY
THOMAS GHBOWDEB CHAMBEBLIN
ProfenoT of Oeology, Unhenity of Chicago
161
THE BEARING OF MOLECULAR ACTIVITY
ON SPONTANEOUS FISSION IN GASEOUS SPHEROIDS.
It is a familiar view that a rotating spheroid of gas may, by cooling and
shrinking, so far accelerate its rate of rotation as to cause its own sepa-
ration into two or more parts. The resulting parts are assigned various
relative values, and the separated masses are given different forms, ranging
from fragments and rings to subequal masses. This view of possible self-
partition has found expression in various cosmogonic conceptions from the
nebular hjrpothesis to the formation of binary stars.
To consider the bearings of molecular activity in a representative case,
let a spheroid of gas be chosen whose mass is comparable with that of the
solar system and whose volume is such as may be hjrpothetically assigned
it. Let its rate of rotation at the outset be such that the value of gravita-
tion at the equatorial surface is greater than the centrifugal component of
rotation. Let cooling follow, in consequence of which the rate of rotation
will be progressively accelerated. Let it be assumed — as has usually been
done — ^that the rate of rotation would at length reach such a velocity that
separation in some form would take place regardless of any question as to
the manner of its realisation. Our question relates to the effect of mo-
lecular activity on the transition from an undivided spheroid to a spheroid
divided in some way, whether by massive fission, into larger or smaller
fractions, or by individual molecules.
In a body whose molecules are boimd together into a coherent mass,
such parts as may be affected by like general stresses are properly treated
as units, within the limits of cohesion, but in a body whose molecules possess
all degrees of freedom, and which act with complete individuality, the treat-
ment may with special appropriateness be based on the molecule as the
unit. Molecular action in a gaseous spheroid consists of encounters or
quasi-encounters, and of rebounds or quasi-rebounds along free paths be-
tween the encounters. Within the mass, the excursions and encounters of
the molecules give rise to an effect equivalent to viscosity which influences
the movement of one part of the gaseous mass upon another part, and may
perhaps have given rise to an impression of coherence; but on the outer
border of the mass — the critical portion in this case — ^this effect becomes a
vanishing quantity, and individuality of action is dominant.
According to the laws of gaseous distribution, the density of a gaseous
spheroid, when controlled solely by its own gravitation, declines from a
maximum at the center progressively toward the surface, where the limit
of gaseous tenuity is reached and an ultra-gaseous state supervenes. The
transition from the gaseous to the ultra-gaseous state is the critical factor
in the case, since it is at the extreme surface of the gaseous mass that the
11 163
164 THE TIDAL PROBLEM.
centrifugal component of rotation first comes into equality with the cen-
tripetal force of gravitation and constitutes a condition precedent to the
separation of any mass of gas as a body.
In the depths of the gaseous spheroid the paths between encounters
may be assumed to be relatively short and hence straight, since gravitation
can not sensibly affect paths of brief duration. At higher levels, the free
paths grow progressively longer, and at length horizons may be reached at
which the attenuation permits free paths of such length and duration that
they may be appreciably curved by the gravitation of the spheroid. At
still greater heights the attenuation reaches such a degree that curved
paths come to dominate and, at a certain stage of rarity, a portion of the
molecules rebounding from encounters in outward directions, find no
molecules in their paths, and therefore hold on their coiu'ses until arrested
and turned back by gravitation, if its force be sufficient, or else they pass
on beyond the limit of the spheroid's control. Theoretically, under
the Boltzman-Maxwell law of molecular distribution, a certain small per-
centage of molecules should reach the parabolic velocity of the spheroid
and escape, but for the purposes of the present discussion this fraction need
not be considered independently of a larger class to be described presently,
with which it may be merged as having like influence on the moment of
momentum of the spheroid.
Such of the outward-bounding molecules as are arrested by the sphe-
roid's gravitation obviously turn back toward the spheroid without a re-
versing encounter and thus describe elliptical loops. In this they differ
markedly from the molecules in the depths of the gaseous spheroid, whose
paths are sensibly straight and whose courses are terminated by encounters
at either end.^ In the elliptical courses the outward movement is ter-
minated by a gradual decline in the molecule's speed until its outward
progress is reduced to zero, when there follows a new movement inward
accelerated by gravitation.
If the to-and-fro, collisional activity of molecules constitutes the essen-
tial characteristic of a gas, the outer border of the strictly gaseous part of
the spheroid should be placed at the transition zone where the molecules
cease to-and-fro passages between encounters and begin to describe ellip-
tical loops limited outward by gravitation, but this demarcation is rather
a matter of convenience than an essential in the consideration of the modes
of action.
In the course of their outgoing and incoming movements, the molecules
pursuing elliptical paths are subject to collision with one another. The
phases of such encounters may vary indefinitely and the velocities of the
rebounding molecules may represent an indefinite variety of interchanges
of kinetic energies. Inspection shows that some of these molecules must
rebound toward the gaseous spheroid, that some must take distinctively
new elliptical paths, while a certain proportion will inevitably be thrown into
courses more or less tangential to the surface of the spheroid, and some of these
may have sujfficient velocities to assume orbits aboiU the spheroid, and thus form
^ This distinction has been drawn by G. Johnston Stoney, Astrophys. Jour., vol. XI,
1900, pp. 251 and 325, and elsewhere.
BEARING OF MOLECULAR ACTIVITY ON FISSION. 165
a revolvtional system of molecules dynamically, independent of the spheroid,
except that they act as miniUe satdlites.
In a stationary spheroid the rebounds that give rise to revolutional
courses are as likely to take one direction as another, and if the mass of the
spheroid be great, the number of molecules which will acquire revolution
from molecular activity alone may be neglected in this discussion.
But in a spheroid in a state of rapid rotation, especially a spheroid
approaching the critical stage of centrifugal separation, the molecules shot
outwards in the direction of rotation will start with the sum of the common
velocity of rotation and the individual velocities acquired from the last
encounter, while the molecules shot in a direction opposite to the rotation
will have only the difference between the common velocity of rotation and
the velocity acquired from the last encounter, the meridional component
in each case being neglected as immaterial here. It follows from this that
when the velocity of rotation is high, the molecules starting from encoun-
ters in the direction of the spheroid's rotation will much more largely pass
into orbital paths than molecules starting in the opposite direction.
In a spheroid having the mass of the solar system and a radius equal to
the radius of Neptune's orbit, the equatorial velocity required for separa-
tion by mass is above 5 kilometers per second, while the average molecular
velocity of all known molecules, at a temperature of 2000^ C. and standard
terrestrial pressure, falls below this. The average molecular velocity of
most known substances falls much below this even at 4000^ C. It seems
clear therefore that, for most of the known molecules, the effect of molecu-
lar velocity directed backward is merely to destroy a part of their rotational
speed, and that they still move forward relative to the center of the sphe-
roid. With a spheroid having the solar mass and a radius equal to the dis-
tance of the earth from the sun, and hence a separation-speed of nearly 30
kilometers per second, only a very small fraction of the molecules could
acquire velocities sufficient to neutralize their rotational velocities at the
critical stage of separation. The number of molecules that could acquire
the 60 kilometers per second required to neutralize their rotational veloci-
ties and add sufficient velocity to give them an orbital course in a retro-
grade direction must obviously be negligibly small in a case of this kind.
Practically all molecules must be regarded as having forward courses with
velocities which are either enhanced by being shot forward or retarded by
being shot backward.
The velocity of centrifugal separation is practically identical with the
velocity of circular revolution about the spheroid in a minimum orbit.
Larger orbits involve lower velocities but require additional potential en-
ergy and moment of momentum. When the rate of rotation of the spheroid
is very near, or essentially at, the critical stage of centrifugal separation, a
slight addition to the velocity of an outer molecule in a forward direction,
arising from molecular interaction, will give to it a velocity greater than
that required for the minimum circular revolution; and before the critical
state has been actually reached, all molecules on the equatorial periphery
which receive forward impulses of any appreciable amount will have more
than the requisite velocity for minimum circular revolution. If all mole-
166 THE TIDAL PROBLEM.
cules whose projections have forward components are regarded as fulfillmg
these conditions, nearly half the molecules given outward projections will
be included.
Encounters and rebounds in gases under familiar terrestrial conditions
range into the billions per second, but encounters are much less frequent
in rare gases. Excluding outward flights that do not properly belong to the
true gaseous state, the maximum period between encounters is less than
the rotation-period of the spheroid, and the average period is much leas
than that. Since interchanges are thus frequent, all molecules are liable to
receive a forward impulse within a brief period.
Molecules which simply receive a forward and outward projection
greater than that requisite for free circular revolution about the spheroid
do not, however, enter upon free orbits, directly, in most cases, because the
paths on which they enter, the orbital in type, normally lead back to their
starting-points, and in nearly all cases they cut the spheroid before they
return to these points and thus complete a free orbit. If this were univer-
sally and inevitably true, the way to free orbits along this line of evolution
would be effectually barred. There are three lines of escape from this
result, the first and second of which are probably unimportant; the third
is probably effective.
1. The first is the case in which molecules receive impulses from molec-
ular interaction in lines tangent to the points of impact, and hence take
elliptical paths about the spheroid which return tangentially to the points
of impact as their peri-spheroidal climax. Their liability to encounter the
spheroid is thus limited to these tangential touches which, in the rare con-
dition of the gas at these vanishing points of gaseous organization, will
not necessarily involve capture. Such molecules will not, however, be free
from collision with the molecules pursuing elliptical loops above the gaseous
spheroid.
2. By hypothesis, the spheroid is shrinking, and if the rate of shrinkage
is appreciable during the free flight of molecules whose paths only slightly
cut the surface of the spheroid such shrinkage may leave these paths free,
so far as the gaseous spheroid is concerned.
3. The two cases just named are perhaps more serviceable in defining
conditions where gradations rather than sharp limits prevail than as
sources of free orbital paths. The most important case is built upon the
action of the ultra-gaseous molecules outside the gaseous spheroid, as lim-
ited above. These molecules start from the outer part of the spheroid—
strictly, from all depths from which there is an open path outward in the
line of their projection — and pursue elliptical courses with return to the
spheroid, except in the cases just noted. We have seen that, in the rep-
resentative case of the solar system, the rotational velocity is so great,
relative to the average molecular velocity, that most of the molecules will
pursue forward courses, even when directed backward, and hence will be
moving in harmonious directions. Considered as independent molecules,
they constitute a corona of particles rising in curves, predominantly at low
angles, and descending at similar angles to the spheroid. They are liable
to collide in these courses and a certain percentage of collisions is inevi-
BEARING OF MOLECULAB ACTIVITT ON FISSION. 167
table. Collisions in the rising parts of the curved courses take precedence
in time, and hence in probability, over collisions in the declining courses;
for if collision is realized in the first part of the course the molecule is likely
to lose its chance in the latter part by being either thrown back to the sphe-
roid by the first collision or else thrown outwards where collisions are less
imminent. In any case, the collisions probably result either in an earlier
return of the molecules to the spheroid, or in throwing them into new
paths, of the orbital type, which will bri'ng them back to this point of last
collision and not to the spheroid. This point of collision lies above the sphe-
roid, and does not require the orbit to cut any part of the spheroid, though
it may do so in a portion of the cases. The predominant effect will appar-
ently be to drive the outer molecules into larger orbits and throw the inner
ones back to the spheroid. Apparently this will be a self-adjusting process,
so far as frequency and efficiency are concerned, for the number of molecular
flights per unit of time will be cumulative as the acceleration of rotation
approaches the critical stage when, as we have seen, any molecular incre-
ment forward will lead to quasi-orbital ffight. This will increase the
contingencies of collision, and hence a cumulative number of molecules
will be driven into independent orbits.
Now the most significant element in this process is the partition of mo-
ment of momentum that is involved. Each molecule that passes into a
free orbit necessarily takes with it more than a mean portion of moment of
momentum. Those molecules which make elliptical ffights and return to
the spheroid without collision carry back whatever moment of momentum
they took out, but those thrown into permanent orbits retain, as a rule,
not only what they took out but also the additional moment of momentum
gained from the collisions which gave these free orbits. It follows that every
molecule that goes into a free orbit takes a disproportionate amount of the
moment of momentum of the spheroid and thus reduces its rotation, or else
retards its increase of rotation, to that extent.
If the quantitative value of this loss of moment of momentum by the
spheroid could be compared with the increment of rotation assignable to
shrinkage, it would be possible to determine whether the spheroid could
ever, tmder these conditions, reach the critical stage requisite for the sep-
aration of any portion of its mass bodily. A mode by which a rigorous
demonstration can be reached has not yet been found, but, from the nature
of the case, I entertain, with others, the view that the separation must take
place molecule by molecule, and it seems to me inevitable that these mole-
cules must go into orbits each carrying an excess of moment of momentum
at the expense of the spheroid, and hence that the critical stage of exact bal-
ance between the centrifugal and centripetal factors of the spheroid is never
reached. If so, bodily separation is excluded by the conditions of the case.
The conviction that such rotating gaseous spheroids must shed portions
of their matter molecule by molecule, if they do so at all, has long been
held by students of the subject, but I am not aware that the loss of moment
of momentum from the spheroid has been urged as a reason why the crit-
ical state prerequisite to bodily separation may not be attainable.
OONTEIBlinONS TO COSMOGONY AND THE FUNDAMEFTAL PROBLEMS OF GEOLOGY
GEOPHYSICAL THEORY
UNDER THE PLANETESIMAL HYPOTHESIS
BY
ABTHTJB C. LUNN
Iruiructor in AppHed McUhemaHa
IM
GEOPHYSICAL THEORY
UNDER THE PLANETESIMAL HYPOTHESIS.
SYNOPSIS.
This paper is devoted mainly to a quantitative study of that portion of
the earth's internal energy which is supposed to have been derived from the
mechanical energy of a primitive system of planetesimals, of its transforma-
tion into thermal form during the epoch of accretion, and its subsequent
redistribution by conduction.
In Part I a theory initiated by Fisher is developed on the basis of the La-
placian law of density, together with certain auxiliary assumptions. Form-
ulas and tables are given showing the variation of dimensions and internal
densities of the mass during the epoch of accretion, the differential effect of
deposit of a stratum on the size and moment of inertia of the mass, and the
deformation of mass-elements accompanying the resulting compression. De-
termination, under alternative secondary postulates, of the original distri-
bution of temperature produced by the compression and its redistribution
by conduction shows the existence of a characteristic zone of rising temper-
atures during the earlier stages.
Part II comprises an inquiry as to what changes in the results of Part
I are produced by changes in the secondary hypotheses employed and a
critical examination of the latter. The computed masses of the nucleus at
various stages of accretion are compared with the observed masses of the
smaller planets in the solar system. The previous theory is reviewed, with
the substitution of Roche's formula for the density; and to serve as basis
of comparison, certain other laws of density are deduced to satisfy special
conditions.
Criticism in the light of general thermodynamics leads to a recognition
of the theory given as possibly an extreme view, referring to a substance
where the work of compression is mainly frictional.
In Part III is outlined a contrasting theory for the case of a substance
such that the work of compression is done mainly against volume-elasticity,
under the assumption that the successive strata deposited at the surface are
reduced to uniform entropy by free radiation while exposed. The thermal
phenomena in this case are compared with those under the conditions of
Parti.
171
172 GSOPHTSICAL THEORY UNDKB THB PLAN8TE8IlfAL HYPOTHESIS.
INTRODUCTION.
The following studies were undertaken at the suggestion of Prof. T. C.
Chamberlin, as auxiliary to the development of the hypothesis put forward
by him as to the origin of the earth by planetesimal accretion, a main
object being to secure quantitative inferences which might aid in forming %
judgment regarding the probable efficiency of thermal energy, whose source
is gravitational, as an agent of geological importance under the restrictions
imposed by the hypothesis. Experimental evidence regarding the behavior
of substances under the enormous temperatures and pressures met with in
the interior of cosmic bodies must be considered almost wholly lacking, and
derived by highly uncertain extrapolation from determinations made within
the limited range accessible to laboratory measurement. Precise conclusions
based on accurate observed data could therefore not be looked for, but
it was felt that it should be possible to deduce with some confidence at least
the order of magnitude and general features of the thermal phenomena of
gravitational origin, under the conditions assumed by the hypothesiB in
question, when supplemented by certain minor hjrpotheses.
The general hypothesis assumes that the earth, in common with other
bodies of the solar system, was formed by the accretion of planetesimil
masses, more or less similar in chemical composition, at least when consid-
ered on a large scale, so that the more important local differences in the body
of the resulting planet are to be ascribed to differences in physical condition,
chiefly in pressure and temperature; and that the history of the earth in
this aspect comprises two main epochs, the earlier one of growth by accretion,
first at a rapid, and later at a declining rate, shading into the subsequent
longer period of relative quiescence and constancy of mass, accompanied
by a gradual redistribution and partial loss of a store of thermal energy
derived from the primitive mechanical energies of the system.
According to the mode of transformation this energy may be treated as
mainly of three kinds: (1) that which is stored in the underlying mass,
through the progressive static compression which accompanies the deposi-
tion of the successive layers at the surface; (2) that derived directly from
the kinetic energy of the masses impinging on the surface, through the vis-
cous damping of waves and vibrations due to the impacts; (3) that derived
in a similar way from motions which arise from the continual disturbance
of equilibrium produced by the surface accretions, independently of the
momentum of impact.
To trace out exactly the final distribution of the second and third kinds
would be a matter of forbidding difficulty, even if an acceptable assumption
could be made regarding the precise law of accretion. No attempt will be
made here to account for the third, but from estimates made in the sequel,
where attention is confined to the first two, it would appear to be a rather
small portion of the whole, being in fact under a certain special set of con-
ditions strictly zero.
As to the second kind, however, the energy of impact, a useful estimate
can be made very simply if it be assumed that the single ideal substance
contemplated is highly viscous. For in such cases the motions produced by
INTRODUCTION. 173
an impact do not travel far from their point of origin before being practi-
cally wiped out by frictioni thus exhibiting a kind of mechanical radiation
of the energy brought in kinetic form by the impinging mass, but in such a
way that f rictional absorption confines the distribution of that energy to the
neighborhood of its source. This sort of process would be closely imitated
if the surface structure were that of a loose aggregation of small masses,
even if the latter were perfectly solid. It is clear, therefore, that after the
transformation the impinging mass retains only a small part of its own orig-
inal energy, but it secures a certain compensation from the masses whose
deposition occurs in time and place near its own. Now, when the average
or normal energy of impact per unit-mass does not vary sensibly during the
time required for depositing a layer whose thickness is somewhat in excess
of the radius of influence implied above, then the compensation may be
regarded as practically exact, except as affected by direct radiation into
space during the time that the mass remains exposed.
With this interpretation it seems a fair equivalent to assume that each
planetesimal mass retains its own primitive kinetic energy after impact in
thermal form, but immediately loses a portion by ordinary radiation before
it is covered up. This setting of the matter will be accepted hereafter, and
it will be further assumed that the process of accretion, though slow enough
to permit the loss of a large portion of the heat of impact by immediate
radiation, is yet sufficiently rapid so that internal conduction has not time
to modify sensibly the distribution of heat arising from compression before
the growth is complete.
It may also be supposed that in connection with high viscosity the mass
would possess sufficient plasticity to enable its own gravitation to keep it in
a condition approaching hydrostatic equilibrium, with an approximately
spherical form, aside from the secondary effects of rotation and consequent
polar flattening; for it is supposed that in the long run the accretion would
be practically equable over the whole surface and that the effects of tem-
porary inequalities of serious magnitude would be quickly obliterated.
It is well understood, by analogy with the behavior of such materials as
wax and pitch, that the combination of plasticity and viscosity, such as
here contemplated as appearing under slow changes, is in no way inconsis-
tent with the appearance of extreme rigidity under the action of sudden
or rapidly varying forces. It should be noted, however, that a satisfactory
theory as to the history of the earth's dominant surface features seems to
require that to the earth-substance be attributed a rigidity sufficient to allow
the alternate accumulation and subsidence of shearing strains, deep in the
body of the earth, to such an extent that the periods involved, though short
perhaps in comparison with the durations implied in phenomena of thermal
conduction in bodies of cosmic size, are nevertheless of higher order than
the periods of precession, nutation, and tidal phenomena, which have
hitherto furnished the chief data pointing to the practically perfect extreme
rigidity of the rotating earth. The hypothesis of practical fluidity tmder
slow deformation must therefore be understood only as a crude first approx-
imation from a geological point of view. But on account of its simplicity,
and because of its occurrence in previous theories of the earth's constitution.
174 GEOPHYSICAL THEOBT UNDKB THB PLANETESIlfAL HYPOTHESIS.
it is desired to develop first the consequences of this assumption, leaying for
later study the question of the modifications needed to allow for rigidity.
The following developments refer entirely to the simple ideal case of s
spherical body, radially symmetric in every essential feature, originating by
deposition of successive spherical layers, and maintained under its own grav-
itation in hydrostatic equilibrium which is at least approximate during the
period of growth and practically exact thereafter. All variables represent-
ing the physical magnitudes concerned are therefore considered as functions
of the time and of the distance from the center of the mass. It is desired
to study the primitive distribution of thermal energy due to compression
and impact, together with the character and rapidity of the modificatioDB
brought about by conduction and radiation.
The term " thermal energy " used in the foregoing refers to the entire in-
trinsic energy of the substance as depending on pressure, density, and tem-
perature; part being the stored or latent energy of the compressional strain,
the remainder appearing in a corresponding augmentation of temperature.
It is the latter portion only which is subject to direct transfer by pure con-
duction, though its redistribution in that way, through alteration of the
geometric distribution of the mass by thermal expansion and contraction,
may lead to the redistribution also of the energy of strain, accompanied in
general by further transformation of energy from gravitational to thermal
form. The character of the phenomena might easily vary radically with
variations in the relative importance of these two portions of the intrinsie
energy, without inconsistency with the general hypothesis.
It is thus essential, for the construction of a definite theory, to include
further assumptions as to the thermodynamic properties of the earth-sub-
stance, which should cover three main points: (a) the characteristic equa-
tion of the substance or relation between the thermodynamic coordinates—-
pressure, density, temperature; (b) the form of the intrinsic energy as a
function of these coordinates; (c) the value of the thermal conductivity in
terms of the same variables. Auxiliary coefficients such as specific heats,
thermal expansion, and compressibility can then be deduced and the as-
sumptions checked or numerical parameters determined by means of obser-
vations or estimates of these physical magnitudes in the case of substances
at the earth's surface. It is evident, from the number of these secondary
hypotheses needed, that any sharply crucial test of the main hypothesis
from the present point of view is out of the question; all that can be done
is to form a judgment as to its plausibility in accounting for the play of
thermal and gravitational forces in geologic history, by developing several
alternative suppositions on these secondary points.
INTBODUCnON. 175
OBNBRAL EQUATIONS; NOTATION; NUMERICAL CONSTANTS.
The following notations will be used throughout, and special values of
the variable referring to the center and surface of the earth denoted by sub-
scripts 0, 1, respectively:
t»time. ^»los8 of potential energy in cou-
rts distance from center. traction from infinity.
x^rjr^. B = total energy of compression.
/t> = density. c= energy of compression per unit-
p= pressure. mass,
m » mass within radius r. J^ mechanical equivalent.
A; = constant of gravitation. JJ» modulus of cubic compression.
g = acceleration of gravity. a » specific heat.
V = gravitation potential. 8 = temperature.
^ = conductivity.
Numerical values are given, unless otherwise stated, in terms of the units
centimeter, gram, second, and centigrade degree, for convenience in using
published data on the absolute values of the physical constants; and are
based chiefly on the following assumed constants, unless expressly stated:
Total radius r^ = 6.370 X IV mean density p^ = 5.516
surface gravity g^—QSl
From these are obtained the following:
volume = 1.083 X 10" total mass m^ = 5.972 X 10*^ * = 6.665 X 10"^
Further are assumed :
^1=2.70 to 2.75 (estimated average for surface-rock)
J = 4.2 X 10' ff J = about 4 X 10" a^ = about 0.2 X^ = about 0.005
As stated above, it is necessary to supplement the general hjrpothesis by
certain assumptions as to the physical properties of the earth-substance, in
particular the form of the characteristic thermodynamic surface:
F(p,^,(?)-0 (1)
and the form of the intrinsic energy, conveniently an expression of the type:
eHiP^O) (2)
The fundamental equations forming the basis of the theory may then be
grouped in two classes, related to two curves, not necessarily identical, on
the surface (1). The first curve belongs to the actual distribution of the
physical magnitudes within the earth at a given time, its projection in the
P'p plane being determined in parameter form by the values of p and p as
functions of r. The second curve corresponds to the path of compression
traversed by a particular mass of the substance from the time of its deposi-
tion to the time when the growth of the planet is complete, and is practi-
cally an adiabatic curve if the accretion and, consequent compression are
relatively rapid, as is here supposed.
176 OEOPHYSICAL THEORY UNDEB THE PLANETE8I1CAL HTPOTHB8I8.
In the first class come the equations:
where
and
-^ gp withp,-0 (3)
km ,-.
m— 4;r / pr^dr (^^
which express the condition of hydrostatic equilibrium and yield the im-
portant relation:
subject to the conditions
P.-0 , %)-0
To these must be added the expression for the potential energy ex-
hausted during contraction from infinity:
(7)
2 / " / ""
where
+4ff / prdr W
the latter being equivalent to
with
(9)
Fo=47r / prdr or V^
Let
u = F-7i (10)
then equations (9), (3) give
^(,«^)+4.^.-0 (11)
where p may be considered expressed in terms of u or r, since u is necessarily
and p most probably a monotonic function of r in the concrete case.
The preceding equations then suffice to determine in terms of r all vari-
ables involved, if supplemented by a single hypothetical equation, such as
an expression for p in terms of r or a relation between p and p.
INTRODUCTION. 177
The second class of equations, in addition to (1), (2), includes the equa-
tions of the particular path of compression in question, together with the
expression for the work done in compressing unit-mass:
e- / -S.dp (12)
which gives through integration by parts the relations
ep+p —/OS d{ep) "sdp (13)
the useful auxiliary variable 8 being defined by
P (H)
where the elastic bulk-modulus H is
The limits of the integrals correspond to the supposition that the mat-
ter is compressed from surface density and zero pressure. With e so deter-
mined, and expressed in terms of r by means of equations of the first class,
the total energy of compression is
B= / edm-47r / epr'dr ^ / 8f^^dr (^6)
the last form being obtained by integration by parts and equation (13). It
may be noticed that the quantities ku and 8 are of the same physical dimen-
sions, but with distinct theoretical setting; in case, however, the two paths
on the thermodynamic surface are identical, they are equal at every point
in the body. In this special case the value of E may be given as
An essential feature of the present hypothesis is the necessity of suppos-
ing that most of the energy of impact is wasted by immediate radiation, so
that the compressional energy whose total is E plays the main part in the
succeeding phenomena. It is therefore important for purposes of compar-
ison to determine what ratio the quantity E bears to the total energy
transformed.
An idea on this point may be obtained by considering the case of a planet
formed by condensation of a primitive homogeneous sphere of density p^
Suppose that a particle at distance r from the center in the completed planet
lay momentarily at distance / when deposited at the temporary surface,
and in the homogeneous sphere would lie at distance fj^ these being subject
to the inequalities rj^>r'>r. It may be considered that the particle fell
178 GEOPHYSICAL THEORY UNDER THE PLANETS8IMAL HYPOTHESIS.
from distance fj^ to distance r*^ striking the surface, and then being covered
up more and more deeply, settled finally to distance r. The diagram gives
an idea of the distribution of density at the instant of deposition of that
particle, A being the center, the part CD along the radius referring to the
I ^ i
At BCD
homogeneous spherical shell which is being drawn up to supply surface
deposits at JB, BC the empty region through which the particles are falling,
and AB the radius of the partially formed planet, the variation of density
within which is determined by the condition of equilibrium under a definite
law of compressibility.
Since the mass already deposited is m, the energy of impact per unit-
mass is under these special conditions:
Ct^kfn
(k-K) <•"
where m, /, Tj^ are to be thought of as functions of r; then the total energy
of impact is
E.^^nk I pm(^~~-)r'dr (18)
The quantities E and £| are portions of the potential energy # — #j^ ex-
hausted during condensation of the homogeneous sphere to a condition of
density matching that of the completed planet, ^^^ being the value of ^ for
a sphere of assigned mass m^, and uniform density p^, and determined by
♦.-l*^ rA-^ (19)
The remainder of the energy ^ —^h, if any, is to be treated as of the third
kind named above; but in case the equilibrium in the mass already depos-
ited at each moment of the process is adiabatic, so that the two thermo-
dynamic curves mentioned are identical, it seems probable that this third
kind does not exist, and the total amount ^ - ^j^ transformed is completely
accounted for as £+£^. This is later proved to be the case for one partic-
ular pressure-density law. In any case it is of course not meant that the
primitive distribution need be at all like that in the homogeneous sphere
mentioned, but any other supposition would modify only the energy of im-
pact, leaving the above point of view still useful as a check on the compu-
tation of the compressional portion E; this is verified by the expression of
£i as the difference
r"
"^^*/'?
INTBODUCnON. 179
the first member being the energy of impact corresponding to parabolic
velocities, the second being identical with ^j^. The former would corre-
spond to formation from a Cfystem of small particles each of density p^t but
infinitely dispersed, so that the potential energy available is <P. It should
be noted also that the part E is not affected by the existence of velocities
in the primitive planetesimals due to attractions other than that of the
nucleus on which they fall, and can therefore be treated independently
of any supposition as to their distribution and motions previous to the
aggregation.
12
Part I.— THE THEORY OF FISHER.
GENERAL EQUATIONS.
A definite form for the development of the theory has been initiated by
Fisher/ on the basis essentially of three main suppositions as to the prop-
erties of the ideal earth-substance. These are: (1) that the path of com-
pression traversed by any particular element of the mass is identical as far
as it goes with that defined by the relation between density and pressure
within the earth in its final state; (2) that pressure and density are related
as specified by the classic law of Laplace; (3) that rise of temperature dur-
ing compression is proportional to increase of compressional energy, or of
work done to produce compression.
For critical purposes it will be necessary to undertake a close scrutiny of
these assumptions, as to their agreement with pertinent observed data, and
also, in the light of general thermodynamic laws, of their consistency with
each other, or at least of the exact interpretation to be accorded them in order
to assure consistency. But in view of their close affiliation with the stand-
points adopted in many previous studies of the constitution of the earth's
interior, it will be worth while to develop their consequences in some detail.
The pressure density law, proposed by Laplace,' has been used by many
writers on geophysics, partly it would seem on account of its mathematical
convenience, being the only one which reduces equation (6) to linear form.
It is based on the condition
-^^hp A=const. (20)
which, when applied to the compression of an individual portion of the
mass, gives
P=|0>'-/>.*) (21)
and for equations (12) and (14) the particular forms
. = *^' (22)
2 p
s-hip-^p,) (23)
Assumption (1), however, which gives s=A;u, allows the use of the relation
(20) in the differential equation (6) or (11), which reduces to
the appropriate solution being
sin n r^^^
P=Po-j^ P=9r » (25)
* Rev. O. Fisher. On Rival Theories of Cosmogony. Am. J. of Sc, xi, 1901, p. 414.
* Laplace, M^canique Celeste, book xi, chap. iv.
180
THB THEORY OF FISHER. 181
leaving the constants p^^, qtohe adjusted according to observation and thus
determine h indirectly.
Formulas for the principal variables are, then/
m - ^ (sin p-p cos p) (26)
47rfc|t?o sin j8— ff cos P .^^^
9-— ^ (27)
also
^{(¥)"-(¥)'} <»'
whence
^-f+f 0>-ft) or 7-ip(«i^-co8ft) (29)
For comparison of the variables p, p, e, with their values at the center, there
are the ratios
p sinff
Po" P
(30)
(31)
(32)
^P ^ / /p-M
while the mean density is
p^^^Zp^mhu^^lA (33)
or in form of power series:
^, ft»V 31 6! "^ 7! /
The expressions for E and 4 take particular forms readily reduced to
E"
giving for these total energy values:
S.*?^V.{l+^'-4(?^»)'+3^«cos/?,| (34)
' Most of these fonnulas occur in Fisher's paper, or in antecedent wiitingi where the
Laplacian law is used.
182 GEOPHYSICAL THEORY UNDER THE PLANETE8I1CAL HYPOTHESIS.
(35)
The constants p^, q, may be determined; for example, so as to give
accordance with any assumed values for mean density p^ and surface den-
sity p^, or the latter may give place to the condition resulting from obser-
vations of precession and polar flattening,^ which, according to somewhat
uncertain theory, indicate for the angle P^ a value in the neighborhood of
140^. The following computations are based on the assumed value 141.8^,
together with the numerical constants as listed in the introduction, giving
the following table under the assumed law:
Ptipt
Pt
p»
2.475-141.8°
0.25
2.717
10.87
5.649X10"'
9
E
3.886 X 10-*
4.096X10"
3.620 X10»
2.466 X10*»
1.768X10**
The resulting value of the surface dendty pi, though perhaps rather
Bmall, accords fairly well with eslimates of the mean density of superficial
strata, and an additional check on the applicability of the density-law in
question is found, as pointed out by Fisher, in the fact that the surface
value H^ of the elastic bulk-modulus ranges close to values found by direct
measurement.*
The accompanying tables, then, show the distribution of the principal
magnitudes at the close of the epoch of aggregation. The columns referring
top,p,g are of course not novel, but are inserted for the sake of having all
computations based on an uniform set of numerical constants.
Tablb 1.
r/n
p
p/pi
P/Po
fi^)
0/ffi
0.00
10.874
4.002
1.0000
1.0000
0.0
0.000
.05
10.846
3.992
.9974
.9975
96.3
.098
.10
10.763
3.961
.9898
.9900
191.8
.196
.15
10.625
3.911
.9772
.9776
286.0
^2
.20
10.435
3.841
.9596
.9604
377.3
.385
.25
10.193
3.751
.9374
.9385
466.1
.474
.30
9.902
3.644
.9106
.9120
548.8
.669
.35
9.564
3.520
.8796
.8813
627.4
.640
.40
9.183
3.380
.8445
.8464
700.3
.714
.45
8.761
3.225
.8057
.8077
767.0
.782
.50
8.303
3.056
.7636
.7656
826.7
.843
.55
7.814
2.876
.7186
.7204
879.1
.896
.60
7.296
2.685
.6710
.6724
923.8
.942
.65
6.755
2.486
.6212
.6221
960.4
.979
.70
6.195
2.280
.5697
.5700
988.6
1.008
.75
5.621
2.069
.5170
.5166
1008.3
1.028
.80
5.039
1.855
.4634
.4624
1019.6
1.039
.85
4.452
1.639
.4095
.4080
1022.3
1.042
.90
3.867
1.423
.3556
.3540
1016.7
1.036
.95
3.287
1.210
.3023
.3011
1002.8
1.022
1.00
2.717
1.000
.2499
.2500
981.0
1.000
' Tisserand, M^. G^l., n, p. 235.
* Fisher, loc. cit., p. 4i7; also Kelvin and Tait, Natural Philosophy, n, p. 415.
THE THEORY OF FISHER.
183
In table 1, fifth column, /(x) stands for (1— ix*)^ tabulated for com-
parison because of its occurrence in an approximate formula for the density
used later. The maximum value of g occurs at x« 0.8411, or about 630
miles below the surface.
Tablb 2.
r/ri
lO-Mp
P/Po
l(H»»H
B/Hi
1(H««
e/eo
6e/J
.00
3.076
1.000
6.661
16.02
169.8
1.0000
20,210
.06
3.069
JuvO
6.627
16.93
169.0
.9967
20,120
.10
3.009
.978
6.428
16.69
166.9
.9830
19,870
.16
2.927
.962
6.266
16J29
163.3
.9619
19,440
.20
2.816
.916
6.042
14.76
168.4
.9329
18,860
.26
2.677
.870
6.766
14.07
162.1
.8961
18,110
.80
2.616
.818
6.440
13.28
144.6
.8620
17,220
.36
2.333
.769
6.076
12.39
136.0
.8011
16,190
.40
2.136
.694
4.679
11.42
126.3
.7441
16,040
.46
1.926
.626
4.269
10.40
116.7
.6814
13,770
.60
1.708
.666
3.826
9.34
104.3
.6141
12,410
.66
1.489
.484
3.388
8.27
92.2
.6434
10,980
.60
1J272
.414
2.963
7.21
79.7
.4697
9,490
.66
1.061
.346
2.632
6.18
67.0
.3946
7,970
.70
.860
.280
2.129
6.20
64 J2
.3191
6,460
.76
.672
.218
1.763
4.28
41.6
.2462
4,960
.80
JKX)
.162
1.409
3.44
29.7
.1749
3,630
.86
.346
.112
1.100
2.69
18.7
.1106
2,230
.90
.210
.068
.830
2.03
9.6
.0669
1,130
.96
.096
.031
.699
1.46
2.7
.0162
330
1.00
.000
.000
.410
1.00
0.0
.0000
The second and fourth columns give the values of the pressure and bulk-
modulus in millions of atmospheres, an atmosphere being taken as one
megadyne per sq. cm., while the seventh column gives the specific com-
pressional energy in billions of ergs per gram. The last column is added
for the sake of vividness of interpretation in terms of temperature, being
the measure of the thermal equivalent of e in centigrade degrees, under
specific heat one-fifth that of water.
Of the total store of potential energy ^, it appears that 72 per cent is
accounted for by ^j^, which would be the energy exhausted if a system of
planetesimals, originally infinitely dispersed, united to form a homogeneous
sphere of density p^; this portion is to be thought of as energy of impact.
Of the remaining portion ^— ^j^, the compressional energy E accounts for
53 per cent.
It remains to be determined whether the excess of ^ - ^j^ over E is also
to be ascribed to energy of impact. That a part at least must be so con-
sidered follows from the assumed character of the actual aggregation. For
when a given particle impinges on the momentary "surface, the mass already
gathered has condensed, under its own gravity, to smaller dimensions than
it would occupy in the homogeneous sphere; the radius being smaller, the
velocity of impact is greater than would be the case under the attraction of
the same mass in homogeneous form with the minimum density p^. For a
particle having any assigned position in the completed planet the attracting
mass within can be computed by (26). But the velocity of impact depends
also on the momentary radius of that mass, before it has been further con-
184 GEOPHYSICAL THEORY UNDEB THE PLANETE8I1CAL HYPOTHESIS.
densed by deposition of the remaining layers, so that in order to determine
the total amount and distribution of the energy of impact it is necessary to
know how the compression advances in the existing nucleus as material is
deposited at its surface.
HISTORY OF THE COliPRESSION.
It is possible, under the foregoing assumptions, to trace such a history
of the accretion, if it be supposed that the equilibrium of the mass is main-
tained at each stage of the process. For if the preceding equations can be
accepted as approximate representation of the action of the earth-substance
under compression, to a similar approximation equation (20) must be viewed
as representing a physical property of that substance, independent of the
dimensions of the mass into which it is aggregated, the constants A, q being
physical constants of the material. The distribution of density in a body of
any size is then represented by equation (25), without change in the value
of q, but with the maximum value of j9 chosen to agree with the total radius,
and with central density such as to give the surface density the fixed value
p^. The distance of any particle from the center is then fixed by the angle
fi, which is the measure of that distance in terms of a unit of 1/g centi-
meters, or about 1,600 miles.
Let r/ be the radius of the nucleus already formed at a certain epoch,
and r^ its radius when compressed under the total load afterwards depos-
ited; let /, r be the central distances of any interior particle at the same
epochs; then r/ is to be determined as a function of r^, and r' as a function
of r and r,; the latter function describing the history of the condensation
in the sense that it fixes the position of any assigned particle at any epoch,
if the epoch is specified by the final location of the particles which then lay
at the surface. A translation into time-relations could then be made for any
postulated law of variation in the rate of accretion. Let p^' be the central
density at the earlier epoch. For brevity put
JB=^ C^ainP-pcosP (36)
to be indexed, like the various values of j9, by analogy with the values of r
to which they refer, as in equation (26) .
By equation (26) the identity of the masses within the radii r/, r, at the
respective epochs gives the condition
p,'C/^p,C. (37)
similarly for the corresponding radii r', r;
PoC'^p,C (38)
and from the constancy of density at the momentary surface
P,'B:^P,B, (39)
These equations are to determine the values of p^'j /?/, ^y for any assigned
values of /?, and /?, for instance first ^9/ from (37) and (39) which give
THE THEORY OF FISHER. 185
then p^' from (37) and finally ^ from (38). The values of j3/ and p^'
as functions of j9^ are given in table 3 below. The former were obtained
by interpolation from an extended table giving C/JB and C/B| as functions
of j9. To trace the positions of interior particles at different distances and
different epochs would require a double-entry table, which could be supplied
by equation (38) after p^' has been computed.
The specific energy of impact, or kinetic energy of a unit-mass falling
from infinity to the surface of a nucleus of mass m, and radius r/ is, then,
e.-^ (41)
where by (26) the attracting mass is
wi,=— 3-0,
The values of e^ are given in table 4, in terms of j9., which determines the
ultimate position of the particle.
The total energy transformed by impact is, then,
^«-4;r / \,.^^.r*dT,
which, since p,^p^B^ is equivalent to
But by equation (40)
A'-A-cotA'-?i^A^^2»A
ixrhence
Use of this relation to transform the variable of integration from j9, to
P,' gives
^' ^ j[ Wb7\P7W'^^' r'
where the integrand is expressed entirely in terms of j9/, the upper limit
of^the integral being unchanged according to (40). The result of the inte-
gration is
len^kp.'B,' r(l-ff,coti90» l-i8,coti8^ Pr\
which may be written:
^^,^»^ I Wi._ 6 sinff^,cos A +|eos» ^-1} (43)
186 GEOPHYSICAL THEORY UNDER THE PLANETE8I1CAL HYPOTHESIS.
Gomparison with (34) and (35) gives the fundamental relation
E+Et"* (44)
showing that under the conditions here assumed the original store of poten-
tial energy is entirely accounted for as transformed by impact and static
compression. The proof refers to a primitive condition of infinite disper-
sion, but a similar conclusion would hold for any initial configuration and
distribution of velocities provided ^ stands for the entire primitive store of
energy, potential and kinetiC; a variation in which would make an equal
change in £^ but leave E unaltered.
As represented by the preceding equations the character of the process
of accretion, upon a nucleus composed of material of definite compressibility,
implies that the deposition of a new layer of given thickness brings about a
certain increase of compression of the nucleus and a corresponding sinking
of the former surface toward the center; only part of the thickness of the
new stratum is thus effective in producing actual increase of geometric
dimensions. In order to specify this differential depression numerically, let
a factor of depression D be defined as the ratio, to the total thickness of
stratum, of the part which sinks below the level of the former surface;
then 1 — D is the ratio of geometric increment of radius to total thickness of
stratum.
For simplicity of notation let p now stand for the angle equivalent of
the radius of the nucleus at a given epoch, the mass being by (25) and (26) :
m
-^{i9-^cot^}
where p^^ is the fixed density of surface rock, so that
dm=^|(l-/9cot^)»+^}d/9
dm being the increment of mass and dp the increment of radius in the angu-
lar units defined. If, however, a stratum of the same mass dm were laid
down without producing compression of the nucleus beneath, the relation
to the total thickness dp of the stratum would be given by
From these follows for the depression factor
dp ^ ^ /pBinpy (45)
where C is defined as in (36). The factor D is tabulated in column 6 of
table 3, in terms of /?/, which there represents the momentary radius of the
free surface.
THB THBOBT OF FISHBB, 187
A similar situation occurs with respect to the effect of accretion on the
moment of inertia and period of rotation of the planet. The moment of
inertia is
-g-TT / pr^dr
or, measured for convenience in terms of a unit equal to -5^* G. G. S.
units, pP
which reduces to
2/J*-J(6-^ (46)
Now, in the same units the stratum itself adds to the moment of inertia
the quantity
91
.^a)9-/9«|l+(^)'|d^ (47)
But this is in excess of the true increment, because the differential conden*
sation diminishes the moment of inertia of the underlying mass. The true
increment by (46) is
d/-{^+?^+(^-6)(^y}d^
which may be written:
dl -W-2i9»K (^- 1) d^ (48)
where
giving for comparison:
which ia tabulated in column 7 of table 3. Then by combination of equa-
tions (45) to (49) may be computed
rflog/ j8» 1 dl .„v
which gives the ratio of the percentage increase of the moment of inertia to
the percentage increase of the radius. This would indicate also the percent-
age change in the length of the day if the stratum were deposited entirely
under normal incidence, or if the moments of momentum of the planetes-
imals with respect to the existing axis of rotation exactly compensated
each other. If, however, it be supposed that variations in the rotation, or
even the existence of the rotation, were brought about by lack of such com-
pensation, then the equations allow this effect to be distinguished from that
due merely to changes in the moment of inertia.
188 GEOPHYSICAL THEORY UNDEB THE PLANBTE8IMAL HYPOTHESIS.
As illustration may be considered the case of superficial strata deposited
when the earth had attained practically its final dimensions, and corre-
sponding therefore to the last entry in the tables. A stratum, for instance,
exerting the same pressure as a mercury column of 760 mm. would have a
thickness of 3.8 meters; the depression factor D is 0.737; hence that stra-
tum would depress the former surface 2.8 meters, giving actual increase to
the radius of only 1 meter. This increase bears to the total radius a ratio
^-—X 10~*, while the value of ., ^ is 7.08; hence, for the case of normal
impact, the corresponding increase of the period of rotation is about
-Q X 10~* of the whole, or 0.096 of a second for a day of the present length.
To increase the period just one second out of 86,164 would require a
stratum 39.7 meters thick. The illustration also indicates that under the
assumed law of compressibility the mean pressure of the atmosphere is
responsible for a diminution of the earth's radius amounting to 2.8 meters.
The preceding computation of pressures and densities has been based
on the condition of hydrostatic equilibrium, which for any substance not
completely fluid can be considered strictly applicable only in case each por-
tion of the mass is subjected to compression in such a way as to avoid any
<listortion of shape, which would call into play reactions against shearing
stresses. It will therefore be instructive to determine what kind of defor-
mation in the elementary portions of the mass is implied in the foregoing
account. In view of the radial symmetry assumed, the distortion at any
point in the completed planet may be expressed in terms of a distortion-
factor S, defined as the ultimate ratio of vertical to horizontal dimensions
of a mass which when first deposited at the surface was cubical.
The ratio of final to initial horizontal dimensions is PJP/, being the
same as the ratio of radii of two spheres passing through the same particles
at the respective epochs. The vertical or radial ratio in the same sense is
if dp', dp represent the respective thicknesses. These give for the distor-
tion-factor
pi
^^3-^, since conservation of the mass of a stratum implies
^(IT
whose value at various depths is given in column 5 of table 4, which shows
that the vertical compression throughout exceeds the horizontal, the differ-
ence being most marked about one-fourth of the way to the center, where
the ratio is 0.824. This would be a violent deformation for a body with
perceptible rigidity, but may be admitted in the present theory if it be
supposed that under extremely slow changes the substance is practically
plastic, even if highly viscous. The chief uncertainty would then relate
to the energy of compression, which should take account of the work done
against viscosity under shear. Further comment on this matter will be
reserved for another place.
THE THEOBY OF FISHER.
189
Tablb 3.
fit
^•'
Po'
w«
P*'
IV
dl
dlogl
^l
fii
mi
9i
x»r
U
dlogfi
0.00
0.000
2.72
0.00
0.000
.000
1.000
6.00
.06
.079
2.73
.000246
.003
.004
.999
6.00
.10
.168
2.79
.00196
.012
.017
.993
6.01
.16
.235
2.89
.00666
.027
.038
.985
6.02
.20
.311
3.00
.01638
.060
.067
.973
6.06
.26
.384
3.17
.02963
.077
.102
.968
6.10
.30
.464
3.38
.06036
.111
.144
.940
6.14
.36
.620
3.64
.07834
.161
.191
.920
6.20
.40
.682
3.96
.1142
.196
.241
.897
6.27
.46
.641
4.31
.1583
JZil
.294
.873
6.35
.60
.696
4.72
.2107
.303
.348
.847
6.45
J66
.744
6.19
.2711
.364
.402
.821
6.66
.60
.789
6.71
.3390
.430
.456
.794
6.69
.66
.829
6.28
.4136
.499
.604
.768
6.83
.70
.866
6.90
.4938
.671
.660
.744
6.99
.76
.896
7.66
.6782
.645
.693
.720
6.16
.80
.924
8.23
.6662
.720
.630
.699
6.34
.86
.948
8.91
.7630
.796
.664
.679
6.63
.90
.968
9.69
.8394
.867
.692
.662
6.72
.96
.985
10.26
.9226
.936
.717
.647
6.91
1.00
1.000
10.87
1.0000
1.000
.737
.634
7.08
Table 3, taking as argument the ultimate distances from the center of
what were the surface particles at the respective epochs, shows the varia-
tion of the relative radius, central density, mass, and surface gravity, of the
growing planet, together with the depression factor for differential accretion,
and the factors used in determining the effect of differential accretion on the
moment of inertia; in every case in terms of j9/, which represents the actual
momentary radius.
Tablb 4.
ei«lO-»
5C|«
J
tee
S
0.00
.05
.10
.16
.20
J26
JSO
.85
.40
.45
M
J66
.60
.66
.70
.75
.80
.85
.90
.96
1.00
0.0
1.9
7.7
17^
80.9
48.3
69.4
94.2
122.6
164.4
1B9JS
227.7
268.6
311.9
866.9
408 J2
460.1
496.6
541.9
586.0
624.9
280
920
2,070
3,680
5,760
8,260
11,210
14,690
18,380
22,660
27,110
31,980
37,130
42,490
48,000
63,660
69,120
64,510
69,640
74,390
0.00
.89
.77
1.16
ia»
1.93
2.81
2.70
8.06
8.45
8.88
4.19
4J66
4.91
bJ26
bM
6.90
6.19
6.47
6.72
6.95
1.000
.998
.993
.986
.976
.963
.948
.931
.914
.896
.877
.861
.846
.834
.826
.824
.880
.845
.874
.928
1.000
190 GEOPHYSICAL THEORY UNDER THE PLANBTE8IMAL HYPOTHESIS.
Table 4 is a continuation of table 3, giving in terms of the same argu-
ment the specific energy of impact for a particle falling from infinity, in
billions of ergs per gram, the same in centigrade degrees imder specific heat
0.2, the surface parabolic velocity in miles per second, and the distortion
factor indicating the permanent deformation of elements of the mass at the
various points along the radius.
THE THERMAL PROBLEM.
It has been supposed in the foregoing that the distribution of heat was
not sensibly affected by conduction during the relatively short epoch of
accretion. If, now, it be supposed that the subsequent changes in distri«
bution are determined by conduction only, in accordance with Fourier's
laws, then the form of the temperature curve d^d(r,t) at each instant
may be determined by the differential equation
provided the form of the curve at the initial instant, say <»0, appropriate
conditions relating to the surface, and the values of the conductivity X and
specific heat a be assigned. Since the variations of X and a under changes
in the physical condition of the substance are almost purely matters of
conjecture, the chief value of such an inquiry might well be considered to
lie in the determination of features of the thermal phenomenon which seem
to persist imder varied assumptions on these points. Since, however, the
method of superposition of special solutions is practically the only known
way of obtaining general solutions of equations like (52), it wiU be supposed
that the latter is linear, X and a being assigned in each special case as func-
tions of r but independent of 0, and that the surface equations are linear
and homogeneous in and its derivatives.
According to Fourier's method, of expansion into an infinite series each
term of which is a solution of (52) and satisfies the surface condition, the
solution may then be sought in the form
<?(r,0-T^»«"^»t/.(x) (^3)
n=l
where
r,^^-fll (64)
and y^ {x) or y (jx^ x) f or n = 1 . . . oo are the appropriate fundamental
functions, which are to be determined from
where ^(x)=^/^, <p(x)='oplo^Q, and the successive values of /x employed
are those which allow the individual terms of (53) to satisfy the surface
condition. The coeflScients in (53) will be given by
r a:' f (x) y„{xy dx
4/0
THE THEORY OF FISHER. 191
if the primitive temperature curve is given in the form
e(rfi)^d,F(x) (57)
0^ being the initial temperature at the center.
To determine the form of this initial curve it is necessary to know in
what way the storage of the compressional and impact energy is mani-
fested in rise of temperature, which is again a matter of hypothesis. Fish-
er's third assumption, as stated above, implies that the energy stored in
unit-mass stands in a definite ratio to the increment of temperature. That
ratio is, then, of the nature of a specific heat, expressed in mechanical imits,
but is subject to question as to its identification, as is tacitly done by Fisher,
with the specific heat in the ordinary sense, as relating to rise of tempera-
ture due to heat transferred by conduction or radiation. It will be shown
later, however, in connection with a detailed criticism, that the theory thus
developed may fairly be considered self-consistent, in the light of thermo-
dynamic laws, when associated with what appears as a certain extreme view
regarding the thermodynamic properties of the earth-substance. An effort
will then be made to develop an opposite extreme view, to permit comparison.
For the present, then, the primitive temperature wiU be supposed de-
fined by
^-^' (58)
where v is the fractional value of that part of the energy of impact which
remains after the loss by dissipation at the surface, and a is the specific heat
identical with that occurring in equation (52). A variation of a with the
density need not be excluded, but any possible variation with the tempera*
ture will be disregarded, not only because of the practical value of keeping
the linearity of equation (52), but also becaiise any imcertainty from this
source would be bound up with the inevitable obscurity of the very notion
of temperature under circumstances so far beyond the range of laboratory
tests.
The multiplier y may also vary for different portions of the mass, its
value depending on the rapidity of the accretion. It could be unity as one
extreme, for deposits made with such rapidity that each stratum is covered
up before its loss of heat by radiation is sensible; or zero as the other ex-
treme, for accretion so slow that the cooling of the momentary surface-
stratum by radiation is practically complete.
One curious possibility may be noted in passing. With sufficient veloc-
ities of impact and appropriate values of y, it would be possible for the
expression e+y«< to have the same value for all strata; if then a were like-
wise constant, this would indicate a primitive temperature uniform through-
out the body. For example, if Ci should be for each stratum that derived
from impact at the corresponding parabolic velocity, as listed in table 4,
and y should range from 0.38 for the central portions to 0.27 for the surface
layer, then e+yc< would have everywhere the value 1.7X10" ergs per gram
mass, and for a =0.2 this would give a primitive temperature of about
20,000°. Since this value of a may be too low, and it is practically certain
192 GEOPHYSICAL THEORY UNDER THE PLANETE8IMAL HYPOTHESIS.
that not all of the energy can be considered as manifested in the raising
of the temperature, this number is probably somewhat too high. But in
any case it appears that the planetesimal hypothesis could thus plausibly
assign an origin for precisely the kind of initial thermal condition of the
mass postulated in Kelvin's famous theory, which would then serve to
indicate the subsequent thermal process, and in particular the probable
age of the earth reckoned from the close of the epoch of formation.
It seems, however, more in accord with the spirit of the general hypoth-
esis to assign no such relative importance to the energy of impact. For
the value of e^ should probably be assumed much less than its paraboUe
value, being due to the relative velocities simply of bodies which might
partake more or less of a common motion; for instance of bodies moving
in similar directions around the sun in intersecting orbits of moderate eccen-
tricity. Moreover, even with what might be considered a rapid rate of
accretion, when compared with astronomical processes in general, by far the
greater part of the heat so generated must be expected to escape quickly
by radiation, thus making v a small fraction. The computations following
refer to the case when the value of ve^ is insignificant, so that the compres-
sional energy alone is effective in producing the primitive temperatures,
which are then to be computed from column 6 of table 2 and the assumed
values of a.
The same relative freedom of dissipation into space would tend to keep
the surface at a low and equable temperature after the accretion had
ceased. It will, therefore, be assumed that the surface temperature keeps
a fixed value, which may be taken as the zero of reckoning, and whose
place in an absolute scale would depend largely on the thermal influence
of the atmosphere. It is further supposed that the rise of temperature
produced by compression can be reckoned from this point. With these
stipulations the solution will be determined if definite hypothesis is made
regarding the conductivity and specific heat.
If X and a are uniform throughout the mass, then ^(x) = 1, and ^(z)
=^plPo- Since great precision is needless in these computations, the ratio
plPf^will be replaced by the convenient expression (1— ix*)*, whose close
accordance with it appears from columns 4 and 5 of table 1, and which
simplifies the determination of the solutions of equation (55), which then
reduces to
the required solution of which may be written:
2/(/i,x)=2*(-iya,x^* (60)
i=0
where the a's, which are functions of /i, are all positive and determined by
the simple recursion-formula
(63)
THE THEORY OF FISHER. 193
The values of fi required for the expansion (53), with the surface con-
dition named, are the roots, infinite in number, of the equation
y(/i,l)-*ir(-l)'a,=0 (62)
These roots may be determined as far as required by a trial and error tab-
ulation of y(ji, 1), the computation yielding also the coefficients a^ but
increasing rapidly in length with each succeeding root. The first two roots
and the corresponding fundamental functions are:
/ij- 13.0689
y,(a;)-l -2.17648 x»+2.07407x*- 1.39934 a;*+.72867a:»-.31419x*»
-f. 11658 x«-.03812x" + .01120x**-.00300x^"
+ .00074 x»-.00017x» + .00004x"- .00001 x*« ....
/c, -55.313
y,(x) - 1 - 9.21889 x» + 28.26204 x*- 49.69103 x* + 61 .65731 «•
-59.54423 x*«H-47.37952 x"-32.22346 x" + 19.21509 x**
- 10.23514 x«H- 4.93949 x»- 2. 18394 x» + . 89259 x"
-.33971 x» + .121 14 x»-. 04068 x« + . 01292 x»
- .00390 x»* + .001 12 x«»- .00031 x» + .00008 x«
-.00002x^ ....
The number of terms to be included and the magnitude of the individual
coefficients increase rapidly with the index of the component, so that if
many components are of sensible influence in the representation of the
primitive temperature curve, the accurate determination by this method of
the transformations produced by conduction would require computations
of serious length. But the effect of the higher components on the general
features of the cooling process can be conjectured, with the aid of certain
general properties of the fundamental functions which are obvious in the
light of the theory of linear differential equations of the second order.
The constants fiv p^t * - * will show a rough proportionality to the
squares of the natural numbers. The solutions y{pL^ x) or yjjt) will be
oscillatory in such wise that y^ has n — l roots between and 1, the inter-
mediate half-waves having amplitudes all less than the central amplitude,
which is unity, and decreasing in order when counted from x=0 toward
x« 1, exhibiting these features in a more marked fashion than the functions
, because of the fact that the coefficient tplx) decreases as x goes
from to 1.
For constant a the primitive temperature curve is obtained from the
curve for e/e^, which has unit amplitude at the center, by multiplying
all ordinates by the central temperature, -j; which for <r = 0.2 is about
20,000^, the temperatures being then given by column 8 of table 2.^ It
* The temperatures given by Fisher are larger than those here listed in the ratio of the
number of pounds in a cubic foot of water, presumably through some inconsistency in the
units used m the equivalents of formulas (21) and (22), which however does not occur in
the corresponding npressions in his "Physics of the Earth's Crust," p. 29.
194 GEOPHYSICAL THEORY UNDEB THE PLANETB8IMAL HYPOTHESIS.
will be convenient however to express the temperatures and coefficients
A^ in terms of the central temperature as unit, to leave free choice as to
the numerical value of a. The coefficients A^, A^ then have the values
named below, computed according to (56) by mechanical quadrature.
The influence of the first two components appears from table 5. The
second and third columns give the first two fimdamental functions, the
fourth the primitive temperature curve, and the next two the terms out-
standing after subtraction of the first and of the first two components re-
spectively.
Taslm 5.
z
»i
Vt
F(x)
Fiix)
Ffix)
0.00
1.0000
1.0000
1.0000
-0.0537
-0.0372
—10.00
.05
.9946
.9771
.9957
— .0523
— .0362
— 9.97
.10
.9784
.9106
.9830
— .0480
— .0330
— 9.89
.16
.9521
.8063
.9619
— .0413
— .0280
— 9.76
.20
.9162
.6734
.9329
— .0326
— .0214
— 9.67
.25
.8717
.5230
.8961
— .0225
— .0139
— 9.33
.30
.8199
.3667
.8520
— .0120
— .0050
— 9X»
.35
.7621
.2158
.8011
— .0019
+ .0017
— 8.67
.40
.6996
.0798
.7441
+ .0070
-f .0083
— 8J24
.45
.6338
— .0343
.6814
-f .0135
-f .0129
— 7.74
.50
.5662
— .1222
.6141
+ .0175
+ .0155
— 7.16
.55
.4981
— .1826
.5434
+ .0186
+ .0156
— 6.60
.60
.4305
— .2164
.4697
-f .0160
-f i)124
— 6.78
.65
.3647
— .2266
.3945
-f .0102
+ .0065
— 4.82
.70
.3024
— .2176
.3191
+ .0005
— .0031
— 8.76
.76
.2414
— .1937
.2452
— .0092
— .0124
— 2.47
.80
.1849
— .1599
.1749
— .0199
— .0225
— 0.86
.85
.1325
— .1205
.1105
— .0291
— .0311
+ 1J21
.90
.0842
— .0790
.0559
— .0328
— .0341
-h 4i)l
.95
.0401
— .0382
.0162
— .0260
— .0266
+ sm
1.00
.0000
.0000
.0000
.0000
.0000
+14.22
Here
F (x) = W„ the primitive temperature curve;
F,(x) "F (x)- Ajy,(x) A^ =1.0537
Ft(x) =Fj(x) -il,y,(x) A, = - 0.0165
In the seventh column -rr is the initial rate of change of temperature
in convenient units according to equation (65) below; these entries, to be
reduced to absolute units, must be multiplied by
0,^
31
as they stand the
umt of change would be about 3i^ in a billion years, with the numerical
constants used.
The results show that the first component is by far the most important;
its amplitude at the center differs from that of the temperature curve by
only 5i per cent, and the divergences fall below that percentage over a
range of more than eight-tenths of the radius from the center. If this com-
ponent alone were present the temperature at any time would be represented
by
THE THEORY OF FISHEB. 195
exhibiting the aimple type of cooling described by Fourier, where conduc-
tion produces simply a progressive diminution in the ordinate scale of the
temperature curve, according to the time-exponential law, without change
in the ratios of the ordinates. Since y^ix) is everywhere positive, this
would mean an actual decline of temperature at each point proportional
to the temperature itself. The gradient at the surface would be initially
1^ in 417.5 meters, for <7""0.2, and would decline according to the same
law as the temperatures themselves. This shows that under the condition
for the moment assumed only a small part of the present observed gradient,
about 1^ in 30 meters, could be ascribed to this component, unless the
specific heat were taken very low.
The time-rate of the process is specified conveniently by means of the
interval r, which is the time required to reduce the amplitude of the corre-
sponding component to - or 0.368 of its primitive value, and is to be deter-
^ k
mined by (54), when the value of — , the "thermometries conductivity,
ap
is assigned. If the latter be taken in the neighborhood of 0.01 for surface
rock, the value of -~^ being one-fourth of this, the value of r^ is about
4 X 10" years. The time required to reduce the amplitude of the first com-
ponent by 1 per cent would be about four billion years.
Any higher component dies out in a similar way, at a rate indicated by
its value of r; but because of the alternation in sign of the fundamental
function, would, if occurring singly, indicate falling and rising of tempera-
ture for successive zones in alternate order along the radius, the number
of zones being equal to the index of the component, with the central tem-
perature falling or rising according to the positive or negative sign of the
coefficient A. Thus the second component has a negative coefficient, in
magnitude less than one-sixtieth that of the first, but with r, somewhat less
than iv^; since y, is everywhere numerically less than y^, this means that
with respect to changes of temperature the second component simply modi-
fies the e£Fect of the first nowhere to an extent more than one-fifteenth of
the total e£Fect due to the latter alone. In the zone extending 0.43 of the
radius from the center the temperature falls somewhat more slowly and
thence outwards more rapidly with only these two components included
than would be the case with the first alone.
The influence of each further component could be traced in a similar
way, and many would doubtless be foimd to be sensible within the range
of accuracy of the tables above, if the computation to that degree of ac-
curacy should prove to be feasible. But in the absence of simple analytic
expressions for the functions involved it would be necessary to do this by
numerical calculations of extreme length on accoimt of the greater and
greater number of the coefficients a^ needed, and the insufficiency for deter-
mining the coefficients A^ of a tabulation of the fimctions with a moderate
number of entries.
The residuals in column 6 show that the influence of the higher compo-
nents is meager in the central portions, but relatively serious in the more
13
196 GEOPHYSICAL THEORY UNDER THE PLANETESIMAIi HYPOTHESIS.
superficial zones. It is accordingly in the latter region that such general
considerations as above are insufficient to give a just idea of the complete
thermal process.
Conjecture may, however, be made with some confidence as to the prob-
able character of the modifications produced by the higher components, by
direct inspection of the primitive temperature curve, which by (22) and
(58), with the substitution of the approximate expression used previously
for the density, becomes
e =i?o(i -^T(i - \xy{i~x')-^ (64)
which is accurate enough for the purpose. Then, according to (52), the
initial rate of change of temperature becomes:
bO ex
dt a^jr
r.(-«-f--T^--f--T''4'"}-{'4-r
(65)
/9 X
in which the coefficient of — ^-S is that tabulated in column 7 of table 5.
For the part of-rr due to any single component, say the nth, the corre-
'^ at
spending coefficient would be fi^A^^.
These figures indicate that from the center outwards over a distance of
about eight-tenths of the radius the thermal process is not very different
from that represented by the first component alone, except that the latter
exaggerates the rate of decline somewhat in the more central portions on
account of the opposing effect of the higher components. But in the outer
zone of about two-tenths of the radius the process is in the earlier stages
totally different; here the temperature actually rises for a certain interval
of time, which would be different for different depths, very short for points
extremely near the surface because of the constancy of the surf ace-tempera*
ture, and for points near the boundary between the regions of rising and of
falling temperature, but presumably of considerable length at intermediate
depths. Since this trend of temperature in the outer zone is brought about
by the higher components, which practically die out in a time sufficient to
produce only a relatively small change in the amplitude of the first com-
ponent, the whole process may be conceived to occur in two epochs, an
earlier one of gradual accommodation of the temperature-curve to the slowly
declining first component, and a later one where that component is left
practically isolated. During the latter epoch the temperature would decline
steadily at all points at rates nearly proportional to the existing tempera-
tures. But during a large part of the earlier epoch the heat lost from the
central portions is conducted through an intermediate zone, whose thermal
condition is nearly stationary, and thence outwards to produce an exaltation
of temperature in a zone a few hundred miles thick just below the surface.
Dissipation through the surface in the earlier stages is very slight, owing to
the smallness of the gradient, the primitive temperature-curve being tan-
gent to the X-axis at the point a;= 1, because of the occurrence of the squared
factor (p—py in formula (22).
THE THEORY OP FISHER. 197
The zone of rising temperature, which would be narrowed down and
finally disappear as the conduction progressed, extends at the start from
the surface to a depth of about 700 miles. The total rise of temperature
would be trivial near the extremes of this zone, but more marked toward
its interior. According to the residuals in column 6, table 5, the rise may
be expected to be most significant in the neighborhood of a depth of about
400 miles, reaching in that region, with allowance for even a considerable
percentage of decline in the first component, a value of probably at least two
one-hundredths of the central temperature, or 400^, which is the increase
over initial temperatures ranging about 1,200^. Such a change might
carry the substance through its temperature of fusion, even under the high
pressures there sustained.
The foregoing sketch of the thermal process lacks, of course, the pre-
cision which could be reached through a computation extended to include
all components of sensible influence; this also would alone suffice to yield
an accurate estimate of the time-intervals implied, which would probably
be counted in billions of years for the epoch during which rising tempera-
tures occur. It must be noted also that a small outstanding portion of the
energy of impact might alter the features of the thermal process seriously,
especially in the strata near the surface, where the very fact of the oc-
currence of the rising temperatures may be said to be due to that relative
deficiency in the heat from the purely compressional source which is rep-
resented by the upward concavity of the initial temperature-curve.
It is desirable, however, to know more precisely the result of the complete
computation, under at least one set of reasonable assumptions, which, in
the absence of experimental information, may fairly be conditioned by the
practicability of the calculations; for instance, through the use of a suitable
alternative hypothesis regarding the specific heat. This has thus far been
treated as a constant, but there would seem to be some reason, under a
molecular theory, for supposing it to decrease with increase of density, since
the consequent diminution of the intermolecular spaces might tend to throw
more of the energy into the " unordered " kinetic or thermal form, by inter-
fering with the " ordered " movements which have been conceived by Hertz
and others to accoimt for the storage of energy apparently in latent or
potential form. It will be of interest to see how far the thermal process
described above is modified by supposing the specific heat thus variable.
A simple supposition on this point, hardly more arbitrary than any other
that could be made and having at least the merit of yielding tractable form-
ulas, is that a is inversely proportional to /o, or ap^a^^. With constant
" calorimetric " conductivity k this makes the "thermometric" conductivity
— also constant, and reduces equation (55) to the form:
giving for the fundamental fimctions:
smnnx (87^
^ nnx
198 GEOPHTBICAL THEORY UNDEB THB PLANSTESIMAL HTPOTHBSI8.
and for the expansion (53) the ooefficienta:
r
r I zF(z) ai
il»->2n;r I zF{z)amnKxdx (68)
if the initial temperature-curve is y^dj^ix) and the coefficients are ex-
pressed in terms of 0^ as unit. The intervals r are determined by
'-'-^ w
and are thus in this case strictly proportional to the squared reciprocals of
the natural numbers.
To determine F{x), it may be noted that the initial temperatures can be
determined from those of the previous hypothesis by multiplying by pipi]
giving for the central temperature
which for a « 0.2 is about 81,000^; and for the curve in terms of the central
ordinate
i-n-^-i^) (7.)
For convenience in determining the coefficients A^ the last equation will
be replaced by the formula
F(a:) =(1 -a:»)»(l-y x«+|x*)
or
F{x) =l-^x«+||x*-gx*+|a:» (72)
which, in comparison with (71), leaves residuals at most 3 units in the
fourth decimal place. By (68) the expansion has then the coefficients
where
r
Sn^'^ — I f ^ sin n^d^
which can be reduced through integration by parts, giving finally
A Of nn-iJ 136 562 6480 . 72576 ) .„.
The first few of these coefficients are:
^1= +0.70590 i42= +0.36797 4^= +0.05354 ilo= +0.02240
^4,= -0.10765 i45= -0.03284 A^^ -0.01632
THB THEOBT OF FISHER.
199
The alternation of the sign begins with the second and the steady decline
in numerical value continues throughout.
A convenient unit of time for exhibiting the changes produced by con-
duction in the temperature-curve is the interval T required to reduce the
amplitude of the first component by 1 per cent, which with the same con-
stants as before, for the surface-stratum, would be about 1^ billion years.
The successive changes in the temperature-curve are shown in table 6,
abbreviated from an extended computation covering an epoch of 35 such
intervals, the number of terms of the series included ranging from 100 for
the earliest entries to 5 for the latest. The unit of temperature is the initial
temperature at the center.
Tabub 6.
*
•(0)
• (6D
• (lOT)
• (lAT)
• (2or)
• (262*)
• (SOD
• (867)
0.00
1.0000
0.9210
0.8496
0.7849
0.7263
0.6734
0.6256
0.5823
.05
.9982
.9149
.8440
.7798
.7218
.6693
.6219
.5790
.10
.9781
.8967
.8276
.7649
.7083
.6571
.6109
.5690
.16
.9402
.8669
.8006
.7406
.6863
.6372
.5928
.5527
.20
.8956
.8265
.7640
.7075
.6564
.6101
.5683
.5305
.25
.8404
.7765
.7188
.6666
.6194
.5767
.5381
.5031
.90
.7762
.7186
.6663
.6191
.5765
i»79
.5080
.4712
.85
.7049
.6540
.6080
iM64
.5288
.4948
.4639
.4357
.40
.6286
JS849
.5456
J5100
.4778
.4486
.4220
.3976
.45
.5492
.5182
.4808
.4514
.4248
.4006
.3783
.3577
.50
.4691
.4408
.4153
.3923
.3713
.3519
.3389
.3171
.55
.8903
.3697
.3511
.3841
.3185
.3038
.2899
.2766
.60
.3149
.3016
.2896
.2784
.2677
.2573
.2471
.2370
.05
.2447
.2383
.2323
.2268
.2200
.2138
.2062
.1990
.70
.1816
.1812
.1806
.1789
.1761
.1724
.1680
.1630
.76
.1266
.1316
.1352
.1368
.1366
.1351
.1826
.1294
.80
.0810
.0903
.0966
.1001
.1015
.1015
.1003
.0980
.85
.0453
.0574
.0648
.0688
.0708
.0715
.0712
.0702
.90
.0200
.0326
.0389
.0423
.0441
.0449
.0449
.0445
.95
.0049
.0143
.0178
.0197
.0207
.0212
.0213
.0211
1.00
Here, as in the previous case, the earlier stages of the conduction are
marked by the division of the mass into two zones, an inner where the
temperature falls and an outer where it rises. The spherical surface of
boimdary between these lies initially at a depth of not quite 1,200 miles,
and as the conduction progresses rises toward the surface. Its passage
through any particular horizon marks the attwiment of the maximum tem-
perature which occurs at that depth; while its ultimate coalescence with
the free surface, simultaneous with the occurrence of the maximum surface
gradient, indicates the final establishment of a downward trend of temper-
ature throughout the body. This decline during the further progress of the
conduction is in the outer portions first accelerated and later retarded,
and is ultimately everywhere of a character more and more closely approx-
imating to the simple type represented by the first component alone.
Table 7 gives numerical data, with the epochs reckoned from the begin-
ning of the conduction:
200 OEOPHTSICAL THEORY UNDER THE PLANETE8IMAL HTPOTH£81S.
Tabui 7.
»
Depth
in miles.
Epoch of
maximum
tempemture.
Initial
Total rin.
0.76
.80
.85
.90
.95
1000
800
600
400
200
16.9 T
22.2 T
25.6 7
27 AT
29.1 T
10,240**
6,650
3,660
1,620
400
1,680
2,130
2,020
1,330
The zone of rising temperatures vanishes at epoch 29.7 T, correspond-
ing to the maximum surface gradient of 1^ in about 180 meters, about one-
sixth of that obtained from observations at the present time. The great-
est total rise of temperature occurs at depth about 480 miles, where it is
2,200^, with initial temperature 2,340°.
As compared with the former case sketched above, the results of the
present assumptions, though giving considerably larger primitive temper-
atures, show a close qualitative similarity in the thermal changes. But the
zone of rising temperatures is at first somewhat deeper, the temperature-
increments within it are greater, and the period of its existence longer. This
is to be ascribed largely to the influence of the second component of the
series, whose coefficient is here large, while in the former case it is relatively
trivial. In both cases the conductivity has been treated as constant. It has
often been supposed, however, that it would probably increase considerably
with increase of density. The effect of this would be to facilitate the trans-
fer of heat from the central parts to the outer zone, presumably increasing
the total temperature-increments there, perhaps shortening the time dur-
ing which the temperature rises. It would be likely, also, to increase the
value of the maximum surface-gradient, but whether this could be brought
up to the present observed value through any reasonable assumption of
this character remains to be determined.
The magnitude of the thermal changes in the zone of rising tempera-
tures, resulting from the arbitrary special conditions developed above, seems
to make it probable that the existence of this zone should be considered an
essential feature of the thermal process under the planetesimal hypothesis.
Part II,— CRITICAL AND SUPPLEMENTARY.
In the development of the theory in Part I, the attempt was made to
give explicit statement, though without critical setting, of the more impor-
tant of the secondary hypotheses which were used as auxiliary to the gen-
eral hypothesis in order to give the theory a sufficiently definite form. But
since the main purpose is to seek such features of the geophysical phenom-
ena as seem to be essentially consequences of the general hypothesis, it is
necessary further to inquire how far the results are peculiar to the special
conditions adopted and how far they seem to persist under variations of
these secondary assumptions; and also to what extent these assumptions
are subject to obscurity or positive objection through the accessibility of a
direct or indirect test by observation or well-established theory.
The Laplacian law of density has been assumed chiefly because of its
analytic convenience, though it seems doubtful whether any geophysical
theory is likely to be sufficiently trustworthy in detail to afford more than
a crude test of any assumption on this point. Nevertheless, by inspection
of some of its consequences, it is possible to surmise the probable character
of its departure from the true law of compressibility of the average earth-
substance. It appears that the modifications which seem to be needed
from the standpoint of the planetesimal hypothesis agree, at least in kind,
with those familiar from the indications of general geophysical theory.
Colunm 7 of table 3 shows that in the earlier stages of the growth of the
planet the thickness of a stratum deposited is nearly all effective in enlarg-
ing the geometric radius of the mass, while toward the last not much more
than one-quarter of a new stratum remains above the former horizon; as
the mass grows larger the less significant becomes the actual increase of
dimensions produced by a new stratum of given thickness. Moreover, with
a strict interpretation of the law of compressibility assumed above, there is
a definite limit to the possible radius of the planet, no matter how much
material might be laid down. For in equation (25) the factor 9 is a defi-
nite constant, depending solely on h, which is determined by the compressi-
bility as a physical constant of the material, independent of the dimensions
of the mass into which it may be aggregated; while if there are to be no
meaningless negative densities introduced, the angle /9 can not surpass the
value K, at which the density at the center becomes infinite. This means
that no amount of accretion could produce a mass with radius greater
than ^ centimeters, or about 5,000 miles.
Now, even independently of any supposition as to the actual origin of
the planets, there seems to be little reason for supposing that if deposits of
indefinite extent could be brought about at the surface of an existing planet
there would be such a limit of growth, at least of such comparatively meager
dimensions. This objection has not much force, for the reason that the law
might be practically accurate for the range of densities contemplated and
seriously in error for the higher densities; but its suggestion is that the true
201
202 GEOPHYSICAL THEOBT UNDER THE PLANETS8IMAL HTPOTHBSIS.
compressibility most probably diminishes, as the density increases, more
rapidly than is postulated by the Laplacian law. This would allow a larger
part of the compression to take place during the earlier stages, giving to the
nucleus, when it reaches a given radius, a larger mass than is assigned it
above.
In the light of the planetesimal hypothesis a similar concluaon can be
reached in another way. For in view of the like origin predicated of the
earth and other bodies of the solar system from the primitive planetesimals,
composed of more or less similar materials of definite compressibility, it may
be expected that at the successive stages when the earth-nucleus reached
dimensions equal to those of various planets at the present time its mass
should show some agreement with the observed masses of those planets.
The supposition of the small upper limit of diameter just mentioned is of
course negatived by the existence of the planets of the Jovian group. But
such comparison is futile, partly because of the uncertainty as to their
true dimensions, brought about by their extensive atmospheres, pardy
because of the wide difference in physical condition as compared with the
earth, illustrated in particular by mean densities smaller than even the
surface-density assumed for the earth. Moreover, it is conceivable that
the discrepancies of an assumed pressure-density law might become serious
only outside of the range met with in smaller bodies like the earth. But a
comparison with the other planets of the terrestrial group should prove
instructive. Table 8 gives their observed radii and masses as compared
with the earth, and the hypothetical masses computed by interpolation
from columns 2 and 4 of table 3.
Tabub 8.
Relative
radius.
m
obieryed.
m
compated.
Moon
0.273
.382
.634
.972
0.0121
(?)
.107
.78
0.0103
.029
.086
.86
Mercury
Mara
Venufl
To interpret this table it is necessary to observe that if the masses cor-
responding to assigned dimensions be computed according to two different
laws of compressibility, using the mean and surface-densities of the present
earth as given constants, then those masses must approach agreement, on
the one hand for small bodies where the compression is slight and the aver-
age density therefore not very different from the assigned surface-density,
on the other hand for bodies approximating the earth in size, the average
density then approaching the assigned mean density. Thus it appears that
the divergence of result between two such assumptions would be likely to be
most marked for bodies of intermediate size, like Mars, Mercury, and the
moon, while the agreement should be close for Venus on the one hand and the
asteroids on the other — as, for instance, if the masses resulting from the
Laplacian law be compared as above with the observed masses, supposing the
latter to correspond to a definite, though unknown, law of compressibility.
CBITICAL AND SUPPLEMENTABT. 203
The masses of the asteroids are purely conjectural, while that of Mer-
cury is too uncertain to be used. For Venus the computed mass exceeds
the observed; but at least part of the difference may be ascribed to the
influence of atmosphere and irradiation in bringing about an overestimate
of the planet's true diameter; for example, if the observed radius were
corrected by — 2 per cent of its value, the computed and observed masses
would agree. For the moon and Mars, the computed masses fall below
the observed by 15 per cent and 20 per cent, respectively.
From this, as before, the suggestion is that the assumed law of com-
pression should be modified in the direction of allowing greater compressi-
bility at the lower densities, and less at the higher, which would have the
effect of assigning greater mass to the nucleus at intermediate sizes, and to
the present earth a steeper density-gradient near the surface, together with
a relatively more nearly homogeneous central portion. This agrees with
conclusions which have been drawn from observations of precession and the
transmission of seismic disturbances.
In view of this comparison it seems quite conceivable that a law of com-
pressibility might be constructed, agreeing with the data furnished by the
earth in its present condition, and such that the observed masses of the
planets now existing in its neighborhood would prove to be the same as
those computed for the nucleus at epochs when the din^ensions correspond.
It wiU accordingly be of interest to review the previous theory, with the
substitution of a density-formula whose variations from that of Laplace
have the general trend indicated.
A formula of this character for the density is the simple one proposed
by Roche: ^ ,- .v . ,^-v
jO— ft(l— car) c ""Const. (74)
From this, according to the general equations previously used, are derived:
4 / 3 \
m - g^ ;r/>o r» ^1 - y cx^J (75)
whence H is determined by (15); also:
p.d|(«»_«^») + (a._«j.)l (77)
A , .. a+2«,+2 ,_„
'-^(*-^«) •— T— ^78)
/o«-ft(l-|c) p,~p,(l-e) (79)
E=8itAr*'^e(l~e) (80)
#
87rAr,«(l-|c+jC») (81)
* Roche, Acadteie dee Sdenoes et Lettres de Montpellier, v. 8, 1848, p. 235.
1204 GEOPHYSICAL THEORY UNDER THE PLANBTE8IMAL HYPOTHESIS.
in which are put
^ "16 ^^P%W
ft
1-c
The constants p^j c may be determined by assigned values of the mean
and surface densities, which will here be supposed to be in the ratio 2:1,
giving c = ^, whence results
/Oo = 9.653
/t)i- 2.758
£-3.110X10"
Hi -3.45X10"
*- 2.426X10"
and for the quantities p, p, H at various depths the values listed in table 9,
which for the present density-formula replaces the corresponding columns
in tables 1 and 2. This value of c is somewhat less than that used by Roche,
which seems to give rather too small an estimate for the surface-density,
when the more modem determinations of p^ are used.
Comparison with the tables of Part I shows that the range of values in
density, pressure, and specific compressional energy is in each case some-
what less than under the previous conditions, but that the modulus of com-
pression, while somewhat less at the lower densities, is decidedly greater at
the higher, showing that the departures of formula (74) from (25), which it
replaces, have quaKtatively the character shown to be needed. Of the
energy-totals, tfj^ is necessarily the same as before, while # and E are respec-
tively 1.2 per cent and 14.1 per cent less than their former values, so that
relatively more of the primitive energy is transformed by impact and lost
by radiation. This is obviously due to the greater mass of the nucleus, at
a given radius, and the correspondingly larger velocity of impact in thb as
compared with the former case.
Table 9.
»
p
lO-Wp
lO-^H
be
J
0.00
9.66
2.80
7.39
16,610
.06
9.64
2.79
7.36
16,640
.10
9.68
2.76
7.26
16,360
.16
9.50
2.68
7.08
16,060
.20
9.38
2.69
6.86
16,630
.26
9.22
2.48
6.66
15,090
.80
9.03
2.36
6.22
14,430
.86
8.81
2.20
6.83
13,670
.40
8.55
2.03
6.44
12,810
.46
8.26
1.86
4.94
11,860
.60
7.93
1.66
4.46
10,800
.65
7.57
1.46
3.96
9,680
.60
7.17
1.27
3.45
8,600
.65
6.74
1.07
2.95
7,260
.70
6.27
.88
2.47
5,990
.76
5.77
.69
2.01
4,720
.80
5.24
.52
1.58
3,460
.85
4.67
.36
1.19
2,260
.90
4.07
.22
.86
1,190
.95
3.43
.10
.57
360
1.00
2.76
.00
.34
CRITICAL AND SUPPLEMENTARY. 205
Table 9, based on the formula p^Poil-j^x*), gives the values at va-
rious depths of the density, of the pressure and bulk-modulus in millions of
atmospheres, and the thermal measure for (7 = 0.2 of the specific compres-
sional energy.
To estimate the modifications in the thermal process due simply to the
change in the formula for density, it will be well to repeat the former alter-
native hypotheses regarding the specific heat.
If a, X are constant, the primitive temperature-curve by (78) is
(?-(?,(l-x»)»(l-ix»)^(l-|a:») <?,-^ (82)
which is tabulated in column 5 of table 9, while equation (55) reduces to
il(^l)+^<i-''^)''-o («3)
giving for the fundamental functions
y-2*(-l)'a,x« (84)
where the coefficients a^ are determined successively from
o^-l ai=|^ g<" 2t(2i+l) ^^*~^"^^^^^ ^^^ ... .00 (85)
with the values of /i which make y vanish at a;» 1. The complete computa-
tion to sufficient accuracy to determine the details of the conduction is
again hardly practicable, but the first two fundamental functions with their
parameters /i, r, and coefficients A determined from (56) are:
yi-1 -2.04360 x« + 1.69080 a:*- .91977 x«+.36231x»-. 11362 a:**^
+ .02927 x« - .00645 x" + .00123 x»« - .00021 x"
+ .00003 X**— ....
/ii - 1 2.2616 A 1 = 1 .089 0o r^ - 400 billion years
2/a - 1 - 8.59393 x» + 23.99826 x' - 36.99908 x« + 38.77346 x»
- 30.56380 X** + 19.25673 x"- 10.08879 x" + 4.52008 x»«
- 1.76799 x»»+.61344 x»- .19120 x« + .05409 x»*
-.01400 x»*+.00334x«-.00074x» + .00015x«
-.00003 x»*+ ....
/£, =51.5636 A, - -.0705 6. t, =96 billion years
These are given in table 10, columns 2 and 3, together with the residuals
from the primitive curve due to components higher than the first and second
respectively in columns 4 and 5, the tabulation in all cases using as unit the
primitive central temperature 0^^, which for a =0.2 is about 16,600®. The
last column of that table gives the initial rate of change of temperature,
computed from
dd eji { 312^1268 , 132 ,,,^ . 100 .) f, 5 ^^ ,^7.
^"^r^"^"49-^--^^+'2"-i9^ri'-7^1 ^^^^
(86)
206 OEOPHTSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS.
which follows from (52) and (82), the tabulation using, however, as unit of
e^x
measure the value of — ^, equivalent to a change of about SJ® in a billion
years. ^^* ^
Table 10.
X
Vx
Vt
F{z)
n{x)
i%(«)
u
0.00
1.0000
1.0000
1.0000
—0.0890
—0,0185
— 8.91
.06
.9949
.9787
.9963
— .0871
— .0181
— 8.90
.10
.9797
.9164
.9862
— .0817
— .0171
— 8A5
.16
.9649
.8184
.9668
— .0731
— .0154
— 8.76
.20
.9209
.6924
.9411
— .0618
- .0130
— 8.64
.26
.8787
.6482
.9085
^^ .U4o4
— .0098
-— 8.48
.30
.8291
.3963
.8691
— .0338
— .0059
— 8.29
.86
.7734
.2473
.8232
— .0190
— .0016
— ao6
.40
.7128
.1103
.7712
— .0050
+ .0028
— 7.77
.46
.6484
— .0074
.7134
-f .0072
+ .0067
— 7.45
.60
.6817
— .1009
.6606
-f .0170
+ .0099
— 7.06
.65
.6138
— .1679
.5830
-f .0234
-h .0116
— 6.62
.60
.4460
— .2083
.6117
+ .0260
-f .0113
— 6i)8
.66
.3792
— .2243
.4373
4- .0243
+ .0065
— 6.45
.70
.3144
— .2194
.3609
+ .0185
+ .0030
— 4.66
.76
.2525
— .1981
.2840
-h .0091
— .0049
— 8.64
.80
.1938
— .1663
.2082
— .0029
— .0145
— 2.27
.86
.1392
— .1255
.1861
— .0154
— ,0242
— 0.26
.90
.0886
— .0827
.0718
— .0246
— .0304
H- 8.00
.95
.0422
— .0401
.0219
— .0240
— .0268
4- 9.01
1.00
.0000
— .0000
.0000
.0000
.0000
+22.40
In table 10, based on the hypothesis c^c^f ^^Kf P^P$0'~"n^f ^'^
functions y^ y^ are the first two fundamental functions; F{x) gives the
primitive temperature-curve, Fi(a:)=F(a;)— A^y,, F^{x)^F^{x)'-A^^; in
each the temperatures and coefficients A are expressed in terms of the unit
= 16,600^. The last column gives the initial rate of the change of temper-
ature in terms of a unit corresponding to a change of 3}° in a billion years;
the entries in this column must be multiplied by — ^—^ to be reduced to
absolute units. ^* *
Comparison of this table with table 5 shows that the change in distri-
bution of the density makes no striking change in the thermal process. The
second component has a larger coefficient, and the residuals left by the first
two components are somewhat smaller. But these residuals have in gen-
eral the same trend, and together with the values of -rr indicate the same
initial partition of the mass into inner and outer regions of falling and rising
temperature, respectively. The outer zone is, however, shallower than under
the former case, extending at the start to a depth of not quite 600 miles.
The increase in the tabulated values of the initial rate -rr and of the gra-
dient of the first component for points near the surface, together with the
coefficient A,, is offset by the decreased value of d^j in terms of which the
entries are expressed, so that the maximum temperature-gradient at the
surface occurring when the zone of rising temperature disappears, would be
CBITIGAL AND SUPPLEMSNTABT.
207
probably about as before. The time-scale of the phenomena can be con-
sidered practically the same as before, especially in view of the uncertainty
as to the appropriate values of the thermal constants.
Under the other supposition, that X and ap are constant, the funda-
mental functions are again defined by (67), while the primitive tempera-
ture is 1
5-5,(l-x«)»(l— ^x«)
where now
A
^ f5
a^J
Pi
(88)
(89)
or about 58,000^. The coefficients of the series are given by
32
1008)
(90)
A,.2(-l)-{~^
the first few of which are :
Ai« 0.80006 A, -0.29146 A^= 0.08054 A, -0.03598
A,- —0.14122 A,- —0.05174 A,- —0.02646
the alternation in sign and steady decline in numerical value continuing
throughout.
The time-interval T, during which the first component declines by 1
per cent, is the same as with the former density formula, or about 1} billion
years, with the thermal constants assumed. The effect of the conduction
in modifying the distribution of temperature is indicated in table 11, for
epochs differing by five such intervals. To the order of accuracy for this
table the numbers of components of sensible influence range from about 100
for the earliest epoch to 4 for the latest given.
Tabud 11.
X
•(0)
• (5r)
• (lor)
• (153*)
• (2or)
• (253*)
• (SOT)
• (853*)
0.00
1.0000
0.9349
0.8741
0.8174
0.7646
0.7166
0.6703
0.6286
.06
.9946
.9298
.8693
.8129
.7604
.7118
.6668
.6252
.10
.9782
.9146
.8660
.7996
.7481
.7003
.6662
.6165
.16
.9612
.8893
.8316
.7777
.7278
.6816
.6388
.5994
.20
.9142
.8647
.7993
.7477
.0999
.6668
.6161
.6776
.25
.8679
.8116
.7690
.7103
.6663
.6238
.5866
.5604
.80
.8132
.7604
.7116
.6663
.6246
.6862
.5609
.6184
.36
.7612
.7026
.6578
.6166
.6787
.5439
.6119
.4825
.40
.6830
.6393
.6992
.6624
.5287
.4979
.4696
.4436
.45
.6102
.6718
.6368
.5050
.4760
.4494
.4249
.4022
.60
M44
.6018
.4728
.4456
.4215
.3993
.3787
.3595
.65
.4671
.4306
.4070
.3858
.3666
.3488
.3321
.3164
.60
.3801
.3601
.3426
.3270
.3126
.2990
.2860
.2736
.66
.3063
J2919
.2807
.2706
.2807
.2509
.2412
.2316
.70
.2346
.2281
.2228
.2175
.2117
.2053
.1985
.1914
.76
.1698
.1702
.1704
.1691
.1664
.1627
.1682
.1533
.80
.1130
.1198
.1242
.1258
.1253
.1234
.1207
.1174
.86
.0669
.0781
.0848
.0876
.0883
.0877
.0862
.0841
.90
JOSOS
.0463
.0616
.0546
.0566
.0664
.0647
.0636
.96
.0078
.0201
.0239
.0256
.0262
.0263
.0260
.0256
1.00
208 QEOPHT8ICAL THEORY UNDER THE PLANETE8IMAL HYPOTHESIS.
Here the zone of rising temperature extends at first to a depth of a little
over 1,000 miles, and the chief features of its history may be thus sum-
marized:
Table 12.
X
Depth
in miles.
Epoch of
maximum
temperature.
Initial
temperature.
Total rise.
0.76
.80
.85
.90
.96
1,000
800
600
400
200
8.3 r
16.3 r
20.1 T
222 T
23.6 r
9,870«»
6,570*»
3,830*»
1,760*»
460*»
40*
740*
1,300*»
1,470**
1,080*
The rising temperatures finally cease at epoch about 24 T, with the at-
tainment of a maximum surface gradient of 1^ in about 220 meters.
To complete the sketch of the modified theory, in a way parallel to that
of Part I, there might be traced an analogous history of the nucleus during
the period of accretion, supposing the compressibility of the substance to
correspond to equation (76). But in contrast with (25) it must be noticed
that this could not be done by treating (74) as the general expression for
the density at every epoch. For by (76) the modulus of compression has
*''*'°™ H^hp^hy (91)
and if this is to be definite and characteristic of the substance, not depend-
ent on the dimensions of the mass into which it may be gathered, the
coefficients A, h' must be numbers depending only on the units of measure
used, which on account of the manner of th^ir dependence on p^ and c can
be so only if the latter also are particular numbers in the same sense,
not arbitrary parameters depending on the size of the body. The general
expression for the density would have to be sought by integration of equa-
tion (6) with the condition (91), which would yield a family of curves cor-
responding to the nucleus at various stages, but of which only the final one
has the simple parabolic form defined by (74). The general integral would
contain two constants, but one may be considered eliminated either to
dp
make the density finite at the center or by the condition that -r- vanish
at the center. For this particular law of compressibility the general inte-
gral does not appear to be known in form sufficiently simple to make the
computation practicable.
This suggests that in connection with the planetesimal hypothesis it
would be convenient to have a series of examples of curve-families, each of
which could be considered to represent the distribution of density at the
various stages of aggregation of a substance with a certain definite law of
compressibility — equations simple enough to allow the detailed computa-
tions, but so varied as to admit of choice in adapting to other accepted
data. As a help towards the discovery of such it may be asked what general
condition a family of density-curves must satisfy in order that a definite
compressibility may be deduced.
CRITICAL AND SUPPLEMENTARY.
209'
This condition means that if the curves be supposed given in the form
P -nr,a) (92>
where a is the parameter of integration, whose value corresponds, for
instance, to the total radius or the central density, then the modulus H
deduced therefrom must be a function of p only, independent of a. Equa-
tions (3), (4), (5), (15) show this to be equivalent to the condition thai
171
— -j- shall be a function of p only, or in terms of the functional determinant
dr
in which is put, as before,
3 / m \
r
m=«4;r I pr^dr
da
dp
dr
dp
da
(93)
By differentiation and elimination of m this fundamental condition can be
reduced to a partial differential equation for p, in rather cumbersome form.
Trial shows that (74) is not a solution for any manner of dependence of p^
and c on a; but (25) is a solution, if ^^ be a function of a, and q a numerical
constant.
Another special solution of (93) in simple form is found to be
.a*(l+~a^/iV)
2
/I » const.
for which
m-|;rT*a»(l+^aVv)-T
(94)
(95)
so that for various values of a it would give the density curves for masses of
different dimensions, if the substance were to satisfy the condition
dp
dp
1^
hpb
47ck
bp?
(96>
With such a substance the compressibility at various densities would be such
that not only would there be a definite limit to the dimensions which the
mass could attain, but this would be reached with a finite mass, and beyond
that point any further addition of strata at the surface would result in an
actual decrease in size, as appears from the following analysis.
If p^, PifT^he the mean and surface densities and total radius at any
chosen epoch, and a^ the corresponding value of a, then
^«-ai*(l+c)"
2
(97>
210 GSOPHTSICAL THEOBT UNDER THB PLANETESIMAL HYPOTHESIS.
where
whence
^ = l+c (99)
Pi
Then since the surface density is not to change, the total radius r/ and the
parameter a at any epoch are related by
which, combined with (98), gives
/icA*-(l+c)A+l=0 (101)
a quadratic equation for il, in which are put
Hi)' '-id'
The condition for reality of the roots shows that the maximum possible
radius is
, r. 1+c
2 • JJ (102)
as may be deduced also from the transformed equation
B . (1+y-l (103)
From which it is seen that i2 as a function of A has a maximum value
n -f c)* 2
-R =» . at il = .. y As A increases indefinitely from its smallest
admissible value, which is t~7~; the mass also increases from zero without
' 1+c'
limit, but the total radius increases to the maximum value indicated and
then decreases toward zero. For any assigned radius less than the maximum
there would thus be two possible distributions of density, giving masses
less than and greater than the critical mass, and with mean densities less
and greater than double the surface density respectively.
If, however, the constants a^, c be determined by (97), (98), from the
values of p^, p^ heretofore assumed for the earth, the computation ac-
cording to (94) shows that for some distance downward from the surface
the density would increase more slowly than with either of the formulse
used before, but then rise more rapidly and at the center reach a value of
15 to 16. In view of the probable corrections mentioned above as needed
to change the Laplacian formula into one agreeing better with the data
available, it appears that in comparison with (25) the formula (94) is much
less satisfactory, as an approximation to the general distribution of density
CRITICAL AND SUPPLEMENTABT. 211
within the earth. It is here dwelt on briefly for the sake of the comparison
it affords with still another now to be considered, which also belongs to the
class for which ,
f =V (104)
of which (20) and (96) are examples.
It was shown in Part I that, with the particular law of compressibility
there postulated, the progressive condensation under the increasing load of
the material gathere4 at the surface would be accompanied by a deforma-
tion of the elements of the mass whose final amount is indicated by the
distortion-factor tabulated. Attention was called to the consequent uncer-
tainty introduced into the determination of the work of compression if it
be supposed that the substance offers appreciable resistance, either elastic
or viscous, to shearing stresses, so that the working pressure is not purely
hydrostatic. It is conceivable, however, that this deformation might be
widely different in character and amount under another acceptable pres-
sure-density law, so that the acceptance of the special formulas (20) and
(25) would lead to no just estimate of the essential obscurity in the theory
from this source.
As a guide to conjecture on this point it will be of interest to determine
whether there could be a law of compressibility assumed of such nature
that the condensation of the mass would lead to no such deformation, but
rather that the compression would be at all points purely cubical.
If a; be an auxiliary variable, determining the location of a given par-
ticle at some chosen epoch, for instance as before the ratio of the ultimate
distance from the center to the total radius, then the distance r from the
center at any epoch may be considered as a function of a and x:
r:^(p{a,x) (106)
Let x', x^' be the values of x corresponding to two chosen particles; then
the mass of the spherical shell whose bounding surfaces pass through those
particles will be ^
m{x^,7f)»4^n I pr^^dx (IM)
Jx' ^^
As the condensation progresses the spherical surfaces will shrink, but the
mass between them must remain constant. This means that the integral
(106) must be independent of a, whatever the values of x', x"; this gives
the condition
^H^-t^y^^w^-o
which reduces to
in which is put
[dp d{pP)]
Ida"^ dr J
+2^P-0 (107)
P^<^^r)^^ (108)
14
212 GB0PHT8IGAL THBOBT UNDEB THE PLANETE8IMAL HTPOTHBSI8.
obtained from (105) by differentiation and elimination of x, which is admia-
dblei since r is a monotonic function of x.
Moreover, the variations in horizontal and vertical dimensions of a
given element of the mass are proportional to the variations of r and of
-z— respectively as functions of a. The condition of no distortion demands
1 dr
therefore that — -r- shall be independent of a, which gives
~--^-0 (109)
BO that P may be written in the form
P'^-rQia) (110)
showing that, to satisfy the conditions named, a differential accretion at
any epoch must depress each particle an amount proportional to its dis-
tance from the center, but so that the factor of proportionality depends in
an indetermined way on the momentary total dimensions of the body.
The last equation reduces (107) to
of which the general solution is
/>-^/(C) ("2)
where
C=logr+ / Qda (113)
The mass within the radius r is then
m«4;r^(C) (114)
provided that in the function f , whose derivative is the arbitrary function
^' in (112), the additive constant be chosen suitably, which the finiteness
of the mass would show to be possible.
It remains to impose the condition that the substance have a definite
compressibility. Equations (112) and (114) yield
and this must be a function of p only. This is equivalent to the condition
9
then r""'* and r""V' niust be dependent functions of r, ^; or in terms of the
functional determinant ^ ^^ ^,
1
20 0'
Zip' f"
CRITICAL AND SUPPLEMENTARY. 213
This could be obtained from the general equation (93) by substitution of
(112) and change of variables from r, a to r, {;. It shows that, with C an
arbitrary constant, the relation must have the form
giving for determination of the function f
^_3^'«C^'l (116)
on account of which (115) gives for the equation characteristic of the sub-
stance
f-v *-f am
80 that C must be negative. This is, then, the condition which it is neces-
sary that the substance satisfy in order that the condensation under increas-
ing mass may not be accompanied by distortion of the mass-elements.
Conversely, if the substance satisfy this condition, the compression will
take place without deformation. For, with the substitution of (117), equa-
tion (6) takes the form
The solution which is finite at the center has the form
R^RJiiu) R^^'^p. u^R^ (119)
where £ is a definite function, which for sufficiently small values of u can
be expanded as a power series with numerical coefficients of alternate signs
fl(u)=-l-BiU»+B,u*- ....
If the radius of convergence be too limited, the function may be considered
as determined for any value of u by analytic continuation, since equation
(118) has no singularities to prevent. The density is then
p -^^(u) w(u) -[fl(u)]» (120)
so that io also is expansible for small values of u as an alternating series
with numerical coefficients
(jd{u) -» 1 — bju' + fcjW* —
Then
...
whence
dr
tn-y^(u) (121)
where B is the definite function
v?(ji){u)du
(122)
214 GEOPHYSICAL THEORY UNDER THE PLANETSaiMAL HYPOTHESIS.
or as power-series
Since /i is in effect a physical constant of the material, equation (121)
shows that any particular value of u will determine a spherical suiface,
of varying radius according to the varying value of p^, but always passing
through the same material particles, so as to inclose the mass determined
by that equation. If, then, u, v be two values of u corresponding to two
definite particles, a!, ^ and o^, ^^ their distances from the center at two
stages of the compression when the central densities are p^\ P^ respectively,
then the definition of u gives
whence
a' of
showing that the distances of two particles from the center remain in the
same ratio as the condensation progresses. This is precisely the kind of
contraction by which distortion of elements is avoided.
The constant /i would be determined by the mean and surface densities
at a given epoch from the equations
Pi^pM^i) Pm'^^P.-r-r ^I'^P^^f^t (124)
which would yield also the central density ^o &^ ^^^^ epoch; then for any
epoch the first of these would determine the central density for any assigned
total radius.
The determination of the exact distribution of density under these con-
ditions would rest on the computation of the coefficients so that R satisfies
the differential equation (118); but a sufficient idea of the curve may be
gathered simply by comparison of formula (117) with (96) and (20). All of
these belong to the class (104), with exponents «-,-^, and 1 respectively, so
that the first is a sort of average of the other two. With the same assigned
mean and surface densities, it may be inferred that the hypothesis now
considered would result in densities less in the outer strata and greater in
the central portion than those given by the Laplacian formula (25), but
deviating in that way to a less extent than those computed from (94).
Such departures are opposite to the corrections which the Laplacian law
seems to require, though the data from observation are but meager; so
that (117) must probably be rejected, as not giving a satisfactory approx-
imation to the actual distribution of density within the earth. The only
satisfactory curves of class (104) would seem to be those with exponents
somewhat larger than unity.
The intermediate character of (117) as compared with (20) and (96)
appears also in the phenomena which would attend unlimited accretion.
CRITICAL AND 8UPPLBMENTABT. 215
Equations (117) and following, combined with the hydrostatic equation (3),
du u'
and (122) gives
du
from combination of which comes
dz z—v
_-_J^ y.0 s^ua (126)
This equation determines a family of curves in the y-z plane, of which the
one required for the present purpose passes through the origin, at which
the 2;-axis is an inflectional tangent. In the neighborhood of the origin
this curve is given by the expansion
with coefficients all positive. The arc required is that l3ring in the quad-
rant where y and z are positive. » ^
In the half-quadrant where y>z, the inequality '■^>*^ gives y>«-,
which shows that the curve reaches the line y^z %X some point {a,€)
such that 0<a<^3. Crossing this line, with tangent parallel to the y-
azis, the curve passes into the half-quadrant where y>£, where the in-
dz €L — y
equality -r- < ^ gives «* < a* — 2{y - a) *, which shows that the curve must
cross the y-axis, with tangent paralld to the i-axis, at a point (z'^0,yP)
such that
a' 1 r-
j9<aH — 7== or /9<a(l+2"\6)- The curvature does not change
sign in this quadrant, since the di£ferential equation gives
0- -l3(«-»)^-f (f-y) +«^]+«^
which is constantly negative where y and i are positive. This part of the
curve is therefore a simple arch concave to the y-axis and crossing per-
pendicularly at y-"0 and at y»)9.
This means that there is a finite critical mass which is reached only
when the function a)(u) becomes zero and consequently the density at the
center infinite. The corresponding value of u is then infinite, but the
radius is finite, since the density must be everywhere greater than p^.
Thb result may be compared on the one hand with the case n « -^i where
the limiting radius is reached while the mass and central density are both
finite; on the other hand with the case n»l, where the limiting radius ii
approached only asymptotically as the mass increases indefinitely.
From the rejection of the hypothesis last developed it appears, there-
fore, that under the law of compressibility which would result from any
216 QBOPHTSICAL THEORY UNDER THE PLANBTE8IMAL HYPOTHESIS.
acceptable density-formula the compression in the interior as the mass
increased would inevitably be accompanied by local deformations, result-
ing from the inequality of vertical and horizontal compression, and prob-
ably in character and magnitude similar to those described in Part I. Thus
the theory developed from the hydrostatic equations remains subject to
the uncertainty from this source in the computation of the compressional
energy, to an extent not easy to estimate.
The thermal process has thus far been outlined on two alternative
suppositions regarding the variation of p, and two regarding a, but in all
cases the conductivity was treated as constant. This condition might be
replaced by the supposition often made that it increases from the surface
toward the center on account of the increasing condensation of the mate-
rial. Such a variation would have the effect of facilitating the transfer of
heat from the interior to the superficial strata, probably raising the maxi-
mum gradient of temperature attained near the surface and perhaps short-
ening the time during which the temperature has anywhere an upward
trend. The effect on the depth of the region where -r^ is at first positive
is seen by inspection of equation (52), which can be written
dt
'ap\r^dr\ dr)^ X dr dr J
In the second member, the first term, which alone has occurred hitherto,
is positive in the outer region and negative in the interior; ^ is every-
where negative, while if X increases toward the interior -r- is negative,
making the second term positive. This shows that the zone of rising tem-
perature would be deeper than with constant X, with the same original
temperature curve. The more improbable supposition that X decreases
toward the interior would have the contrary effect, but, as will be seen
presently, could by no means eliminate the outer zone entirely.
The fact that under a variety of suppositions regarding the thermal
coefficients ^, a, X there occurs a thermal process marked by the same gen-
eral features, even with no radical differences in order of magnitude in the
numerical data, suggests that those features are not dependent on such
special hypotheses, but due to the general properties of the original distri-
bution of temperature and characteristic of its mode of origin. The follow-
ing general considerations show why they may be expected to persist under
any hypotheses on p, a, Xy not differing too radically from those developed
above.
Whatever may be the actual variations of pt g, o, X, any equations
assuming to represent them as functions of r only can hardly be treated as
more than interpolation-formulas, representing the gross features of the con-
crete situation in the sense of averages, and disregarding the relatively trivial
local variations on account of which it is only to a certain degree of accuracy
that there can be said to be, for instance, a definite law of density at all.
It seems, therefore, practically general to assume in the neighborhood of the
\
CRITICAL AND SUPPLEMENTABY. 217
surface a representation in series ascending powers of «*"ri—ri in the form
p^Pi+a^8+€^8^+ .... g»gi+b^8+h2^+ ....
a»ai + Ci8 + c^+ .... X'^Xi+di8+di^+ ....
such that a few terms suffice to give all the precision which has useful mean-
ing under the circumstances. From these come
-^''9P'^9iPi + (9fii+pfii)9+ • • • •
^ Pi 2ft*
....
-fji-
^'"V+
2ft
• • . •
so that the temperature-curve in the neighborhood of the surface has the
form
(?-(7s»+ C'-s^
2piaiJ
and is consequently tangent to the x-axis at the surface-point :r«>l. The
tangency is of ordinary parabolic type, since the vanishing of Oj would mean
that the surface material was incompressible. The temperature at first
changes at rate
-—mm— . -- . -_Mr»--_.)=. -^ +terms with factor (r—fi)
dt ap r* dr\ drj a^p^ ^ *'
Since the first term here is essentially positive, there is necessarily a region
just below the surface where the temperature rises.
On the other hand, at the center there occur maxima of the curves for
p, Pf and consequently for e. This would most probably happen also for
Of so that the appropriate expansion would be of the form
e^e.-Cr^ ....
with which the initial rate of change is
-rr ■■ -*+ terms with factor r*
the first term of which is essentially negative, so that the temperature
falls in the neighborhood of the center. The only case in which this would
not happen would be when a would have a maximum at the center, strong
enough to throw the maximum of temperature to a point further out,
which is highly improbable, a necessary condition for this being that with
expansions of the type
/9—/9p(l— &,r* . . . .) a— ap(l — Cif* . . . .)
the coefficients satisfy the inequality -^ ^
P%%
218 OSOPHT8ICAL THSOBT UNDSB THB FLAmnmnCAL HTPOTHB8I8.
Thflse resulta do not ezdudo the oeeumnee of moio than two waomg
alternating with fall and rise of temperature, but inqpectlon of the anal-
ogous expansions for intermediate points seems to indicate that this would
demand variations cS p, a, i whieh differ ^ddy in eharaeter from those
thus far postulated, and whieh are perii^ps improbable, but whoae pcoba-
bility it is difficult to estimate with the meager data at hand.
Comparison of the various eases carried out in the computationB gtwm
a f airiy definite idea of a thermal process which, from the general point
of view hitherto adopted, can be conadered characteristic of the ptanetcs*
imal hjrpothesb so far as. concerns independence of particular hypotheses
respecting tibe density, conductivity, and specific heat. Under the basic
assumptions made, the balance of evidence scans to favor its sabetantisl
correctness in qualitative features, and even in general order of numerical
magnitudes, since tibe chief allowances to be made in the latter can be esti-
mated with confidence from the theory itself. For example, the irnigAa of
time involved may be overrated, about in tibe ratio that the true oondnc*
tivity in the interior surpasses, as it probably does, the value used, whidi
is that obtained by observations on the rather loosely aggrq^ated Bialerial
accessible at tibe surface.
There seems to be sufficient reason for supposing the energy of impact
to have little influence in determining the primitive temperature, espe*
dally if the impinging partides be retarded serioudy by an atmoaphere.
But adde from this tiiere have been made certain geiMral aasnmptionsb
whieh, though fundamental in the foregoing theory, seran to be aibitraiy
rather than essential to the generd hjrpothesis, and thus to demand critl-
cd examination. This will be attempted in Part m, in connecticm with
the development of an dtemative theory intended to dd in estimating the
allowance which should be made for posdble modifications of some of these.
Part III.— THERMODYNAMIC THEORY.
As a strict theory, the foregoing deductions imply a sort of ideal earth-
substance, with respect to which certain assumptions are made, in a form
convenient for the purpose in hand, but not sufficient to define completely
its thermodynamic properties. How closely they represent the actual be-
havior of the substances composing the earth's interior is largely conjecture,
in view of the meagemess and limited range of direct experimental informa-
tion; but they may be examined as to their consistency with accepted
thermodynamic laws regarding the interplay of thermal and mechanical
processes.
It has been supposed that a general idea of the thermal process, after
the earth was completely formed, could be obtained by treating it as a
matter of pure conduction and accompanying radiation at the surface.
But the significance for geological theory of the redistribution of heat lies
largely in the resulting expansions or contractions in different portions of
the mass, and these geometric changes would in general involve the passage
of energy between the mechanical and the thermal form, in amount perhaps
by no means negligible in comparison with the heat conducted. In par-
ticular, if the temperature should on the average fall, the energy thus lost
would be partly compensated by that developed out of gravitational work
during the contraction.
As to the possible relative magnitude of these, as it were, opposing move-
ments of energy, a summary estimate may be gathered from the case of a
homogeneous sphere at uniform temperature, contracting so as to remain
such, and supposed to have specific heat and coefficient of expansion con-
stant throughout. In this case the energy developed by ingathering from
infinite dispersion is
6 r
while the thermal content is
from which come
Q^maOJ
'^^U^ S-w
while
dr 6 r* d0
where a is the volume coefficient of expansion, or three times the linear
coefficient; so that
_d# kma /227)
dQ brcJ
which is the ratio of the gain of energy from the gravitational store, to the
loss by decline of temperature; and with an assigned density is proportional
to the square of the linear dimensions. For small bodies it would be negli-^
219
220 GEOPHTSIGAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS.
gible; for examplOi in the ease of a planetoid a mile in diameter, composed
of rock similar to the earth's surface strata and with a— 2X10^, it would
be only two parts in a billion; but for a sphere having the same mass
and dimensions as the earth, with the same value of a, it would be over
one-fourth; in the latter case the treatment of the conduction of heat
independently of its mechanical effects could hardly give more than a crude
approximation. It seems conceivable that a planet considerably larger
than the earth, even though practically solid, might exhibit the phenomenon
described by Lane as occurring in gaseous bodies, of contraction accompanied
by rise of temperature.
As compared, however, with the simple case just mentioned, there is
an essential contrast shown with a distribution of temperature like that
described above as characteristic of the mode of origin postulated by the
planetesimal hypothesis. Here not only are the initial temperatures near
the surface small in comparison with those developed in the interior, but
the changes are widely different at different depths, the interior steadily
shrinking as the heat is conducted outwards, while the outer strata tend at
first to expand under the rise of temperature which continues until the
maximum surface-gradient is reached. In the early stages the surface-
gradient is slight; the thermal energy is for the most part simply redis-
tributed within the earth, while comparatively little is lost through the
surface. Thus any gain of heat from potential energy on the whole could
come only through a preponderance of the internal shrinkage.
To estimate the nature of these movements let it be supposed that there
is continual accommodation of the density of each portion of the mass to its
temperature, while the accompanying variations of the pressure affecting a
given particle are relatively negligible, so that there may be considered to
be a definite coef&cient of expansion a, a function of r.
If then 9 denote variations in time, the adjustment for equilibrium is
determined by the conditions
ddv^advdO 8v='4:7n^8r
where 9r is the change in the central distance r of a given particle, so that
-"/.'
dv^ix I adOr^dr
The equation of expansion may also be written
ddv ^^ a
dv ap
in which a is in effect the coefficient of specific volume expansion referred
to the variation in the heat content Q instead of the temperature 0, Then
the rate of radial motion is
r* / dr\ dr /
^1^1. I a^(ir'^)dr (128)
THERMODYNAMIC THEORY. 221
dd
by substitution of the value of -rr from the equation of conduction, while
the total rate of work of gravity is
r'dr (129)
which by equation (3), through integration by parts, takes the alternative
forms ^ y.
This gives simply the total rate at which energy is transformed from the
potential form, while the manner of its localization remains undetermined.
The last integral, which contains only thermodynamic quantities, suggests
that the rate of transformation per unit volume might be >l -^ ^ , but
any such special interpretation is purely arbitrary as long as the thermo-
dynamic substance is so incompletely defined.
The rates of specific linear expansion horizontally and vertically are
1 dr d /dr\ .. ,, .
The sum of the latter and twice the former gives by (128) the rate of spe-
cific volume expansion a -^ as it should. The moment of inertia is
I-'^Tt I 'pT-dr (132)
and since d[pi^dr) »0 its variation is
which gives
-i'L "
dl^-^n I pi^drdr
*-3-, v*r.i(^g)* ('»)
dl
Any supposition as to how a varies with the depth would appear to be
wholly gratuitous, but it may be worth while to follow out a simple one
suggested by the form of the equations, namely, that a is constant. Under
this condition the radial motion is
dr ,dtf
which is constantly negative at all depths, under all of the hypotheses en-
tertained above in the computation of the temperature-curves, so that the
222 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS.
mass everywhere shrinks toward the center, even while the temperature is
rising in the outer strata. In particular the surface falls at a rate propor-
tional to the temperature-gradient there, so that its shrinkage is first accel-
erated and later retarded, the most rapid fall occurring when the tone of
rising temperatures is disappearing.
In this case equations (131) become
aXdd d fsM
The former is everywhere negative; but the latter, if ^ is constant, is nega-
tive where the temperature-curve is convex upward and positive where it
is concave; moreover, where €, is negative it is numerically less than €|.
Thus the adjustment of density to temperature demands, in the interior,
both horizontal and vertical contraction, with the former more marked; in
the superficial strata, as long as the temperature rises, horizontal contrac-
tion and vertical expansion. To the extent to which the mass resists defor-
mation there are therefore developed at all depths a horizontal thrust and
a vertical tension, which accumulate at rate proportional to €^ — e^; this
may be written a n de\
Numerical or graphic differentiation from the tabulated temperatures shows
that at any one epoch the shearing stress thus indicated is roughly propor-
tional to the square of the distance from the center.
The maximum possible shortening of the total radius, corresponding to
a reduction of the temperature from its initial value down to zero through-
out the mass is
■4/'°""*
Jr.
which reduces to
rr
aE
At
,-4jJ .^*
AnJr^^
The value of ^r, between any two epochs can be found conveniently by
comparison of this last integral with that coming from (128) integrated
with respect to ( by substitution of the appropriate value of dO, The
computation shows that this total shortening, with — constant, would be
something less than 10 per cent of the whole radius; and graphic integra-
tion from table 6 shows that about one-third of this would be accomplished
during the epoch of rising temperatures, so that up to the time of maximum
surface-gradient the circumference would diminish by about 800 miles.
This number would, however, vary considerably under the different hypoth-
eses, of which the one adopted for the moment is such as rather to exag-
gerate the influence of the shrinkage in the central parts, and unless the
conductivity has been seriously underrated must probably be held to refer
THEBMODTNAMIC THEORY. 223
to a period much longer than the whole of geological history. A measure-
ment of the actual shortening indicated by the crumpling of the strata
would allow an estimate of the length of time elapsed, or the " age of the
earth. " Near the beginning the shortening is roughly proportional to the
square of the time, and as an indication of the order of magnitude a graphic
integration from the same table gives about 100 miles in a billion years, the
time being, moreover, inversely proportional to the numerical value used
for the conductivity.
There remains to be considered what is perhaps the principal point of
obscurity in the theory — ^the way in which the initial temperature is deter-
mined from the work of compression. This was done above through the
supposition that between these two there existed a definite proportionality
indicated by the specific heat, while the latter term was not sharply defined,
but for numerical illustration was assigned the value 0.2, an average value
of the ordinary specific heats of certain rocks. Now if e represent the work of
de
compression per unit mass, the ratio -yg for any path of compression is of
the nature of a specific heat in physical dimensions, but its identification
with a in any definite sense of the latter (except of course that which might
be defined as --jg for the given path) amounts to a condition on the ther-
modynamic properties of the "working substance" whose import there is
need to determine.
As concerns the relation of work and temperature, there may be con-
sidered to be two extreme cases conceivable, illustrated by the simple
mechanical example of a weight in frictional contact with a horizontal
plane and drawn by a spring. If the spring is very stiff it is only slightly
extended, and the greater part of the work of the impressed force is done
against the friction at the area of contact; if the spring is weak, the dis-
placement of the point of application of the force comes largely from the
extension of the spring and the corresponding work is stored as elastic
energy.
Corresponding to one extreme there is the fiction of a substance whose
resistance to compression is purely frictional, its transformation of energy
pure hysteresis — ^having at each density a certain critical pressure, the
maximum it could sustain without further crushing, and as a function of
the density to be used in formula (12) in computing the work of compres-
sion. Such a substance would show no tendency to restoration of volume
on relief of pressure; and though the manner of transformation be obscure
it seems natural to treat the heat derived from friction during compression
as equivalent to heat obtained by conduction or radiation, so that the ratio
of temperature to work would depend simply on the value, at the ultimate
density, of the specific heat in nearly the ordinary sense as related to con-
duction at constant volume or constant pressure, at least if the coefficient
of expansion be relatively small.
Such an interpretation read into Parts I and II would give a more
definite and perhaps reasonably self-consistent theory; but the conditions
described would fail to represent the behavior of surface-rock under the
224 GEOPHTSIGAL THEORY UNDEB THE PLANETESIMAL HYPOTHE8I8.
first moderate increments of pressure, and though they might be more
closely followed under pressures beyond the observable crushing point, yet
the agreement, for instance, of the theoretical and empirical values of the
modulus of compression would be little more than coincidence.
The other extreme corresponds to what may be taken as the definition
of a perfectly elastic substance in the thermodynamic sense, including for
the present purpose not only fluids but perfectly elastic solids, since the
work of shearing forces has been left out of account. Here Uie density
depends in a definite way on temperature and pressure, independently of
what series of changes the substance may have passed through; every path
of change is strictly reversible, and in any closed cycle the excess of mechan-
ical work is exactly accounted for by conduction and radiation, so that the
work of adiabatic compression may be considered as stored elastic energy.
The actual materials composing the earth may be judged to partake to
some extent of the properties of both extremes. Observed cases of the
flowage of rocks would seem to be concerned chiefly with permanent change
of shape, with little change of volume, but it is known that the equilibrium
of a body as large as the earth could not be purely that of an elastic solid,
unless it should possess elastic moduli much greater than those of known
substances, so that most probably the violent pressures occurring even at
moderate depths would lead to some permanent diminution of volume, or
such as partly to persist in the event of removal of the pressure. On the
other hand, (Urect experiments on the compressibility of rocks, under what
must here be considered small ranges of pressure, show approximately per^
feet elasticity, with a relatively trivial amount of hysteresis.
It may well be that both extremes could represent acceptably the be^
havior of the same substance under different circumstances; for instance,
according to the intervals of time involved. A bell made of pitch may
sustain well-developed vibrations counted by hundreds per second and yet
in a few hours flow into a permanently altered shape. Similarly the interior
of the earth may be capable of sustaining seismic tremors and tidal oscilla-
tions like an elastic solid, and yet under steady and long-continued stresses
yield in such a way that the expenditure of energy must be counted almost
wholly dissipative.
Thus in view of the great length of time which must be assumed for the
epoch of aggregation, the notion of compression with purely frictional re-
sistance, accompanied by the production of permanent set or non-reversible
diminution of volume, may be the appropriate one under the circumstances
postulated by the planetesimal hypothesis, and would seem to demand no
material modification in the essential features of the theory given in Parts
I and II. For example, under a steadily progressive compression the pres-
sure actually occurring with a given density at the corresponding depth
would be at every epoch the critical pressure for that density, so that the
density-curve for any epoch would necessarily, as was assumed without
comment by Fisher, determine the path of compression traversed by a defi-
nite element of the mass.
It is likely that the phenomena of dynamical geology may themselves
ultimately furnish the material for the most satisfactory estimation of the
THERMODYNAMIC THEORY. 225
propertieB of matter here in question. Their indication seems to point dis-
tinctly to the possibility of accumulation of truly elastic strains over periods
of time much greater than those involved in the oscillatory movements
commonly pointed to as witness to the existence of true elasticity. A
reasonably complete theory would doubtless have to include the simul-
taneous contemplation of both elasticity and viscous plasticity, of volume
and of shape, so as to complicate the theoretical deductions enormously.
To allow comparison it may, then, be of use for the present purpose to
inquire what modifications are needed to give the previous theory the
added definiteness which may come from a complete definition of the ther-
modynamic substance, but on the supposition that this possesses the oppo-
site extreme property, of perfect elasticity of volume under all conditions.
There remains, of course, the same possibility as before of variety in the
secondary features; the following developments give in some detail a
single form as illufftration, one which has the advantage of relative sim-
plicity in the analysis.
Let e now represent the total intrinsic energy per unit-mass; then a per-
fect fluid in the thermod3mamic sense, or a substance which can do work
only through hydrostatic pressure, and has perfect volume-elasticity in the
sense described, finds its complete description conveniently in the analytic
form of p and e as functions of d and v<"— . The condition of conservation
of energy as embodied in the first law ^
dO=|d» + (p+^)di, (134)
gives the following determinations of auxiliary quantities
dp
dv
where a, & are the specific heats, at constant volume and at constant pres-
sure; also ap
dv IL
where K, H, are the isothermal and isentropic bulk-moduli and a the co-
efficient of volume-expansion. The existence of a definite entropy-function
c imposes the condition of integrability
1?1„A^P^ (137)
0'dv d0\0)
in which case
^.f. ^-5-Xa (138)
dd 9 dv 60 ^
so that the condition of integrability is equivalent to
I da ^ d(Ka) (I39)
dv dO
226 GEOPHYSICAL THEORY UNDER THE PLANBTBSIMAL HYPOTHESIS.
In addition to special hypotheses as to the form of the functioiis p, $,
it would be necessary to specify the exact path of the compression in order
to determine the rise of temperature produced by a given amount of me-
chanical work. It has, however, been assumed from the beginning that in
view of the low conductivity of rock the compression might be considered
as relatively instantaneous and therefore adiabatic; under this condition
the path of compression would be a curve of constant entropy, and the
ratio of mechanical work to rise of temperature would be determined by
de \ de p
Id/r ^ ' . de (140)
de\
Comparison of this with (135) shows that -jgj^t which was treated pre-
viously as a specific heat, can be identified with the specific heat at constant
de
volume only at points where "^'O. This latter condition is satisfied
identically by a perfect gas, which the substance might perhaps resemble
in this respect, while di£fering widely in the relation between pressure,
density, and temperature.
de
If, however, -Tp""0 is satisfied everywhere, so that the intrinsic energy
is a function of the temperature only, equation (137) shows that p would
have the form p=0V and consequently by (136) that Ka*^V; where V
is some function of v only. Since at the surface the pressure vanishes, V
would be zero for the argument v^, hence the surface material would have
to be either isothermally incompressible or have a zero coefficient of expan-
sion. Now the observed compressibility and expansion of surface rock are
enormously less than for gases, but the existence of an appreciable value
for both of these is a necessary element in the application to d3mainical
geology, so that a correction is called for if the above be taken as the mean-
ing of the specific heat used in equation (58). Though there is nothing to
impose this special interpretation, the result still suggests one way in which
a coherent theory can be constructed, as a modification of the previous one,
but such as to take account of the measured values of all the thermal and
dynamical coefficients.
Let it be supposed that the specific heat a at constant volume is a con-
stant, understood henceforth as measured in mechanical units; that the
intrinsic energy, instead of depending on the temperature only, has the form
e-<7tf+^(v) (141)
where ^(v) is a function of v to be determined; and that the isentropic lines
have the form .
P-/i(0-y+/a(0 (142)
V'
The latter condition results from the Laplacian equation (21) by treating
h and p^ as functions of the entropy, and is suggested as a condition in view
of the assumption hitherto made, that the path of compression is deter-
THEBMODTNAMIG THEORY. 227
mined by the relation between pressure and density as exhibited at various
depths within the earth at any single epoch. For since the compression has
been treated as adiabatic, this identification of the two thermodynamic
paths means that as long as the effect of the conduction is insensible the
nucleus would be in a condition of isentropic or convective equilibrium
such as described by Bitter and Kelvin. The generalization of equation (21)
consists in supposing that it would give the form of the pressure-density
curve corresponding to convective equilibrium for any value of the entropy
by proper choice of the constants h and p^. It is, then, required to deter-
mine the functions /i,/,, ^, so as to satisfy the given conditions.
Equation (134) shows that
de-^jdO+^^^dv (143)
in which the coefficient of dv must therefore be a function of v only, which
for convenience may be written
and then the integral of (143) is
dip «C -•^ (• = Naperian base) ^^^)
in which the constant of integration is considered to be absorbed in the
undetermined function ^.
Elimination of from (144) and (145) gives
P-<^-^' (146)
which must be identical with (142), and thus gives
Hence
/iW -<^Ac+a /,(f) aBc-b (147)
and
from which come
— 1
^-(1+Bv+c) (148)
^^^+bv (149)
The constant of integration in ^ is omitted, since only differences in the
intrinsic energy are in question, so that there remain five undetermined
constants. In tenns of these auxiliary functions, then, the properties of the
substance are described by equations (141), (144), and (145), from which
may be deduced the following:
15
QBOrBTSICAL l!HXOBT UMDnt IBB VLAKtmSIUAh HTTOIBBBIB.
Ka~»K H -^ («Ac+a) (ISl)
»^1+E^ (152)
IF
Blnee the preBsuie-deiiflity curve at the eloee of the epoeh of eompreeBon k
•apposed to be an adiabatio line, the variation at different depths of all the
magnitades eoneemedi for a planet of definite masBi win be determined in
terms of the constants introduced, when a value is asrigned for the entropy,
which may as well be taken equal to sero, since the levd of reefconing is
arbitrary. The various surface-values are then fixed by the eondition]ig>>iOt
with tt""0| so that the constants are subject to the c<mditionB:
«,-4+B»i+C (154)
5ffl2l«i._B (185)
^-aB+fr (IM)
If then K„Vu9u«u't ""^ eonndered to be obtained by dinet meuuze-
ment, c and Hi may be computed from
o-Ot'-Kivfi^* (157)
ff^.x/l+^iE^*! (168)
and then the wlution of (168) to (166) gives
t>i* o
(IW)
a~?^-aA (160)
b^^-aB (161)
C~0t~-Bvi (162)
leaving the single parameter A undetermined. The Laplacian constant
*" A-H.V (163)
in terms of which the distribution of pressure and density are still given by
(21) and (26), while the excess of intrinsic energy and of temperature over
their surface-values are
THBRMODTNAMIG THEOBT. 229
*^-*^'"^» (fi-^r+^^C".-") (166)
the former corresponding exactly to (22) above.
In the equation (165) for the temperature the first term corresponds,
except as to the value of the coefficient, to that which alone occurred in
the analogous equation in Part I for the case of constant specific heat;
while the second term is new and may be considered as the corrective term
needed to take account of the coefficient of expansion, which occurs as one
of its factors. The numerical value however of this corrective term is
small; for instance, with JiC| = 4XlO^Sd|>"300 (the absolute temperature
of the surface), e4«"2XlO~* and a^-^J, its maximum value, which occurs
at the center where the density is greatest, is about 80^, while it contributes
an initial surface gradient of 1^ in 14,000 meters. The value here used for
Ki comes by comparison with the value deduced for H^ in Part I, which, as
is seen by (158), exceeds it by only one four-hundredth part and is known
to range well with the results of direct measurement on surface rock.
It appears thus that to take account of all the measured mechanical-
thermal constants relating to surface-rock it is sufficient to include only
this trivial variation from the distribution of temperature specified by the
first term of (165), which in algebraic form is identical with that occurring
in the corresponding case in Part I. But the absolute value of this prin-
cipal term depends on the coefficient A, which is here left undetermined,
instead of having the definite value hl2a as under the former hypothesis.
So far as concerns consistency with the conditions thus far assumed, this
constant might be chosen so as to give any assigned value to the central
temperature for example, or to the surface-gradient at some particular
epoch of the conduction. But to make even the maximum surface-gradient
match the present observed value of 1^ in 30 meters it would be necessary
to choose A so that the initial temperatures, while similar in relative mag-
nitude at the various depths, would be in absolute value several times as
large as those listed in column 8 of table 2. This would perhaps be a rather
extreme supposition, though there seem to be no definite data to the contrary.
A suggestion from another source as to a fair mean value to be taken
for A is the following:
In the assumed expression (141) there are the two terms: c0, which may
be considered for vividness as the kinetic portion, and ^(v) the potential
portion, of the intrinsic energy e. The latter part, depending conceivably
simply upon the mutual distances of the constituent particles, may by
analogy with the dynamics of particles plausibly be supposed always to
increase as v increases. Now from (149), transformed by substitution of
the constants as determined in terms of the surface-data, comes
,'(.)-(.A-^)(^-i,)+XM (166)
)
230 GEOPHYSICAL THEORY UNDER THE PLANETEBIMAL HYPOTHESIS.
80 that as v decreases from the value v^, the sign of ^' is ultimately that of
H v^
a A ^, unless this be zero. If then ^ is to be an increasing function of
V, the most moderate value of A which can be assumed is
which as appears from (22) and (58) would give from the first term of (163)
a temperature distribution precisely the same also in absolute values as
that deduced under the first hypothesis in Part I.
This result shows that there is not necessarily any radical difference
between the temperature-distributions deducible from the two alternative
extreme suppositions described, though variations in the special hypotheses
may be expected to produce here, as under the opposite view, considerable
differences in the quantitative conclusions.
It has been pointed out that the surface-gradient of temperature as
deduced from the theory would probably never reach the value at present
found by observation, at least unless the conductivity increase consider-
ably from the surface downward. But the present theory has taken ac-
count only of the heat obtained from gravitational energy, to which must
be added that obtained from several other assignable sources. These might
be relatively unimportant, so that their effect could be included in the
sense of corrections to the above results; but if of sufficient magnitude they
might alter the features of the thermal process completely.
In particular, if heat were steadily generated in each unit of volume at
rate ^(r), the equation of conduction would be
the solution of which may be written
0^u+ I ^ I i^f(T)dT (169)
where u satisfies the equation
dii 1 d /, ,au\
dr)
and is to be determined so that d satisfies the given initial and boundary
conditions. For example, if ^ is constant or the rate of generation per
unit volume the same throughout the mass, the steady component, defined
by the integral in (169), is, with X constant:
whence ^ ^
80 that with the flux corresponding to thermal equilibrium a surface gra-
dient of 1° in 30 meters would mean a central temperature of 100,000**,
THSBMODTNAiaC THBOBT. 231
whatever be the values of specific heat and conductivity. An effect of such
magnitude, produced by radio-activity or otherwise, is not to be lightly
assumed. It is perhaps more likely that the effect of these agencies is
largely confined to the superficial strata.
Another possible explanation of the discrepancy respecting the tern*
perature-gradient is that the assumed density laws may not properly repre-
sent the variation of density near the surface. For, independently of any
special hypothesis, it is plain that in descending from the surface the den-
sity would tend to increase on account of the increase of pressure, and to
decrease by virtue of the increase in temperature; which effect controls
depends on the values of elastic modulus, coefficient of expansion, and
thermal gradient.
Let the density p through its dependence on p and be thereby a func-
tion of the distance b below the surface. Then
but since
this may be written:
dp ^ dp dp dp dd
d$'' dp da 60 ds
^P -/in ^P^ P
de dp
^--^(gp-Kar) (170)
With the values /r-4X10", a«2X10-^ used above, which appear to be
well supported by observation, this would be positive or negative accord-
ing to whether the gradient is less than or more than 1^ in 30 meters, or
just about the observed value. If the gradient were greater than this there
would be at first an actual decrease of density from the surface downward,
because of the preponderant effect of temperature over pressure; if less
there would be an immediate increase in density such as is implied in the
formulas (25) and (74). In the intermediate case, where Kay^gp, as ap-
pears to be nearly realized at the present time, the density-curve would have
a horizontal tangent at the surface-point. It would be useful if a family of
such density-curves could be foimd, expressible in tractable analytic form,
and satisfying the conditions named in connection with equation (92), so
that the effect of such a modification could be determined, through a review
of the preceding theory with the appropriate changes.
For a discussion of the bearing of these results on geological theory;
reference must be made to the papers of Professor Chamberlin, to whom
the writer is deeply indebted, not only for the opportunity of co-operation
in this work, but for suggestion and assistance freely given during repeated
conferences.
OONTRIBIITIONS TO OOSMOGONT AM THE FUin)A][EllTAL PROBLEMS OF OEOLOeT
THE REUTIONS OF EQUILIBRIUM BETWEEN THE CARBON
DIOXIDE OF THE ATMOSPHERE AND THE CALCIUM SUL-
PHATE, CALCIUM CARBONATE, AND CALCIUM BICARBON-
ATE OF WATER SOLUTIONS IN CONTACT WITH IT
BY
JULIUS STIBGLITZ, PH.D.
ProfenoT of ChemMtry^ Unwermiy of Chicoffo.
288
THE REUHONS OF EQUILIBRIUM BETWEEN THE CARBON DIOXIDE OF THE
ATMOSPHERE AND THE CALCIUM SULPHATE, CALCIUM CARBONATE, AND
CALCIUM BICARBONATE OF WATER SOLUTIONS IN CONTACT WITH IT.
When two difficultly soluble salts, such as barium sulphate and barium
carbonate, are formed in a given medium, for instance by the addition of
barium chloride to a mixture of potassium sulphate and carbonate, they are
precipitated, as is well known, in the order of their insolubility. In the
given case barium sulphate, the less soluble salt, is precipitated first, the
more soluble carbonate last. We apply this principle in the familiar case
of the volumetric determination of chlorides by titration with silver nitrate,
in which potassium chromate is used as indicator; silver chromate is very
difficultly soluble, but silver chloride is less so, and is precipitated almost
completely and within the limits of exact quantitative analysis, before any
solid red silver chromate can permanently be formed, the first persisting
appearance of the latter being taken, indeed, as the evidence or indication
that the precipitation of the chloride has just been completed.
The application of the laws of physical chemistry, especially those of
chemical and physical equilibrium, to such cases of precipitation shows, how-
ever, that there must be a limit to this principle of the order of precipitation;
the principle itself may be derived by the application of these laws and is
always, of course, subject to them. The existence of such a limit and its rela-
tions to the solutions in contact with given salts have been investigated
experimentally in a number of cases, the most notable investigation being
the classical one of Guldberg and Waage ' on the conditions of equilibrium
between barium sulphate and carbonate, and potassium sulphate and car-
bonate. The limiting values in all such cases are of extreme interest, as
from them certain definite conclusions may be drawn as to the nature of
precipitates formed in given cases; or, vice versa, from the nature of the pre-
cipitate, conclusions as to the composition of the medium may be drawn,
and the existence of a limiting value leads to the possibility of a complete
reversal of the usual order of precipitation under given conditions.
As has just been mentioned, and as will be shown in detail presently,
the limiting value is for two given, little-soluble salts most intimately asso-
ciated with the composition of the liquid medium from which precipitation
occurs; the composition of the medium may in turn be dependent for one
or more of its essential components, e.g., dissolved carbonic acid, on the
nature of the atmosphere above the solution, with the result that the com-
position of the atmosphere may become a fimction in the mathematical
expressions deduced by the application of the laws of physical chemistry
to the facts of precipitation.
> Joumsl for Praktisehe Chemie (2), 10, 60 (1870).
235
236 EQUILIBRIUM BETWEEN CARBON DIOXIDE OF ATMOSPHERE
Considerations of this nature led the author in 1903/ in the course of a
conversation with Dr. T. C. Chamberlin on the remarkable freedom from
calcium carbonate of a deposit of gypsum/ to suggest that possibly there
might be some connection between the purity of the gypsum and the carbon
dioxide content of the air at the period of the gypsum formation, since the
carbon dioxide in the atmosphere is one of the most important factors
influencing the solubility of calcium carbonate.
Inasmuch as climatic conditions, according to recent theories enter-
tained qualifiedly by Dr. Chamberlin, are also dependent to a certain degree
on the carbon dioxide in the atmosphere, it seemed possible that a study
of the precipitation of calcium sulphate and calcium carbonate from the
point of view of the laws of equilibrium might lead to conclusions which
would be of some use as a further source of information relative to condi-
tions existing at a remote period.
Partly on account of the possibility of obtaining results of some such
specific geological value for the work on the climates of the earth; chiefly,
however, in order to test the possibility of exploiting this method of investi-
gation for geological purposes, Dr. Chamberlin asked me to undertake, in
collaboration with himself, the work of making the necessary calculations
on the conditions of equilibrium determining the precipitation of calcium
sulphate and calcium carbonate.*
As is usual in these cases, it was decided to consider first the ideal case
of solutions containing only the two salts in question. But the work has
been extended to estimate proximately the influence of the presence of
other sulphates in the solutions in the proportions found in ocean waters of
the present day; and finally an attempt has been made to consider the
effect of concentrated salt (sodium chloride) solutions on the conditions
studied. The results obtained for the simpler case of equilibrium for dilute
solutions in the presence of sulphates and moderate amounts of sodium
chloride may, it is hoped, also prove useful in connection with present-day
problems on the calcium carbonate content of the oceans and of fresh waters.
The study of the equilibrium conditions controlling the precipitation of
calcium carbonate and sulphate falls naturally into two parts — a study, on
the one hand, of their relative solubilities, and the consideration, on the
other hand, of the conditions of equilibrium for saturated solutions of cal-
cium carbonate, calcium bicarbonate, and carbonic acid in equilibrium with
an atmosphere of some given content of carbon dioxide, the latter being a
determining factor in the total solubility of carbonate. This second and
more complex part of the study will be taken up first.
* Year Book No. 2, Carnegie Institution of Washinjgton, p. 269.
' The observation was made by Mr. F. A. Wilder in the course of an investigation for
a doctor's thesis submitted to Professor Chamberlin.
* Pressure of other work made it necessary to postpone the calculations to the year 1907
AND CALCIUM CARBONATE, ETC., OF WATEB SOLUTIONS. 237
EQUILIBRIUM IN AQUEOUS SOLUTIONS OP CALCIUM CARBONATE,
CALCIUM BICARBONATE, AND CARBONIC ACID IN CONTACT WITH
AN ATMOSPHERE CONTAINING CARBON DIOXIDE, AND THE SOL-
UBILITY OP CALCIUM CARBONATE IN WATER CONTAINING PREE
CARBONIC ACID.
The most reliable and complete experimental determinations of the
solubility of calcium carbonate in water containing free carbonic acid were
made by Schloesing.^ The theoretical treatment of his results from the
point of view of equilibrium conditions we owe to Bodlaender.' The latter's
work, correct in its theoretical treatment/ shows an error of moment in
the calculation of the solubility constant of calcium carbonate, the constant
most important to us, an error due largely to an error in one of the inves-
tigations from which Bodlaender drew his data.^ McCoy's more recent work
on the equilibrium in aqueous solution for sodium carbonate, sodium bicar-
bonate, and carbonic acid' gives the necessary material for the correction of
the above error in the following pages.
The complex conditions of equilibrium involved when water is saturated
with calcium carbonate and with carbon dioxide under any given partial
pressure may be developed as follows: For a saturated solution of calcium
carbonate, say in contact with the solid carbonate, we have *
CaCO« ^ CaCO, (1)'
Calcium carbonate in aqueous solution is very largely ionized according
^ CaCO, ^ Ca- +C0,^ (2)
At a given temperature in a saturated solution of a difficultly soluble
salt of this nature the product of the concentrations * of the ions is a con-
stant,* which is called the solubility product or the ion product of the salt:
CoaXCcc-Koioo, (8)
That the product of the ion concentrations is equal to a constant for
saturated solutions of difficultly soluble salts must at present be considered
an empirically established fact. As is well known, the law of mass-action
does not give constants when applied to the ionization of strong electro-
lytes, such as salts are. For instance, for the reversible reaction
Naa±?Na'+a^ (4)
> Comptes RenduB. 74, 1552 (1872); 75, 70 (1872).
'Zeitschrift fQr Physikalisehe Chemie, 35, 23 (1900).
' See below, this page, in regard to a contested question of theory.
« McCoy, American Chemical Journal, 29, 437 (1903).
* Ibidem.
* Underscoring of a symbol is used to indicate that the substance is in solid form.
* See below in regard to the conclusion usually based on this equation that at a given
temperature the concentration of the dissolved non-ionised calcium carbonate molecules,
CaCO|, or tiie molecular solubility, has a constant value.
* The term concentration, for which C is used in all the equations, is taken, in accord
with chemical practice, to designate the number of gram molecules or moles of substance
in 1 liter of a solution or of a gas.
* Ostwald, Scientific Foundations of Analytical Chemistryj Nemst, Theoretical Chem-
istry, p. 531 (1904); Bodlaender, loe. eU,; A. A. Noyes, Zeitschnft fQr Physikalisehe Chemie,
16. 126 (1895), 26, 152 (1898), 42, 336 (1902); Report of the Congress of Arts and Science,
vol. IV, 322 (1904); Le Blanc, Zeitschrift fOr Anorganische Chemie, 51, 181 (1906).
238 EQUILIBRIUM BETWEEN CARBON DIOXIDE OF ATMOBPHSBE
we find as a matter of experience that the proportion
is not a constant for different concentrations, as the corresponding expres-
sion is for all other kinds of reversible reactions, including the ionixation of
weak bases and weak acids. The above proportion for a strong electrolyte
grows larger with increasing concentration — the most promising explana-
tion of this apparently abnormal behavior being perhaps that the ionizing
power of the solution is changed by the presence of considerable numbers
of ions (the "salt-e£fect" of Arrhenius).
This result, applied to the ionization of calcium carbonate (equation 2) ,
would mean that the proportion
is not a constant. M)iOO»
It is ordinarily assumed that for the saturated solutions of diflleultly
soluble salts the molecular concentration of the salt (here Ccmx>^ u a
constant at a given temperature in all aqueous solutions: we would expect
then, from what has just been said, that the product Cq^XCq^qc}, could
not have a constant value, since it is the numerator in the variable propor-
tion (6). As a matter of experiment, however, A. A. Noyes, Findlay, and
others found in a number of carefully studied cases of similar salts that
the solubility products are constants, whether the given salt is present alone
or in the presence of another salt modifying its ionization, such as a salt
with an ion in common with it.
The peculiar discrepancy between the empirical results expressed in (5)
and (6) on the one hand, and the empirical result expressed in the constancy
of the ion product on the other hand, as just explained, seems to find a
satisfactory explanation in the work of Arrhenius on the solubility of
salts in salt solutions.^
Arrhenius shows that the old view must be abandoned that in the pres-
ence of excess of the solid salt, the solubility of the molecular or non-ionized
salt is a constant in different saturated solutions (i.e., in solutions saturated
in the presence of other salts). His experiments on the solubility of the
silver salts of various organic acids (silver acetate, valerate, etc.) in the
presence of varying amounts of the sodium salts of the same acids show
that the solubility of the non-ionized silver salt grows smaller with increased
concentration of the sodium salt present. In other words, there is no con-
stant molecular solubility of a precipitate, as has so long been assumed.
The case is analogous to the decreased solubility of gases, such as oxygen or
carbon dioxide, in salt solutions as compared with pure water — a fact which
is used below in the discussion of the equilibrium conditions. |^
Arrhenius uses his data as an argument against the correctness of some
of Noyes's conclusions. No attempt, however, seems to have been made^by
Arrhenius in this paper to study the ion-products for the silver salts in his
own experiments, and calculations were therefore made with his material to
'Zeitschrift fUr Physikalische Chemie, 31, 221 (1899).
AND CALCIUM CARBONATE, ETC., OF WATER SOLUTIONS.
239
ascertain whether, in spite of (or rather because of) the variability of the
molecular solubilities, the silver salts give constant solubility products or
not. As is seen from the tables given below, the extremely interesting
result was obtained that, according to Arrhenius's own experiments, the
silver salts, whether present alone or with varying amounts of sodium salts
with the acid ion in common with them, give rather good constants for
the solubility products.
As the question of the constancy of the solubility or ion product for a
difficultly soluble salt is of particular importance in the investigation this
paper treats of, space will be taken here to report the calculations made
with Arrhenius's data for three of the silver salts, the acetate, which is the
most soluble salt studied, the valerate, and the butyrate, which are very
much less soluble than the acetate.
In table 1, giving the solubility of silver acetate at 18.6° in the presence
of varying amounts of sodium acetate, column 1 gives the molar concentra-
tion of sodium acetate used; column 2 gives the degree of ionization of the
sodium acetate in the mixture; column 3 gives the concentration of the
ionized sodium acetate; column 4 the total solubility of silver acetate;
column 5 the degree of ionization of the silver acetate; column 6 the con-
centration of the ionized part of the silver acetate; column 7 the concentra-
tion of the non-ionized part, which represents therefore the molecular
solubility of the silver salt. Columns 1, 4, 5, 6, 7 are taken from Arrhenius's
tables; the degrees of ionization of the sodium acetate as given in column 2
were calculated with the aid of the isohydric principle, whose reliability has
been amply demonstrated.^ Column 3 is derived from columns 1 and 2.
For a saturated solution of silver acetate
CH3C0,Ag±5CH,C0/-l-Ag-
the solubility product would be
Ca«XCchkx)i"K
(7)
(8)
in which Cgh^oo, represents the total concentration of acetate ions irrespec-
tive of their origin from silver or sodium acetate; in each of the experi-
ments it is the sum of the values given in a line under columns 3 and 6. The
values of the solubility product constant K, as calculated, are given in the
last column.
Table 1.— ^SOvar AalaU.
1
2
t
i.
6
6
7
8
N».AO0t.
100«i
lO*Acet.'
Ag-Acet
lOOaa
lO^Aoet''
101 Mol.
KxlO»
• • • •
0.0508
72.0
42.7
16.6
182
0.033S
SOi)
26.6
0.0474
68.1
32.3
15.1
190
0.0667
78.0
52.0
0.0384
64.4
24.7
13.7
190
0.1338
76.0
100.0
0.0282
58.4
16.4
11.8
191
0.2667
69.6
1S6.7
0.0208
50.4
10.2
10.1
200
OJMQO
63.0
816.0
0.0147
42.8
6.3
8.4
202
* Noyes, CongreflB of Arts and Science, St. Louifl, vol. iv, p. 318, gives an excellent sum-
mary on this question.
240
EQUILIBBIXJM BETWEEN CABBON DIOXIDE OF ATMOSFHKBB
Tables 2 and 3 correspond to table 1.
Tablb 2.— Silver VaUrate.
1
2
8
4
6
6
7
S
N».Val.
lOOft]
lOi.VaL'
Ag'Vnl.
lOOaa
10»VmL"
lO^MoL
KxlO*
....
• • • •
0.0095
87.3
8.3
1.2
6.9
0.0175
86.5
15.1
0.0047
81.1
3.8
0.9
7.2
0.0349
84.0
29.3
0.0030
75.5
2.2
0.8
6.9
0.0689
80.1
55.9
0.0018
68.1
1.2
0.6
6.9
0.1395
75.8
105.7
0.0015
59.7
0.9
0.6
(M)
Tablb Z.— Silver BvtyraU.
1
2
8
4
5
6
7
8
Na-Batyr.
lOOiil
10«Batyr.
Ag-Batyr.
lOOot
lOiBatyr.
101 MoL
KxlOi
....
• • • •
0.0224
81.1
18.2
4.2
33.0
0.0066
86.0
56.8
0.0199
79.6
15.8
4.1
(49.8)
0.0164
84.9
13.9
0.0169
77.3
13.1
3.8
35.4
0.0329
83.0
27.3
0.0131
73.5
9.6
3.6
35.4
0.0658
80.0
52.6
0.0091
67.7
6.2
2.9
36.4
0.1315
75.9
100.0
0.0060
59.9
3.6
2.4
37.3
0.2630
70.1
184.4
0.0040
51.1
2.0
2.0
36.8
0.4980
63.3
312.1
0.0027
43.2
1.2
1.5
37.6
It is quite obvious from a consideration of the values in the last column
of each table that rather satisfactory constants are obtained in all cases for
the ion products of the silver salts in saturated solutions. It may be added
that of the two bracketed irregular values, the one in table 2 (last line)
corresponds to an experiment which shows the probability of some experi-
mental or other error in considering the seventh (next to the last) column,
where 0.6 remains unchanged, while it should grow smaller. And the
irregular value in table 3 (line 2) corresponds to an experiment the figures
for which, as given by Arrhenius, show a decided divergence from the prin-
ciple of isohydric solutions — so that it also is unreliable.
Since, then, the results of Arrehnius's determinations, as thus calculated,
agree with the conception that for a difficultly soluble salt the solubility
product is a constant at a given temperature for saturated solutions, and
since this conclusion was also reached experimentally by A. A. Noyes,
Findlay, and others, as explained above, we may accept this now as an
empirically-established fact. The values calculated below for the solu-
bility product of calcium carbonate on the basis of Schloesing's data also
show excellent agreement.*
The value of the " solubility product " for calcium carbonate is particu-
larly important for the study of the precipitation of calcium carbonate
under varying conditions, and its calculation is one of the first objects of
this investigation. Its significance lies in the fact that since in saturated
solutions of calcium carbonate in different mixtures the concentrations of
the calcium and the carbonate ions are variable, they need not and usually
* Vide Stieglitz, Journal of the American Chemical Society, 30, 946 (1908), for a more
complete discussion of this question.
AND CALCIUM CARBONATE, BTC., OF WATER SOLUTIONS. 241
would not have equal valueSi but they are dependent on each other to the
extent that the product of the two concentrations has a constant value for
the saturated solutions. For instance, if to a saturated solution of the
carbonate any acid is added, either a strong one like hydrochloric acid or
a weak one like carbonic acid, the hydrogen ions of the acid must imme-
diately combine to some extent with part of the carbonate ions, CO,^, to
form more or less bicarbonate ions, HCO,': as the CO/ ions disappear, more
molecular calcium carbonate must ionize (equation 2) to re-establish the
equilibrium, leaving the solution undersaturated with molecular carbonate,
and the solid carbonate, if present, must dissolve. AU this can be briefly
expressed in the equation
Coa X (Coo.^-«) < Koaoo. (9)
The solution is undersaturated when the product of the concentrations
of the calcium and carbonate ions is smaller than the solubility constant;
similarly oversaturated (precipitation resulting) when the product is greater
than the constant; and just saturated when the product equals the constant.
When the solubility of calcium carbonate is thus increased by the addi-
tion of an acid (say carbonic acid) owing to the formation of bicarbonate
ions, HGO/, calcium bicarbonate Ga(HC03)] is formed; it remains, as most
salts do in dilute solutions, very largely ionized. In equations (3), etc.,
Cqi^ means, of course, the total concentration of calcium ions, irrespective
of their origin from calcium carbonate or bicarbonate.
These relations which we have been discussing in a qualitative sense
may be developed quantitatively as follows:
The formation of calcium bicarbonate by the action of carbonic acid on
calcium carbonate is a reversible reaction:
Ca-+CO/+H+HCO/ ±? Ca- +2HC0/ (10)
and
Ca- +2HC0/ ±* Ca(HCO,), (11>
The calcium ions appearing on both sides of equation (10) in equal
quantities evidently do not affect the equilibrium, and we have more
^^^ ^ CO,^ + H- +HCO/ ±5 2HC0/ (12)
and consequently:
Coo.XCHXCHoo.-iXC?Hoo. (13)
Canceling ChoOi o^ both sides, we have
Cco»XCh ji-^ f^As
— p -K'^Iooi^tioii (14)
The study of these equations shows that, except for the formation of some
non-ionized calcium bicarbonate, the formation of bicarbonate and the
equilibrium between bicarbonate and carbonate are largely independent of
the nature of the metal ion present, particularly since all salts (like calcium
bicarbonate) are largely ionized in dilute solutions and all similar salts are
about equally ionized in equivalent solutions. The above conclusion has
242 EQUIUBRIUM BBTWEBN CARBON DIOXIDB OF ATMOSPHERE
been confirmed by the work of Bodlaender on calcium and barium bicar-
bonate and of McCoy on sodium bicarbonate.
Equation 14 represents the equilibrium condition between carbonate ions,
hydrogen ions, and bicarbonate ions, or the secondary ionisation of carbonie
acid expressed in ^^.q^, ^ ^. ^^q,, ^^gj
and the constant of the equation may be called the second ionisation oon-
stant of this acid.
The primary ionization of carbonie acid, which must supply the major
portion of hydrogen ions, is expressed in
H,C0,±5H+HC0/ (16)
and we have then:
Ch X ChCO> __ XT/ /17\
K'loniiaUon ^^Yf then, be called the first ionization constant of carbonic
acid.
The concentration of carbonic acid in solution is, according to Henry's
law, at a given temperature, proportionate to the concentration or partial
pressure of the gaseous carbon dioxide in the atmosphere with which the
solution is in equilibrium, viz:
CH«co.=igMXC3oo, (18)
This completes the equations involved in the complex condition of
equilibrium we are considering. Summarizing our conclusions, we have
the following four mathematical equations expressing the conditions of
equilibrium in saturated solutions of calcium carbonate and bicarbonate in
contact with the atmosphere, all the constants of which must be simul-
taneously fulfilled for the condition of equilibrium:
I* Coft X Ceo, ■" Ksolub. Prod.
II' ChsOOs " *ga8 X Ceo,
IIP Ch X Cneo, = K'lonUatioB X Chkx),
IV* ChX Ceo, »K^iom«ti«iX Cneo.
The solubility of calcium carbonate, either as carbonate or as bicarbon-
ate, reaches a limit when the solubility product of equation I is reached,
and the value Ceo,; ^^^ concentration of the carbonate ions, in this equa-
tion is in turn a function of equation IV; two values of this equation are
functions of equation III, in which in turn the concentration of the dis-
solved carbonic acid CHjeo, is dependent on the atmospheric carbon dioxide
as expressed in equation II.
The constants of equations II, III, and IV are known, and the constant
of equation I, which we wish to determine, can be obtained from the other
constants with the aid of Schloesing's experimental work on the solubility
of calcium carbonate.
* The solubility product of a saturated solution.
' The solubility of carbon dioxide under varying partial pressures.
' The primary ionization of carbonic acid.
* The secondary ionization of carbonic acid.
AND CALCIUM CARBONATE, ETC., OF WATEB SOLUTIONS. 243
The solubility factor of carbon dioxide in water, kg„ of equation II, has
been determined by Bunsen^ for different temperatures. The solubility is
usually given in liters (a) of gas, reduced to 0° and 760 mm., absorbed by
1 liter of water; then to reduce this to terms of gram molecular concentra-
tions as required for the application of the mass-action law, we make
(19)
22.4
since a gram molecule of a gas occupies 22.4 liters under normal conditions.
For instance, at IQ"", 0.9753 liter is absorbed or 0.9753/22.4 « 0.04364 mole,
if the carbon-dioxide gas has the pressure of 1 atmosphere. For any other
pressure P of the gas, expressed in atmospheres, the molar concentration
of dissolved carbonic acid is, then, according to II
Ch.co,-*«mXP (20)
In salt solutions the solubility is considerably less than in pure water,
as determined by Setchenow* and more recently by Ge£fcken/ and, when
salts are present, corrections made on the basis of these determinations
wiU be used.
The first ionisation constant of carbonic acid (equation III) has been
determined by Walker and Comack^ from the conductivity of aqueous
solutions of the acid. The secondary ionization is so small (see below)
that it scarcely contributes to the conductivity of carbonic acid and con-
sequently it can be neglected in the determination. The value found for
the first ionisation constant is:
K'loni^tion -3.04 X 10-' (21)
The second ionization constant of carbonic acid (equation IV) was cal-
culated by Bodlaender ' from Shields's * experiments on the hydrolysis of
sodium carbonate, but, owing to an error in Shields's calculations, found by
McCoy,' the value given by Bodlaender (1.295 X lO^'^O can not be accepted.
It was recalculated by McCoy from Shields's results and foimd to be 12.0 X
10~" or ten times as large as Bodlaender's value. McCoy then determined
the constant by a study of the condition of equilibrium between sodium
bicarbonate, carbonate, and carbonic acid, and from the values for deci-
normal solutions of bicarbonate he obtained 6.0X10~~^", which was only
half as large as the constant calculated from Shields's data; 0.3 normal and
normal bicarbonate solutions gave still other values.
A recalculation of this constant, which is important for our work, was
made from McCoy's data, and corrections which were indicateci^^by the
latter, but not carried out, were made for changes in the solubility of carbon
^ liebig's Annalen, 93, 20 (1855); Dammer, Handbuoh der Anoiganisohen Chemie
II, 372. See also Geffcken, Zeitschrift fQr PhysikaliBche Chemie, 49, 257 (1904).
> Dammer, Handbuoh der Anorganischen Chemie, n, 1, 367.
« Journal of the Chemical Society, London, 77, 8 (1900).
§ Yiiui cit*
* Zeitschrift fOr Physikalische Chemie, 12, 174 (1893).
"^ Loe, cU,
16
244 SQXnUBRIUM BETWEEN CABBON DIOXIDE OF ATM08PHSRE
dioxide in salt solutions and for the changes in the ionixation of salts in
mixtures. While the recalculation did not change the order of the results,
it removed the uncertainty as to the effect of these corrections.
From the two equations for the ionisation of carbonic acid. III and IV,
we get by dividing III by IV:
ChcO« _ K'lcmimtioB _j^ 722)
The constant K can be readily calculated from any of McCoy's results,
and then K^iooi«aioo 3.04 X lO"^ ,«.,
^ lonintioo^ j? " g K^)
In 0.0999 normal solution of sodium bicarbonate according to table 2,*
under a partial pressure of 0.00161 atmosphere (P), McCoy found 68.2 per
cent bicarbonate and 31.8 per cent carbonate. The degrees of ionisation
of these salts can be put equal to those of sodium acetate and sodium
sulphate, respectively, similar salts ionising very much alike, and the deter-
minations of the ionisation of the acetate and sulphate from their conduc-
tivities being far more reliable than the estimations from the conductivitifiB
of bicarbonate and carbonate solutions which really represent complex
mixtures. A consideration of the curves for the conductivities for sodium
carbonate and sulphate shows them in fact to be practically parallel for
more concentrated solutions in which the hydrolysis of the carbonate is
small, but for more dilute hydrolysed solutions the curve for the carbonate
bends and cuts the sulphate curve, which is an indication of hydrolysis.
Hence it was deemed safer to determine the degree of ionisation from the
sulphate and acetate curves. Then in the experiment ' mentioned we have
Choo, -0.0999 X 0.682 X 0.783 -0.05335
Ceo, -0.0999X0.5X0.318X0.687 -0.01091
The solubility of carbon dioxide in salt solutions is smaller than in
water. A correction was made for this by putting the solubility equal to
that of 0.1 normal potassium chloride solution. From the results of Geff-
cken,* whose work is the best on the subject, the decrease in the coefficient
of absorption is practically proportionate to the concentration of the salt,
the coefficients for pure water, for 0.5 molar and 1.0 molar solutions of
potassium chloride forming practically a rectilinear curve, the concentra-
tions and the coefficients being used as coordinates. By interpolation on
the curve, the coefficient of absorption for 0.1 molar potassium chloride
at 25®, the temperature at which McCoy worked, is 0.742, reduced to
' Loc. cU,
'The degree of ionixation of 0.1 molar sodium acetate, Kohlrausch und Holbom,
Leitffthigkeiten, pp. 159 and 200, is calculated as 0.783.
The degree of ionization of 0.1 equivalent sodium sulphate, ibid,, is 0.687. In a mix-
ture of the two, the degrees of ionization would be very slightly modified, but a calculation
of the change made on the basis of the principle of isohyorie solutions showed the correc-
tion to be negligible.
' Loc. cit.
AND CALCIUM CABBONATB, ETC., OF WATBB 80LUTI0NB. 245
00.760 mm. and Jb^.^-0.03313*
Then
Cbuco, -0.03313X0.00161 -5.33 X10-*
Inserting the values determined for the functions in equation (22) we have
C^HCO, „ 0.05335' ^igQ5-K
Ch.co.XCco. 0.01991 X5.33X10-* ' *"
The second ionization constant of carbonic acid is, then, according to
equation (23) ^^^ 3.04X10-^
4895
K^H^- :.\^. 6.21 X 10-"
A similar calculation for experiment 6, which McCoy considered as
presenting the most favorable conditions for accuracy and in which the
partial pressure of carbon dioxide was 0.00404 atmosphere, gave 6.205 X 10^".
For experiment 1, table 3,t 0.3 normal bicarbonate was used, 57.9 per
cent remaining as bicarbonate, 42.1 per cent being converted into carbon-
ate when equilibrium was established under a partial pressure of carbon
dioxide of 0.00319 atmosphere. We have t && before
Chco, -0.3X0.579X0.70 -0.1216
Coo, -0.3X0.5X0.421 X 0.584 -0.0369
By interpolation of Ge£fcken's results, the coefficient of absorption of car-
bon dioxide by 0.3 normal potassium chloride at 25° is 0.720 and Ag^ ^
0.03214. § Hence
Chkx).-0.03214x0.00319-1.02X10-^
and
C^HOO. 0.1216» ^qoiQ^K
ChkxhXCoo, 1.02X10-^X0.0369 '''''^"
According to equation (23) we have, then,
K^ic»i«tioa-7.8X10-"
We will use the mean of the two constants or 7 X lO""^^ for the calcula-
tions made in this investigation.
THE DBTBRMINATION OP THE SOLUBILITY CONSTANT FOR
CALCIUM CARBONATE.
With the aid of the constant for the secondary ionization of carbonic
acid, of the constant for its primary ionization as determined by Walker
and Comack, and the experiments of Schloesing on the solubility of calcium
carbonate under the influence of varying partial pressures of carbon dioxide,
* McCoy used 0.0338 without correction for the changed solubility.
t Page 456, he, eU.
t The degree of ionisation of 0.3 normal sodium acetate, loe. eU,, is 0.70; of 0.3 equiva-
lent sodium sulphate, he, eU,, it is 0.584.
i 0.0338 for pure water.
246 EQUILIBRIUM BETWEEN CARBON DIOXIDE OF AT1C08PHERB
we can now determine the value for the solubility product of calcium
carbonate^ n \yn xr
to which the precipitation of calcium carbonate is subject. As explained
above, the solution may be considered just saturated whenever the product
of the ion concentrations is equal to this constant. Excess of carbonic
acid increases the solubility through its hydrogen ions, which form bicar-
bonate ions HCO3' ^^^b ^^6 carbonate ions CO/ of the calcium carbonate.
Cdft X (CcJO,— «) < KoaOOa
the solution is no longer saturated with calcium carbonate and the solid
carbonate will go into solution until we again have:
C'oa X C'cOi = KcaCO*
the concentrations (Vof^ and G'cx>, being imequal now, calcium ions being
in excess.
As stated on page 242, the solubility will depend on all four of our
fundamental equations I to IV (on p. 242), the functions being dependent
on the various constants. They may all be combined as follows: Divid-
ing equation III by IV, we have first:
/.
C^HCOa _ K lonitttUm ^ojx
CcOs X ChsOOi K^Ioaintion
We may substitute in this equation for Ceo, ^^ value as obtained from
equation I, viz. ^y^ , and have
CcaXC*HC0« ^ K^IonUation
Ko»CO« X ChsOOs K^'ioniaatioo
or
CcaXC^HCOa xT \^ ^'loniiation /ok\
p ='rLCmO0tXj^ (25)
Now, when calcium carbonate is dissolved under the influence of excess
of carbonic acid in the absence of any other calcium salts, as was the case
in Schloesing's experiments, practically all of the calcium is present as
bicarbonate, the quantity of carbonate being minimal and quantitatively
negligible in comparison with the bicarbonate. Since
Ca(HC03), ^ Ca- + 2HCO3' (26)
the concentration of the calcium ions will be half that of the bicarbonate
'''''^ Cca = iCHCO, (27)
and by substituting this value in (25) we have
C3 TTf
IICO3 oxr vy lonisation /oo\
7S = ^^CaCOa X -jT^ (Zb)
^HjCOs ^ loniaation
For the concentration of the dissolved carbonic acid we have, according
^°"- CH^o,°fc«.»XP (29)
^Vide Bodlaender, loc, cit.
AND CALCIUM CABBONATB, ETC.^ OF WATBB SOLUTIONS. 247
where Jbg^ is expressed in molar terms and P is the partial pressure of the
carbon dioxide above the solution (see p. 243).
Equation (28) may be transformed then into
f
lonisation
2Xk^X ^J^"""^ XKcoo. (30)
an equation which holds for saturated solutions of calcium carbonate and
bicarbonate in equilibrium with solid calcium carbonate and gaseous car-
bon dioxide of any pressure P at a given temperature. The composition
of such saturated solutions has been determined by Schloesing, and since
from his data the values of ChcOs ^^^ ^^ ^ ^^^ ^ ascertained, and since
all the constants excepting the solubility product K(:MX)t ^^^ ^^^ known;
the value of this constant can now be determined.'*
The value of the first ionization constant of carbonic acid is 3.04 X 10~^;^
the value of the second ionization constant is 7 X 10~".^ Ge£fcken's^ recent
very exact determinations of the absorption of carbon dioxide by waterj
made with an improved apparatus, give somewhat higher values than found
by Bunsen years ago. At 16^i the temperature at which Schloesing's deter-
minations were made, by interpolation of the values found by GefFcken for
15^ and 25°, the coefficient of absorption is found to be 0.9890 (reduced to
0^.760 mm.) instead of 0.9753. The change in solubility produced by the
presence of the small amount of salts present (in Schloesing's experiments
this is only 0.01 equivalent, as an average) is negligible. The coefficient
0.9890 corresponds to a constant Jbg^ equal to 0.04415 in molar terms.
Substituting the values of our three known constants in equation (30)
we obtain i /^ o /> ^ v •■ n i
^.2X0.04415x5^^ XKo^
- 383.4 xKcitCO.
and
Chco. (31)
\Kck00i
7.264 X V i"
Schloesing's results give the total amount of calcium carbonate dissolved:
it is present almost exclusively as calcium bicarbonate. The degrees of
ionization, a, of the calcium bicarbonate in the various solutions may be
put equal to the degrees of ionization of calcium acetate in equivalent con-
centrations;^ then, in any given solut'on
CHCOi"«XiCoft(Hoo8)j (32)
o Bodlaender, loe. cii,
»Page 243.
«Page 245.
*Loe, eU.
'Bodlaender used calcium chloride and nitrate for this purpoee. As salta of organic
acids usually are somewhat less ionixed than salts of stron^r morganic acids, it seemed
better to ascertain the degree of ionixation bv comparison with calcium acetate. The con-
ductivities for calcium acetate are ^ven in Kohlrausch and Holbom's Leitf&higkeiten on
pase 161 and the degrees of ionization are calculated in the usual wa^. There is an average
difference of about 4 per cent between these coefficients of ionization and those used by
Bodlaender.
248
EQUILIBBIUM BETWEEN CABBON DIOXIDE OF ATMOSPHERE
In table 4 the data are tabulated and the calculated values for ^ EQ^oOk
given. Column 1 gives P the pressure of carbon dioxide in atmospheres,
column 2 the solubility of calcium carbonate as determined by Schloesing
and expressed in gram equivalents of calcium bicarbonate I — ^-^ — ^jper
liter. Column 3 gives the degree of ionization a of the corresponding
calcium acetate solutions, column 4 the concentration of acid carbonate
ions, ChgOsi ^ calculated from the numb ers give n in columns 2 and 3.
In the last column we have the values for -v Koaoot ^ calculated according
to equation (31).
Table 4.
00,
Atm.(P)
*Oft(HCO,),
100*
^00.
10* X
^•=cw».
1
2
3
4
5
6
7
8
9
10
11
12
0.000504
.000808
.00333
.01387
.02820
.05008
.1422
.2538
.4167
.5533
.7297
.9841
0.001492
.001700
.002744
.004462
.005930
.007200
.01066
.01327
.01676
.01771
.01944
.02172
90.9
90.2
88.8
87.1
85.9
85.0
82.8
81.0
79.8
79.1
78.4
77.5
0.001356
.001533
.002437
.003886
.005097
.006120
.008828
X)1074
.01257
.01402
.01525
.01684
235
227
225
228
231
229
233
284
282
235
233
233
The mean value for y^ KckOOt is 0.002325.
The agreement among the values found for A/KoaoOt under a partial
pressure of carbon dioxide ranging from ^^ atmosphere to 1 atmosphere
is excellent, and this agreement forms a very good test of the correctness of
the whole theoretical treatment, such a constant resulting from a consider-
ation of the conditions of equilibrium on the basis of the theory of ionisa-
tion.*
From the value for -v/Kcacos= 0.002325, we find the solubility product
^^^^" Cca X Ceo, = Kcaco, = 1 .26 X 10"^ (33)
It may be pointed out that this method of determining the solubility
product constant of calcium carbonate must be far more reliable than a deter-
' As Bodlaender (loc, cit,) points out, it is clear from the theoretical treatment that the
same equation applies to the effect of carbon dioxide on the solubility of other difficultly
soluble salts, sucn as barium carbonate, the only difference being that the solubility pro-
duct constant of bariimi carbonate (Cbr X CoOs = KBaCOs) is substituted for the constant
for calcium carbonate. All the other expressions remain the same. Bodlaender has used
the results of Schloesing on the solubility of bariu m ca rbonate as affected by carbon diox-
ide to calculate in the same way the value for iX KBaCOa and obtained again an excellent
agreement among the values for the constant. Only the numerical value of the constant,
not the constancy of the results, is affected by the corrections made in this paper, namely,
in the value of the second ionization constant for carbonic acid and in the method of calcu-
lation of the degrees of ionization a. As the value for v"^ KBaCOs ^ not of interest in this
investigation, the data were not recalculated.
AND CALCIUM CARBONATE, ETC., OF WATEB SOLUTIONS. 249
mination based on the direct solubility of calcium carbonate in water.
The latter method gives excellent results with salts b'ke silver chloride,
calcium or barium sulphate, but calcium carbonate is largely hydrolyzed
by water into the hydroxide and carbonic acid and bicarbonate, and this
hydrolysis, together with the possibility of absorption of carbon dioxide
and consequent change in the equilibrium conditions, affects these direct
determinations. Bodlaender has estimated that about 80 per cent of the
calcium carbonate is decomposed by water in its saturated solution in pure
water. In Schloesing's experiments the hydrolsrsis is practically completely
overcome by the measurements being made in the presence of an excess of
carbon dioxide. No calculations were made by Bodlaender as to the extent
of any hydrolysis in these experiments of Schloesing, but we can readily
determine it as follows: In experiment 1, in which we have the smallest
pressure of carbon dioxide and therefore the most favorable conditions for
hydrolysis, we have as the pressure of carbon dioxide (P) 0.000504 atmos-
phere. Consequently, according to equation (20)
CHrfX),-0.04415X0.000504-2.225Xl0-»
According to the table we have
Choo,- 0.001356
Now, for carbonic add we have (equation III, p. 242),
ChXChoQi-3.04X10-'xChkx),
and inserting the given values for Chcos ^^^ ^HiCOti ^^ ^^ •
3.04X10-^X2.225X10-^ ^
^" 0.001356 -4wxnr'
Now, for the ionisation of water at 16°, we have:
ChXCoh-0.55X10-"
and consequently
0.55XlO-'« , .^.f^
^«" 4.99X10-^-^^^°
Calcium bicarbonate, the chief salt in solution, is hydrolyzed according
*^ iCa(HCO,), + HOH±* JCa(OH), + H,CO,
and since the equivalent concentration of the bicarbonate ions in experi-
ment 1 is 0.001356, the part hydrolyzed is 1.1 X 10~^/0.001356, or about 0.08
per cent. So, even in this first, least favorable experiment, the hydrolysis
is negligible. In experiment 5, we find in a similar way
Ch=7.43x10-» Coh = 7.40X10-»
and the part hydrolyzed is 7.4X10~*/0.0051, or 0.0014 per cent.
Hydrolysis is reduced therefore to almost nothing, and the value found
for Kc^ioo,, 1.26X10^, needs no correction from this source.
250 EQUIUBSIUM BETWEEN CARBON DIOXIDE OF ATMOBPHSBE
EQUILIBRIUM BETWEEN CALCIUM CARBONATE AND GYPSUM.
THE BOLUBILITT PRODUCT OF CALdUli SULPHATE.
For a saturated solution of gypsunii ionised according to
CaS04±5Ca'+SO/ (34)
we may put, as we did for calcium carbonate,
Cca X CSO4 « KGtes64 (35)
The value of this solubility product constant can be determined from
the solubility of calcium sulphate in water and its degree of ionization in
the saturated solution, since the salt dissolves without any hydrolytic de-
composition. Kohlrausch and Rose^ give the solubility of gypsum at 18^
as 2.07 g. (anhydrous calcium sulphate) per liter of water. This represents
a concentration of 2.07/136.1 or 0.0162 gram molecule and 0.0304 gram
equivalent of calcium sulphate. The degree of ionization is best ascertained
from its conductivity in the saturated solution; the specific conductivity
of the solution is given ' as 0.001891 reciprocal ohms at 18^; its equivalent
conductivity is, then
, 0.001891X1,000 ^^^
'*^^* o:o3oi *^-^
The conductivity of calcium sulphate at extreme dilution' is 123 and its
degree of ionization in the saturated solution therefore 62.2/123 or 50.6 per
cent. Then Cca =Cso4- 0.0152 X 0.506 « 0.00769
and
Kcd304 «= Cca X Cs04 - 0.00769* = 5.92 X lO"* (36)
The solubility product of calcium sulphate at 18^ is therefore 5.92 X 10~*.
CALCIUM SULPHATE AND CALCIUM CARBONATE.
If we have a solution saturated both with calcium sulphate and with
calcium carbonate, for instance in contact with both solid salts, we have
in the saturated solution simultaneously
Cca X CcOs ■= KctoCOs (37)
and
Cca X C!so4 = KGteso4 (38)
The value for C(^ is the same now in both equations, representing as it
does the total concentration of calcium ions, irrespective of their source
from sulphate, carbonate, or bicarbonate. Then, dividing equation (38)
by equation (37), we have for a solution in equilibrium with both salts at
C8O4 _ Kcaso* _ 5.92X10-* _ ^ 7^ .oo\ 4
Cc^3 " "Kcac03 " 1 .26 X 10^ " ^-^^ ^
'Zeitschrift fQr Phvsikalische Chemie, 12, 241 (1893).
' KohlrauBch and Holbom, loc. cU,, p. 77.
* Ibid., p. 200, table 86.
* The value for KcaCOs determined at 16° is used ; no correction is made for the differ^
•nee of 2^.
AND CALCIUM CARBONATE, ETC., OF WATER SOLUTIONS. 251
This means that the concentration of sulphate ions must be about 6,000
times as large as that of the carbonate ions in a solution saturated with
both salts, and consequently one or the other salt will come down first pure,
when precipitation from a mixture is effected, until this ratio is reached.
Under ordinary conditions obtaining in nature or in laboratory experiments,
the excess of sulphate ions is not as large as this, and so the carbonate is
precipitated first until the concentration of carbonate ions has fallen to
about one five-thousandth that of the sulphate ions. On the other hand,
if in such a solution, which is in equilibrium with both solid salts, the con-
centration of the sulphate ions is by any means increased, e.g., by addition
of some sodium or potassium sulphate, the solution will be oversaturated
with calcium sulphate [Gq^ X G^so4 ^ ^^Q^isoj; gypsum will be precipitated, and
owing to the loss of calcium ions the solution will now be undersaturated in
regard to the carbonate [(Cca— «)XCoo,<Kc^ooJ> ^^^ solid calcium car-
bonate must dissolve until the above ratio is reached. In the same way
the order of precipitation may be reversed if by any means the concen-
tration of the carbonate ions in a solution is persistently kept below one
five-thousandth that of the sulphate ions during the process of precipitation
or crystallization, for instance, by the addition of an acid.^
For an aqueous solution saturated with gypsum and calcium carbonate
at about 18®, the concentration of calcium ions may be taken as 0.00769,
as practically all of the calcium ions are derived from the sulphate. The
concentration of carbonate ions in such a solution b, then,
p Kciico. ^ l-26X10-* g. ^> .^Qv
^* Coa 0.00769 == 1-^X1"^ ^*">
With the aid of this value we can calculate, for varying partial pressures
of carbon dioxide, the maximum proportions of calcium carbonate and
bicarbonate which can be present in solutions saturated with gypsum and
calcium carbonate at approximately 18®.
CALCIUM SULPHATE, CARBONATE, AND BICARBONATE, WITH REFERENCE TO
OIVEN PARTIAL PRESSURES OF CARBON DIOXIDE.
The present average partial pressure of carbon dioxide in the atmosphere
is 0.0003 atmosphere. We may ask what is the maximum amount of cal-
cium present as carbonate and bicarbonate in 1 liter of a solution which
is in equilibrium with this partial pressure of carbon dioxide and which is
saturated both with gypsum and with calcium carbonate. The significance
of this quantity will be discussed presently; it can be calculated as folio ws,^
with the aid of the equations developed above. We have:
CHXCH0Ot"=K'ioiiU»tion XChK»i (m)
Ch X Coo, = K^Ioni-tioa X ChOO, (I V)
GhiCOs — *gM X Ceo, (II)
> Analogous relations have been developed experimentally and theoretieaUy in eon-
neetion with Guldbei^ and Waage's claflsical work on barium sulphate and carbonate.
Vid€ Nemst, Theoretical Chemistry, p. 633 (1904), and Findlay, Zeitochrift fOr Phynkalische
Chemie, 34, 409 (1900).
262 EQUILIBBIUM BBTWEBN CARBON DIOXIDE OF ATM08PHEBB
Combining the three equations by dividing m by IV and multiplying
by 11, we find: j^,
C?HC!0.-4;;52=^=^X*,„XCoo.XCoo. <«)
•^ IciniMition
and
Now, for a solution saturated with calciimi carbonate and gypsum at
about 18^1 Coo, ^^7 ^ ^^ ^ & ^^ approximation to have a maximum
value of 1.64 X 10~* as shown above (equation 40), the calcium ions produced
from the bicarbonate being neglected in this first approximation and only
those from the gypsum being considered.^ At 180 the solubility constant
for carbon dioxide,' k^,^ is 0.04183 if Coot ^ expressed in atmospheres.
In the given case Coot ^ 0.0003 atmosphere. Inserting all these values
and the two known ionization constants of carbonic acid into equation (42),
we find I
and
Chco,- 0.0003
For the calcium ions belonging to the bicarbonate we have
Coa-iCHOO,- 0.00016
We have found then the ionized portion of the calcium bicarbonate in
the saturated solution. To determine the total dissolved bicarbonate its
degree of ionization in the mixture must be ascertained. Its degree of
ionization will depend not on its own concentration alone, but, according to
the principle of isohydric solutions, also on that of the calcium sulphate
present. We may imagine, according to the method of Arrhenius, the water
divided between the two salts in such a way that each in its portion yields
the same concentration of the common ion calcium. Since there is 50 times
as much sulphate as bicarbonate, the latter will secure only about 2 per
cent of the water, the sulphate about 98 per cent, and the sulphate will
ionize practically as if it were present alone. Its degree of ionization is then
50.6 per cent (p. 250), and its concentration of calcium ions 0.00769 or
0.01538 calcium ion equivalent. This, then, must also be the concentration
of the calcium ion equivalents in the isohydric bicarbonate solution, and so
^® ^^"^^ C^Ca(HC0a)2Xa= 0.01538
' The amount of calcium bicarbonate found in solution by this first approximation
corresponds to 0.00015 gram ion of calcium. The calciiun ions from the carbonate are
negligible and therefore the total concentration of calcium ions from sulphate and bicar-
bonate IB 0.00769+0.00015 or 0.00784, and the maximum value for CcOa is, corrected,
1.26XlO-«/0.00784 (equation 40) or l.iexiO"' in place of 1.64 XlO-*. No correction was
made for this small difference, the results of the first approximation bein^ considered suiffi-
ciently accurate, especially in view of the facts that the solubility of calcium sulphate will
be slightly affected by the presence of the bicarbonate in such a way as to counterbalance
this error and that the decrees of ionization of salts are uncertain.
' Geffcken, loc. cU. The total salt concentration (0.03 mole) is too small to require a
correction for the changed solubility of carbonic acid.
AND CALCIUM CABBONATB, XTC.| OF WATSB SOLUTIONS. 253
By consideriiig the bicarbonate to ionize like the acetatCi we easily find
by trial that ^GaCHCO,), has a concentration of 0.0196 equivalent, whose
degree of ionization is found from the conductivity curve for calcium ace-
tate to be 78.5 per cent
0.0196X0.785 -"0.01538 calcium ion equivalent,
as required by the isohydric principle.^
Thus we find the degree of ionization of the calcium bicarbonate to be
78.5 per cent. The ionized portion of the bicarbonate is 0.00015 gram
molecule per liter, as was found above; consequently the total amount of
calcium bicarbonate in 1 liter is 0.00015/0.785 or 0.00019 gram molecule.
A solution saturated with gypsum and calcium bicarbonate and carbon-
ate at about 18^ under a partial pressure of 0.0003 atmosphere would contain
therefore 1.9 X 10^ gram molecule of calcium bicarbonate. Such a solution
by evaporation would deposit an equivalent amount of calcium carbonate,
or 0.019 gram of calcium carbonate per liter,' and gypsum deposited by
evaporation from such a solution would be contaminated with 0.019/2.07
or 0.9 per cent of calcium carbonate (referred to the anhydrous gypsum).'
For a solution saturated at 18^ with gypsum and with calcium carbon-
ate and bicarbonate under a partial pressure of carbon dioxide 10 times as
great as the present average value in the atmosphere, the proportion of
calcium bicarbonate in solution can be calculated in a similar fashion. As
a first approximation we have
Choc - y^^^S? X 0.04183 X 0.003 X 1.64 X lO-*
10X0.0003=0.00095
aJTo
Since Cq^ is }CH00sy ^^® concentration of calcium ions in solution would
be close to 0.00769+0.00047 or 0.00816, and consequently in the saturated
solution the maximum value for the carbonate ions is
p 1.26X10-^ - KAvirv-*
^«° 0.00816 -^'g*X^^
Introducing this corrected value into the above equation, we find
Choo,- 0.00916 and Coa- 0.00046
The degree of ionization for the calcium bicarbonate is found by the
method used above to be 78 per cent and the total calcium bicarbonate,
ionized and non-ionized, in 1 liter, is 0.000458/0.78 or 0.00059 gram molecule.
A solution saturated with gypsum and calcium carbonate and bicar-
bonate at 18® under a partial pressure of 0.003 atmosphere would contain
therefore 5.9 X 10~~^ gram molecules of calcium bicarbonate per liter, and by
' 20 c.c. of a 0.0196 equivalent or 0.0098 molar solution of caldum bicarbonate would
contain 0.000196 gram molecule and 0.000196X0.785-0.00015 Cca, the total concentra-
tion of calcium ions derived from the bicarbonate in solution. The bicarbonate ionises as if
it were all dissolved in about 20 c.c. water.
' The amount of calcium carbonate in solution in 1 liter is so minute as to be n^ligible.
' Cf. Findlay, loe. cU.
254
EQUIUBSIUM BETWEEN CARBON DIOXIDE OF ATM08PHEBE
evaporation such a liter would deposit 0.069 gram of calcium carbonate as
a contamination of the gypsum or 0.059/2.07, or 2.85 per cent (referred to
the dehydrated gypsum). Briefly the amount of calcium carbonate pro-
duced under a tenfold increase in the partial pressure for carbon dioxide is
very closely proportionate to the square root of 10.
For a solution saturated at 18® with gypsum, calcium carbonate and
bicarbonate under a partial pressure one-tenth as great as its present
average value, viz, 0.00003 atmosphere, we may put
Chco, = V 0. 1 X 0.0003 = 0.000096
and Cca 0.0000475. The degree of ionization is 78.5 per cent, and the total
dissolved bicarbonate is 0.0000475/0.785 or 0.00006 gram molecule per liter.
A solution saturated with gypsum, calcium carbonate and bicarbonate
at 18® under a pressure of 0.00003 atmosphere of carbon dioxide would
contain 6 X 10~~* gram molecules of calcium bicarbonate per liter, and would
deposit by its evaporation 0.006 gram of calcium carbonate, producing a
contamination of 0.006/2.07 or 0.3 per cent (referred to dehydrated gypsum).
In table 5 the results are summarized for the calcidations made for
aqueous solutions under the conditions named. Column 1 gives the carbon
dioxide pressure in atmospheres; column 2 gives weight in grams of calcium
sulphate present in one liter; column 3 shows similarly the calcium bicar-
bonate present, but expressed in grams of carbonate per liter. Ck>lumn8
4 and 5 give the values of columns 2 and 3 in terms of gram molecules per
liter. The last column gives the proportion of calcium carbonate which
would be found in gypsum separating from such an ideal solution by evapo-
ration, the proportion referring to dehydrated gypsum.*
Tablb 5.
CO,
(atmosp.) .
CaSOf
(grama).
CaCOs
(grams).
CaSO.
(moles).
CaCO>
(moles).
Percent
CaCO,.
O.OOOOS
.0003
.003
2.07
2.07
2.07
0.006
.019
.069
0.0162
.0162
.0162
0.00006
.00019
.00069
0.30
0.90
2.85
DISCUSSION.
The significance of these results is as follows: For an ideal condition,
if the natural waters of the earth were supposed to contain only lime salts,
that is the sulphate, carbonate and bicarbonate in equilibrium with the
carbon dioxide of the atmosphere, then by evaporation they would deposit
first, as is now the case, until the solution became saturated with gypsum,
all the calcium carbonate in solution in excess of the amounts given in
* Cameron and Seidell, Journal of Physical Chemistry, 6,652 (1901), got experimentally
0.037 g. calcium carbonate or 1.9 per cent from a solution containing 1.93 grams calcium
sulphate (0.0143 mole) in equilibrium with air. The partial pressure of the carbon dioxide
in the air is not specified (laboratory air is notably richer in carbon dioxide) and no tempera-
ture is mentioned, hxit it probably was between 22** and 25® (see pp. 643 and 650). Only
one determination is reported, and under the circumstances we must be content with the
fact that the result is of the order calculated. (Vide McCoy, American Chemical Journal,
29, 461 (1903), who also questions the quality of the air used by Cameron.)
AND CALCIUM CARBONATE, ETC.^ OF WATER SOLUTIONS. 255
table 5| in columns 3 and 5, depending on the partial pressure of the atmo-
spheric carbon dioxide. When the solution becomes saturated with gypsum
this will, by continued evaporation, crystallize out, but no matter whether
it is deposited in the same locality as, or in some other locality^ than, the
first great deposit of calcium carbonate, the g3rpsum must inevitably be
continuously contaminated with some calcium carbonate, varying from 0.3
to 2.85 per cent, according to the partial pressure of the carbon dioxide in
the atmosphere within the limits given. Perfectly pure gypsum would not
be formed imder such conditions. Vice versa, a very exact determination
of the amount of calcium carbonate present in g3rpsum prepared under
such ideal conditions could be used as a criterion of the carbon dioxide con-
tent of the atmosphere under which the gypsum was formed.
Obviously such a hypothetical ideal condition as to constancy of tem-
perature and purity of the solutions never existed on the earth, and the
presence of other salts, notably of sulphates even in smaller quantities and
of chlorides in larger amounts, modifies decidedly the numerical values deter-
mined above: the foundation for the study of the relations of gypsum and
calcium carbonate having been laid for this ideal condition, it seemed desir-
able to pursue the inquiry to ascertain, at least roughly, the influence the
presence of other salts would have, especially the sulphates of magnesium,
potassium, and sodium, and the chloride of sodium. In the following pages
an attempt has been made to estimate only roughly the influence of the
presence of other sulphates.
CALCIUM SULPHATE, CARBONATE, AND BICARBONATE IN THE
PRESENCE OP SULPHATES.
The solubility of a difficultly soluble salt like calcium sulphate depending
on its solubility product
Cok X Cs04 — KcaS04
the presence of other sulphates in solutions that are not too concentrated
to interfere with the application of the laws of solution would, through the
increase in the concentration of the sulphate ions, have as its chief effect a
decrease in the concentration of the calcium ions, according to the equation
just given. Since about half of the gypsum is ionized in its saturated solu-
tion, a decrease in the concentration of its calcium ions would imply a
decrease in the solubility of gypsum. The decreased solubility of the gyp-
sum or the decrease in the concentration of calcium ions would, vice versa,
increase the solubility of calcium carbonate and bicarbonate; so, from both
causes, gypsum crystallizing from a not too concentrated sulphate solution
under the conditions we are stud3ring might be expected to be more contami-
nated with calcium carbonate than was found for aqueous solutions.
In order to study these effects, it was decided to examine from this
point of view the probable effect of the sulphates of magnesium, potassium,
and sodium in the proportion in which they are found with calcium sul-
phate in the present sea-water, determining also the effect of var3ring partial
pressures of atmospheric carbon dioxide, as was done before.
' Such a change of locality was stiggeeted to the author by Dr. O. Willcox.
256 EQUILIBBIUM BETWEEN CARBON DIOXIDE OF ATMOSPHERE
At present we have in sea-water in 1,000 parts 1.239 grams or 0.009
gram molecule of calcium sulphate, 1.617 grams or 0.0135 gram molecule
of magnesium sulphate, and 0.860 gram or 0.005 gram molecule of potassium
sulphate.* Such a solution would contain 0.0275 mole or 0.055 gram equiv-
alent of total sulphates. The degrees of ionization of calcium and magnesium
sulphates are practically the same,' and while potassium sulphate ionises
far more readily, it forms a relatively small component of the system, and
we can, with sufficient accuracy for our purpose, consider the degree of
ionization for the sulphates to be that of a 0.05 equivalent magnesium sul-
phate solution or 48 per cent. Then
qso4 - 0.0275 X0.48 - 0.0132
Go»-0.0091 X0.48:-0.0044
and
CoaXCso*- 0.000058
Since for a saturated solution the solubility product constant is 0.000059
(EcuaoJ/ ^6 s^ that this sea-water would be almost saturated with gyp-
sum were it not for the modifying influence of the presence of a large pro-
portion of sodium chloride and other salts in it.^
A purely aqueous solution of the above composition should be nearly
saturated with gypsum and a large part of the latter should crjrstallize out
during its concentration say to one-quarter its original volume. If at the
beginning, when the solution is practically saturated with gypsum, it also
at the same time be considered to be saturated with calcium carbonate and
bicarbonate in equilibrium with a partial pressure of 0.0003 atmosphere
carbon dioxide, we would have (see p. 250) a maximum concentration of
carbonate ions q, q q^qo
^^" i;7oo " i;7oo""^-* ^ ^°^
Then, according to equation 42 (p. 252) we must have sufficient calcium
bicarbonate in solution to give a concentration of acid carbonate ions
/3.04xi0-r
Ch(X). - V ^^^^3^=^ X 0.0003 X 0.0^^
=0.00039
and the concentration of the ionized calcium carbonate would be 0.000195
gram molecule. Its degree of ionization is 74 per cent, calculated by the
method used before, so the total calcium bicarbonate in solution is 0.000264
gram molecule per liter.
* Ghamberlin and SaliBbury, Geology, p. 309.
' Kohlrausch and Holbora, loe. eit.f p. 200. For 0.01 equivalent the degrees of ionisa-
tion are 63 and 65 per cent respectively.
* Page 250.
* In this calculation we are not including a study of the effect of sodium chloride,
but are limiting ourselves to an examination of the effect of sulphates alone in an aqueous
solution. It is imderstood that the calculations are only for a rough orientation, the sim-
plicity of the laws for dilute solutions being lost as solutions become more concentrated and
more complex. Vide also E. C. Sullivan, Journal of the American Chemical Society. 27,
529(1905).
AND CALCIUM CASBONATB| STC., OF WATER SOLUTIONS. 257
If such a solution is now allowed to evaporate under a partial pressure
of 0.0003 atmosphere of carbon dioxide to one-quarter its original concen-
tration, we should have left a solution containing approximately 0.08 gram
molecule of sulphates per liter, whose degree of ionisation may be taken as
39 per cent.* Then Qgo^^ 0.08X0.39 -0.0312
and the maximum concentration of carbonate ions would be
Consequently
^-W-«"^^^
Choo,-0.00039X^^^
and the concentration of ionised calcium bicarbonate is 0.0003 mole. Its
degree of ionization is 63.4 per cent and the total concentration of calcium
bicarbonate 0.00047 mole. Comparing this result with the original solu-
bility of the bicarbonate (0.000264 mole), we find that while the solubility
of the gypsum is decreased (see below) by the accumulation of the other
sulphates, the solubility of the bicarbonate increases, in consequence of the
decreasing concentration of the calcium ions.
The total original gypsum in solution was 0.0091 gram molecule per
liter. By evaporation of the solution to one-quarter of its volume, C8O4
has become 0.0312, and consequently
p 0.000059
^" 0.0312 "°°°^®
which we may consider as derived entirely from the ionization of the gypsum.
Its degree of ionization in a 0.08 molar solution of sulphates is about 39
per cent, and therefore the total concentration of the gypsum left in solution
is about 0.0019/0.39 or 0.0049 gram molecule. A quarter of a liter would
contain 0.0012 gram molecule, and therefore of the total original 0.0091
gram molecule 0.0079 gram molecule or about 90 per cent would have
crystallized out. The weight of this would be 1.07 grams (calculated as
calcium sulphate).'
The total original concentration of calcium bicarbonate was 0.000264
gram molecule, and we have left a quarter of a liter with 0.00047/4 gram
molecule. Consequently, 0.000147 gram molecule or 0.0147 gram of cal-
cium carbonate should be deposited with 1.07 grams of calcium sulphate,
corresponding to a contamination of 1.37 per cent.
A further evaporation to one-eighth of the original volume should de-
posit according to similar calculations 0.101 gram calcium sulphate with
0.0047 gram calcium carbonate, representing 4.7 per cent.
With a partial pressure of 0.00003 atmosphere of carbon dioxide the
original solubility of the bicarbonate would be 8.3 X 10~~* moles. By concen-
tration of 1 liter to one-quarter of a liter, the solubility would be increased
* The degree of ionization of magnesium sulphate, the chief sulphate left.
> During the evaporation to one^uJf liter, we should expect a deposit of 0.72 gram sul-
phate with 0.0104 gram carbonate or 1.44 per cent, carbonate would be present.
258
BQUIUBBIUM BETWEEN CARBON DIOXIDE OF ATMOSPHEBE
to 14.8 X 10~* gram molecule per liter and only 0.0046 gram of calcium car-
bonate would be deposited with 1.07 grams of calcium sulphate crystalliz-
ing out as gypsum by the concentration! representing a contamination of
0.43 per cent.
With a partial pressure of 0.003 atmosphere of carbon dioxide the
original solubility of calcium bicarbonate would be 10 times as great as in
the case just discussed, amounting therefore to 8.3 X 10^ moles. Evapora-
tion of the solution to one-quarter its volume would increase the solubility
to 14.8 X 10~^ gram molecules per liter and 0.046 gram of calcium carbonate
would be deposited with 1.07 grams of sulphate, representing 4.3 per cent.
The results are summarized in table 6.^
Tablb 6.
00,
(atmoB.).
CaSOi
(giama).
CaCO.
(gimm).
GaSO^
(mole).
OaCO.
(mole).
Percent
Ca(X)s.
0.00003
.0003
.003
1.24
1.24
1.24
0.00S3
X)264
.083
0.0091
.0091
.0091
0.000083
.000264
.00083
0.44
1.43
4.40
Column 1 gives the carbon-dioxide pressure in atmospheres; column 2
gives the weight in grams of calcium sulphate in 1 liter; column 3 the solu-
bility of calcium carbonate and bicarbonate (expressed in grains of calcium
carbonate) ; columns 4 and 5 give the values of the two previous columns
in moles per liter. The last column gives the contamination of gypsum with
sodium carbonate (referred to dehydrated gypsum) produced by evapora-
tion of three-quarters of the solution.
This table can be regarded only as a rough approximation, as it was
not considered desirable to make more involved calculations at the present
time, and certain factors modifying the results were therefore not considered,
such as the change in solubility of carbon dioxide, the influence of the
primary ionization of potassium sulphate, etc. It suffices, however, as an
orientation as to the effect of the presence of sulphates on the probable
contamination of gypsum with calcium carbonate: the decreased solu-
bility of gypsum in the presence of such sulphates, allowing the calcium
bicarbonate to dissolve more nearly as it does in the absence of gypsum,
would increase the tendency to contamination of the gypsum and so a
perfectly pure or exceedingly pure gypsum crystallizing under such condi-
tions would a fortiori indicate a very low partial pressure of carbon dioxide.
* The calculations involve no greater total concentration than about 0.08 gram mole-
cule total sulphates per liter (p. 257). Cameron and Seidell, Journal of Physical Chemis-
try, 5, 650 (1901), have determmed the solubility of gypsum in the presence of sodiimi sul-
phate and found that 0.1 gram molecule sodium sulphate at 22^ reduces its solubility from
2.1 grams calcium sulphate to 1.4 grams, which is qualitatively in accordance with the
above calculations. No attempt was made by them or by the author to determine to what
extent the ion product for calcium sulphate remains a constant in these determinations.
In more concentrated solutions of the sodium sulphate, as Cameron and Seidell point out,
double salt formation increases the solubility of the gypsum.
AND GALCnrM CABBONATB^ ETC., OF WATEB SOLUTIONS. 259
CALCIUM 8ULPHATB, CARBONATB, AND BICARBONATE IN THE
PRESENCE OP SODIUM CHLORIDE.
In both the previous cases discussed it was found that the absence of
calcium carbonate in gypsum crystallizing from solutions containing cal-
cium carbonate in equilibrium with the carbon dioxide of the air would be
considered an indication of a very low partial pressure of the carbon dioxide.
Even if the great mass of the excess of calcium carbonate in solution were
deposited first, in the same locality or elsewhere, before the point of satura-
tion for gypsum were reached, the requirements for equilibrium would be
such as to keep so much of the carbonate in solution as to form an easily
discernible contamination of the gypsum formed by further concentration.
In nature the crystallization of gypsum is supposed to occur usually by
the concentration of waters containing a large excess of other salts, notably
of sodium chloride, and the last question we shall try to consider now is the
effect of sodium chloride on the conditions discussed in the preceding parts.
A salt like sodium chloride which has no ion in common with gypsum
should, according to the law of mass-action, increase the solubility of the
latter up to a certain point; the chloride and sulphate must react to a con-
siderable extent to form calcium chloride and sodium sulphate:
CaS04-l-2Naa ±5 Caaj + NajSO^
As the ionization of the new salts in moderately concentrated solutions is
by no means complete, considerable amounts of calcium and sulphate ions
must be suppressed to form these salts in non-ionized form, and this would
lead to an increased solubility of calcium sulphate according to equation
(35), page 250. In fact, a rough calculation of the result of Cameron's '
determination of the effect of sodium chloride in 0.017 molar solution on
the solubility of gypsum, made with the aid of Arrhenius's principle of
isohydric solutions, led to a value for the ion or solubility product for
calcium sulphate
CoaXCsO4-9.5X10-«
in the presence of the salt, as compared with 6.5 X 10~^, the value of the solu-
bility product at 23® in the absence of salt.
For concentrated salt solutions the conditions, as is always the case,
become more and more complex; we are more likely to have complex ions,
such as NaS04, and double salts formed in large quantities and leading
not so much to abnormal changes in solubility as to changes which we have
no means of estimating at present.' Cameron, however, has given us a
large amount of empirical data which will be useful for the consideration of
our subject. In regard to gypsum and sodium chloride^ he finds the solu-
bility of the former is rapidly increased, rising from 2.37 grams calcium
sulphate (0.0174 mole) per liter to 7.50 (0.0555 mole) in the presence of
130 grams of sodium chloride, after which there is a gradual decrease in
* Loc, cU,, 5, 660.
' As to the lowering of Bolubilitv of the non-ionised calciiun sulphate in salt solutions,
comparable with the decreased solubility of carbon dioxide, see pp. 237, etc.
17
260
EQUILIBRIUM BETWEEN CABBON DIOXIDE OF ATMOSPHEBE
solubility again, which Cameron ascribes to the condensation of the solvent
in aqueous solutions of electrolytes.^
Entirely analogous effects must be anticipated for the action of sodium
chloride on calcium carbonate and bicarbonate, their ions also being sup-
pressed, for instance, according to
Ca(HCO,),+2Naa±>Caa,+2Na(HCO,)
calcium chloride and sodium chloride being formed in considerable quanti-
ties according to the principle of isohydric solutions.'
Again, the molecular solubility of calcium carbonate is liable to be
decreased, as was discussed for the sulphate, and we have in this case also
the fact that the solubility of carbon dioxide is considerably less in salt
solutions than in pure water, and this decreased solubility will reduce the
amount of bicarbonate dissolved approximately proportionately to the cube
roots of the change in the coefficients of absorption.' So we have forces
tending to increase the solubility as well as such as tend to decrease the
solubility of calcium carbonate. As a matter of fact, Cameron found that
at 25® for calcium carbonate and bicarbonate in equilibrium with air,^ the
solubility was increased from 0.1046 gram bicarbonate per liter to 0.2252
gram by 51 grams (0.87 mole) sodium chloride, and then it was decreased by
additional sodium chloride. There is, therefore, at first an increase in solu-
bility and then a decrease, exactly as for gypsum. The characteristic bend in
the curve occurs earlier than in the case of gypsum, which was to be expected,
as the molecular solubility of carbon dioxide is also affected in this case.
Cameron * also determined the effect of sodium chloride on the solu-
bilities of calcium sulphate, carbonate, and bicarbonate simultaneously,
i.e., in mixtures in equilibrium with solid gypsum, solid calcium carbonate,
and the air. This is the work that is of most interest and importance for us,
and as we shall use the data the table is reproduced here (table 7).
Table 7.
CaSO«
CaSOi
(mole).
Ca(HC03),
Ca(HCO,),
NaCl
1
NaCl
(grams).
(gram).
0.0603
(mole).
(grams).
(molea). |
1
1
1.9298
0.0143
0.000375
'
2
2.7200
.0201
.0724
.000450
3.628
0.0625 1
3
3.4460
.0255
.0885
.000550
11.490
0.1979 1
4
6.1560
.0381
.1006
.000625
39.620
0.6824 1
5
6.4240
.0475
.0603
.000375
79.520
1.3696
6
5.2720
.0390
.0563
.000350
121.900
2.0995
7
4.7860
.0354
.0482
.000300
193.800
3.3379
8
4.4620
.0330
.0402
.000250
267.600
4.6090
!
^ hoc. cit.j by 576.
' This has been shown to be reliable up to 0.4 normal solutions. (Summary by A. A.
Noyes, loc. cU,)
' When no other calciiun salt is present, the solubility changes according to the cube
root of the concentration of the carbon dioxide (equation (30), p. 247). When another
calcium salt, e.g., gypsum, is present in large excess, so that Cca may be considered constant,
the solubility changes approximately as the square root of the concentration of carbon
dioxide (equation (25), p. 246, and (42), p. 252).
^ No mention is made as to the partial pressure of the carbon dioxide, whether *' labora-
tory air" or pure country air was used.
» hoc. cU,, 5, 653.
AND CALCIUM CABBONATE, ETC., OF WATER SOLUTIONS.
261
We note, first, the interesting fact that the solubility of the calcium
bicarbonate is very much reduced by the presence of the gypsum, as required
by the theory developed above. Cameron did not report any experiments
on the efifect of a change in the partial pressure of carbon dioxide, but it
will undoubtedly be entirely analogous to that discussed in the previous
parts of this paper; it is probable that the solubility will increase roughly
proportionately to the square root of the pressure of carbon dioxide.^
We may now raise the question whether, owing to the displacement of
the bend in the curve of the solubility of bicarbonate as compared with
that of gypsum in the presence of sodium chloride, by a process of concen-
tration gypsum would crystallize out of any of the above solutions free
from carbonate, and if not what the contamination would be.
If we go from solution 2 to solution 3, the sodium chloride concentration
is increased from 0.0625 to 0.1979 mole, which would result if 100 liters of
solution 2 were concentrated to 31.54 liters of solution 3. Of the 272 grams
of calcium sulphate in 100 liters of solution 2, 108.7 grams would be retained
in solution and 163.3 grains deposited. At the same time, of the 4.5 grams
calcium carbonate in solution (as bicarbonate), 2.77 grams would be de-
posited, that is 1.7 per cent referred to pure calcium sulphate.
Table 8 gives the results of the calculations of this efifect of progressive
concentration from solution 2, through solutions 3, 4, etc., to solution 8.
In column 1 the number of the solution used is given, referred to the num-
bers in table 7. In column 2 the number of liters used for concentrating is
given, and in the next column we have the number of liters to which the
solution has been concentrated. Column 4 gives the weight in grams of
ealcium sulphate deposited, and the last column gives the proportion of
calcium carbonate to anhydrous sulphate deposited.
Tabls 8.
Solution
Volumt
Volume
GrainiCe804
Percent
used.
uaed.
left
deposited.
OrCOi.
No. 2
100.00
31.54
16^.3
1.7
No. 3
31.54
9.12
61.7
1.9
No. 4
9.12
4.54
17.8
2.3
No. 6
4M
2.96
13.6
0.5
No. 6
2.96
1.86
6.7
0.7
No. 7
1.86
1.36
2.8
0.8
We see from table 8 that the most favorable point for the deposit of
pure gypsum is in going from solution 5 to solution 6, which is the point
where the curve of solubility of gypsum bends, its solubility now decreas-
ing and a relatively large amount of it being deposited; the bend in the
curve of the solubility of the bicarbonate has already been passed and there
is now no corresponding increase in the formation of carbonate, this increase
having occurred before and produced the strong contamination in the
previous series.
^ See note 3, p. 260.
262 EQUILIBRIUM BETWEEN CARBON DIOXIDE OF ATMOSPHERE
Nothing is stated in Cameron's paper whether the air used was "labo-
ratory air'' with an excess of carbon dioxide in it or pure country air.^ If
laboratory air was used, the amount of calcium bicarbonate obtained in
solution and subsequently precipitated would be excessive and the results
in the last column of table 8 might be materially lower for air with a carbon-
dioxide content of 0.0003 atmosphere.
It is interesting to note that according to Usiglio's ' work on Mediter-
ranean water, calcium sulphate began to be deposited when the water
reached a density of 1.13. This corresponds to a chloride content of about
17 per cent, or about 3 gram molecules of sodium chloride per liter, a con-
centration reached in Cameron's experiments for solution 7, from which,
going to solution 8, gypsum would be obtained with about 0.8 per cent car-
bonate. At this concentration, the concentration of other sulphates is
still so small that they would tend to increase the contamination with
carbonate, as described above, rendering the calcium sulphate less soluble
and the bicarbonate more so.
At a lower partial pressure of carbon dioxide than 0.0003 atmosphere,
the proportion of carbonate would be reduced approximately in the ratio of
the square roots of the ratios of the partial pressures,* i,e,, a partial pressure
of only 0.00003 atmosphere would reduce the carbonate to about ^ parts of
the values given in the last column of table 8, or to still less on the prob-
able assumption that the experiments on which the table is based were not
carried out with pure air, and would produce a very pure deposit.
THE EFFECT OF TEMPERATURE CHANGES.
Reliable data on the solubility of calcium carbonate at temperatures
other than 16^, the temperature at which Schloesing's experiments wer&
carried out, are not at hand, and so the effect of changes of temperature om
the conditions we are studying can not be estimated. It may be pointed out,
however, that aside from a probable increase in the solubility of calcium
carbonate, a higher temperature would affect chiefly the solubility coefli-
cient of carbon dioxide, and through it would reduce the formation of
calcium bicarbonate. The solubility * of gypsum is about the same at 50**
to 65^ as at 18^ and ionization constants are usually not changed greatly
by changes of temperature; but the coefficient of absorption * for carbon
dioxide at 65.5^ is just about one-ninth as large as the coefficient at 18^.
This would result, according to equation (42), in reducing the formation of
calcium bicarbonate to -J i or one-third of the value found for 16®. In other
words, a rise of temperature of some 50® would probably have about the
same effect on the solubility of calcium bicarbonate as a decrease in the
* Results obtained by McCoy (American Chemical Journal, 29, 461 (1903), in repeating
other work of Cameron on conditions of equilibrium involving the carbon dioxide of the
air and showing decided discrepancies, form a very strong indication that pure air was not
used by Cameron.
* Encyclopsedia Britannica.
» See note 3, p. 260.
* Comey, Dictionary of Solubilities, p. 422.
^ Dammer, Handbuch der Anorganischen Chemie, n, 1, p. 371.
AND CALCIUM CARBONATE^ BTC.^ OF WATEB SOLUTIONS. 263
partial presBure of carbon dioxide to one-ninth its present value. In view,
howeveTi of the unknown change of solubility of calcium carbonate (the
ehange in the ion product constant), it must remain undecided whether a
rise of temperature would be a favorable or an unfavorable factor in the
crystallisation of gypsum free from carbonate. The formation of bicarbon-
ate would also be directly proportionate to the square root of the solubility
eonstant according to equations (40) and (42).
SUMMARY OP RESULTS.
(1) From Arrhenius's data on the solubility of silver acetate, valerate,
and butyrate in the presence of the sodium salts of the same acids, it was
shown that the solubility or ion products are approximate constants, as
calculated on the basis of the well-established principle of isohydric solu-
tions. This, with the results of others,^ removes the discrepancy existing
in the relation between the solubility product and the ionization of strong
electrolytes and gives us a safer empirical foundation for the consideration
of the equilibrium conditions existing between two precipitates, one which
is in harmony with the fundamental work of Guldberg and Waage.
(2) The second ionization constant of carbonic acid may be taken as
7.0 X 10^'^ as calculated from McCoy's data with the aid of corrections sug-
gested but not carried out by the latter.
(3) The solubility product of calcium carbpnate is found to be 1.26 X 10~*
on the basis of Schloesing's experiments on the solubility of calcium car-
bonate at 16^ under varying partial pressures of carbon dioxide.
(4) The solubility of calcium carbonate and calcium bicarbonate is ap-
proximately proportionate to the square root of the partial pressure of
carbon dioxide in the presence of a large excess of calcium sulphate.
(5) The theory of the equilibrium conditions between calcium sulphate
and calcium carbonate and bicarbonate has been developed and may prove
useful in the study of the natural waters of the present day.
(6) Considering the results given in tables 5,' 6,' and 8,^ we find that the
favorable factors for the crystallization of pure gypsum should be:
(a) The absence of other sulphates which in moderate proportions
render gypsum less soluble and consequently enable solutions to take up
more calcium bicarbonate than pure aqueous saturated solutions of gypsum
can dissolve.
(b) The presence of sodium chloride in the proportion of about 8 to 25
per cent.*
(c) A very low partial pressure of carbon dioxide, the solubility of cal-
cium carbonate varying approximately as the square root of the partial
pressure of carbon dioxide, according to (4).
(d) An increase of temperature by decreasing the coefficient of absorp-
tion of carbon dioxide would possibly, but not certainly, be a favorable fac-
tor, the formation of calcium bicarbonate being proportionate to the square
root of the absorption coefficient of carbon dioxide, which falls with increase
of temperature.
> A. A. Xoyes, loc. cU,; Findlay, loc. eU. < Page 254. < Page 258. « Page 261.
* Solutions 5, 6, 7, table 7, p. 260.
■ >
- «•
264 SUMMARY OF RESULTS.
It is conceivable that the conditions (a), (b), (c), and (d) should lead to
a primary deposit of exceptionally pure gypsum, especially when acting
jointly. The considerations developed make it desirable to. examine such
and other deposits of gypsum very carefully and exactly for even very
small quantities of carbonate.
(7) Even if the great mass of an excess of calcium carbonate in a solu-
tion were deposited first in some other locality before the point of satura-
tion for gypsum were reached, the requirements for equilibrium would be
such as to hold carbonate in solution and to make the question of the place
of deposit of the excess of carbonate in the first instance one of no moment.
^
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