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K<OU_1 60556 >m 

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Gall No. Accession No. 

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Made and Printed in Great Britain by 
Butler & Tanner Ltd , Frame and London 


In this work the Author has endeavoured to cover as wide 
a field as possible and it has necessarily been somewhat 
difficult to decide upon what to include and what to omit, 
especially when dealing with the most recent developments. 
It is hoped, however, that the book will serve to give the 
reader a sufficiently complete view of the present state of 

Mathematics has been almost entirely excluded in order 
that the volume may appeal to the amateur, no less than the 
student. Where possible mathematical methods of reasoning 
have been followed, which it is hoped will prove of value to 
students of elementary mathematical astronomy in clothing 
the symbols with more descriptive matter than is given in 
the text-books on Mathematical Astronomy. 

Considerable trouble has been taken to procure a repre- 
sentative series of Astronomical photographs, and these have 
been obtained from various sources which arc indicated on 
the plates. The Author desires to express his thanks for 
permission to reproduce these photographs, and great pains 
have been taken to ensure accurate reproductions. 

The Author is greatly indebted to his friends, Mr. D. J. R. 
Edney and Mr. J. Jackson, both of the Royal Observatory, 
Greenwich ; to the former for help in the preparation of 
many of the line diagrams, and to the latter for reading through 
the proofs in his absence at Christmas Island while in charge 
of the Expedition to observe the total Eclipse of September 21, 









V THE SUN 108 










INDEX e ........ 387 




I a. The Moon. Aged 12 days 80 

6. Copernicus 

II Portion of Moon, showing Plato, Copernicus and the 

Apennines . . . . . . . .100 

III Portion of Moon round Mare Nubium . . .106 

IV a. Sun-Spot Group. 1920 March 21 . . . . 116 
6. Sun-Spot Vortices. 1908 October 7 

V Solar and Sun-Spot Spectra with Iron Arc and Furnace 

Comparison Spectra . . . . . .122 

VI a. Sun Photographed in K 3 Light. 1910 April 11 .130 

b. Sun Photographed in Ha Light. 1910 April 11 
VII Large Solar Prominence of 1919 May 29 . . .132 

a. Sobral, Brazil. 1919 May 29 d. Oh. 2m. G.M.T. 

b. Principe. 1919 May 29 d. 2h. 13m. G.M.T. 

VIII a. Solar Corona, 1886 August 29 . . .134 

b. Solar Corona, 1901 May 18 

IX Yerkes Observatory 40-inch Refractor . . . 160 

X Mount Wilson Observatory 100- inch Reflector . .170 

XI Dominion Astrophysical Observatory 72-inch Reflector . 186 

XII a. Mars. 1899 January 30 and February 1 . . 238 

6. Jupiter. 1891 October 12 

c. Saturn. 1894 July 2 

XIII a. Mars. 1909 October 5 248 

6. Jupiter. 1891 October 12 

c. Saturn. 1911 November 19 

XIV a. Three Minor Planet Trails 256 

6. Halley's Comet and Venus, 1910 

XV a. Comet Brookes (1911 c.) 266 

6. Comet Delavan (1914) 

XVI Comet Morehouse 27 

a. 1908 November 10 d. 6h. 14m. 

b. 1908 November 25 d. 5h. 55 rn. 

XVII a. Parallel Wire Diffraction Grating .... 294 
b. Star Images Obtained with Grating 

XVIII Star Clouds in Sagittarius 330 

XIX The Large Magellanic Cloud 352 

XX a. The Pleiades 356 

b. Tho Orion Nebula 
XXI a. Great Nebula in Andromeda .... 362 

6. Cluster M13 Herculis 

XXII a. Spiral Nebula, M101. Ursa) Maj. . . .370 

b. Spiral Nebula, " Whirlpool," Canum Venat 

XXIII Region round the Southern Cross . . . .376 

XXIV Nebulous Region round p Ophiuchi . . Froiriispiece 



1. The Celestial Sphere. Suppose an observer to be 
viewing the heavens on a clear night. He will see a large 
number of stars of different degrees of brightness, but he will 
have no reason to suppose that a given star is nearer to or 
farther from him than its neighbours. Although their distances 
may actually vary very considerably, they will appear to 
be at the same distance. The appearance will be just as 
though he were surrounded by a vast sphere, to the surface 
of which the stars are attached, the observer himself being 
at the centre. After some time he will notice that a change 
has taken place : some of the stars will have disappeared 
from view beneath the horizon on one side and new ones 
will have appeared above the horizon on the opposite side, 
but he will not notice any change in the relative positions 
of the stars which remain visible. As a result of careful 
observation, he might conclude that the whole sphere was 
turning round an axis, carrying the stars with it, and he 
would be able to locate very approximately the direction 
of the axis by noticing which stars appeared not to change 
their positions. 

This imaginary sphere, at the centre of which the observer 
seems to be placed, is known as the Celestial Sphere. Although 
the sphere has no material existence, the conception is of 
fundamental importance in astrononry. This is due to the 
fact that astronomical measurements are not directly concerned 
with distances but with angles. Two stars are said to be 
at a distance of 5 apart when the directions to the two stars 
from the observer make an angle of 5 with one another. 
This angle can be measured without the observer having 
any knowledge of the actual linear distances of the two stars. 

1 B 


The apparent position of any celestial body can therefore 
be regarded as the point in which the line drawn to it from 
the observer meets the sphere. It is convenient to suppose 
the radius of the sphere to be extremely large compared with 
the distance of the Earth from the Sun, so that wherever 
the observer may be he can always be regarded as being 
fixed at the centre of the sphere. The lines from all observers 
to any given star will then cut the celestial sphere in the 
same point. All straight lines which are fixed in direction 
with regard to the celestial sphere may therefore be considered 
also as being fixed in position, and as cutting the celestial 
sphere in the same point. 

2. It has been mentioned that the whole sphere appears 
to an observer to be in rotation, carrying the stars with it 
as though fixed to it. As will be shown in the next chapter, 
this rotation is only apparent, actually being due to the 
rotation of the Earth on its axis, the observer being carried 
with it. It is nevertheless convenient for the present to 
suppose the observer to be at rest at the centre and the sphere 
to be in rotation. Since only relative motion is involved 
this assumption is permissible. 

The two ends of the axis about which the sphere rotates 
are called the Poles. They are also the points in which the 
axis of the Earth produced will cut the celestial sphere. If 
a star were situated at either of these points, its diurnal 
motion would be zero. The positions of the poles are not 
marked by any bright stars, but the bright star Polaris (the 
Pole-star) is at present only a little more than 1 distant 
from the northern pole. The soutnern pole is not marked 
by any conspicuous star. 

3. The Zenith is the point on the celestial sphere vertically 
overhead. The Nadir is the diametrically opposite point. 

The positions of the zenith and nadir depend upon the 
position of the observer on the Earth's surface. For an 
observer at the north pole of the Earth's axis, the north 
pole of the heavens would be in the zenith. The direction 
to the nadir is, in all cases, determined by the direction of 
gravity at the point. 


The Horizon is the great circle of the celestial sphere which 
has the zenith and nadir as its two poles. The plane of the 
horizon is the plane passing through the observer at right 
angles to the direction of gravity. 

Vertical Circles are great circles passing through the zenith 
and nadir. They are therefore perpendicular to the horizon. 

The Meridian is the great circle passing through the zenith 
and the pole. It meets the horizon in the north and south 

The Prime Vertical is the vertical circle at right angles to the 
meridian. It meets the horizon in the east and west points. 

In Fig. 1 is the observer, Z is the zenith, Z' the nadir, P 
the (north) pole, NESW the horizon, SZPN the meridian, 
EZWZ' the prime vertical, N, S, E> W the north, south, east 
and west points respectively. 

If A denote the position of a star, the great circle ZAQZ' is 
the vertical circle through the star. 

FIG. 1. Altitude and Azimuth. 

0, Observer 

N, 6', North and South Points 
NEtiW, Observer's Horizon 
P, P', The roles 

A, Star 

ZAQZ', Star's Vertical 

LAM, Star's Almucantur 

Z, The Zenith 
Z' The Nadir 
NPZS, The Meridian 
EZWZ' t Prime Vertical 

SQ or SWNQ, Star's Azimuth 

QA, Stai'a Altitude 

ZA, Star's Zenith Distance 

The position of the star can be fixed if its distance from the 
zenith be known and also the angle between the vertical circle 
through the star and the meridian, i,e. if the arcs ZA and SQ 
be known. 


The Zenith Distance of a body is its angular distance from the 
zenith (2 A). 

The Altitude of a body is its angular elevation above the 
horizon (QA). 

Obviously the altitude and zenith distance are comple- 
mentary angles. 

The Azimuth of a body is the angle at the zenith between 
the meridian and the vertical circle through the star. It is 
therefore measured by the arc intercepted on the horizon 
between these two circles. 

Azimuth is usually reckoned from the south point westwards. 
Thus in Fig. 1 the azimuth of the body A is, according to this 
convention, 360 SQ. It is sometimes measured, however, 
either east or west of south up to 180. Then the azimuth of 
A would be E. of S. by an amount SQ. The former method 
is the more convenient, though it is really immaterial what 
convention is adopted provided that it is stated and adhered 
to. In this book, azimuths will be reckoned from to 360 
starting from the south point westwards. 

An Almucantur is a small circle of constant altitude or 
zenith distance (i.e., parallel to the horizon). LAM is the 
almucantur through the star A. 

4. The Celestial Equator is the great circle which has the 
two poles of the heavens as its poles. It is the circle in which 
the plane of the Earth's equator meets the celestial sphere. 

Hour-Circles are great circles passing through the two poles. 
They are therefore perpendicular to the equator. 

The meridian is obviously the hour-circle through the zenith 
arid is therefore perpendicular to the equator as well as to the 

In Fig. 2, is the observer at the centre of the celestial 
sphere, Z is the zenith, P, P' the poles, NESW the horizon, 
ELWM the equator ; A denotes the position of a star and 
PAQP' the hour-circle through A. 

The Declination of a body is its angular distance from the 
equator. It is reckoned as positive for bodies north of the 
equator and negative for those south. 

In Fig. 2, QA is the declination of A. 

The North Polar Distance of a star is its angular distance from 



the Earth rotates on its axis a complete rotation will always 
take place in a period given by the quotient of 24 hours by the 
sine of the latitude. This agrees exactly with Foucault's 
result : the latitude of Paris is 48 50', the sine of this latitude 
being about *75. This value divided into 24 hours gives the 
value of 32 hours, actually observed. The experiment of 
Foucault has frequently been repeated, and provided that 
proper precautions are taken, a result in agreement with the 
preceding rule is always obtained. The rotation of the Earth 
is thereby experimentally verified. 

10. Theory of Foucault's Experiment. Foucault's 
experiment is of such funda- 
mental importance and the 
theory is so simple that it may 
be given here. We will suppose 
for simplicity that the pendulum 
is set swinging in the meridian 
at a Fig. 4). No loss of gener- 
ality is thereby produced, as 
the rate of rotation does not 
depend upon the direction of 
swing, a constant angle between 
two given directions being in- 
volved. The direction of swing 
is therefore along the tangent, 
ac, to the Earth at the point 
a, which meets the axis OP 
produced in c. 

A short time later suppose that the Earth has rotated through 
a small angle from west to east and carried the pendulum to 
the point b on the same parallel of latitude. The tangent to 
the Earth in the meridian at b obviously meets OP in the same 
point c. The pendulum is still swinging in the direction ac, so 
that relatively to the tray it will have apparently changed its 
direction in the clockwise direction through an angle bca. In 
the same time the Earth has rotated through the angle boa, 
where o is the centre of the parallel of latitude on which a and b 
are situated, o is also on OP. The rates of rotation being 
inversely proportional to the angles turned through in the 

FIG. 4. Theory of Foucault's 


same time, it follows that the rate of rotation of the pendulum 
is to that of the Earth as angle bca to angle boa. But 

angle bca __ arc ab . arc ab __ ao 

angle boa ac ' ao ac 

sin acO = sin aOA = sine latitude. 

11. Other Proofs of the Earth's Rotation. There are 

several other methods of establishing the Earth's rotation of 
which two only need be referred to. The first of these methods 
was originally suggested by Newton in 1679. Suppose a heavy 
object to be dropped vertically from a great height, say from 
the top of a high tower. Then, if the Earth is in rotation, since 
the top of the tower must be moving more rapidly than the 
bottom, an object dropped from the top should retain its 
original easterly velocity whilst falling, and should strike the 
Earth a little to the east of the point vertically beneath the 
point of projection. The deviation is relatively small, so that 
slight disturbing causes, such as the effects of air-currents, 
become very important and render the experiment much less 
conclusive than that of Foucault. Some experiments on this 
principle were made in 1831 in a disused mine-shaft in Saxony : 
a free fall of over 500 feet was available, but the theoretical 
deviation for this distance is only slightly greater than 1 inch. 
The results of individual experiments differed considerably 
inter se, but the mean of a large number was in fairly good 
agreement with the theoretical value. 

The second of these methods was also suggested by Foucault 
and performed by him. It depends upon the properties pos- 
sessed by the gyroscope of maintaining the direction of its 
axis invariable in space, unless it is acted upon by any disturb- 
ing forces. The gyroscope consists essentially of a rapidly 
spinning wheel very carefully balanced, which is mounted in 
gimbals with as little friction as possible, so that it is free to 
swing in any direction. If the gyroscope is set in rotation, its 
axis will continue to point in the same direction in space, so 
that the rotation of the Earth will appear to make it rotate, 
the theory being the same as for the pendulum experiment. 
By using a gyroscope, the rotation of the Earth may readily be 
made evident to a large audience. 


12. Possible Variability of the Earth's Rotation. The 

period of rotation of the Earth on its axis provides the funda- 
mental means of measuring time. It is therefore important 
to know whether this period of rotation is constant, in other 
words whether the day changes its length. It would seem 
from mechanical considerations that the length cannot remain 
absolutely invariable : any cause which would tend to change 
the angular momentum of the Earth must change its period 
of revolution. Such possible causes are the friction of the tides, 
transportation of matter from one part of the Earth's surface 
to another by rivers, etc., elevation and subsidence of the 
ocean bottom. It might be thought that clocks would provide 
a means of checking the constancy of the period of rotation, 
but this is not possible, as the period is much more constant 
than the rates of even the best clocks, so that the constancy of 
the period is used to check our clocks. 

It is only by comparing the times at which such phenomena 
as eclipses or occupations occur with the calculated times, or 
by comparing the motions of the Moon and planets with their 
theoretical motions, that changes can be detected. It is found 
in this way that the period of rotation remains remarkably 
constant : there is some evidence that the length of the day is 
at present increasing, at a rate of the order of one-hundredth 
of a second per century, but it is not probable that this rate of 
increase has remained constant in the past or that it will remain 
so in the future. 

13. The Size of the Earth. The problem of determining 
the size of the Earth reduces to the problem of determining 
the number of miles in one degree of the Earth's surface. A 
double operation is involved in this. Two fundamental points 
lying as nearly as possible on the same meridian are chosen 
on the Earth's surface and the actual distance between these 
is measured by a surveying operation which is called a geodetic 
triangulation. The latitudes of the two stations are then 
determined by astronomical observations with as much 
accuracy as possible. The geodetic portion of the observation 
involves the most time and work. The survey work is based 
upon an initial base-line which must be very carefully measured. 
Starting from this base-line a chain of triangles is laid down 


connecting the two points. The corners of the triangles are 
marked by suitable objects for observation, which may be 
either such well-defined marks as church spires or specially 
erected artificial observation posts. With accurate surveying 
theodolites the angles of these triangles are measured in suc- 
cession, so that, starting from the measured base-line, the 
lengths of the sides can be calculated by trigonometry. It is 
then possible to deduce the distance apart of the two stations in 
latitude which can bo compared with the difference of latitude 
deduced from the astronomical observations. The length of 
one degree of the Earth's surface is thus derived and thence its 
circumference and radius. In this way many long arcs of 
meridians have been measured. 

Such observations also give information as to the exact form 
of the Earth. The problem is a highly technical one, and it 
would be far outside the limits of this book to enter into details. 
It seems probable that no simple geometrical solid will 
accurately represent the shape of the Earth, even when local 
variations, such as hills and valleys, are disregarded. It has, 
however, been found that the Earth can be represented with 
sufficient accuracy for most purposes as an oblate spheroid, 
i.e. a figure formed by the revolution of an ellipse about its 
shorter axis. The Earth is flattened at the poles, the diameter 
along the axis being shorter than a diameter in the equatorial 
plane. The most accurate determination is probably that of 
Helmert, which gives for the longer semi-axis 20,925,871 feet 
and for the shorter semi-axis 20,855,720 feet, or approximately 
3,963 miles and 3,950 miles respectively. It follows that the 
nearer the poles the greater is the length of one degree in 
latitude. Thus in the latitude of Sweden, it is necessary to 
travel more than half a mile farther than near the equator in 
order to increase the latitude by one degree. 

It is customary to define the flattening of the Earth or of any 
other oblate body by a quantity called the ellipticity. This is 
defined as the ratio of the difference between the major and 
minor axes to the major axis. It is expressed as a fraction and 
gives a measure of the departure of the Earth from a sphere. 
The ellipticity corresponding to Helmert's values for the axes is 



The ellipticity can bo determined also by other methods, 
as e.g. by pendulum observations which really determine the 
variation of gravity over the Earth's surface and also by certain 
astronomical methods. 

The triangulation method is, however, the most accurate 
and it is, in addition, the only one which determines also the 
size of the Earth. 

14. The Mass of the Earth. The problem of determining 
the mass of the Earth is often incorrectly spoken of as " weigh- 
ing the earth." By the mass of a body is meant the quantity 
of matter contained in it. The weight of a body at the surface 
of the Earth is the force of attraction which the gravitation 
of the Earth exerts on the matter forming the body ; this 
varies for the same quantity of matter according to its position 
on the Earth's surface, whilst on the Sun, for instance, the 
weight of a given body would be about twenty-eight times its 
weight on the Earth. The phrase " the weight of the Earth " 
therefore has no meaning. It is quite possible, however, to 
determine the quantity of matter in the Earth, or the mass of 
it. Another aspect of the same problem is to determine the 
mean density of the Earth since, its size and volume being 
known, its mass can then be calculated. 

The basis of all the methods is the Law of Gravitation, first 
discovered by Newton. This states that any two particles of 
matter attract one another with a force which is directly pro- 
portional to the product of their masses and inversely 
proportional to the square of the distance between them. 
Expressed algebraically, the law may be written in the form 

in which m l9 m 2 are the masses of the two particles, r their 
distance apart, / is the force of attraction and y is a numerical 
and universal constant, called the Constant of gravitation. The 
constancy of y implies that the force of attraction between the 
two particles does not depend upon their physical or chemical 
constitutions or upon their positions in the universe. Its 
value is, of course, dependent upon the units in which the 
masses and distance are expressed. 
The law of gravitation as stated above is valid only for 


" particles," i.e. for masses of very small size. To find the 
attraction between two masses of finite size, it is necessary to 
sum up the attractive forces between every pair of particles 
composing them. The result' in general will be a complicated 
expression, depending upon the sizes and shapes of the two 
bodies. In the special case of a spherical body the result is 
very simple, provided that the sphere is of the same density 
throughout (homogeneous) or is built up of successive con- 
centric spherical layers. The force of attraction between such 
a sphere and any particle is then the same as that between 
a particle supposed placed at the centre of the sphere and of 
the same mass as the sphere and the second particle. This 
result will hold approximately for the Earth. 

15. Determination of the Mass of the Earth. The 

principle of all the methods for determining the mass of the 

Earth is to compare 

5 $ the force of attraction 

2 \, / z i between two known 

masses with the force 
of attraction between 
one of the two masses 
and the Earth. The 
difficulty of the deter- 
Station mination is due to the 
smallness of the con- 

Ficf. 5. The Mountain Method of Deter- afnnf rvf rrrn \ri-f of inn 
mining the Mean Density of the Earth. Stant Ol g ravltatlon - 

If the masses are ex- 
pressed in grams and their distance apart in centimetres, 
then the value of y is 6-658 x 10~ 8 dynes. Thus two 
spheres of lead weighing each .10 kilograms and with their 
centres 12 cms. apart would attract one another with a force 
of only - 2 T 2 -nd of a dyne. Many different methods have been 
used, of which it will be sufficient to refer to three, involving 
somewhat different principles. 

(i) The Mountain Method. This was one of the earliest 
methods^to be used. The principle can be seen from Fig. 5. 
Two convenient stations, A and B, are selected on opposite 
sides of a mountain. Suppose that, in the absence of the 
mountain, a star S would pass at culmination near the zenith 


at the two stations. The mountain mass will attract a plumb- 
bob suspended at A, and since the direction of the plumb- 
line (or the direction of the force of attraction due to gravity) 
intersects the celestial sphere in the zenith, astronomical obser- 
vations will shew the zenith at A to be displaced by the 
mountain mass to Z^ since the star 8 at culmination will 
apparently be displaced away from the zenith and towards B. 
Similarly, the zenith at B will be displaced in the opposite 
direction. The distance apart of the stations A and B 
can be determined by a survey operation and the difference 
of their geographical latitudes calculated, since the dimensions 
of the Earth are known. The difference of latitude determined 
from astronomical observations at the two stations A and B, 
being the angle between the two plumb-lines, will be greater 
than the calculated value. The difference between the observed 
and calculated latitude difference is due to the attraction of the 
mountain. The size of the mountain can bo determined by 
surveying and its density estimated from an examination of 
the rocks composing it. The mass of the mountain can thus 
be approximately obtained, and from the observed deflection 
of the plumb-line the relative masses of the Earth and moun- 
tain can be computed. Thus the mass of the Earth is 

Observations by this method were made in 1740 by Bouguer 
at Chimboraso in South America, and in 1774 by Maskelyne 
at Schiehallien in Scotland. The method is not equal in 
accuracy to the two following methods, since the mean density 
of the mountain mass cannot be determined with sufficient 
accuracy, there being no certainty that the surface rocks have 
the same density as the interior of the mountain. 

(ii) The Torsion- Balance .Method. The principle of this 
method is illustrated in Fig, 6. x, x are two small balls carried 
at the ends of a light rod which is suspended horizontally 
from its mid -point by a fine wire. Two large masses W 9 
suspended at the same level from the ends of another hori- 
zontal rod, are brought into the position W near to the small 
masses x. The attraction between the large and small masses 
pulls the rod carrying the latter out of position, until the 
attracting force is balanced by the force due to the stiffness 
of the suspending wire and arising from the twist in it (position 



XiXj. The masses W are then brought into the position JY 2 , 
near the masses x but on the opposite side, deflecting them 
into the position # 2 - The total angle through which the rod 
moves from the position 1 to the position 2 is accurately 
observed with a telescope, and this angle is four times as great 
as the angle through which the rod 
would be deflected by bringing one 
sphere up to one ball. The torsional 
force in the wire, tending to twist the 
rod back into its original position, is 
proportional to the angle through which 
the rod is turned. The constant of the 
proportion can be easily determined. 
For this purpose, the masses W are re- 
moved and the rod is twisted through a 
small angle ; it will then swing to and 
fro, and if the time of swing be observed 
it is possible to determine the constant. 

p IQ 6 Principle of This enables the attracting force between 

tho Torsion Balance the masses W and x to be calculated 

Mothod of Determining lj(1 /. ,, . . P ,,, 

tho Mass of the Earth, and therefore the constant of gravitation 
to be determined. 

When the coefficient of gravitation is known, the mass of 
the Earth (or its equivalent, the mean density) is readily 
determined. The weight of any mass due to the Earth's 
attraction is known and the mass of the Earth is therefore 
obtained from the formula / = ymjm 2 /r*, f and m x being 
known (weight and mass), r being the radius of the Earth 
and y the gravitation constant. 

This experiment was first performed by Henry Cavendish 
in 1797-8. The apparatus used is shown in Fig. 7, Tih is the 
torsion rod hung by the wire Ig ; x, x are the attracted balls 
2 inches in diameter, hung from its ends ; W, W, the 
attracting masses, which are of lead, 12 in. in diameter. 
The torsion balance is enclosed in a case so as to exclude 
draughts of air and the masses W are suspended outside. 
The masses are turned by a string acting on the pulley P. 
T y T are the telescopes for observing the deflection of the 
masses x, mirrors n> n at the ends of the rod hh reflecting 
light from the lamps at the side of the case. The ease 



surrounding the apparatus serves to prevent disturbances 
arising from air currents. 

An improved form of this experiment was performed by 
Boys in 1895, the apparatus being made more delicate and 
reduced in size. The attracting masses, W, which were leaden 
spheres, 10 cms. in diameter and weighing 7-4 kgms. each, 
were suspended from the top of the case at different levels 
in order not to neutralize each other's effect and the attracted 

FIG. 7. Cavendish's Apparatus. 
hh, tor&ion rod hung by wire lg; x, x, attracted balls hung from its endi ; W, W attracting 

masses were placed at the corresponding levels. The attracted 
masses x, x were of gold, 5 mm. in diameter, weighing about 
1-3 gms., and were suspended by quartz fibres from the ends 
of a small rectangular mirror Q, about 2-4 cms. in length, 
which formed the torsion rod. The mirror was suspended by 
a very fine-drawn quartz thread, possessing great strength. 
It reflected a distant scale, enabling the deflection to be read 
with great accuracy. The attracting masses were moved from 
one position to the other by rotating the top of the case. A 
diagram of the apparatus is shown in Fig. 8 on a scale of 
about -]-. 
The value obtained bv Bovs is probablv the most accurate 



yet determined. He found for the constant of gravitation 
the value 6-658 X 1C" 8 dynes, corresponding to a mean density 
for the Earth of 5-527. The value found by Cavendish was 

(iii) The Common Balance Method. Suppose a spherical 

FIG. 8. Boys' Torsion Balance. 

mass to be hung from one arm of a balance and counterpoised 
by another mass at the end of the other arm. If a heavy mass 
is introduced beneath the suspended mass, it will exert a 
slight additional pull upon it, causing an apparent increase 
in weight which can be determined by adding a small weight 
to the other pan. The weight added determines the attrac- 
tion between the suspended mass and the attracting mass, 



and knowing this, the constant of gravitation can be determined 
as before. 

The apparatus used by Poynting in 1891 in applying this 
method is shown in Fig. 9. Lead spheres A and J3, weighing 
20 kgm. 3 were hung from the two arms of the balance, which 

FIG. 9. Poynting's Gravity Balance. 

had a 4-ft. beam. The balance was supported in a case, to 
exclude draughts, above a turn-table whose axis was below 
the central knife-edge of the balance. On this turn-table 
was the attracting mass M of 150 kgm., which could be brought 
under each of the suspended masses in turn. It was balanced 
by a smaller mass m, which was introduced to prevent a 
tilting of the floor by the heavy mass M . The attraction of 


m was, of course, allowed for. After rotation, a balance was 
again obtained by sliding along a small rider. This was done 
from outside the case and the position of the rider was observed 
with a telescope. Poynting's value for the mean density was 

The mean density of the Earth found from these experi- 
ments corresponds to a total mass of about 6 x 1C 27 grams, 
or about 5 X 1C 21 tons. 

16. The Interior of the Earth. A knowledge of the 
constitution of the interior of the Earth must be derived by 
indirect methods, since it is not possible to penetrate the 
surface sufficiently far to obtain any information of value. 
The deepest mine-shaft in the world, at the Morro Velho Mine 
in Brazil, reaches only to a depth of 6,426 ft., a distance which 
is infinitesimal compared with the radius of the Earth. The 
temperature of the Earth increases rapidly from the crust 
inwards, the average rate of increase in descending a mine 
being about 1 F. per 200 ft. Volcanoes and hot springs also 
provide evidence that the interior is hot, but we have no 
direct evidence as to the temperature gradient at great depths. 

The mean density of the Earth is about twice that of the 
surface rocks, the high value being doubtless due to the great 
pressure to which the interior is subjected. It was formerly 
believed that the high temperature in the interior necessitated 
a liquid core, and the outflow of molten lava from volcanoes 
during their periods of activity seemed to afford corroborative 
evidence of this view, It is not now, however, believed to be 
correct, as serious objections are involved on theoretical 
considerations . 

The tides in the ocean are due mainly to the gravitational 
attraction of the Moon on the water, as will be seen in 
Chapter IV. If the interior of the Earth were fluid with a 
solid exterior shell, the gravitational pull of the Moon on the 
fluid interior would deform the surface shell, with the result 
that it would rise and fall with the ocean waters. The height 
of the tides would therefore be very materially reduced, and 
the observed tides could not be accounted for. If the interior 
of the Earth is not fluid and yet not perfectly rigid, an elastic 
deformation will be produced by the attraction of the Moon, 


but this will not greatly decrease the height of the tides. A 
comparison between the observed and theoretical heights of 
the tides, in fact, enables an estimate to be made of the 
rigidity of the Earth, and it was from such considerations as 
these that Lord Kelvin was led to conclude that the Earth 
as a whole must be more rigid than glass but not quite so 
rigid as steel. It therefore appears probable that the interior 
is solid, with a high rigidity. 

The form of the Earth provides another argument leading 
to a similar conclusion. It has been seen that the Earth is 
not quite spherical but has the form of a spheroid, being 
flattened at the poles. Now it can be shown mathematically 
that a fluid mass of matter in rotation, as the Earth is, must 
assume a spheroidal form, and it is not unreasonable to suppose 
that the cllipticity of the Earth's figure is due to its rotation. 
The actual value of the ellipticity is, however, not that of the 
figure which would be assumed by a fluid mass of the size and 
density of the Earth, rotating in a period of one day. In 
order to account for the observed value, it is again necessary 
to suppose that the interior of the Earth is a solid, possessing 
high rigidity. 

Corroborative evidence is provided by the speed with which 
earthquake waves travel. The interior of the Earth being 
hotter than the surface heat is gradually passing outwards, 
and the interior contracts as it cools. The outer crust is 
supported by the inner nucleus and it is probable that earth- 
quakes are due to the outer crust adapting itself to the con- 
tracting nucleus. When an earthquake occurs, the disturbance 
spreads outwards through the surrounding earth in a manner 
analogous to that in which the discharge of a cannon produces 
a disturbance spreading out in the air, which the observer 
detects as a sound. The earthquake waves cause minute 
oscillations of the surface extending to great distances, and 
these oscillations can be detected with the aid of a delicate 
instrument, called a seismograph, which greatly magnifies 
them. At a great distance from an earthquake two separate 
disturbances are recorded, and it has been shown that one 
of these is due to waves which travel through the interior 
of the Earth and that the second is due to waves which travel 
over the surface. From the ve 1 city of the waves which 


travel through the interior, the rigidity of the Earth can be 
calculated. The results so found are again not compatible 
with the existence of a fluid nucleus. 

17. The Variation of Latitude. Connected with these 
theories is the phenomenon of the variation of latitude. It 
was shown mathematically by Euler that if a body which, 
like the Earth, is symmetrical about an axis is set in rotation 
about that axis and is not acted upon by any external forces, 
it will continue to rotate about that axis with a constant 
angular velocity. But if it is set in rotation about any other 
axis, the axis round which the body will turn in the case of 
a nearly spherical body such as the Earth will always point 
in the same direction in space (i.e. among the stars), but it 
will describe a cone in the Earth, the axis of this cone being 
the axis of figure of the Earth. Euler showed that this cone 
would be described in a period of 305 days. This is equivalent 
to saying that the Earth's axis, instead of being directed to 
the same point in the sky, will describe a small circle amongst 
the stars. Since the elevation of the pole at any station is 
equal to the latitude of the station, the effect would be 
detected by a regular change of latitude of places on the 
Earth, with a period of 305 days. 

The existence of such a variation was first detected by 
Ktistner in 1888, who found a variation in the latitude of 
Berlin. This result was confirmed by Chandler who, from a 
more thorough discussion of several series of observations, 
showed that the period of the variation was about 430 days, 
instead of the predicted 305 days. This result has been fully 
confirmed by later investigations which have shown that the 
movement of the Earth's axis of rotation about its mean 
position is compounded of two motions, one of semi -amplitude 
about 0"-18 with a period of 432 days, and the other of semi- 
amplitude about 0"-09 with a period of exactly one year. 
The latter is mainly due to meteorological causes, involving 
movements of masses of air from one portion of the Earth to 
another. The former is the motion investigated by Euler who, 
however, in deducing the period, had assumed the Earth to be 
perfectly rigid. Other considerations have just led us to the 
conclusion that this is not the case, and by making the appro- 



priate modification in Euler's investigation, the observed 
period can be accounted for. 

The motion of the Earth's pole (i.e. the end of its axis of 
rotation) about its mean position from 1912 to 1920 is illus- 
trated in Pig. 10. In this figure, the origin of the co-ordinates 
corresponds to the mean position of the pole. A displacement 
along the positive direction of the x axis, indicates a movement 

,+0*40 . +0-30 +0-20 . +0-10 . 0-00 -0-JO ~0*20 -0-30 


+0-30 - 

+0-40 +0-30 


- -0-20 

- -0-10 

0-00 - 

- +0-10 

- +0-20 



FIG. 10.- 

-The Movement of the Earth's Pole, 1912-0-1920'0. 
(0"'01 = I foot on Earth's surface.) 

along the meridian of Greenwich ; a displacement along the 
y axis indicates a movement in the perpendicular direction. 
During the period, the pole has described the irregular curve 
shown, the position at intervals of -y th year being indicated. 
When the two components of the motion are in the same 
phase, the total motion is large, as in 1916 ; when in opposite 
phase, the total motion is small, as in 1919. An angular 
motion of 0"-01 corresponds approximately to a movement 
of 1 ft. on the Earth's surface. 


18. The Earth's Atmosphere. The Earth is surrounded 
by an atmosphere which is a mixture chiefly of nitrogen and 
oxygen. At the surface of the Earth, dry air consists princi- 
pally of about 78 per cent, of nitrogen, 21 per cent, of oxygen, 
nearly 1 per cent, of argon, and smaller amounts of carbon 
dioxide, hydrogen, neon, helium and other rare gases. Water- 
vapour is present at the surface to the extent of about 1-2 per 
cent, of the total. 

At great heights, the lighter gases predominate, the percent- 
age of hydrogen and helium beyond 80 kilometres probably 
being very large : at the same time, the density rapidly 
decreases upwards, so that the total amount of hydrogen is 
not large. 

If the atmosphere were homogeneous, having the same 
density throughout as at the surface, it would only extend 
to a height of about 5 miles, but on account of the rapid decrease 
in density upwards the actual height is very much greater 
than this. A lower limit to the height can be obtained from 
observations of meteors (see Chapter XI) which, on coming 
into the Earth's atmosphere from outside with a large velocity, 
are raised to incandescence by friction. If observations of 
the path of the same meteor are made from two stations at 
some distance apart, it is possible to calculate the height of 
the meteor. The maximum heights so obtained are about 
120 miles. The actual height to which the atmosphere extends 
must be very much greater than this, as the meteor will 
penetrate some considerable distance through the outer very 
rarefied layers before its temperature is raised sufficiently to 
render it visible. 

The blue colour of the sky is a result of the Earth pos- 
sessing an atmosphere. When light passes through a medium 
containing numerous small particles, a certain proportion of 
the light is scattered sideways by these particles and the 
shorter the wave-length of the light the greater will be the 
scattering. The blue light is therefore scattered to a much 
greater extent than the reel light. The light as it travels 
onward is thus gradually robbed of its blue portion and will 
appear red. This effect is readily seen by looking at a street 
lamp from a short distance in a fog. The light from the Sun 
which passes through the upper layers of the Earth's atmosphere 


would, in the absence of the atmosphere, pass outside the 
Earth and the sky would therefore appear black and the stars 
would be seen at all hours of the day. The molecules in the 
atmosphere, however, scatter the blue light towards us, so 
producing the blue appearance of the sky. The more free 
the air is from the comparatively large dust particles, the 
purer and deeper will be the blue. To the same cause are due 
the golden tints of sunset. When the Sun is near the 
horizon, the light from it which reaches an observer passes 
through a much greater length of atnmsphere than when it 
is higher in the heavens. A greater proportion of blue light 
being then lost, the light reaching the observer is tinted red. 
The beautiful colours which frequently accompany the setting 
of the Sun are mainly due to dust particles and depend very 
largely upon the amount of dust in the atmosphere. 

Another phenomenon which is due to the Earth having an 
atmosphere is twilight. Were there no atmosphere, darkness 
would result the moment the Sun had set ; but the particles 
in the upper atmosphere continue to reflect light from the Sun 
for some time after it has actually disappeared below the 
horizon. The amount of light reflected becomes inappreciable 
after the Sun lias sunk about 18 below the horizon, and the 
time taken for this to occur is generally used as a measure 
of the duration of twilight. For places near the equator, the 
Sun will move approximately in a great circle which meets the 
horizon at right angles and it will descend 18 below the 
horizon much more rapidly than at a place in a higher latitude, 
for which the Sun moves in a great circle cutting the horizon 
obliquely. The duration of twilight is therefore considerably 
greater in higher latitudes than it is near the equator. 

19. Refraction. Another atmospheric effect, for which 
it is necessary to make allowance in astronomical observations, 
is known as Refraction. When a ray of light passes from one 
medium into another of different density its direction is 
changed. If it passes from a less dense into a denser medium 
it is bent towards the normal to the interface between the two 
media ; conversely, if it is passing into a rarer medium it is 
bent away from the normal. 

If we suppose, for simplicity, that the Earth is flat and 



P f p 

FIG. 11. Elementary Theory of 

the atmosphere homogeneous with a definite upper surface, 
a ray of light coming from a star, or other body, will, on 
entering the denser atmosphere, be bent towards the normal, 
i.e. towards the direction to the zenith. The body will appear 
to an observer to be in the direction from which the ray comes ; 
the effect of refraction is, therefore, to make the apparent 
zenith distance of a body less than it actually is. Thus, in 

Fig. 11, PQ is a ray which 
enters the atmosphere at Q 
and is refracted towards an 
observer at O. The angle 
ZOP' or Z'QP' is a measure 
of the apparent zenith dis- 
tance ; the angle Z'QP is a 
measure of the true zenith 
distance. Refraction, there- 
fore, appears to increase the 
altitude of the body by the 
angle PQP'. The effect of re- 
fraction increases with zenith 

distance, reaching its maximum on the horizon when the body 
is rising or setting. For large zenith distances, the rate of 
increase of the refraction with zenith distance is largo, so that 
in the case of the Sun or Moon near the horizon, the refraction 
is appreciably different for the upper and lower limbs. The 
lower limb is raised by refraction more than the upper, so 
that the apparent vertical diameter becomes less than the 
horizontal. This gives rise to the well-known flattened shape 
of the Sun or Moon at rising and setting. 

An approximate formula for the change in zenith distance 
due to refraction, which is valid for all except large zenith 
distances, can be obtained from elementary considerations. 
The curvature of the Earth is neglected and the assumption 
made that the atmosphere is horizontally stratified, so that 
the density is the same for all points at the same height. 
It then follows from the laws of optics that a ray of light 
will pass through the atmosphere in such a manner that 
^uSinZ is a constant at every point of its path, ^ being the 
refractive index of the air at any point and Z being the angle 
which the ray at that point makes with the vertical. If Z 


is the value of Z when the ray enters the atmosphere, then, 
since in vacuo the refractive index is unity, 

/iSinZ = SinZ . 

In this formula ^ and Z can now be taken as referring to 
the surface of the Earth, and if is the change in the zenith 
distance due to refraction, Z ~ Z + f . Hence 
//SinZ - Sin(Z + f ) 

= SinZ + CosZ, since f is small, 
or f -- (^ 1) tanZ. 

Hence, except for large zenith distances, when the curvature 
of the Earth cannot be neglected, the refraction is proportional 
to the tangent of the zenith distance. 

The index of refraction of a gas depends upon its tempera- 
ture and pressure, and therefore the coefficient of tanZ in the 
preceding formula will depend upon the temperature and the 
barometric height. The refraction decreases with increase in 
temperature and increases with increase in barometric height. 
Tables, such as those of Bessel, have been constructed giving 
the refraction with accuracy for any zenith distance : these 
are based upon a standard temperature and pressure. Auxili- 
ary tables are given containing the corrections to apply for 
other temperatures and pressures. 

For air at zero Centigrade and a barometric height of 76 cm. 
// is 1-000294. The approximate refraction formula then 
gives, expressing C in seconds of arc, 

" = -000294 x 206265 X tanZ = 60"-6 tanZ, 
which is sufficiently accurate down to about 70 Z. D. For 
small zenith distances, the refraction is approximately one 
second of arc per degree of Z. D. 

At an apparent altitude of the mean refraction is about 
35', and at altitude 30' it is about 29'. The angular 
diameter of the Sun or Moon being about 30' it follows that 
when on the horizon the effect of refraction is to shorten the 
vertical diameter to about 24', whilst the horizontal diameter 
remains unaltered. The resultant flattening is therefore very 

Corrections for refraction must be applied to all astronomical 
observations in order to reduce apparent zenith distance to 
true zenith distance. 



20. Dip of the Horizon. Another correction which it 
is necessary to apply to certain astronomical observations is 
that for the " dip of the horizon." In taking observations 
at sea, the true altitude of a body is not observed, but the 
angular distance between the body and the visible horizon 
or sea-line. Owing to the curvature of the Earth, this visible 
horizon does not coincide with the true horizon, but falls below 
it by an amount depending upon the height of the observer 
above sea-level. The angular difference between the true 
and visible horizons is called the Dip of the Horizon. 

If, in Fig. 12, O is an 
observer and C the centre 
of the Earth (supposed 
spherical), then OH which 
is perpendicular to 00 is 
the trace of the astrono- 
mical horizon. If OB is 
a tangent to the Earth 
at the point B, then B 
is the point on the sea- 
line in the direction OB. 
The observed altitude 
SOB, of a star 8, is 
greater than the true alti- 
tude, 80H, by the angle 
HOB, which is denoted by A , and gives the dip of the horizon 
for the observer at O. Since CB, BO are perpendicular and 
also CO, OH, the angle ACE angle BOH = A. Hence, if 
R is the Earth's radius and h the height of the observer above 

CosZl = - 

FIG. 12. The Dip of the Horizon. 

B + /r 

Since A is small and is expressed in circular measure, and 
since h is small compared with B, this formula gives 

! = i -A- 


7 2A 
Expressing ^1 in minutes of arc (3,438 minutes in one radian) 

1 - 

or A ~ 


and h in feet, and putting R = 20,880,000 ft., the formula gives 


A ^ 3231 

An approximate formula for the dip is thus : 
A (in minutes of arc) = Vh (in feet). 

This formula is sufficiently accurate for observations at sea, 
which are not of extreme precision. The dip is subtracted 
from observed altitude to obtain apparent altitude, and this 
\\hcn coi reeled for icfi action, gives true altitude. 



21. The Apparent Motion of the Sun. Although the 
Sun rises and sets and exhibits other phenomena due to the 
diurnal motion of the Earth on its axis, it is at once apparent 
that its motion on the celestial sphere is much more compli- 
cated than the motions of the fixed stars. In 7 it was shown 
that any given star always culminates at the same zenith 
distance. But if the motion of the Sun, as seen by an observer 
in the northern hemisphere, be considered, it is evident that in 
the summer it reaches a much higher altitude at culmination 
than in the winter. If the Sun be regularly observed, starting 
in the spring about the end of March, i.e. at the vernal equinox, 
it will be seen that then it rises approximately in the east point 
of the horizon and sets in the west point. Each succeeding 
day it will be found (the observer being assumed in the northern 
hemisphere) to rise and set a little farther towards the north 
and to reach a slightly higher altitude at culmination, though 
on any one day its path on the celestial sphere is very nearly 
a small circle. Towards the end of June, the altitude reached 
at culmination attains its maximum and the Sun then rises 
and sets at its farthest north. Thereafter, it retraces its course 
and near the end of September, at the autumnal equinox, it 
again rises and sets in the east and west points respectively. 
It continues to move southwards until, near the end of Decem- 
ber, it reaches a minimum altitude at culmination and rises 
and sets farthest south. Thereafter the Sun commences to 
move gradually northwards again and completes one cycle by 
the next vernal equinox, in the period of 365 days. These 
movements should be considered in conjunction with Fig. 2. 

The strong light of the Sun hinders the stars being seen at 
the same time. But we know that the diurnal motion of the 



stars is only apparent and due to the rotation of the Earth on 
its axis and that therefore, at any given place, any one star 
will always rise and set at the same points of the horizon. It 
follows that the Sun moves northwards amongst the stars from 
the winter to the summer solstice and southwards from the 
summer to the winter solstice. Moreover, the stars which rise 
in the eastward horizon as the Sun is setting in the westward 
are not the same in summer and winter. Suppose, for instance, 
that the three bright stars in the belt of Orion are observed 
rising in the east in the winter ; it will be found that they rise 
each evening 4 minutes earlier than the preceding evening. 
If one evening they are observed to be rising just as the Sun is 
setting, then a few weeks later it will be found that they are 
well up in the eastern sky at sunset. It follows, therefore, that 
the Sun moves eastwards amongst the stars as well as north 
and south. This eastward motion continues throughout the 
year, during which period it completes an entire circuit of the 
heavens and at the end of it has returned to its original place. 
If accurate determinations of the Sun's position relative to 
the stars are made with a meridian circle and the positions are 
plotted on a celestial globe, it will be found that the plotted 
points lie on a great circle which cuts the equator at an angle of 
about 23|. This great circle is known as the Ecliptic, being 
originally so called because it was found that eclipses only 
occurred near the times when the Moon crossed this great 
circle. The ecliptic may be regarded as the path of the Sun 
on the celestial sphere and it is from this point of view that 
we have approached it. But it must be remembered that 
it is not possible to say, a priori, whether the relative motion 
of the Sun and Earth is due to the motion of the Sun or to that 
of the Earth. The Earth, if seen from the Sun, would appear 
to move in this same path, though remaining six months behind, 
since lines drawn from the Earth to the Sun and from the Sun 
to the Earth respectively point to diametrically opposite points 
on the celestial sphere. It is known, however, from other 
considerations referred to later, see 116, 119, that in reality it 
is the Earth which is in motion around the Sun. If, then, 
mention is made of the motion of the Sun in the heavens, what 
is really meant is the apparent motion due to the motion of the 


22. The Ecliptic may therefore be defined as the trace on the 
celestial sphere of the plane of the orbit of the Earth round the 

The, Zodiac is a zone extending along the ecliptic. It is 
divided into twelve signs each comprising 30 of longitude 
and known by the name of the constellation included. 

The Obliquity of the Ecliptic is the angle between the ecliptic 
and the equator. Its value, about 23, is the maximum 
distance which the Sun can reach north or south of the equator, 
i.e. the Sun's declination can vary between 23| N. and 23^ S. 

The Equinoxes are the points at which the ecliptic cuts the 
equator. When the Sun is at either of the equinoxes, it is on 
the equator and therefore rises and sets exactly in the east and 
west points. The lengths of day and night are then equal, 
hence the term equinox. 

The Vernal Equinox (or First Point of Aries) is the point at 
which the Sun is passing from south to north of the equator. 
It is the origin for the measurement of right ascension (see 6). 
The Sun is at the vernal equinox about March 21. 

The Autumnal Equinox (or First Point of Libra) is the point 
at which the Sun passes from north to south of the equator. 
The Sun is at the autumnal equinox about September 23. 

The terms vernal and autumnal equinox are also used to 
denote the times when the Sun crosses the equator. 

The Solstices are the points on the ecliptic midway 
between the equinoxes. At these points the Sun attains its 
greatest north and south declinations and reaches its greatest 
and least altitudes in the heavens. It therefore stops moving 
in altitude or " stands " for a few days, hence the term solstice. 

The Tropics are the two small circles on the celestial sphere 
which are parallel to the equator and pass through the solstices. 
The word means " turning " and it is when the Sun is on the 
tropics that its motion turns from northwards to southwards 
or vice versa. The path of the Sun lies between the tropics 

The position of a celestial body may be defined with reference 
to the ecliptic by co-ordinates analogous to right ascension 
and declination which define the position relatively to tho 
equator. The longitude of a star is measured eastwards 
along the ecliptic from the vernal equinox to the foot of the 
great circle passing through the pole of the ecliptic and the 


star. Or, in other words, it is the angle between the two great 
circles through the pole of the ecliptic passing through the 
vernal equinox and the star respectively. The latitude of a 
star is its distance north or south of the ecliptic measured 
along a great circle through the poles of the ecliptic. 

23. The Nature of the Earth's Orbit. A general idea 
of the shape of the orbit described by the Earth around the 
Sun may easily be obtained. Prom observations with the 
meridian circle, the position of the Sun on the celestial sphere 
at various times throughout the year may be determined, 
#1, S 2 , & ... in Fig. 13. 
To an observer on the Sun, 
the Earth would appear at the 
same times in the diametrically 
opposite directions OE l9 OE 2 , 
etc. If the positions of the 
points EI, E 2 ,E 3 . . . on their 
several radii can be found, 
both the shape and size of the 
orbit are determined. By 
simple methods it is possible 
to determine their positions ^ 1( ^,. 

*\ _ M FIG. 13. The Karth's Orbit. 

on any arbitrary scale, but 

the determination of the true scale is a matter of some 
difficulty, the consideration of which will be deferred for 
the present. To determine the relative lengths it is only 
necessary to measure the values of the angular diameter 
of the Sun in the positions S 19 $ 2 . . . which can easily be 
done by projecting an enlarged image of the Sun upon a screen. 
The diameters can be determined either in arbitrary linear 
measure or, by computation from the dimensions of the appa- 
ratus used, can be expressed in arc. 

If the observations are accurately made, it will be found 
that the angle subtended by the Sun is not constant throughout 
the year, but is greater in winter than in summer. Thus, the 
values of the Sun's semi-diameter at certain times of the year 
are approximately : 

January 1 . . 16' IS" July 1 . ,15' 45" 
April 1 . .16' 2" October 1 . 16' 0* 



This variation is due, not to an actual change in the Sun's 
diameter, but to changes in the distance between the Earth and 
the Sun. The distance, in fact, must be inversely proportional 
to the angular diameter. If then, the points E 19 E%, E z . . . 
are chosen on the radii through 0, so that their distances from 
are inversely proportional to the angular diameters of the 
Sun at Si, $ 2 , S 3 . . . , and the points so plotted joined by a 
curve, this curve will represent the shape of the orbit of the 
Earth around the Sun which is at the point 0. 

The curve so found is not quite a circle, being slightly oval in 
shape. It is an ellipse with the Sun in one of the foci. An 

ellipse can be simply con- 
structed by taking two points 
8 and 8^ (Fig. 14), fixing the 
ends of a piece of cotton to 
these points by pins and run- 
ning a pencil round inside the 
cotton, which is kept taut in 
the process. It is obvious 

FIG. U.-Tho Ellipse. that SP + S * P is constant 

for any point on the ellipse. 
If A and A are the points on the curve in 88 1 produced, 

SP + S,P - SA + S,A 

= SA + 8A t (from symmetry) 

A A! is called the major axis of the ellipse, 8 and S are called 
its foci. A line BOB 1 through 0, the mid-point of S8 l and 
perpendicular to it, is called the minor axis. 

The smallest value of the radius- vector SP, joining one of the 
foci to any point P on the curve, is SA, and the greatest value 
is 8 A i. The ratio of SO to OA is called the eccentricity. The 
larger the eccentricity the more the ellipse deviates from a 
circle, which is the limiting case of the ellipse when the foci 
SSt both coincide in the centre 0. The eccentricity e can be 
expressed in terms of the lengths of the major (2a) and minor 
(2b) axes, viz. : 

e = (a 2 - 6 2 )V. 

It must be carefully distinguished from the ellipticity referred 
to in 13, which is equal to (a - b)/a, and which is a much smaller 


quantity. Thus the eccentricity of the Earth's orbit is about 
I/GO, its ellipticity about 1/7200. 

For an ellipse of small eccentricity, the ellipticity is |e 2 . 

If the Sun is at the focus S, then the Earth is at the point A 
(the end of the major axis nearest this focus) on January 1, 
and at the other end AI, six months later, on July 3. These 
positions are called perihelion and aphelion respectively. The 
Earth is therefore nearer the Sun in winter than in summer. 
When the Earth is at perihelion with respect to the Sun, the 
Sun is said to be at perigee, with respect to the Earth ; when 
the Earth is at aphelion, the Sun is at apogee. 

24. The Motion of the Earth in its Orbit. If the Earth's 
orbit as so determined be carefully plotted on squared paper 
and the position of the Earth at various intervals be marked, 
then by drawing the radii from the focus occupied by the Sun 
to these positions and counting the squares included between 
consecutive radii and the ellipse, the relative areas described 
by the Earth's radius-vector in various times will be deter- 
mined. It will be found that the areas swept out are directly 
proportional to the times or, in other words, in equal times 
equal areas will be swept out. This is one of Kepler's laws of 
planetary motion, which will be referred to in 116. 

Thus in Fig. 15, if the Earth 
moves from E t to E 2 in the 
same time as from E 3 to ^E^ 
the area bounded by SE ly 7S 
and the curve is equal to that 
bounded by SE 3 , SE* and the 
curve. If the earth is nearer 
the Sun at E 1} E 2 than at E 3 , 
EI, it follows that the arc E^E^ 
must be greater than the arc FlG - 15. Tho Motion of the Earth 
J? 3 JSJ 4 . Therefore the Earth 
moves faster when near perihelion than when near aphelion. 

25. Diurnal Phenomena Connected with the Earth's 
Motion. We can now proceed to show how the information 
we have gained as to the motion of the Earth around the Sun 
will enable us to explain various phenomena connected with 


the rising and sotting of the Sun, the length of the day, and 
the duration of twilight and also phenomena connected with 
the seasons. The diurnal phenomena will be dealt with first. 

For this purpose, the Sun may be considered as a star of 
variable decimation. At the vernal equinox, the Sun is just 
passing from south to north of the equator. Its decimation 
is therefore zero, but gradually increases day by day until the 
summer solstice, when it reaches its greatest value, 23 27' N., 
this value being equal to the obliquity of the ecliptic. From 
then onwards, the declination decreases until it is again zero 
at the autumnal equinox, when the Sun is passing from north 
to south of the equator. Between the autumnal equinox and 
the winter solstice, the south declination of the Sun increases 
from zero to its maximum, 23 27' S., after which it again 
decreases until the Sun crosses the equator northwards at the 
next vernal equinox. 

The change in the Sun's declination is most rapid at the 
equinoxes, when it amounts to nearly 24' per day, and is least 
rapid at the solstices, when it is only a few seconds of arc per day. 
The rate of change is sufficiently slow to justify the assumption, 
in dealing with the diurnal phenomena, that the declination 
remains constant during one day, but changes from day to day. 

It has been shown in 5 that at the equinoxes, the Sun, being 
on the equator, rises in the east point and sets in the west point, 
and that this is true whatever the latitude of the place of 
observation. The diurnal path of the Sun is then a great circle 
(the equator), of which exactly one half is above arid one half 
below the horizon and the lengths of day and night are equal. 
As the Sun moves north of the equator, its diurnal path becomes 
a small circle, and, since the north polar distance of the Sun is 
then less than 90, it is evident that at places in the northern 
hemisphere more than half the path is above the horizon and 
that the intersections of the path with the horizon are north 
of the east and west points. Between the vernal and autumnal 
equinoxes, tlieref ore, the Sun rises to the north of east and sets 
to the north of west, so that the length of the day is longer than 
t>ie length of the night, for places in the northern hemisphere. 
Obviously, for places in the southern hemisphere, the converse 
holds, night being longer than day. The length of day attains 
its maximum value at the summer solstice. 


For places on the equator (since for such places the Poles are 
on the horizon), the diurnal circles of the Sun's motion all cut 
the horizon at right angles and the lengths of day and night 
are then always equal. The Sun always rises about 6 a.m. and 
sets about 6 p.m. 

In 5, it was also shown that at any place, those stars whose 
north polar distances are less than the latitude of the place will 
never set. It follows that at the summer solstice, when the 
Sun's declination is about 23-|, at all places north of north 
latitude 6G -|- the Sun will not sink below the horizon and there 
is no night, the Sun being visible at midnight. At all places 
south of south latitude 66|, the Sim will not, at the same 
season, appear above the horizon. The two parallels of north 
and south latitude 66| are called the arctic and antarctic circles 
respectively. The higher the latitude, above 66|-, the longer 
will be the period during which the Sun does not sink below or 
rise above the horizon, and at the Poles themselves, since the 
horizon then corresponds with the equator, the Sun will remain 
above the horizon for six months continuously and will then 
disappear below the horizon for six months. 

A reference to Fig. 2 shows that the meridian altitude of the 
Sun, SB, equals SP - PR or (180 - <f>) - (90 - d) or 
90 </> + (5, where d is the declination of the Sun, cf> the 
latitude of the place of observation. If </> ^><5, the meridian 
altitude is less than 90, and in that case the Sun will cross the 
meridian to the south of the zenith. If < d, the Sun will 
pass through the zenith. This is possible only if is not greater 
than 23 1. At places on the two parallels of north and south 
latitude, 23|, the Sun just reaches the zenith, but does not 
pass to the north or south of it respectively. At all places 
between these parallels, d becomes at two periods of the year 
greater than </>, and at such places the Sun will pass at some 
periods of the year to the north of the zenith and at other 
periods to the soiith. The parallel of 23| north latitude is 
called the Tropic of Cancer, because the Sun is overhead near 
the summer solstice when the Sun is in the sign of the zodiac 
called Cancer. Similarly the parallel of 23| south latitude is 
called the Tropic of Capricorn^ the Sun being overhead when 
in the sign of Capricorn. 

At places whose north (south) latitude is greater than 23 1 


the Sun is always to the south (north) of the zenith, but, in the 
northern hemisphere, its midday altitude is greatest at the 
summer solstice and least at the winter solstice, the reverse 
holding in the southern hemisphere. 

26. Duration of Twilight. If the Earth possessed no 
atmosphere, darkness would follow immediately upon sunset. 
The effect of the reflection and scattering by the Earth's atmo- 
sphere is to cause some illumination to reach the observer before 
sunrise and after sunset, this phenomenon being known as 

No precise measure of the diiratioii of twilight can be made. 
It is, however, found that no perceptible twilight remains after 
the Sun has sunk an. angular distance of 18 below the horizon. 
The time taken by the Sun to sink this distance can therefore 
be used as a convenient measure of the duration of twilight. 
This time depends upon the angle at which the Sun's diurnal 
circle cuts the horizon ; the more acute the angle, the greater 
the distance the Sun must travel in its path before it has sunk 
18 below the horizon and therefore the longer twilight will 
last. On the equator, the diurnal circles cut the horizon at 
right angles and the duration of twilight is considerably less 
than in higher latitudes. In high latitudes also, there is a 
marked seasonal variation, twilight being longest at the summer 
solstice and least at the equinoxes. 

27. The Seasons. The Earth completes one revolution 
in its orbit round the Sun in a period of one year, passing 
perihelion on January 1 and aphelion at the middle of the 
year. The ecliptic is divided into four equal parts by the two 
equinoctial and the two solstitial points and the periods taken 
by the Sun in apparently traversing from one of these point u 
to the next are called the seasons. Owing to the varying 
velocity of the Earth in its orbit, the lengths of the four seasons 
are unequal. This will be made clearer by a reference to Fig. 
16, which shows the orbit of the Earth with the Sun in one of 
the foci S. The positions of perihelion P, and aphelion A are 
at the two ends of the major axis of the ellipse. If $T is the 
direction from the Sun to the First Point of Aries and two lines 
are drawn at right angles through S, one of which passes through 


T, these lines will intersect the orbit in the points designated 

T, and M , N. Since the Sun is in the First Point of Aries at 

the vernal equinox, the Earth 

must then be at . It will 

be at T six months later, at 

the autumnal equinox. It 

will similarly bo at N at the 

summer solstice and at M at 

the winter solstice. The orbit 

is described in the direction 

The Earth is at perihelion 
P at the end of the year and 

the portion of the orbit from M FIG. 16. The Seasons. 

to corresponds to winter, 

that from 2= to N to spring, that from N to T to summer and 
that from T to M to autumn. It is evident that the areas MS^=, 
-M$T, NS~, NS^ are unequal, but that they increase in this 
order. In 24 it was stated that the Earth moves in its orbit 
so that the rate of description of areas included between the 
curve and radii vectors to the Sun is constant. It follows 
that the times taken in describing these four areas increase with 
the areas or that, in other words, the seasons are of unequal 
length, winter being the shortest, autumn slightly longer, 
spring longer still and summer the longest of all. As a matter 
of fact, the approximate durations are : 

Spring . . . .92 days 21 hours. 

Summer . . . . 93 ,, 14 

Autumn . . . . 89 18 ., 

Winter . . . . 89 1 

These statements are correct for the northern hemisphere. 
For any place on the Earth, the definition of summer as that 
period of the year taken by the Earth to pass from the point 
N to the point T of its orbit is not correct, as summer is strictly 
that one of the seasons which has the highest average temper- 
ature. For places in the southern hemisphere, therefore, 
summer corresponds to the portion M== of the orbit autumn 
to the portion ^N , winter to the portion Nv and spring to 
the portion T M. It follows that in the southern hemisphere 



_1 Equinox 

autumn and winter are of longer duration than spring and 

The variation in the heat received from the Sun to which the 
seasons owe their importance, is due to the axis about which 
the Earth rotates not being perpendicular to its orbit. We 
have seen indeed that the Earth's orbit lies in the ecliptic and 
that this is inclined at an angle of about 23| to the Earth's 
equator. The axis of rotation of the Earth, being perpendicular 
to the equator, is therefore inclined at an angle of 23J to the 
direction normal to its orbit. Further, this axis always 
remains parallel to itself as the Earth passes round the Sun, 
for the north polar distances of stars remain constant through- 
out the year, except for certain minute changes arising from 

other causes. The 
direction of the axis 
is always towards 
the pole of the 
equator and is 
therefore at right 
angles to the direc- 
tion T , this direc- 
tion being that of 
the intersection of 
the equator and 
ecliptic. At the 
solstices and equi- 
noxes, the direc- 
tions of the axis are shown in Fig. 17. It will be seen 
from an inspection of the figure that in winter time the north 
end of the axis points away from the Sun, the Sun then being 
below the celestial equator, whilst in summer, it is the soutli 
end which points away, the Sun then being above the celestial 
equator. This is represented in a different manner in Fig. 18 
A is any point in the northern hemisphere which has the Sun on 
its meridian, AS the direction to the Sun. is the centre of 
the earth, so that OA is the direction of the zenith at A. In 
winter time the angle between AS and AZ is greater than in 
summer, i.e. the zenith distance of the Sun is greater or its 
altitude is less in winter than in summer. This has already 
been shown otherwise in S 25. 




17. The Inclination of tho Earth's Axis 
to Ecliptic. 


Combining the two effects, whose magnitudes may be 
obtained by calculation, it is found, as illustrated in Fig. 23, 
that the equation of time vanishes four times in a year, on or 
about April 15, June 15, August 31 and December 24. Its 
maximum positive value is nearly 14 J minutes about February 12 
and its maximum negative value is nearly 16 J minutes about 
November 3. In the figure, the thin lines represent the two 
separate components, the thick line the combined equation 

20 m \ . 

30 60 90 120 160* f80 210 240*270 300 330 360' 
FIG. 23. The Composition of the Two Components of the Equation of Time. 

obtained by adding algebraically the ordinates of the two 
curves. At the bottom of the figure are given the solar 
longitudes, at the top the day of the year. 

30. One or two consequences of the mean Sun being some- 
times in advance and sometimes behind the true Sun may be 
noted in passing. 

If mean noon always coincided with apparent noon, then the 
interval between sunrise and noon would be equal to the 
interval between noon and sunset. Expressed otherwise, if 
the times of sunrise and sunset arc given in civil reckoning 
(a.m. and p.m.) the sum of the numbers would be 12 h. Om. 
This is not, in general, the case. At Greenwich the Sun rises, 


for instance, on February 9 at about 7 h. 29 m. a.m. and sets 
at 5 h. 1 m. p.m., the sum being 12 h. 30 m. On November 2, 
on the other hand, sunrise is at 6h. 55m. a.m., sunset at 4h. 31 m. 
p.m., the sum then being llh. 26m. These differences are 
due to the varying sign and magnitude of the equation of time. 
For the Sun is due south at apparent noon and it may be 
assumed that the decimation does not vary during one day ; 
it therefore follows that when the equation of time is positive 
the interval between mean noon and sunset will be longer 
than that between sunrise and mean noon by twice the amount 
of the equation of time ; when, on the other hand, the equation 
of time is negative, the former interval is shorter than the latter 
by twice its amount. 

Another phenomenon well known in high latitudes is that 
the times of latest rising and earliest setting of the Sun do not 
occur on the shortest day. In such latitudes, the latest sun- 
rise occurs some days after the winter solstice, the maximum 
difference being about 3 minutes. This is due to the change 
in the equation of time from day to day. Near the end of the 
year, the equation of time is increasing daily at the rate of 
about 30 seconds per day. In an interval of one week, there- 
fore, it increases nearly 4 minutes. Neglecting for the moment, 
therefore, the variation in the decimation of the Sun and the 
resultant change in the times of sunrise and sunset, it follows 
that after one week the Sun rises and sets 4 minutes later 
than at the beginning. This change, combined with the normal 
change due to the varying declination, produces the observed 

The rate of change of the equation of time is of importance 
in another respect. It obviously provides a measure of the 
excess of the length of the true day (measured from one 
apparent noon to the next) over that of the mean day. This 
excess has its greatest positive value (about 28 seconds) near 
the winter solstice and its greatest negative values (about 20 
seconds) near the two equinoxes. There is a smaller positive 
maximum of about 12 seconds near the summer solstice. The 
two days have equal length near the middle of February, the 
middle of May, the end of July and the beginning of November. 

31. Local Time and Standard Time. We have up to 


the present been concerned with local mean and local apparent 
time. Local mean noon at any place is the moment of passage 
of the mean Sun across its meridian and similarly for local 
apparent noon. At two places, not situated on the same 
meridian of longitude, the times of local mean and also of 
local apparent noon will be different. The difference in time 
will obviously be the time equivalent of the longitude difference. 
Thus at a place whose longitude is I degrees east of Greenwich, 
local mean noon will occur y 1 ^ I hours before Greenwich mean 
noon. This variation in the time of noon becomes of great 
importance in view of the rapidity of modern transport ; if 
local mean time were everywhere adhered to innumerable 
difficulties would result, as it would be far from easy for accurate 
time to be kept. In order to avoid these difficulties, a system 
of standard or zone time has been adopted by most of the 
principal countries of the world. Under the zone system, the 
same time is adopted over the whole of the region on the Earth 
comprised between two meridians of longitude corresponding 
to a longitude difference of 15, the time corresponding to that 
of the central meridian of the zone. At the boundaries of the 
zone the time changes abruptly by one hour. The first zone 
is comprised between longitudes 7| E. and 7 J W r of Greenwich 
and throughout it Greenwich time is used. In successive zones 
east of Greenwich, the times are one, two, three . . . hours fast on 
Greenwich, and in the zones west of Greenwich the times are 
one, two, three . . . hours slow on Greenwich. Thus the same 
time is used over a wide area, but this time never differs by 
more than 30 minutes from local time. Occasionally the zone 
boundaries deviate slightly from the meridians, when simplifica- 
tion results from such deviation : e.g. if on the seaboard of a 
country, a small area only lies in one zone, it is convenient to 
bend the boundary of the adjacent zone so as to include such 
region. Several examples of this may be seen in Fig. 24, 
which shows the boundaries of the time zones at sea and along 
the sea-coast. 

The 180th meridian from Greenwich is called the date line. 
Proceeding eastwards from Greenwich the time in the 12th 
zone will be 12 hours fast on Greenwich, whilst proceeding 
eastwards the 12th zone will be 12 hours slow on Greenwich. 
There is therefore a discontinuity of 24 hours in this zone. 


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The date line runs through the middle of the zone. Between 
172^ E. long, and the date line, the time is 12 hours fast on 
Greenwich. Between 172J W. and the date line, it is 12 hours 
slow on Greenwich. If, at Greenwich, it is midnight on the 
night of, say, August 20-21, the time carried by a ship approach- 
ing at that instant the date line from the west will be noon on 
August 21, but on one approaching it from the opposite direc- 
tion it will be noon on August 20. On crossing the date line the 
date on the former ship will be changed to August 20, one day 
being thus repeated, and that on the latter to August 21, one 
day being thus missed out. Therefore, in going round the 
world eastwards, the number of days occupied on the journey 
will be one more than the number of days reckoned at the 
point of the commencement and finish of the journey, but each 
day on the journey will be less than 24 hours in length. If the 
journey is made, on the other hand, in a westward direction, 
the number of days taken will be one less than the number 
reckoned at the point of commencement, each day, however, 
being longer than 24 hours. This phenomenon was made the 
basis of Jules Verne's story entitled " Around the World in 
Eighty Days," in which the hero started out eastwards on his 
journey, and after the completion of the journey in, as he 
thought over 80 days, he found that he was a day ahead of 
the calendar and that the journey had been completed within 
the prescribed time. He had not, in fact, put his calendar 
back one day when the date line was crossed. 

The system of time zones and the date line are shown in 
Pig. 24. It will be noticed that the date line does not coincide 
throughout its length with the 180th meridian, but that for 
local convenience it is deviated in the neighbourhood of land 
in several places. 

32. Precession of the Equinoxes. When defining sidereal 
time in 28, it was pointed out that the length of the sidereal 
day was measured with reference to a hypothetical star, the 
First Point of Aries or, in other words, the ascending node 
of the ecliptic. The length of clay so determined will not be 
quite the same as that which would be obtained if an actual 
fixed star were to be chosen as the reference body, unless the 
First Point of Aries is fixed relatively to the stars. If the 


ecliptic and equator are fixed in space, the First Point of Aries 
will be fixed relatively to the stars, but if not it should be 
possible to detect a difference in the length of the year when 
observations are made relatively to a star or to the First Point 
of Aries. 

This difference was actually detected by the ancients in the 
following manner. Two methods were used by them to 
determine the length of the year. One method was by the use 
of the gnomon : if, for instance, a vertical pole is set up and 
the length of the shadow observed, it will be noticed that its 
shortest length on any given days occurs at the time of apparent 
noon, the Sun then being at the highest point of its diurnal 
path. If observations be made each day at noon, a gradual 
change in the length of the noonday shadow will be found ; 
the shadow will be shortest at the summer solstice when the 
Sun's decimation is greatest, and longest at the winter solstice 
when the declination is least. By observation of thebe 
phenomena in two successive years the length of year can be 
found, though not with very great accuracy, since the Sun's 
decimation changes so slowly near the solstices. But if the 
observations are continued over a large number of years, this 
inaccuracy will be so much reduced that a very exact value 
of the length of the year will be obtained. Since the equinoxes 
occur midway between the solstices, this value gives also the 
interval between two consecutive passages of the Sun through 
the vernal equinox or First Point of Aries. 

The second method of determining the length of the year 
was by what was termed the heliacal risings of stars. A star 
is said to have a heliacal rising when it rises above the horizon 
exactly at cunrise. At any given place the heliacal rising of 
a bright star near the ecliptic will occur only once a year, on 
the date when the Sun in its passage around the heavens 
passes near that particular star. By successive observations 
of such risings, the length of the year, or the interval between 
two consecutive passages of the Sun past a fixed star, will be 

On determining the length of year by these two methods, 
a slight discordance in the two values was found. The former 
method gave a somewhat shorter year, indicating that the Sun, 
in its apparent motion round the heavens, passes more quickly 


from the vernal equinox along the ecliptic back to the vernal 
equinox than it does from a given star and back to that star. 
The only possible interpretation of this result is that the 
vernal equinox is in motion relatively to the stars and this 
in turn means that either the ecliptic or the equator, or both, 
cannot be exactly fixed. It was discovered by Hipparchus, 
whose result has been confirmed, that the ecliptic is fixed on 
the celestial sphere, but that the equator is not : it has been 
found that, apart from slight variations due to the motions of 
the stars themselves, the distances of the stars from the ecliptic 
do not change, but that their declinations, or distances from the 
equator, show progressive changes. The equator therefore 
moves in such a way that the equinoxes move very slowly 
along the ecliptic, the direction of this motion being opposite 
to that of the Sun's motion. This phenomenon was called 
by Hipparchus the Precession of the equinoxes. In magnitude 
it is not very large, amounting to about 50" per year, though 
this causes a difference of 20 minutes in the lengths of the 
year, as determined by the two methods referred to above. 

33. Physical Cause of Precession. The physical cause 
of the phenomenon of precession is very simple. We have 
seen that the Earth is not quite spherical in shape, but that 
it is flattened at the poles, the polar diameter being less than 
the equatorial diameters. We may regard the Earth as being 
built up of a spherical portion whose diameter is equal to the 
length of the Earth's polar axis and of an outer shell whose 
thickness gradually decreases from the equator to the poles, 
where it vanishes. Since the bulge lies in the equator, it is 
inclined to the ecliptic. 

The gravitational force between the Earth and the Sun 
would, if the Earth were a perfect sphere, merely hold it on its 
orbit. We may investigate the effect of the Earth's ellipticity 
by regarding the pull as being built up of two components, the 
pull on the spherical portion, which will act through the centre 
of the earth, and that on the outer shell. It is the latter which 
has significance in the explanation of precession. In Fig. 25 
is shown a section through the Sun and the axis of the Earth 
at the time of winter solstice. The line to the Sun is the trace 
of the ecliptic, is the centre of the earth ; N, 8 are the ends 


of its axis of rotation, E iy E 2 the ends of an equatorial diameter. 
Then the angle between OE l and the direction from to the 
Sun is equal to the obliquity of the ecliptic. It is evident 
that E! is nearer the Sun than E 2 and that the attraction on 
the NE S portion of the outer shell is greater than that on the 
NE 2 S portion, and there is, therefore, a mechanical couple 
tending to turn the equator of the Earth into the plane of the 
ecliptic. (The attraction on the spherical portion lias no 
turning moment, since it passes through the centre.) If the 
Earth were not rotating, it would be turned in this sense until 
the equator coincided with the ecliptic : but actually it 
possesses turning moments about two axes at right angles, 
viz. about its axis of 
rotation and about an 
axis passing through 
its centre at right 
angles to the axis of 
rotation and in the 
plane of the ecliptic. 
It can bo shown, by ^^^ 

the principles of ele- FlG> 25. To illustrate Precession. 

meiitary rigid dyna- 
mics, that under such circumstances a steady state of 
equilibrium can only be maintained when the axis of rota- 
tion rotates uniformly around the ecliptic in a backward 
direction, the mechanical effect of the change in space of the 
axis of rotation introducing a force couple which just balances 
that which is tending to turn the equator into the ecliptic. 

This is simply illustrated by an ordinary gyrostat. If the 
gyrostat is spinning about its axis and a small weight is 
affixed to one end of the axis, or the finger pressed lightly upon 
it in a vertical direction, that end of the axis is not depressed, 
but the whole gyrostat will commence to rotate about a vertical 
axis, i.e. to precess. 

The effect of the attraction is therefore to cause the pole of 
the equator to move in a small circle about the pole of the 
ecliptic, the radius of the circle being equal to the obliquity of 
the ecliptic, which remains constant. 

The rate of rotation depends upon the relative magnitudes 
of the two force couples. That tending to turn the equator 


into the ecliptic is very small compared with the angular 
momentum couple of the earth's rotation. The rate at which 
the earth precesses is therefore extremely slow. Since the 
precession in one year amounts to 50"-2, a complete rotation 
at this rate would occupy 360/50 // -2 years or 25,800 years. 

As a consequence of this processional motion, it follows that 
Polaris, the Pole Star, which is at present only slightly more 
than 1 distant from the north pole of the equator and which 
serves as such a convenient guide to the position of the pole, 
has not always been near the pole. In the time of Hipparchus 
it was 12 distant from it, and about 13,000 years ago it was 
47 distant. The distance at present is slowly decreasing and 
will continue to decrease to a minimum distance of about 30', 
after which it will again increase. 

Another consequence of precession is that the First Point of 
Aries is now no longer in the constellation of Aries but in that 
of Pisces. 

34. Variation in Precession. The processional motion, 
which was stated in the preceding paragraph to amount to 
50"-2 in one year, is not uniform throughout the year. 
Evidently the magnitude of the couple exerted by the Sun 
tending to turn the equator into the ecliptic must depend 
upon the Sun's declination. Whon the Sun is crossing the 
equator, the couple vanishes ; at the solstice, when the 
declination has its greatest values, the couple is at a maximum. 
In addition, superimposed upon the solar precession, is a 
precession arising from the attraction of the Moon, whose orbit 
is inclined at about 5 to the ecliptic. It is the sum of the two 
precessions which amounts on the average to 50"-2 annually. 
The lunar precession vanishes twice each month, when the 
Moon is crossing the equator. The value of the precession is 
therefore variable throughout the year, being very much 
greater at some periods than at others. 

Even the annual value of the precession is not constant. 
This arises from another phenomenon. The orbit of the Moon 
meets the ecliptic in two points (the nodes) which themselves 
have a westward movement on the ecliptic ; this movement is 
much more rapid than precession, one revolution being com- 
pleted in slightly less than 19 years. At a certain stage in this 


westward motion, tho ascending node of the Moon's orbit will 
be at the vernal equinox and the angle between the Moon's 
orbit and the equator will then be about 28|. After an 
interval of 9J years, the ascending node will have moved 
westwards to the autumnal equinox and the inclination will 
then only be 17|-. The lunar portion of the precession will 
therefore be much greater in the first position than in the 
second, as the couple tending to turn the equator increases 
with the inclination of the equator to the orbit. This is 
connected with the phenomenon of nutation. 

35. Nutation. Precevssion causes gradual changes in both 
the right ascension and declination of a star. The effect on 
declination is most easily seen. If a star has right ascension 
about h., so that the star is near the vernal equinox, it is 
evident that the effect of the westward motion of the vernal 
equinox along the ecliptic is to increase the distance of the 
star from the equator, i.e. to increase its declination. The 
rate of increase is the component perpendicular to the equator 
of the motion of the equinox along the ecliptic. If the star 
has R.A. 12 h., its declination will decrease annually by the 
same amount. For stars, however, whose R.A.'s are 6h. or 
18 h., precession has no effect on their declination since the 
movement of the equator does not in the case of such stars 
either increase or decrease their distance from the equator. 
The declination componentof the precession is, in fact, dependent 
only upon the star's right ascension, being proportional to 
cos a. The effect of precession on right ascension, is somewhat 
more complicated, being dependent upon both the right 
ascension and the declination of the star. 

It was first shown by Bradley that the changes in the declina- 
tions of the stars could not be explained by means of aberration 
(see 36) and precession only. After separating out these 
effects, the careful and long continued observations made by 
Bradley revealed another motion, by which the pole of the 
equator is at some times before and at other times behind the 
position which it should occupy on the hypothesis of a uniform 
precessional motion in a small circle about the pole of the 
ecliptic and, in addition, the distance from this pole is sometimes 
greater and at other times smaller than its mean value. 


Bradley was able to trace the origin of this motion by proving 
that it was periodic in nature with a period of about 19 years. 
This period has already been referred to : it is the period of 
revolution of the nodes of the Moon's orbit around the ecliptic. 
The phenomenon is known as Nutation or " nodding." The 
axis of the Earth, instead of describing a circle, moves in a 
slightly wavy curve, whose mean distance from the pole of the 
ecliptic remains constant. Each wave occurs in a period of 
about 19 years, so that there must be nearly 1,400 waves in the 
entire circumference. 

The phenomenon of nutation is due physically to the exist- 
ence of a small component of 
the attractive force with which 
the Moon tends either to 
accelerate or to retard the 
processional motion. 

A general idea of the nature 
of the orbit of the pole of the 
equator around the pole of the 
ecliptic may be obtained from 
Fig. 26. In this figure, the size 
of the nutations are relatively 
much too great and their nuni- 
FIG. 20. Nutation. ber too small. 

36. Aberration. Another phenomenon connected with the 
orbital motion of the Earth which produces changes in the right 
ascensions and declinations of stars is that known as Aberration, 
which, together with nutation, was discovered by Bradley 
in the eighteenth century. 

When Bradley commenced his investigations, he was endea- 
vouring to determine the distances of stars, which at that time 
were believed to be much nearer than we now know them to be. 
Bradley argued that if a star is at not too great a distance it 
should show an appreciable annual motion in the sky, and the 
nearer the star, the larger would be this motion. In fact, if a 
cone is formed with its vertex at the star and the orbit of the 
Earth round the Sun as its base, the apparent motion of the 
star duo to its finite distance would be given by the section of 
this ccne by the celestial sphere (whose radius is infinitely 


great). The star should therefore appear to describe in the 
sky an ellipse of small angular diameter, the shape of this 
ellipse being similar to the projection of the Earth's orbit on 
the tangent plane to the celestial sphere at the point where the 
radius to the star meets it. The major axis of this ellipse will 
be parallel to the ecliptic and the star at any time will be at 
that point in the orbit which is opposite to the Earth. 

As a suitable star to observe Bradley chose y Draconis, which 
passed very near the zenith of his observatory and whose 
position was therefore practically unaffected by refraction. A 
special zenith telescope, now preserved at the Royal Obser- 
vatory, Greenwich, was used. By careful observation, Bradley 
found an annual displacement of the star which was of the type 
which he had anticipated, except that the position of the star in 
its small orbit was only 90 from the position of the Earth in 
her orbit, instead of being opposite to it. This led Bradley to 
the discovery of the phenomenon of aberration. 

Aberration is the apparent displacement of a star, arising 
from the fact that the velo- 
city of light is not infinite 
compared with the orbital 
velocity of the earth. It 
can be simply explained by 
the parallelogram of forces. 
The simplest illustration 
is to imagine a train 
moving with uniform velo- 
city whilst a shower of rain 
is falling vertically. To an 
observer in the train, fac- 
ing the direction of motion, A 
the rain drops appear to be FIG. ^7. A Lei ration, 
falling*, slantwise towards 

him, since their velocity relative to him is a combination of 
their velocity and the velocity of the train. If light consisted 
of material corpuscles, it is clear that in a precisely similar 
manner the apparent direction of the light coming towards us 
from a star is the direction of the resultant velocity of the light 
corpuscles and the earth. A similar result holds on the wave 
theory of light, although then it is not so self-evident. 


If then, in Fig. 27, OP is the direction of motion of the Earth 
around the Sun, OS the direction to a star, and if OA, OB 
represent in magnitude the velocities of the Earth and of light, 
the apparent direction of the star is OC, the diagonal of the 
parallelogram OACB. 

If 6 is the small angle SO Si or OCA which measures the 
displacement, and a is the angle POSi, between the direction of 
the Earth's motion and the apparent direction to the star, and 
if c and v are respectively the velocities of light and of the 
Earth, the triangle OCA gives 

Sin 6 = sin a. 

The angle, a, between the direction to the star and that of the 
Earth's motion, is called the Earth's way. Also since is 
small, we may replace sin 6 by 0, and the displacement is : 

V 1) 

= Sin a = sin (Earth's way). 
c c 

It has thus been proved that all stars are at any instant 
apparently displaced towards that point of the heavens to 
which the Earth is at that instant moving. The amount of the 
displacement depends upon the angle between the Earth's lino 
of motion and the direction to the star, being proportional to its 

sine. The constant ( J of the proportion is called the con- 
stant of aberration and has a value of 20"47. 

Now the direction of motion of the Earth is along the tangent 
to its orbit, which is in the plane of the ecliptic. Viewed from 
the Earth, this point on the celestial sphere is the point on the 
ecliptic which is 90 behind the Sun. The aberration at 
displacement of a star is therefore always towards the point 
on the ecliptic which is 90 behind the Sun. At the vernal 
equinox, the displacement is towards the winter solstitial point 
and so on. 

In the case of a star situated at the pole of the ecliptic, the 
magnitude of the aberrational displacement is constant through- 
out the year, since the direction of the Earth's motion and the 
direction from the Earth to the star are always at right angles to 
one another. In the case of any other star, these directions are 
only at right angles at the two points on the orbit where a plane 


through the star perpendicular to the ecliptic cuts the orbit. 
The displacement of the star is then parallel to the ecliptic and 
of amount 20"-47. At the two points on the orbit midway 
between these points, the direction of motion is parallel to the 
orbit and the displacement is along a great circle perpendicular 
to the ecliptic and of amount 20"-47 sin S, where S is the star's 
latitude or distance from the ecliptic. 

In general, the aberrational displacement is such that the 
star appears to move in an ellipse, whose major axis is parallel 
to the ecliptic ; this ellipse is similar to the projection of the 
Earth's orbit on to the tangent plane to the celestial sphere at 
the star and is therefore similar to the ellipse which would be 
described on account of parallax if the star were relatively near. 
At any given time, however, the displacements of the same star 
due to parallax and to aberration are in directions which differ 
by 90. There is the further difference that the magnitude 
of the parallactic displacement is dependent upon the distance 
of the star, but that of the aberrational displacement is not. 

After Bradley had discovered aberration, he found certain 
other residual phenomena which required explanation. His 
observations showed that, after allowing for the effects of 
aberration and precession, the north polar distance of y Draconis 
gradually decreased during the years 1728, 1729, 1730, 1731. 
Amongst other stars observed by Bradley was one in Camelopar- 
dalus with opposite right ascension to y Draconis. Bradley 
found that, after allowing for the effects of aberration and 
precession in the case of this star, there was an equal and 
contrary change in north polar distance. This indicated a 
movement of the Earth's axis away from the one star and 
towards the other. By continuing his observations for many 
years more, Bradley was able to connect this " nutation " with 
the motion of the Moon, as already described, so completing an 
investigation which was a masterpiece of careful and patient 

37. Brief reference may be made to the effect of the attrac- 
tion of the planets on precession and that of the rotation of the 
Earth on aberration. 

Careful investigation shows that the distances of the stars 
from the ecliptic (i.e. their latitudes) are not absolutely constant 


but show very slight changes. This indicates a slight motion of 
the ecliptic itself. The phenomenon arises from the attraction 
of the planets on the Earth, the effect of which is slightly to 
disturb its path. A gradual change in the obliquity of the 
ecliptic is thus produced : this is very small, amounting to a 
decrease of only about half a second per year. The change is 
oscillatory but with a very long period many thousands of 
years. The attraction of the planets produces other slight 
disturbances in the Earth's orbit, the principal being a small 
change in the eccentricity, which is at present slowly decreasing, 
and a slow eastward revolution of the apses of the orbit (the 
points nearest to and farthest from the Sun). 

In addition to the aberrational displacement referred to in 
the previous section, there is a further slight effect caused by 
the motion of the observer which has its origin in the Earth's 
rotation. This effect is known as diurnal aberration. The 
constant in the aberration formula is about 0"-3 cos l } I being 
the observer's latitude : it is thus greatest at the equator. 
The effect on the position of a star is greatest when the star 
crosses the meridian, and it then produces an increase in its 
apparent right ascension of amount 0"-3 cos I sec d, d being the 
declination of the star. 

38. The Calendar. A year being the period of revolution 
of the Earth about the Sun relative to a certain body of refer- 
ence, the length of the year will vary according to the reference 
body chosen. 

The natural unit marked out for the use of man is the period 
of revolution relative to the First Point of Aries, since this 
period determines the commencement of the seasons and all 
associated phenomena and ensures that these always take 
place at about the same date in each year. The year so 
defined is called the Tropical Year. It has been found by 
observation to consist of 365-242216 mean solar days. This is 
the year whose length was determined by the ancients by the 
use of the gnomon. 

The period determined when the starting-point is a point 
fixed amongst the stars is called the Sidereal Year. This is the 
year determined by the heliacal risings of the stars. We have 
already seen that it is longer than the tropical year, owing to the 


retrograde motion of y, of 50"- 22 annually. In fact the tropical 
year : 360 - 50"-22 = sidereal year : 360. 

A third year is obtained by taking as starting-point the 
perihelion of the Earth's orbit. In the previous paragraph, it 
Was mentioned that the axis of the Earth's orbit has a slow 
eastward revolution ; this amounts to ll"-25 annually. The 
anomalistic year, as this period is called, is therefore longer than 
the sidereal year, since the Earth has to move through an 
additional ll"-25 before completing one revolution relative to 
perihelion. Thus sidereal year : 360 = anomalistic year : 
360 + H"-25. 

The lengths of the three years are : 

Tropical year 365-242216 days 

Sidereal year = 365-256374 

Anomalistic year = 365-259544 

In addition to these three kinds of year, there is the Civil 
Year, which consists of an exact number of days. It is, 
however, based upon the tropical year in the manner now to be 

There are three natural units of time marked out by nature 
for the use of man, viz. the apparent solar day, the lunar 
month and the tropical year. The first of these may be replaced 
by the mean solar day, which it is necessary to introduce on 
account of the inequality in the length of the apparent solar day. 
The three periods are not commensurable amongst themselves. 
For civil purposes, it is necessary that the civil year should 
contain an exact number of days, as otherwise a portion of one 
day would fall in one year and the rest of it in the succeeding 
year, with obvious inconveniences. 

It becomes necessary therefore to devise a calendar in which 
the civil year shall contain an integral number of days but such 
that the average length of the year shall be very nearly equal to 
the Jropical year. Early calendars did not conform to this 
condition, being based mainly upon the lunar year of twelve 
lunar months. Such a calendar is still in use by the Moham- 
medans, but it suffers from the great disadvantage that the 
seasons fall in different months from year to year, the length 
of the year being only about 354| days. The Roman calendar 
was of a similar nature, but in order to keep the seasons correct, 
days or months were arbitrarily inserted. In order to avoid the 


resulting and inconvenient confusion, the aid of an Alexandrian 
astronomer, Sosigenes, was called in by Jiilius Csesar and it was 
to him that the ingenious suggestion of leap year is due. The 
so-called Julian Calendar was the result : three years of 365 
days were to be followed by one year of 366 days, giving a mean 
length for the civil year of 365-25 days. This is -007784 days 
longer than the tropical year, a difference which only amounts 
in 400 years to somewhat over 3 days. The Julian calendar 
was introduced in the year 45 B.C. At the same time, the epoch 
of the commencement of the year was changed* It had 
previously commenced in March, but the date was now altered 
to January 1 the day of the new Moon following the winter 
solstice, 45 B.C. The year preceding the change was made 
unduly long and is known as the year of confusion. 

The Julian calendar being in error by 3 days in 400 years, the 
error gradually accumulated in the course of centuries. The 
next step towards improvement was made by Pope Gregory 
XIII, on the advice of the Jesuit astronomer, Clavius, with a 
view to bringing the date of Easter nearer to the vernal equinox, 
since the date of Easter was gradually tending to come more 
and more into the summer. The Gregorian Calendar modified 
the Julian calendar by omitting certain leap years : all century 
years are excluded unless their date number is divisible by 400. 
Thus the year 1900 was not a leap year, but the year 2000 will 
be. The effect of this modification is to shorten the average 
length of year : in 400 years there will only be 97 leap years 
instead of 100, so shortening this period by 3 days, and thus 
practically accounting for the error of the Julian calendar. 
The average length of the civil year of the Gregorian calendar is 
365-2425 days : this makes the average civil year too long by 
0-000284 days, so that the amount of error is only 1 day in 
about 4,000 years. In the year 1582, when the change was 
adopted by Roman Catholic nations, the day following October 4 
was called October 15, in order to adjust for the accumulated 
error. The Gregorian calendar was not adopted in England 
until the year 1752, when the difference between the two calen- 
dars had increased to 11 days. The day following September 
2, 1752, was called September 14 ; at the same time, the begin- 
ning of the year was changed from March 25 to January 1. 

In Russia the old style was adhered to until after the 


Revolution and the difference was then 13 days. It had 
become customary before the change for both dates to be used 
for commercial and scientific purposes. 

39. The Reform of the Calendar. The division of the 
year into twelve unequal months is purely arbitrary and in this 
respect our present calendar suffers from many inconveniences. 
The first of January, or any other given date, occurs one day 
later in the week in any given year than in the preceding year, 
except in the case of leap year, when dates after February 29 
occur two days in the week later. Also the quarters of the year 
are of unequal length. In order to avoid these and other 
similar disadvantages, various schemes for the reform of the 
calendar have from time to time been put forward. Of these 
the following scheme, first proposed by Armelin in 1887, is 
probably the simplest and has most to recommend it : 

The year is formed of four equal quarters with the addition 
of one or two supplementary days, according to whether it is 
an ordinary or a leap year. Each quarter consists of two 
months of 30 days each, followed by a third month of 31 days, 
there being therefore exactly 13 weeks in each quarter. The 
nominal year of 365 days consists of four identical three-monthly 
periods of 91 days, the first two periods being separated from 
the last two by an intermediary day, which is undated and is 
placed outside the week. It is proposed to call this day 
Peace Day. In leap years a second supplementary day is added 
at the end of the year and called Leap Day. A simple perpetual 
calendar is thus obtained : 

(1st quarter .... January. February. March. 

\2nd April. May. June. 

Peace Day. 

(3rd quarter .... July. August. September. 

\4tji October. November. December. 

Leap Day (in leap years only). 

/Monday . I 8 15 22~29 6 13 20 27 4 11 18 25 

Tuesday . 2 9 16 23 30 7 14 21 28 5 12 19 26 

For Wednesday 3 10 17 24 18 15 22 29 6 13 20 27 

each -^Thursday 4 11 18 25 29 16 23 30 7 14 21 28 

quarter Friday . 5 12 19 26 3 10 17 24 1 8 15 22 29 

Saturday. 6 13 20 27 4 11 18 25 29 16 23 30 

\Sunday . 7 14 21 28 5 12 19 26 3 10 17 24 31 


Arranged in this way, the last day of each quarter is a Sunday, 
and this is the most suitable day to be followed by Peace Day 
and Leap Day which might conveniently be taken as Bank 
Holidays. The adoption of such a calendar would provide a 
suitable opportunity for fixing the dates of the movable religious 
festivals. The whole calendar could easily be carried in the 
memory. The principal objection advanced against the 
scheme is that the insertion of the supplementary days breaks 
the continuity of the week. The objection is not of very much 
weight, and unless the continuity of the week is broken, it is not 
possible to make the same dates always correspond to the same 
days of the week. 

40. The Julian Date. The Julian Date is a system of 
reckoning extensively employed in astronomical calculations 
for the purpose of harmonizing the various systems of chrono- 
logical reckoning. The system was originally put forward in 
1582 by Scaligcr. The Julian Period consists of 7,980 Julian 
years of exactly 365 J days : the starting-point or " Epoch " is 
4713 B.C. January 1. The date of any phenomenon can be 
expressed without any ambiguity by the number of days which 
have elapsed since the Julian epoch and the interval in days 
between any two events can therefore be at once found when 
their Julian dates are known. The system is particularly 
convenient for expressing by a formula the dates of maxima 
and minima of a variable star. In the Nautical Almanac 
for any year is given the Julian year and day corre- 
sponding to January 1 for each year of the Christian Era. 
Thus : 

At mean noon, 1920, Jan. 1, there have elapsed 2,422,325 

Julian days. 
At mean noon, 1921, Jan. 1, there have elapsed 2,422,691 

Julian days. 
At mean noon, 1922, Jan. 1, there have elapsed 2,423,056 

Julian days. 

41. The Me tonic Cycle. In connection with the calen- 
dar, reference may be made to the Lunar Cycle of Melon, 
discovered by him about 433 B.C. and still used in fixing the 


dates of the movable religious festivals. The rule gives a 
simple relationship between the length of the lunar month and 
the tropical year and was used by the Greeks to predict the 
days on which their religious festivals, dependent on the phases 
of the Moon, should be celebrated. Meton found that after a 
lapse of 19 years, the phases of the Moon recurred on the same 
days of the same months. In fact, 19 tropical years, of 
365-24222 days, equal 6939-602 days, whilst 235* synodic 
months (i.e. from new Moon to new Moon), of 29-53059 days, 
equal 6939-689 days. Hence, after 19 years, the mean phases of 
the Moon recur on the same days within about 2 hours. If the 
dates of full Moon are recorded during one cycle, they are 
therefore known for the following cycle. These dates were 
inscribed in letters of gold upon the public monuments and, for 
this reason, the number of a year in the Mctonic cycle is called 
the Golden Number. The first year of a cycle may, of course, 
be chosen arbitrarily. The year 1 B.C. commences the cycle 
now in use and hence to find the golden number of a year, add 1 
to the date number and divide by 19, then the remainder is the 
golden number : if the remainder is 0, the golden number is 
taken as 19. 

Several rules have been put forward from time to time for the 
calculation of the date upon which Easter Day will fall in any 
year Easter Day being the first Sunday after the full Moon 
which happens upon or next after the vernal equinox. Most of 
these rules are subject to various exceptions, but the following, 
first devised in 1876, is subject to no exceptions. The rule is as 
follows : 

Divide By Quotient Remainder 

The year x . . . .19 - i 

,,,,,,. . . .100 b c 

r b . . . . . .4 d c 

b + 8 25 / 

6 -/ + ! .... 3 (/ 

19a + 6 - d - g + 15 . . 30 - h 

c. ..... 4 i k 

32 + 2e + 2i - h - k . . 1 - I 

a + llfi +221 . .451 m 

h +1 - 1m + 114 . .31 n o 


Then n is the month of the year and o + I the number of the 
day of the month on which Easter falls. 

E.g., to find the date of Easter Day in 1922, we have 
a =3 6-1& c =22 d = 4 

6=3 / = i g = 6 A = 21 

i = 5 fc =- 2 Z = 4 m = 

^ == 4 o = 15 

Therefore Easter Day occurs, in 1922, on the 16th day of the 
4th month, i.e. on April 16. 

For a demonstration of the rule, reference may be made to 
Butchers' Ecclesiastical Calendar, p. 226. 


41. After the Sun, the Moon is to us the most important of 
the heavenly bodies. Her tide-raising force is of vital import- 
ance to mankind and her silvery light is always welcome at 
night. She is much the nearest of our celestial neighbours and 
therefore assumes an importance which would not otherwise 
be warranted by her size. Thus the Moon has been intimately 
associated with the progress and development of astronomy. 
Her motion round the Earth provided Newton with an approxi- 
mate verification of his law of gravitation ; the detailed study 
of her motion has served to vindicate that law to a very high 
degree of accuracy and has occupied the best parts of the lives 
of several famous astronomers. The study of eclipses and of 
the tides have each raised many new problems and developed 
into important branches of astronomy, whilst theories of the 
formation of the Earth-Moon system are closely related to 
general theories of cosmogony. 

42. Apparent Motion of the Moon. The phenomena 
connected with the apparent motion of the Moon are much 
more easily observed than are the corresponding phenomena 
in the case of the Sun : not only is the apparent motion of the 
Moon much more rapid, but also the background of bright 
stars is easily visible, relative to which the motion may be 
observed. When the Moon is in the neighbourhood of a bright 
star or planet, her eastward motion amongst the stars can be 
seen during the course of a single night. It is also evidenced 
by the large retardation in the time of rising of the Moon from 
night to night. 

Owing to the eastward motion of the Moon amongst the 
stars being much more rapid than that of the Sun, the Moon is 



continually overtaking and passing the Sun. In fact, whilst 
the average daily angular motion of the Sun relative to the 
stars is only about 1, that of the Moon is about 13. When 
the Moon overtakes the Sun it is said to be in conjunction. This 
occurs when the longitudes of the Sun and Moon are equal. 
When their longitudes differ by 180, the Sun and Moon are 
said to be in opposition. Both at conjunction and at opposition, 
the Sun, Earth and Moon are practically in one straight line, 
but whereas at conjunction the Moon is between the Earth and 
the Sun, at opposition the Earth is between the Moon and the 
Sun. When the longitudes differ by 90, the Sun and Moon are 
said to be in quadrature. 

The sidereal revolution of the Moon is the period occupied 
by the Moon in passing from a given star back again to the 
same star. Its average length is about 27 d. 7 h. 43 m. 11-6 s., 
or 27-32166 days, but varies from revolution to revolution on 
account of the various perturbing forces which may increase 
or decrease the interval by several hours. 

The period naturally associated with the Moon, however, is 
its period of revolution with regard to the Sun, since it is this 
period which controls the phases. A lunar month may be 
defined as the period from new Moon to new Moon, or from 
full Moon to full Moon, i.e. from conjunction to conjunction, 
or from opposition to opposition. This period is also known 
as the Synodic revolution. It is longer than the sidereal period, 
on account of the eastward motion of the Sun amongst the 
stars which must be overtaken. Its mean length is 29 d. 12 h. 
44 m. 2-87 s. or 29-53059 days. Its actual length may vary 
considerably from this mean value (the total variation is about 
13 h.) on account of the eccentricities and perturbations of the 
orbits of the Moon around the Earth and of the Earth around 
the Sun. 

Another period which may be mentioned is the tropical period, 
i.e. the period of revolution relative to the First Point of Aries. 
Owing to the slow retrograde motion of T, this period is very 
slightly shorter than the sidereal period, the actual difference 
being about 6-85 seconds. The tropical period is 27-32158 days. 

The sidereal and synodic periods of revolution are connected 
with the length of the sidereal year. The mean daily motion 
of the Moon relative to the Sun is equal to the difference 



between its mean daily motion relative to the fixed stars and 
the mean daily motion of the Sun relative to the fixed stars. 
Since the daily motion is inversely proportional to the period 
of a complete revolution it follows that, if all the periods are 
expressed in days, 

1 _ 1 _ 1 

sidereal revolution synodic revolution sidereal year* 

If the apparent diameter of the Moon is measured at different 
times, it will be found to vary only within narrow limits. The 
distance of the Moon from the Earth is therefore approximately 
constant ; measurement has shown that the mean distance is 
about 238,000 miles or about sixty times the Earth's radius. 
The Moon is therefore a companion of the Earth in its annual 
motion around the Sun. 

43. Phases of the Moon. The most striking phenomenon 
connected with the Moon is its waxing and waning, i.e. the 
variation of its visi- 
ble outline, to which 
we give the name of 
phases. The explana- 
tion of these phases 
is very simple. The 
Moon is not self- 
luminous like the 
Sun, but owing to 
the high reflecting 
power of its surface 
it is able to reflect 

back Some of the FIG. 28. Explanation of Phases of Moon. 

light from the Sun 

which falls upon it and it is by means of the light reflected 
from that portion of the surface which is illuminated by 
the Sun that the Moon becomes visible to us. In general, 
only a portion of this illuminated surface is visible from the 
Earth and to the variation of the amount visible with change 
of the relative positions of the Earth, Moon and Sun is due 
the phenomenon of the phases. 

In Fig. 28, suppose ACBD represents a section of the Moon 
in the plane containing the Earth and the Sun, is the centre 



of the Moon and OS, OE are the directions towards the Sun 
and Earth respectively at any time. Then the hemisphere 
whose trace by the plane of the paper is ACB, AB being per- 
pendicular to OS, is illuminated by the Sun, whilst the hemi- 
sphere CBD, CD being perpendicular to the direction OE, faces 
the Earth. The only portion of the lunar surface therefore 
visible to the Earth is a lune, symmetrical about the plane 
SOE, whose trace is CB. What then is the shape of the portion 
of the Moon's surface actually visible under these circum- 
stances ? Referring to Fig. 29, PCQD represents the 
hemispherical portion of the Moon's surface facing the Earth, 

O is the Moon's centre and 
the great circle PBQ forms 
the boundary between the 
parts of the surf ace which are 
illuminated by the Sun and 
the parts whcih are not illu- 
minated. This bounding circle 
will appear in projection on 
the plane PCQD as a semi- 
ellipse PLQ, in which L is the 
foot of the perpendicular BL 
drawn from B on to CD. 
The semi-axes of this ellipse 
are OP, the Moon's radius and 
OL. The latter is equal to 
OB cos BOD or to OB cos SOE (Fig. 28), i.e. to the radius of 
the Moon multiplied by the cosine of the angle subtended by 
the Sun and Earth at the centre of the Moon. The semicircle 
PCQ forms the other bounding surface of the illuminated 
portion of the surface visible from the Earth. The apparent 
outline of the Moon is therefore formed of a semicircle and a 
semi-ellipse, the semicircle portion being the boundary facing 
the Sun : the common diameter of the semi-ellipse and the 
semicircle is perpendicular to the plane containing the Sun, 
Earth and Moon. 

A portion of a sphere, such as PGQBP, intercepted by two 
great circles, is called a lune, the angle of the lune being COB. 
This angle is 180 BOD and is the angle between the direc- 
tions from the Earth to the Moon and Sun respectively. The 

FIG. 29.- 

-Shape of Visible Lunar 


visible portion of the Moon's surface is therefore a lune whoso 
angle is equal to the angle subtended by the Sun and Moon at 
the Earth. 

When the Earth is between the Sun and the Moon the angle 
BOD becomes zero. In that case the whole of the illuminated 
surface is visible and the Moon appears as a complete circle. 
This is called full Moon and occurs therefore when the Sun 
and Moon are in opposition. After opposition, as the angle 
SOE gradually increases, the illuminated visible portion of the 
Moon's surface correspondingly decreases. After an interval 
of one quarter of the lunar month, the angle SOE will be a 
right angle and then the minor axis of the elliptical portion of 
the boundary, PLQ, vanishes and the ellipse becomes a straight 
line POQ. At quadrature, therefore, the Moon appears as an 
illuminated semicircle. This is called the last quarter. The 
illuminated area continues to decrease until at conjunction, 
when the Moon is between the Sun and the Earth, only the 
unilluminated hemisphere faces the Earth and the Moon 
becomes invisible. This is called new Moon and occurs at an 
interval of half a lunar month after full Moon. After new 
Moon the bright area gradually increases again, becomes a 
semicircle at the next quadrature, when it is called first quarter, 
and a complete circle at the next full Moon. 

Age in days 2 7J 11 14fr 18 21J 27 

FIQ. 30. Successive Phases of Moon. 

The successive appearances of the Moon from new Moon to 
new Moon are shown in Fig. 30. 

The age of the Moon is the time, expressed in days, that has 
elapsed since the previous new Moon. Plate 1 (a) shows the 
Moon at the age of 12 days. 

The elliptical portion of the boundary PLQ is called the 
terminator. Owing to the mountainous nature of the Moon's 
surface, and to the gradual shading off from light to dark, it 
does not appear as a sharply-defined semi-ellipse. In Plate I (a) 


the terminator is the right-hand limb : the gradual tran- 
sition from light to darkness is well shown. The points at P 
and Q are called the cusps. The line joining the cusps is per- 
pendicular to the plane passing through the observer and the 
centres of the Sun and Moon. The angle between this plane 
and the observer's horizon is very variable, so that, for a given 
age of the Moon, the lino joining the cusps will be inclined at 
different angles to the horizon in different months, at some 
times being nearly vertical and at others nearly horizontal. 
Shortly after new Moon, the angular distance between the 
Sun and Moon as seen from the Earth is small, the Moon being 
slightly east of the Sun, owing to its more rapid easterly motion 
amongst the stars. When the age of the Moon is small, it 
therefore sets soon after sunset. During the first half of the 
lunar month, the Moon sets later each night, but always crosses 
the meridian between noon and midnight. The western limb 
is then the bright semicircular limb. At full Moon, the Moon 
is diametrically opposite to the Sun in the heavens and there- 
fore crosses the meridian near midnight. After full Moon it 
crosses the meridian after midnight, but before the next noon 
and the eastern limb becomes the bright limb. It still con- 
tinues to set later each night and to rise later until shortly 
before the next new Moon, the rising occurs only a little before 
the rising of the Sun. 

44. To an observer on the Moon, the Earth would present 
phases which are the exact counterpart of the phases of the 
Moon as observed from the Earth. This will be evident from 
Fig. 28. Thus " new earth " will occur to the lunar observer 
at the time of " full moon " and " full earth " at the time of 
" new moon," while the first and last quarters of the Moon and 
Earth will occur together. The phenomenon of " Earth- 
shine," or as it is popularly called " the old Moon in the arms of 
the new," is connected with the phenomenon of these com- 
plementary phases. When the Moon is very young and the 
slender bright crescent is visible, the remainder of the portion 
of the Moon facing the Earth is often seen faintly illuminated. 
This faint illumination is due to reflected sunshine from the 
Earth's surface falling on the Moon, it being then " full earth " 
to a lunar observer. 


45. Orbit of the Moon. The motion of the Moon relative 
to the Earth is much more complicated than that of the Sun 
and its detailed consideration would be far outside the range 
of this book. It will therefore only be possible to sketch the 
principal phenomena connected with it. 

A first approximation to the motion can be derived, as in 
the case of the Sun, by measuring the apparent diameter of 
the Moon from day to day. The diameter will be found to 
vary in a manner which is approximately consistent with the 
hypothesis that the Moon moves around the Earth in an 
elliptical orbit, of which the Earth occupies one of the foci. 
When the Moon is at its least distance from the Earth it is 
said to be in perigee, and when at its greatest distance, it is 
said to be in apogee. The eccentricity of the orbit is much 
greater than that of the Earth's orbit, being 0-0549 or roughly 
jj L . This eccentricity is sufficiently large for the non-uniformity 
of the motion to be easily observable and it was a phenomenon 
well-known to the ancients. If we imagine a mean Moon to 
move in the Moon's orbit, starting with the true Moon at 
perigee and completing one revolution in the same time, then 
7 days after passing perigee, the true Moon will be about 6 17' 
in front of the mean Moon ; this difference will gradually 
decrease until apogee is reached by the two bodies at the same 
instant, after which the true Moon lags behind the mean Moon, 
the difference again reaching a maximum of about 6 17' at 
about 7 days before the next perigee passage. This inequality 
in the motion, arising from the eccentricity of the Moon's orbit, 
is called the equation of the centre. It is analogous to and may 
be compared with that component of the equation of time 
which is due to the eccentricity of the Earth's orbit ( 29). 

The plane of the Moon's orbit is inclined to the ecliptic at 
an angle of about 5 8' 43". The two points in which the orbit 
cuts the ecliptic are called the nodes, and that node at which 
the moon passes from the south to the north of the ecliptic is 
called the ascending node, the other node, at which the Moon 
passes to the south of the ecliptic, being called the descending 
node. The plane of the orbit is not fixed in space, a fact which 
has been known from very early times. This may be made 
evident in the following way : the position of the Moon at any 
instant can easily be fixed relatively to neighbouring bright 



stars and the passage of the Moon across the ecliptic can there- 
fore be determined, as the line of the ecliptic through the con- 
stellations is marked on any good star map. If the Moon 
crosses the ecliptic in the sign of, say, Gemini at a certain time, 
it will be found to be crossing it about 18 months later in the 
sign of Taurus, i.e. the node retrogrades through one sign, or 
about 30 in longitude, in a period of about 18 months. A 
complete revolution of the nodes in a retrograde direction 
relative to the fixed stars is completed in 6793-5 days, or 
approximately 18f years. The period of revolution relative 
to the First Point of Aries is somewhat greater, as the equinoxes 
themselves are also in retrograde motion, though at a much 
slower rate, and this motion has to be caught up. The Moon 
can therefore be regarded as moving in a plane which is mean- 
while retrograding so as to complete one revolution around 
the ecliptic in about 18f years. 

The inclination of this plane to the ecliptic is not absolutely 
constant, although its variation is slight. The inclination can 
be measured, after the position of the nodes has been found, by 
determining the Moon's latitude when it is 90 from the nodes, 
i.e. its maximum latitude. It is thus found that the inclination 
oscillates with a period of about 173 days and a total amplitude 
of about 18'. 

There is also a slight inequality in the rate of retrogression 
of the Moon's nodes ; when the Sun is in the nodes or 90 from 
them, the rate of retrogression has its mean value, but when 
the Sun is 45 from a node, the rate has its maximum or 
minimum value, the greatest inequality between the mean 
movement of the nodes and the true movement being 1 40'. 

These inequalities in the inclination and the rate of retro- 
gression were discovered by Tycho Brahe, who showed that 
they could be explained by supposing that the pole of the lunar 
orbit moves uniformly on a small circle of radius about 9' in a 
period of 173 days, whilst the centre of this circle moves in a 
small circle of radius 5 9', with its centre at the pole of the 
ecliptic, in a period of about 18f years. 

46. In the preceding section, the movement of the plane of 
the Moon's orbit has been discussed and also the equation in the 
motion due to the eccentricity of the orbit itself. There are 


other inequalities connected with the orbit of which mention 
must be made. 

If the position of perigee be determined, by noting when the 
apparent diameter of the Moon is greatest, it will be found 
that the position of the perigee is not fixed relatively to the 
fixed stars, but has a direct motion of about 401" per day. 
Relatively to the fixed stars one revolution is completed in 
about 3,232 days, 11 hours, 14 minutes, or about 8 years, 311 
days : relatively to the First Point of Aries a complete revo- 
lution occurs in the somewhat shorter period of 3,231 days, 
8 hours, 35 minutes, as the equinox moves backwards and 
meets the perigee before the latter has completed a sidereal 

Both the rate of motion of the perigee and the value of the 
eccentricity are variable, their variations being connected and 
having the same period, viz. half the time between two con- 
secutive passages of the Sun through the perigee. The latter 
period is 412 days, so that the period of variation of the eccen- 
tricity is 206 days. The eccentricity is greatest when the Sun 
is in the line of apses of the lunar orbit (i.e. in the line joining 
perigee and apogee), and the motion of the perigee then has its 
mean value. The maximum inequality in the longitude of the 
perigee is dz 12 20', whilst the eccentricity varies between the 
limits 0-0549 0'0117. 

47. The Evection. It was stated in 45 that the equation 
of the centre, or the maximum distance between the true Moon 
and a mean Moon moving in the orbit in the same period, 
amounts to 6 17', the equation being due to the eccentricity 
of the Moon's orbit. But owing to the variation of the eccen- 
tricity of the orbit, to which reference has been made in the 
preceding section, there will be a corresponding variation in the 
equation of the centre between the limits 5 3' and 7 31'. It is 
customary to represent this variation by an inequality which 
is called the Evection, and the difference in distance between the 
true Moon and the mean Moon is then obtained by adding to 
the mean equation of the centre (6 17') a variable term repre- 
senting the evection. The value of the evection at any instant 
depends upon the distance of the Moon from perigee and also 
upon the distance between the Moon and the Sun, and it involves 


not only the variation in the eccentricity to which reference 
has been made, but also the related variation in the longitude 
of the perigee. The combined effect can be represented by the 
expression 76' sin (2E 0), in which E represents the mean 
distance between the Sun and Moon at any instant, i.e. the 
elongation, and is the angular distance from perigee of the 
mean Moon. The maximum value of the evection is therefore 
1 16'. The first three terms of the series for the distance of 
the true Moon from perigee are 

+ (6 17') sin + (1 16') sin (2E - 0), 

the first term giving the position of the mean Moon, the second 
term the correction to its position on account of the equation 
of the centre and the third term the correction on account of 
the evection. 

Before determining the period of the evection, reference may 
be made to two further lunar periods of revolution, additional 
to those mentioned in 42. 

The anomalistic period of revolution is the interval between 
two consecutive passages of the Moon through perigee. Owing 
to the forward movement of the perigee, this period is longer 
than the sidereal period and is equal to 27 d. 13 h. 18m. 33s. 
= 27-55455 days. 

The Draconic period of revolution is the interval between two 
consecutive passages of the Moon through one of its nodes. 
Owing to the rapid retrograde motion of the node this period 
is the shortest of the various periods of revolution associated 
with the Moon and is equal to 27 d. 5 h. 5 m. 36 s. = 27-21222 

The period of the evection depends upon 0, whose period is 
the anomalistic period (27 -554 days) and upon E, whose period 
is the synodic period (29-531 days). The angular change of 

^ / 2 1 \ 

(2E 0) in unit time is therefore 2n ( ), 

V 29-531 27-5547 

and this must equal , T being the period of the evection. 

This period is therefore found to be 31-81 days. 

It may be noticed that when the Moon is in perigee or apogee 
(0 = or 180), the evection vanishes if the Sun is either in 


conjunction, opposition or quadrature (E = 0, 90, 180, 

48. The Annual Equation .The Moon is held in its orbit 
around the Earth by the force of their mutual attraction. 
Both bodies are also attracted by the Sun and this attraction 
has to be taken into account in determining the motion of the 
Moon. Now if the Sun attracted both the Earth and the 
Moon with the same force acting in parallel directions, its 
attraction could obviously be disregarded in so far as the 
relative motion of the Moon and Earth is concerned. In effect 
it is only the difference of the attractions that requires to be 
considered. Now when the Sun and Moon are in conjunction, 
the intensity of the Sun's attraction at the distance of the Moon 
is greater than at the distance of the Earth, owing to the Moon 
being then nearer to the Sun than is the earth. The residual 
effect of the solar attraction on the Earth-Moon system at 
conjunction is therefore a force tending to draw the Moon 
away from the Earth. At opposition, on the other hand, 
the attraction is greater at the Earth's distance and is equiva- 
lent to a force tending to draw the Earth away from the 
Moon. As far as the Earth-Moon system alone is concerned, 
the forces in the two cases act in such a direction as to in- 
crease the distance apart of the two bodies. At quadrature, 
on the other hand, the intensities of the attraction at the Moon 
and Earth are equal although acting in slightly different 
directions. The resulting inequality in the motion of the 
Moon, termed the variation, is discussed in the next section. 

The annual equation is a small inequality in the Moon's 
motion with a period of an anomalistic year which is due to 
the variation of the Earth's distance from the Sun. At peri- 
helion, when the Earth-Moon system is nearest to the Sun, 
the residual effect of the solar attraction just discussed is 
greater than at aphelion when the system is at its greatest 
distance from the Sun. At perihelion, therefore, there is in the 
mean a greater force arising from the Sun's attraction tending 
to draw the Earth and Moon apart than there is at aphelion : 
the effect is obviously the same as would be produced if the 
Earth's attraction on the Moon was somewhat less at perihelion 
than at aphelion. The result is that for the six months of the 



year around perihelion (October 1 to April 1), the mean radius 
of the lunar orbit is greater and the Moon's angular velocity 
in its orbit is less than their annual mean values, whilst for the 
six months around aphelion (April 1 to October 1), the mean 
radius is less and the angular velocity greater than the average. 
On account of this inequality, a correction is necessary to the 
mean longitude of the Moon to obtain the true longitude, the 
correction having its largest negative value on April 1, and its 
largest positive value on October 1, and vanishing at perihelion 
and at aphelion. It can be stated in the form 

True longitude mean longitude (11' 16") sin 0, 
where 6 is the Sun's longitude measured from perihelion, 
which is on January 1 and increases at the rate of about 
1 per day, being 180 on July 1. 

49. The Variation . In addition to the equation of the 
centre, the evection and the annual equation, there is a fourth 

principal inequality in 
C the motion of the Moon. 

This is called the varia- 
tion, and as explained in 
the preceding section, it 
is due to the variation in 
the magnitude of the 
residual solar attraction 
on the Earth -Moon system 
during a synodic month. 
Referring to Fig. 31, if 
represents the position of 
the Earth, ACBD the 
orbit of the Moon described in the direction of the arrow, 
A, B the positions of the Moon when in conjunction and 
opposition respectively, (7, D the positions of quadrature, then 
at any other point E of the orbit, the effect of the residual 
solar attraction can be represented by the arrow El. This is 
greatest at A and B. El can be resolved into two components, 
Em normal to the orbit and En tangential to the orbit. We 
shall neglect the normal force. The tangential component 
obviously vanishes at A and B, also at C and D. Between D 
and A, and again between C and B, it acts in such a way as to 
accelerate the motion, whilst between A and C. B and D it acts 

FIG. 31. The Lunar Variation. 


so as to retard the motion. The variation in velocity pro- 
duced in this way reaches its greatest value at A and B and its 
least value at C and D, and if Off, OF, OG 9 OH make with 
AB and CD angles of 45, it is apparent that the velocity will 
be greater than its mean value between H and E y F and O and 
less than the mean value between E and F, G and H. The 
inequality in motion so produced has therefore a period of half 
the synodic month, or 14-77 days, and its magnitude has been 
found to be 39' sin 2E, E being the angle subtended at the 
Earth by the directions to the Moon and Sun. Thus we have 
True longitiide = mean longitude + 39' sin 2E. 

50. Tables of the Moon's Motion. The preceding dis- 
cussion of the four principal inequalities in the Moon's motion 
will give a small indication of the complexity of the motion of 
the Moon. The problem of determining the motion of the 
Moon is a particular case of the celebrated problem of three 
bodies, which may be stated in the following form : Three 
bodies of masses m ly m 2 , m 3 which severally attract each other 
with forces proportional to the products of their masses and 
inversely proportional to the squares of their distances apart, 
are set in motion from certain points, with given velocities in 
given directions ; to determine the subsequent motion. The 
problem is not capable of a general solution, but in certain 
cases an approximate solution can be obtained whose accuracy 
will depend upon the degree to which the approximation is 
carried and this, in general, is conditioned by the labour 
involved. In the case of the system Earth-Moon-Sun the great 
distance of the Sun simplifies the problem to some extent, 
but this is largely offset by the greater mass of the sun. 
The perturbing effects of the major planets have also to bo 
taken into account. The solution of the problem is required in 
order to predict, with the accuracy required, the position of 
the Moon for some years ahead. 

Newton, in his immortal Principia, published in 1686, was the 
first to attempt to explain on dynamical principles the motion 
of the Moon, but the first tables of the Moon's motion, con- 
structed so as to enable the position to be readily obtained at 
any required time, were given by Clairaut in 1752. The 
problem continued to attract the attention of the foremost 


mathematicians until Hansen succeeded in developing the 
theory in a form adaptable to numerical computation. As the 
outcome of many years' work, his Tables de la Lune were pub- 
lished in 1857 by the Admiralty and have served until the 
present time as the basis for the computation of the Moon's 
positions, which are given in the Nautical Almanac. The 
observed position of the Moon has, however, gradually deviated 
from the position computed from the tables. In the French 
Connaissance des Temps, the places of the Moon are computed 
from tables published in 1911 by Radau and based on 
Delauiiay's Theory. This theory is the most general theory of 
the motion of the Moon yet given, but, unfortunately, it is not 
very well adapted to numerical computation. In 1920 a new 
set of tables, prepared by E. W. Brown from his own lunar 
theory, were published. They are the most complete taJbles 
ever computed, and are the outcome of chirty years' work. 
The theory includes 1,500 separate terms, of which the equation 
of the centre, evection, etc., are the principal, and the tables 
enable the position to be obtained without the enormous 
labour of computing each time these 1,500 terms. Brown's 
tables will in future be used in the computations for the Nautical 
Almanac, commencing with the year 1923. 

51. The Secular Acceleration of the Moon. There is 
one term in the motion of the Moon to which reference must 
be made, as its origin is to some extent obscure. If a represents 
the Moon's mean motion in longitude in unit time, then 
apart from the various periodic terms, the longitude of the 
Moon at time t can be represented in the form 

L=L +al 9 

L being the longitude at the time chosen for origin. 

It is found, however, that an equation of this type cannot 
be made to satisfy both modern observations of position and 
ancient observations of times of eclipses. For long intervals 
of time a formula of the type 

L = L + at + bt* 

is found to be necessary. If the unit of time is taken as a 
century of 36,525 days, b has the value of approximately 10" 
or 11 " and this quantity is generally called the lunar secular 


acceleration. It is, in part, due to purely gravitational causes : 
it was shown by Adams that the slow diminution of the eccen- 
tricity of the Earth's orbit will produce an apparent secular 
acceleration of the Moon's motion of amount 6"-l. The 
difference between this quantity and the observed amount is 
not due to gravitational causes. It is now generally attributed 
to a very minute and gradual lengthening of the period of the 
Earth's rotation, which serves as our basis of time determin- 
ation. The only means available for testing the assumption 
that the period of rotation of the Earth is invariable is by 
observations of the members of the solar system and the 
comparison of the results with gravitational theory. But the 
change is actually so minute that observations extending over 
a long period are necessary. The change in the period of 
rotation is generally attributed to dissipation of energy by 
friction between tidal currents and the sea bottom and recent 
investigations show that the probable dissipation so produced 
is of the right order of magnitude to account for the residual 
lunar acceleration. 


FIG. 32. The Path of the Moon around tho Sun. 

52. Path of the Moon with respect to the Sun. We have 
hitherto been considering the orbit of the Moon with respect 
to the Earth ; but the Moon along with the Earth moves 
around the Sun and it is of interest to inquire what is the actual 
path of the Moon relative to the Sun. Since the Moon goes 
round the Earth about 12| times in one year, the Moon's path 
would cross that of the Earth about 25 times, if the two paths 
were in the same plane. It might therefore be anticipated 
that the path of the Moon relative to the Sun would consist 
of a series of waves or even loops. Actually, however, the path 
is everywhere concave to the Sun, a fact of the truth of which 
it is not at once easy to satisfy oneself. It is obvious that at 
full Moon, when the Moon is on the side of the Earth remote 



from the Sun, its path will be concave to the Sun, but it might 
have been anticipated that at new Moon its path would be 
convex. Actually the paths of the Earth and Moon are some- 
what as shown in Fig. 32. E, M denote the positions of the 
Earth and Moon at corresponding instants, the suffixes , i, 2, 3 
denoting new Moon, first quarter, full Moon and last quarter 
respectively. It will be seen that the path of the Moon is a 
sort of distorted oval, everywhere concave to the Sun. 

That the path of the Moon at new moon is concave to the 

Sun can be shown from ele- 
mentary considerations. In 
Fig. 33, S is the Sun, E , 
MQ the positions of the 
Earth and Moon respec- 
tively at the time of new 
moon. A short time later 
suppose the Earth to have 
moved to E ly and the Moon 
toMi. It will be sufficiently 
accurate for the present 
purpose to suppose both 
orbits to be circular and in 
the same plane, in which 
case E S = EiS = R, the 
radius of the Earth's orbit 
round the Sun, and E M = 
EiMi = r, the radius of the 

Moon*s orbit round the Earth. If M L is the tangent to the 
Moon's orbit at M , then the orbit relative to the Sun will be 
concave if M t is on the same side of M L as is the Sun. 

Calling the angle E^SE^O, and R being the point in which the 
line through E l parallel to E S cuts the orbit of the Moon 
relative to the Earth when the Earth is at E l9 it is seen from the 
figure that if the line EM always pointed in a fixed direction, 
so that the Moon did not revolve relative to the fixed stars, 
the Moon would be at R when the Earth was at E l and the 
motion of the Earth would have carried it a distance equal to 
E P (the projection on E M Q of EoEj} below M L, i.e. an 
amount R (1 cos 6). Actually the Moon moves from R to 
MI, and since its average daily angular motion relative to the 

FIG. 33. To illustrate the Concavity 
of the Moon's Path. 



stars is about 13, the angle M JS^R will be 130. The Moon is 
thereby lifted a distance r (1 cos 130) in a direction perpen- 
dicular to MJL above ft, and the resultant distance of M l 
below M L will be R (1 - cos 0) - r (1 - cos 130). 

O 2 
If is a small angle, we can put 1 cos 0= , and since 


R = 93,000,000 miles r = 240,000 miles, R = 390r approxi- 
mately. The above expression is therefore 

02 ^ 390r - (13) 2 r } - HOr 

and is therefore positive. Therefore, after new moon, the Moon 
moves below M L, i.e. towards the Sun, and its orbit is there- 
fore concave to the Sun. 

53. The Rotation of the Moon. Does the Moon rotate 
on its axis as well as travel round the Earth ? This is a ques- 
tion which is easily answered, 
although the answer is sometimes 
found puzzling. Observation of 
the Moon with a low telescopic 
power, such as a pair of pris- 
matic binoculars, will suffice to 
show the principal surface de- 
tails. Continued observations will 
reveal that these markings are 
permanent, and that the Moon 
always turns the same face to- 
wards the Earth. This means 
that the Moon rotates on its axis 
in the same time that it takes to 
make an orbital revolution about 

the Earth. That this is so will be readily seen from Fig. 34. 
E denotes the position of the Earth, M the centre of the Moon 
at any instant and A the point on the Moon's surface in the 
line EM Q . At any subsequent instant suppose the centre of 
the Moon to have moved to M l and let B be a point on its 
surface such that M B is parallel to M Q A, and C another 
point such that M^GE is a straight line. 

If the Moon did not rotate on its axis, the point A would 

FIG. 34. Rotation of Moon 
on its Axis. 



have moved to B when M Q moved to M t . Observation shows, 
however, that A moves to C. Therefore, whilst the Moon has 
moved relatively to the Earth through an angle M EM ly it 
has turned on its axis through an angle BM^G. But since 
M B and M A are parallel, these angles are equal, so that the 
two rates of rotation are identical. 

54. The Librations. The statement that the Moon always 
presents the same face to the Earth is only approximately 
correct, for sometimes a little more of one portion of the 
surface and a little less of the diametrically opposite portion 
is seen. This phenomenon is called Libration and is due to 
a variety of causes. There are three principal librations to 
which reference may be made. 

The Libration in Longitude. This libration is due to the 
rotation of the Moon on its axis being at a uniform rate whilst 

its orbital revolution 
around the Earth takes 
place, owing to the 
eccentricity of its orbit, 
at a rate which is not 
uniform. The result is 
that sometimes a little 
more of the eastern limb 
and sometimes a little 
more of the western limb 
may be seen. This libra- 
tion can be discussed and 
its approximate magni- 
tude very simply calculated owing to a remarkable theorem in 
dynamics, viz. : that if a body is moving in an elliptic orbit 
under the gravitational attraction of another body which is 
in one of the foci, then the line joining the body to the 
other focus will rotate at a constant rate which is equal to 
the mean rate of rotation of the body about the former focus, 
provided that the eccentricity of the orbit is small. The 
theorem is an approximation, which is justifiable if the square 
of the eccentricity can be neglected. 

Referring to Fig. 35, E represents the Earth in one focus 
of the Moon's orbit, F the other focus, M 1} M 2 , H z . . , 

FIG. 35. The Moon's Libration in 


successive positions of the Moon's centre. Let AI be the 
point on the Moon's surface which is in the line FE produced. 
Then when the centre of the Moon is at M 2 , A l will have 
moved to A 2 , on the line M 2 F, because the rate of rotation 
of the line FM and the rate of rotation of the Moon on its 
axis are equal, both being equal to the mean rate of rotation 
of the Moon about the Earth. Therefore, the point on the 
Moon's surface which previously pointed to E now points to F, 
and when the Moon is at M 2 more of the surface on one side 
has come into view, the amount being measured by the angle 
FM Z E. More of this side will remain in view until the Moon 
has come to the position M t on EF produced ; thereafter, 
for the second half of its revolution, more of the surface at 
the other limb will come into view, as at M 5 . The libration 
will be a maximum when the Moon is at the end of the minor 
axis of its orbit, as at M 3 , and then FM 3 = EM 3 = a, the 
semi-major axis of the orbit, since the sum of the radii to the 
two foci always equals 2a. But the distance between the 
foci of an ellipse is 2ae, so that since the distance EF is really 
very small compared with EM 3 , it follows that the angle 
EM 5 F = EF/EM 3 = 2e. Since the eccentricity of the orbit 
is 0-0549, this gives 6 15' as the maximum libration in 
longitude. On account of the inequalities in the Moon's 
motion, the actual value of the libration is found to be 
somewhat larger than this value. 

The Libration in Latitude. This libration is due to the axis 
of rotation of the 
Moon being in- 
dined from per- 
pendicularity to 
the plane of its 
orbit by about & ^ s 

6|. The lunar FIG. 30. The Moon's Libration in Latitude. 

equator is there- 
fore inclined to the orbital plane at the same angle. The 
axis of rotation remains constantly parallel to the same direc- 
tion in space throughout the entire orbital revolution, just as 
does the Earth's axis. From Fig. 36 it will be seen that the 
result of this is that now more of the region about the Moon's 
south pole will be seen and now more of the region about its 


north pole, the portions visible in the two positions shown 
being the hemispheres acb, def. 

The Diurnal Libration. The statement that the Moon 
presents the same face always to the Earth holds (in the 
absence of librations) for the centre of the Earth. But 
obviously, if there are two observers at different points of the 
Earth's surface, each will see a small portion of the Moon's 
surface which will be invisible to the other. If, instead of 
two observers, we consider one observer whose position changes 
on account of the Earth's rotation, it follows that as he is 
moved round he will gradually see slightly different portions 
of the surface. This libration effect is called diurnal or 
parallactic libration. Its amplitude can amount nearly to 1. 

55. The Distance of the Moon. The principle of the 
method of determining the distance of the Moon is very 

simple. It consists in 
making simultaneous ob- 
servations of the position 
of the Moon relative to the 
stars at two observatories 
widely separated on the 
Earth, such as Greenwich 
and the Cape of Good Hope, 
and using the known dis- 
tance between these obser- 
vations as a base-line. In 
order to make the theory 

-V OT T-k + -4.- * 4-1 njr , as simple as possible we 
FIQ. 37. Determination of the Moon s . * r 

Distance, will suppose the two ob- 

servatories to be on the 

same meridian of longitude, at A and B in Fig. 37. C is 
the centre of the Earth and M the position of the Moon 
at the instant of crossing the meridian. The latitudes of A 
and B being known, the distance AB and the angles of the 
triangle ABC can be calculated. Astronomical observations 
give the angles MAZ A and MBZ R which are the zenith- 
distances of the Moon at A and B respectively at the instant 
of meridian transit. The angles MAB and MBA are then 
determined, since, for instance, MAB = 180 MAZ* BAG 



and both the latter angles are known. In the triangle MAB, 
the base AB and the two adjacent angles are now known and 
therefore the distances MA, MB can be calculated. The 
distance, MC, from the centre of the Earth can then be 
derived from either the triangle MAG or the triangle MBC. 
The mean distance of the Moon is about 240,000 miles or 
about thirty times the Earth's diameter. On account of the 
eccentricity of the orbit, the distance can vary between the 
limits 222,000 and 253,000 miles. 

The position of the Moon given in the Nautical Almanac 
is referred to an observer supposed to be situated at the centre 
of the Earth. To determine the apparent 
position for an observer at any point of 
the Earth's surface a correction must be 
applied for what is called the Moon's 
" parallax." The horizontal parallax of 
the Moon is the angular semi-diameter 
of the Earth as seen from the Moon, i.e. 
r/Ry r being the radius of the Earth and 
R the distance from the centre of the 
Earth to the Moon. If A is a point on 
the Earth's surface, C the Earth's centre, 
and M the Moon (Pig. 38), then to .re- 
duce the zenith-distance of the Moon as Flo . sg.Tho Parallax 
observed at A to its value for the f th Moon. 
centre of the Earth, the angle AMC 
must be subtracted. But angle AMC = rSinZ/JS or co = 
o> SinZ, where c5 is the horizontal parallax, or the value of 
co when Z = 90, i.e. the correction for parallax is propor- 
tional to the sine of the zenith-distance. The constant of 
proportionality is the "horizontal parallax," and its value 
can be calculated when the distance of the Moon has been 
determined. The horizontal parallax of the Moon can vary 
between the limits 53'-9 and 61/-5. 

56. The Size of the Moon. To determine the size of the 
Moon it is only necessary to know its angular diameter and 
its distance. The former quantity can be measured directly, 
and the method of determining the latter has just been 
explained. In this way it is found that the diameter of the 


Moon is about 2,200 miles or rather more than one-quarter 
of that of the Earth. The volume of the Earth is about 
fifty times that of the Moon. If the average density of the 
Moon were the same as that of the Earth, this would also give 
the ratio of their masses, but, as we shall now show, the actual 
ratio is somewhat greater than this figure. 

57. The Mass of the Moon. The method of determining 
the mass of the Moon involves an interesting application of 
the law of gravitation. We have heretofore supposed the 

Earth to move around the Sun 
in an elliptic orbit : this is not 
strictly accurate. The Earth and 
Moon form a compound system, 
in motion around the Sun, and 
the law of gravitation requires 
that their centre of gravity should 
describe the elliptic orbit, so that 
the orbit of the Earth will not 
be strictly elliptical. If E, M 
represent the masses of the 
Ort T, , ... ~7 i Earth and Moon respectively 

Fro. 30. Dot orrni nation ot the L * 

Mass of the Moon. (Fig. 39), then the centre of 

gravity of the compound system 

is in the line joining their centres and at a distance 


- R from the centre of the Earth. Whilst the centre 
M +E 

of gravity describes its elliptic orbit, the Earth moves around 
it in an approximately circular orbit of radius MR/(M + E) ; 
the Moon describes a similar orbit of larger radius ER/(M +E), 
the line joining the two bodies always passing through their 
centre of gravity. The period of this motion with respect to 
the Sun is a lunar month. Its effect is to produce an apparent 
displacement of the Sun as viewed from the Earth, the Sun's 
apparent motion being slightly accelerated during one half of 
the lunar month and slightly retarded during the other half. 
The total change in the apparent place of the Sun is small, 
amounting only to about 12". The distance of the Sun from 
the Earth is known in other ways to be about 93,000,000 miles 
and at this distance 12" in angular measure corresponds to 


about 5,760 miles. That is, the radius of the orbit of the 
Earth about the centre of gravity of Earth and Moon is 2,880 
miles, and this must be equal to M/M + E times the Moon's 
distance. The latter's distance having previously been deter- 
mined, it is deduced that the mass of the Moon is about 1/81 
of that of the Earth. The mean, density of the Moon must 
therefore be less than that of the Earth, for equal densities 
would have given a ratio of 1/50. The ratio of the densities 
is about 1-6 : 1. 

58. Rising and Setting of the Moon. The phenomena 
connected with the rising and setting of the Moon are more 
complicated than in the case of the Sun on account of the 
large diurnal variations in declination to which the Moon is 
liable. These arc due to the rapid daily motion in its orbit 
of about 13 11' ; when the Moon is near the intersection of 
its orbit with the equator, the diurnal change in declination 
is given by Sin 13 11' Sin i =0-228 Sin i, where i is the 
inclination between the plane of the Moon's orbit and the 
equator. Since the ecliptic is inclined to the equator at an 
angle of 23 27' and the lunar orbit is inclined to the ecliptic 
at 5 9', the inclination of the lunar orbit to the equator has 
the limits 23 27' 5 9' or 18 18' and 28 36', corresponding 
to diurnal motions in declination of 4 6' and 6 16' respectively. 
These values are for a mean angular motion of the Moon : 
when the lunar perigee is in the equator, which occurs twice 
in nine years, the daily motion is greater than the mean and the 
daily variation in declination can then exceed 7. On the other 
hand, when the Moon is at a distance of 90 from its point of 
crossing the equator, the declination changes very slowly, 
the motion of the Moon being at such times nearly parallel 
to the equator. 

The effect of the changes in declination on the times of 
rising and setting have now to be considered. We will neglect 
at first the Moon's motion in right ascension ; an increase in 
declination without any alteration in right ascension would 
lift the Moon nearer the north pole, along a great circle ; 
the length of time it would stay above the horizon would then 
be increased at places in the northern hemisphere, but the 
time of crossing the meridian would be unaltered, as this 



depends solely upon the right ascension. It follows that an 
increase in decimation will cause the time of rising to become 
earlier and the time of setting to become later ; the reverse 
results will follow from a decrease. If, on the other hand, 
the right ascension increases without any change of declina- 
tion, the times of rising, of crossing the meridian and of 
setting will all be retarded equally. 

Combining the two effects, there is a normal retardation 
from day to clay in the times of rising and setting due to the 
progressive increase in the right ascension of the Moon ; this 
is increased or decreased by the changes in declination. When 
the Moon crosses the First Point of Aries, the declination is 
increasing most rapidly and this tends, therefore, to counter- 
act to some extent the normal retardation in the time of 
rising ; the time of setting, on the other hand, is retarded 
on both accounts, so that the normal retardation in the time 
of setting is increased. Similarly, when the Moon passes 
through the autumnal equinox, the declination is decreasing 
most rapidly and the retardation in the time of rising is then 
greater than usual, and in the time of setting is less than 

The Moon passes through the equinoxes once in each 
revolution, so that the phenomena of a small and of a large 
daily retardation in the time of rising and setting must occur 
once every month. The mean retardations of about 52 minutes 
must therefore show considerable variations throughout the 
month. The phenomenon is most noticeable at the autumnal 
equinox, for the Moon when in the First Point of Aries will 
then be in opposition to the Sun and will therefore be full, 
so that the rising will take place near sunset. For several 
nights in succession, therefore, near full moon the Moon will 
rise approximately the same time. This is the phenomenon 
known as the Harvest Moon. In southern latitudes the same 
phenomenon will occur at the vernal equinox. 

The times of moonrise or set are given in ordinary almanacs 
such as Whitaker's ; an examination of these times will 
illustrate the phenomena described above. One point in 
reference to these figures deserves mention : the almanacs 
give only the times of moonrise or moonset, never both as 
in the case of the Sun. This is because either the rising or 


the setting occurs during daylight and is not observable. 
Occasionally, there will be a day for which neither the time 
of rising nor the time of setting will be given. If, for instance, 
the Moon rises just before the beginning of a certain solar 
day, it may not rise again until just after the end of the same 
day, the lunar day being longer than 24 hours. 

One other feature in connection with the path of the Moon 
in the heavens may be mentioned. The full moon always 
appears at the point in the heavens which is opposite to the 
Sun. It follows that near the time of winter solstice the full 
moon must be near the summer solstice point of the ecliptic 
and gets much closer to the zenith than it does near the time 
of summer solstice, when full moon occurs at the winter 
solstice point. For this reason, the Moon is said to " ride 
high " in the winter. 

59. The Tides. It is mainly to the attraction exerted by 
the Moon on the waters of the oceans that the tides are due. 
The Moon, therefore, is of enormous economic importance to 

To explain the production of the tides we shall, for the sake 
of simplicity, suppose the Earth to be a sphere uniformly 
covered with a relatively shallow ocean. The Moon exerts 
an attraction on the Earth and on the waters, as a result of 
which we should expect the water to be heaped up at the 
point of the Earth directly under the Moon, the attraction 
of the Moon on the water being greater than its attraction on 
the land ; and also at the diametrically opposite point, because 
the attraction of the Moon on the land will there be greater 
than on the water and will therefore pull the Earth away 
from the water. 

Actually, the phenomena are somewhat more complicated, 
the actual heaping up of the water which constitutes the tides 
being due to causes of a less simple nature/ Consider the 
attractive force on a small volume of water, not directly under 
the Moon. The force will act in the direction towards the 
Moon and can be resolved into two components, one normal 
to the surface and one tangential to it. It is the latter which 
is the more important, as it causes the particles of water to 
flow over the Earth's surface, towards the point where the 



Moon is overhead. When the Moon is east of a given place 
the water particles will be pulled towards the east ; when it 
is west of the same place, they will be pulled towards the west. 
There is therefore an oscillation of the water particles in a period 
of half a lunar day. The tangential forces are therefore 
responsible for a heaping up of the water at the point of the 
Earth nearest the Moon, the particles on all sides being pulled 
towards this point. The crest of the tidal wave so produced 
follows the Moon round as the Earth and Moon rotate. 
Actually there is a lag in the effect, so that the crest is not 
directly under the Moon and a further complication is intro- 
duced by the irregular contour of the ocean boundaries. 
Approximately, the crest is at about 90 distant from the 
point under the Moon, and there is a second crest diametrically 
opposite to it, causing two high and two low tides per day. 
These two tides are, in general, of unequal height. This 
is due to the fact that the Moon does not move in the plane 

of the Earth's equator : the 
attraction of the Moon on the 
water covering our simple 
model will pull the spherical 
boundary of the water into 
the shape of a spheroid, but 
the axis of the Earth will 
not be the axis of the sphe- 
roid. This will be made 
clear by Fig. 40, in which 
the heaping up of the water 
is much exaggerated. The 
high tide at any point A may be represented by the line 
Aa ; it will also be high tide at the same instant at the 
point B, the height there being represented by the line 
Bb, which is obviously smaller than at A, with the Moon, 
as shown, north of the equator. But after 12 hours, 
A will have come to B, owing to the rotation of the Earth, 
and the height of the next high tide at A is therefore repre- 
sented by Bb. The two tides will only be equal when the 
Moon is on the equator and this occurs twice per month. 
The phenomenon is known as the diurnal inequality of the 

Fid. 40. Tho Diurnal Inequality 
in the Tides. 


There are two other causes which operate to produce 
inequalities in successive high tides. The first is the solar 
attraction which operates in an exactly similar manner to 
that of the Moon. Although the Sun is much larger than 
the Moon, its greater attracting power on this account is more 
than counterbalanced by its much greater distance, so that the 
solar tide is only about 5/1 1th of that produced by the Moon. 
At full and new moon, however, the tidal force due to the 
Sun is added to that due to the Moon, whereas, at quadratures, 
the forces are in opposition. The high tides at new and full 
moon are called spring tides, and are therefore much higher 
than those at first and third quarters, which are called neap 
tides. The ratio in the heights is 11 + 5 to 11 5 or 8 to 3. 
The second cause of the inequalities in the high tides is the 
large eccentricity of the lunar orbit, the resulting variation 
in the Moon's distance causing the tide-raising force to be 
about y greater at perigee than at apogee. When perigee 
occurs at the time of new or full moon, the high tides will 
be particularly high and the low tides correspondingly low. 

For an exact tidal theory, the actual contours of land and 
sea need to be taken into account. The preceding very 
simple theory will serve to illustrate the manner in which 
the tides are produced and the general qualitative effects 
which result. 

60. Surface Structure on the Moon. The more important 
features on the surface of the Moon can be revealed by a small 
telescope of, say, three or four inches' aperture with eye-pieces 
giving magnifications up to about 200 diameters. For the 
minor details larger instruments are necessary, but the magni- 
fication cannot, in general, be increased beyond about 1,000 
diameters with any advantage on account of atmospheric 
irregularities. With this magnification, the Moon is in effect 
brought to a distance of about 240 miles from the Earth and 
objects only 500 ft. in diameter can be seen. 

A small telescope is sufficient to reveal the rugged nature 
of the lunar surface ; the details can be most easily seen 
several days before or after full Moon, as the Sun's light then 
falls on the surface obliquely, throwing shadows, which indi- 
cate the relief. Near full Moon, the Sun's light falls nearly 


in the direction of our vision and then no shadows can be 
seen from the earth. 

On the only side of the Moon which we can see, there are 
ten mountain ranges, besides numerous isolated lofty peaks, 
more than 1,000 cracks or rills and at least 30,000 craters : 
most of these objects have been given names. There are also 
several large areas, almost devoid of craters, which appear 
darker than the rest of the surface : from long established 
usage, dating back to the time of Galileo, these areas have 
been called maria or seas, although it has long been known 
that they are not seas. The system of lunar nomenclature 
and many of the names of the principal objects date back 
to the seventeenth century : the first map of the Moon was, 
in fact, constructed by Hevelius in 1645. These dark areas 
are well shown in Plate L (a). The contrast in brightness 
between the dark and bright areas is increased in a photo- 
graph, owing to the light from the dark areas being actinically 
weaker than that from the bright. In Plate 2, showing a 
portion of the Moon photographed with the 100-inch reflector 
of the Mount Wilson Observatory, one of the dark areas, the 
Mare Imbrium, can be seen in the centre of the plate. 

The most numerous objects to be seen are the so-called 
craters which are to be found all over the visible surface. 
Plate III gives an indication of the large number of craters 
011 the Moon's surface. They vary greatly in size, from the 
great walled-plains such as Archimedes, which appear as 
circular mountain ranges surrounding more or less level plains, 
and which may be more than 100 miles in diameter, to minute 
craters which require the highest telescopic power available 
to render them visible. Many of the craters, e.g. Copernicus, 
have a lofty mountain peak as their centre. Copernicus is 
shown in Plate I (6), in which the central peak is easily dis- 
tinguished. The outer walls and also the central peaks may 
reach enormous heights, some exceeding 15,000 ft. These 
heights can be calculated from a measurement of the angular 
length of the shadow cast, combined with a knowledge of the 
angle at which the Sun's light is falling on the surface and of 
the distance 9f the Moon. From some of the craters, under 
favourable conditions of illumination, bright rays or streaks 
can be seen radiating radially in all directions, sometimes 


extending to very great distances and passing over many 
craters in their course. Such streaks can be well seen, for 
instance, in Plate I (a) radiating from the large crater, Tycho, 
towards the top of the photograph, and in the case of Coper- 
nicus, the large crater in the right-hand top corner of Plate II. 
The nature of these streaks is not known. 

The origin of these craters has been the subject of much 
discussion and is by no means settled. The term " crater " 
suggests a volcanic origin and is, on that account, perhaps as 
unfortunate and misleading as is the description of the dark 
areas as seas. One theory of their formation does, however, 
attribute them to volcanic origin. It is supposed that many 
ages ago matter was ejected from the central mountain and 
that this matter gradually piled up and formed the outer 
ring. Although the theory has gained wide acceptance, it 
is not without serious difficulties. Owing to the force of gravity 
on the Moon being only about one-sixth of that on the Earth, 
a given eruptive force would produce a much greater effect 
on the Moon than on the Earth. But it is difficult to believe 
that some of the enormous lunar craters could have been 
produced by volcanic forces similar to those which have 
produced most of the craters on the Earth. Moreover, there 
are no volcanic formations on the Earth which resemble at 
all closely the lunar craters. Signs of lava flow on the Moon 
which might have been anticipated from the violence of the 
supposed disturbance, are almost or entirely lacking. 

The only serious rival theory supposes that the craters 
were produced by the bombardment of the lunar surface by 
numerous meteors. Thus, whilst the volcanic theory attributes 
their origin to the action of internal forces, the meteoric theory 
attributes it to the action of forces from outside. The objec- 
tions to this theory are even more serious than in the case 
of the volcanic theory. Meteors which could have produced 
the larger craters must have been of enormous size and it 
is impossible to believe that the Moon could have been so 
bombarded without many of the meteors having also struck 
the Earth. Signs of meteoric bombardment might therefore 
reasonably be expected to be found on the Earth. There 
is, indeed, a crater in Arizona which is supposed to have been 
formed as the result of a large meteor striking the Earth : 


it is very similar in structure to many of the lunar craters, 
although its size is insignificant in comparison, its dia- 
meter being only about three-quarters of a mile, and the 
height of the walls above the surrounding plain only 
about 150 ft. The craters produced by bombs dropped from 
aeroplanes are also generally similar to the lunar craters, and 
this fact has been advanced in support of the meteoric hypo- 
thesis. It might be argued that the later processes of sedi- 
mentation on the Earth would have concealed craters similar 
to those on the Moon, but it does not seem probable that all 
traces would have disappeared and no evidence be found in 
rock strata. No such evidence has been discovered by 
geologists. A further very strong objection to the meteoric 
theory is that, to form craters in this way, the meteors must 
all have fallen vertically. But if a stream of meteors coming 
from outside had struck the Moon, many of the impacts must 
have been oblique and many merely glancing impacts. The 
formations which might have been so produced are con- 
spicuously absent. There is, indeed, a long straight valley 
in the range of lunar mountains cabled the Alps, but this appears 
to be an exceptional formation. 

The cause of the origin of the craters must therefore be 
regarded as still an open question : no theory yet advanced 
can account satisfactorily for their existence on the Moon 
and not on the Earth. Equally puzzling are the systems of 
bright streaks radiating from Tycho and a few other craters. 
These cast no shadows and are therefore neither elevations 
nor depressions. They pass on in straight lines over craters, 
rills, or whatever object lies in their track. No satisfactory 
explanation of them has yet been given. 

The mountain ranges on the Moon are extremely rugged 
and very lofty, but in view of the smallncss of the force of 
gravity at the surface of the Moon, there is no difficulty in 
supposing them to have been produced by forces similar to 
the forces which have given rise to the mountains on the 
Earth. In the upper half of Plate II can be seen the Apennines, 
the finest range of mountains on the Moon. 

61. Physical Conditions on the Moon. As far as is 

known with certainty, the Moon is a dead world which shows 


no evidence of change. There is little doubt but that it has 
no atmosphere. When the Moon in its motion eastward 
amongst the stars overtakes a star, the disappearance or 
" occupation " of the star as the Moon passes in front of it 
takes place with remarkable suddenness. If the Moon pos- 
sessed an atmosphere, the rays of light from the star passing 
through the atmosphere would be bent or refracted, and the 
nearer the rays approached the limb of the Moon, the longer 
would be their path through this atmosphere and the greater 
the amount of their bending. The star would therefore dis- 
appear gradually. There are other arguments which support 
this conclusion : the limb of the Moon, projected upon the 
Sun's disc during a solar eclipse, is perfectly sharp, and the 
outlines of the shadows of the lunar formations are very sharp, 
with no gradation at the boundary between light and dark. 
The theory of gases leads to the same conclusion ; a gas con- 
sists of a large number of molecules which are in motion to 
and fro with very large velocities : at the confines of the Earth's 
atmosphere there are molecules continually flung outwards 
with velocities of such magnitude that the force of the Earth's 
gravitation cannot hold them back and they escape into 
space. This process on the Moon would be much more rapid 
owing to the reduced gravitation, and it is probable that, if 
the Moon ever had an atmosphere, the mass of the Moon was 
too small to enable it to be retained. 

It is probable also that there is no water on the Moon. 
There is little or no- appearance of erosion or weathering on 
the lunar mountains or craters, and there are certainly no 
clouds to be seen at any time. In the absence of an atmosphere 
and of water, the existence of any vegetation is very improb- 
able, although competent observers have occasionally seen 
patches of a greenish hue which have been thought to show 
signs of change during the course of the lunar month, and 
have been corijecturally interpreted as vegetation. The 
inference seems at present hardly justifiable. Any evidence 
of change on the Moon's surface must be accepted with 
caution as, owing to the variable angle under which the sun- 
light falls and the change in the length and position of the 
shadows, apparent changes in the appearance of the craters 
and peaks may easily be misinterpreted as actual physical 


changes. At present, the existence of any physical change 
on the Moon has not been established to the general satis- 
faction of astronomers, though the possibility of slight changes 
should not be excluded. 

62. The Origin of the Moon. According to the theory 
proposed by G. H. Darwin and now generally accepted, the 
Moon and the Earth were formerly one body which had prob- 
ably been thrown off from the central body of the solar system. 
The entire mass had then a high temperature, and was in a 
fluid or plastic condition and in rapid rotation about an axis. 
Such a mass would gradually cool, contracting meanwhile 
with a corresponding increase in its rate of rotation. The 
course of evolution of the mass can be traced out by mathe- 
matical reasoning, and Darwin showed that the configuration 
would at first be that of an ellipsoid, rotating about its short 
axis. This would, in the course of time, give place to a 
pear-shaped figure, and then to a dumb-bell shape, the neck 
of which would gradually contract, until eventually the mass 
would split into two unequal masses almost in contact and in 
rapid rotation about their centre of gravity. On account of 
their plastic nature, each body would raise on the other large 
tides. It can be shown that tidal protuberances thus produced 
will act in such a manner as to accelerate the motion of the 
bodies in their orbits ; it follows from mechanical principles 
that this acceleration of the motion will result in an increase 
of the orbital radii and in the periods of revolution. The 
Moon and the Earth, which were in close contact and rapid 
rotation originally, therefore gradually separated and there 
was a corresponding increase in the lunar period. As the 
plasticity of the bodies decreased and their separation increased, 
the effect of the tidal forces gradually diminished and finally 
the two bodies reached their present condition. It can further 
be shown that the effect of tidal action would be to slow down 
the period of rotation of each body on its axis until this period 
became equal to the period of rotation of the bodies the one 
about the other. In the case of the Moon, as we have already 
seen, this process is completed. It is known that the period 
of rotation of the Earth on its axis is increasing, though very 
slowly. According to Darwin's theory this process should 


continue and, if the theory is correct, the last stage of 
equilibrium of the Earth-Moon system will be one in which 
the terrestrial day and the lunar sidereal day will each be 
equal to the period of revolution of the two bodies about one 
another and this period will equal 55 of our present days. 
The rotations of the two bodies will then take place exactly 
as if they were rigidly connected, the Earth turning always 
the same face to the Moon and the Moon the same face to the 


63. The Distance of the Sun. The distance of the Earth 
from the Sun may be regarded as the fundamental distance 
in astronomy. As we shall see later, when discussing Kepler's 
laws governing the motions of the planets around the Sun, 
if the periods of these motions are known, it is only necessary 
to know the mean distance of the Earth from the Sun. in. order 
to be able to determine the mean distance of every planet. 
It is possible, in fact, from observations of the angular motions 
of the planets and the application of Kepler's laws of planetary 
motion, to draw a correct map to scale of their orbits, but 
the scale-value of this map will remain arbitrary until any 
one distance has been determined. The determination of 
the distance from the Sun. of any other member of the solar 
system will suffice therefore to determine the mean distance 
of the Earth. This distance serves also as the base-line from 
which the distances of the stars may be determined. 

Instead of the Sun's distance, we may alternatively use 
the Sun's parallax, this term having a meaning analogous 
to its meaning when applied to the Moon ( 55), i.e. the solar 
parallax is the angle subtended by the radius of the Earth 
at the Sun. It is usually expressed in seconds of arc, having 
a value 8"-80. If this is converted into circular measure and 
divided into the radius of the Earth, expressed in miles, the 
quotient gives the Sun's distance also in miles. 

The solar parallax is closely related to the constant of 
aberration. In 36, it was explained how Bradley discovered 
the apparent displacement of a star due to aberration ; obser- 
vations made for the purpose of measuring these displacements 
determine the aberration constant, which is equal to the ratio 




of the mean velocity of the Earth in its orbit to the velocity 
of light. The velocity of light can be measured experimentally 
and it follows th^t a determination of the aberration constant 
in effect gives the mean orbital velocity of the Earth. Multi- 
plying this by the number of seconds in the year gives the 
circumference of the Earth's orbit and hence its mean radius. 
The following table gives the values of the solar parallax 
corresponding to various values of the aberration constant : 

Aberration Constant. 

20"- 46 

Solar Parallax. 


The determination of the Sun's distance may therefore 
be made by a direct method, in which the distance of any 
member of the solar system is found, or by an. indirect method, 
involving the prior determination of the constant of aberration. 

64. The Transit of Venus Method. Although this 
method is not capable of giving results of a high order of 
accuracy, it is of considerable interest historically, Halley 
having shown in 1716 how observations of the transit of Venus 
could be used to determine the solar parallax. 

The orbit of the planet Venus lies within that of the Earth, 
and being inclined at a small angle to the ecliptic, it sometimes 
happens that the planet comes directly between the Earth 
and the Sun, and it is then seen as a dark spot moving across 
the Sun's disc. Such an occurrence is called a transit of 
Venus. The transits occur at irregular and distant intervals 
which are alternately short and long ; the short ones are 
always 8 years, the long ones alternately 121| and 105| years. 
The following are the dates of the transits between 1600 and 
2200 : 

Date. Interval. 

2004, June 7. 121 J years. 

2012, Juno 5. 8 

2117, Doc. 10. 105J 

2125, Dec. 8. 8~ 


























Only five transits have yet been seen, viz. those of 1639, 
1761, 1769, 1874 and 1882. The first of these was predicted 
by a poor and unknown English curate named Horrocks who, 
at the time but 22 years of age, had been able to correct an 
error in Kepler's writings and to calculate the date of the 
occurrence. It so happened that the predicted date fell on 
a Sunday, and Horrocks was torn between his desire to make 
the observation, which at that time was a unique one, and 
to perform his duty at Church : the predicted time was 
uncertain within a few hours and a continuous watch was 
necessary in order that the transit might not be missed. He 
decided to put duty first and to observe in the intervals between 
the services, and was rewarded by seeing the black dot on 
the Sun's disc in the afternoon, shortly before sunset. A 
tablet in Westminster Abbey, with a quotation from Horrocks' 
work, Venus in Sole Visa (1662), " Ad majora avocatus quoo 
ob hsec parerga negligi non decuit," commemorates the 

The transit of 1769 was observed with a view to the deter- 
mination of the solar parallax, the value obtained being 
8"-57. The two transits of the nineteenth century, in 1874 
and in 1882, were extensively observed with the best appliances 
available, in the hope that a value would be obtained which 
could be accepted without question as correct. 

The theory of the method will now be briefly explained. 
When the transits occur, Venus is at " inferior conjunction," 
i.e. between the Earth and the Sun, and therefore at its nearest 
to the Earth. Its distance from the Earth is then only about 

two -sevenths of the 
Sun's distance, and 
a displacement of 
the observer on the 
Earth will cause a 
FIG. 41. Transit of Venus : Haiioy's Method, much greater dis- 
placement r e 1 a- 

tively to the stars of Venus than of the Sun : the circum- 
stances of the transit will therefore vary according to the 
position of the observer on the Earth. 

Suppose two observers on the Earth are situated at the 
points A and B, which are widely separated in latitude (Fig. 

THE SUN 111 

41). F is the position of Venus : then the apparent paths 
of Venus across the Sun during its transit as seen from A 
and B respectively are represented by aa,i and bbi. If both 
observers are provided with accurate clocks and observe the 
time taken during the transits from a to a l and 6 to 61, it is 
possible to deduce the lengths of these two arcs. 

The synodic period of Venus is the period of one revolution 
of Venus with respect to the line joining the Earth and the 
Sun and is known. If Venus actually moves from F to V l 
(Fig. 42) whilst appa- 
rently moving from a to 
a l9 the angle VSV l can 
be calculated if the time 
taken is known, for angle 

VRV - timf nf -frnnqit FlG ' 42. Theory of Determination of 

VbV i . time 01 transit Earth's Distance from Transit of 

= 360 : synodic period. Venus Observations. 

Also the ratio of the 

distances EV, SV is known, for, as has been explained, 
planetary observations enable the orbits of all members 
of the solar system to be drawn to scale. The angle 
VEVi can then be deduced, and this is equal to the length 
of the chord aa { in angular measure. Similarly bb l can be 
determined in angular measure. Since the angular dia- 
meter of the Sun is known, the distance pq between the 
mid-points of the two chords can be calculated in angular 
measure. But since the linear distance AB is known and the 
ratio of the distances Vp, VA, pq can also be obtained in 
linear measure. The knowledge of the same length in both 
angular and linear measure at once gives the distance between 
the Sun and Earth. Many refinements have to be taken 
into account in making the calculations, but the general 
principle of the method is as explained above. For the 
transits of 1874 and 1882 extensive preparations were made 
and numerous expeditions, which were organized with great 
thoroughness, were dispatched by the Governments of Great 
Britain, the United States and other countries. Although 
the transits were widely observed, the results were disappoint- 
ing and did not greatly increase our knowledge of the solar 
parallax. It was found impossible to state with certainty 
what was the exact moment at which Venus touched the 



Sun's disc, the difficulty probably arising from the existence 
of an atmosphere on Venus. The uncertainties in the recorded 
times of transit were therefore as great as 10 seconds of time. 

65. Observations of Mars or Minor Planet. The 

principle of this method is very simple, involving the measure- 
ment of the relative displacement of a planet as seen from 
two different points on the Earth whose distance apart can be 
calculated. In order to make the displacement as large as 
possible for a given base-line it is desirable to use a planet 
as near the Earth as possible. The planet Mars was used 

in 1877 by Sir David Gill, who 
observed from the Island of 
Ascension. The orbit of Mars 
has a high eccentricity and the 
most favourable time for se- 
curing the observations is there- 
fore when Mars is at its closest 
approach to the Sun (i.e. at 
perihelion) and the Earth is 
near aphelion, i.e. at its greatest 

FIG. 43. Favourable Opposition 
of Mars. Mars at Perihelion, 
Karth at Aphelion. 

distance from the Sun, the two planets being at the same time 
in opposition, so that Mars is on the meridian near midnight 
(Fig. 43). The observations may then be made in one of two 
ways : (i) Simultaneous observations may be made from two 
observatories (Fig. 44), A and J5, widely separated in latitude, 
the observations consisting in the measurement of the angular 
distance of Mars from 

one or more neighbour- A ^_ ^ To Star s 

ing stars. These enable 
the angle AMB to be 
calculated, since the 
star is so distant that 
it is seen in the same 
direction from A and 
B, so that the angle 

AMB is simply the sum of the two angles MAS, 
and then, the base-line AB being known, the distance 
between Mars and the Earth can be calculated : all other dis- 
tances in the solar system can then be deduced, (ii) The 

To Star S, 

FIG. 44. -Observation of Mars from 
Solar Parallax. 

THE SUN 113 

observations may all be made from one station. Mars being in 
opposition crosses its meridian near midnight and is there- 
fore visible throughout the night. The diurnal rotation of 
the Earth then provides the base-line, observations being 
made at A, and again, after an interval of several hours, when 
the station has moved round to B. A correction must then 
be applied for the orbital motions of Mars itself, and also of 
the Earth between the two observations. 

There are advantages and disadvantages attaching to both 
methods : the second method eliminates to a very large extent 
all errors of a personal or of an instrumental nature and 
involves less interruption on account of unfavourable weather. 
In the first method, it is easier for personal and instrumental 
errors to enter, and the weather conditions may be unfavour- 
able at one station when they are favourable at the other. 
On the other hand, if several observatories can co-operate 
the accumulation of observations should give a result of 
greater accuracy. 

Great accuracy in the observations are required, for the 
angles to be measured are small. Sir David Gill used a helio- 
meter for measuring the angles, this instrument enabling 
angular distances in the sky to be determined with a high 
precision. He obtained the value 8"-78 for the solar parallax. 
The chief source of error lay in the difficulty of measuring 
accurately the distance between the planet, whose image 
possesses a definite disc and a star. Gill therefore decided 
later to repeat the observations, using certain of the minor 
planets which, owing to the large eccentricities of their orbits, 
come sufficiently near to the earth for the purpose. He 
selected the planets Victoria, Iris and Sappho : these small 
objects appear in the telescope as star-points and the error 
referred to is thus avoided. He made an extensive series of 
observations at the Cape of Good Hope in the years 1888 and 
1889 and secured the co-operation of observers at New Haven, 
Leipzig and Gottingen ; similar instruments being used at 
each place. The final result of the whole series was to give 
a value for the solar parallax of 8"- 80, which is the value 
accepted at present. 

After the completion of this work, a small planet was dis- 
covered by Dr. Witt of Berlin, to which the name of Eros 



was given. This planet, only 28 miles in diameter, has an 
orbit with a high eccentricity and at times comes to within 
a distance of 14 million miles from the Earth : it is therefore 
admirably adapted for observation for the determination of 
the Sun's distance. One of the close approaches of Eros to 
the Earth occurred in 1900 and a very extensive series of 
observations were undertaken in co-operation by many obser- 
vatories. Long focus telescopes were employed and obser- 
vations secured by photography. The results confirmed the 
value 8"- 80 for the solar parallax, which corresponds to a 
distance from the Sun to the Earth of 92,800,000 miles. 

66. Other Methods for Determining the Sun's Distance. 

The methods described in the preceding section and the 
indirect method depending upon the determination of the 
aberration constant, to which reference has already been 
made, provide the most accurate means of determining the 
Sun's distance. Other methods have also been used, and 
although not susceptible to the same high degree of accuracy, 
they serve to confirm the result. As this distance is funda- 
mental in astronomy, it is not desirable that any avenue for 
determining it should remain unexplored. 

One method depends upon the disturbance caused by the 
Earth in the motion of Venus. To a first approximation 
the motion of Venus is in an ellipse, but when Venus and the 
Earth are near their distance of closest approach, the gravi- 
tational attraction of the Earth on Venus causes the latter 
to depart somewhat from true elliptical motion, and the 
observation of these perturbations provides a means of deter- 
mining the mass of the Earth and through it, by Kepler's 
laws, of the Earth's distance. Another method depends 
upon the inequality in the motion of the Moon, which is termed 
the parallactic inequality. Referring to 49, the disturbing 
effects of the Sun to which the variation is due are greater 
at any point in the portion D A C of the orbit than at the 
corresponding point in the portion DBG, owing to the Moon 
being nearer to the Sun in the former position. This second 
order effect is known as the parallactic inequality. It 
involves the Earth's mean distance from the Sun and compari- 
son of the calculated with the observed value enables the 
distance to be determined. 

THE SUN 115 

A third method has also been applied which depends upon 
different principles. As will be explained in 69, the light 
from a star when passed through a spectroscope is split up 
into separate lines whose positions are shifted slightly to the 
red or blue according as there is relative motion of the star 
and the observer away from or towards one another. The 
method provides a means of measuring the relative velocity 
of the star and observer in the line of sight. If then the 
Earth's orbital motion is directed at one time of the year 
towards a certain star, spectroscopic observations will give 
the difference between the velocity of the star away from the 
solar system and the velocity of the Earth : observations 
made six months later, when the Earth's motion is directed 
away from the star, will give the sum of these two velocities. 
The difference between the two measures enables the velocity 
of the Earth in its orbit to be determined and hence its distance 
from the Sun. 

67. The Size and Mass of the Sun. Having determined 
the mean distance of the Earth from the Sun, it is a simple 
matter to determine the size of the Sun. It is only necessary 
to measure its mean angular diameter. This is found to be 
32' 4". Expressing this in circular measure and multiplying by 
the Earth's distance, we find that the Sun's diameter is about 
865,000 miles or about 108 times the diameter of the Earth. 

The determination of the mass of the Sun must naturally be 
based upon the previous determination of the mass of the 
Earth. The method of determining the latter, or its equivalent 
the Earth's mean density, has already been explained ( 15). 
Newton's law of gravitation, together with a knowledge of 
the distance of the Earth from the Sun, suffice to connect 
together the masses of the Sun and Earth. Since the Earth 
attracts a body on its surface with the same force as it would 
if its mass were concentrated at its centre, it follows that 
the ratio of the forces per unit mass acting on such a body 
due to the attractions of the Sun (/) and Earth (g) respectively 
is given by ,. __M !m 


where m, M are the masses of the Earth and Sun respectively, 
r is the Earth's radius, R the Sun's distance. The attractive 
force /is readily calculated. Assuming the orbit of the Earth 


to be a circle, its acceleration towards the Sun at any point 
is known from dynamical principles to equal V*/R, where 
V is the velocity of the Earth in its orbit. But since this 
acceleration is simply a measure of the force per unit mass, 
it must equal /, i.e. f=V 2 /R. R is known and V can be 
found from the fact that the Earth completes one revolution 
(distance 2nR) in one year. V is thus found to be about 
18-5 miles per second, and R being nearly 93 million miles, 
f= 0-233 inch per sec. per sec. In this way, by substituting 
this value of / in the above formula, it is found that the mass 
of the Sun is about 332,000 times that of the Earth. 

This value can be obtained in another way. It can be 
shown, by dynamical principles involving Newton's law of 
gravitation, that for any satellite moving around a central 
body, the period of revolution is proportional to the distance 
multiplied by the square root of the distance and divided 
by the square root of the mass of the central body. Com- 
paring the period of the rotation of the Earth about the Sun 
with that of the Moon about the Earth (the Earth being 
then regarded as the central body), the distance of the Earth 
from the Sun is about 400 times that of the distance of the 
Moon from the Earth and the period of the Earth about the 
Sun is about 13J times that of the Moon about the Earth. 
The ratio of the masses of the Sun and Earth is therefore 
approximately (400 x y" 400 / 13 !) 2 or about 350,000. An 
exact calculation, allowing for disturbing factors, would give 
again a ratio of 332,000. 

It is now possible to compare the mean densities of the 
Earth and Sun : the diameter of the Sun being 108 times 
that of the Earth, and its mass 332,000, its density is 332,000 
-f-(108) 3 times that of the Earth, or approximately only one 
quarter as dense as the Earth. The mean density of the Earth 
being about 5-5, that of the Sun is only about 14 times that 
of water. 

The force of gravity at the Sun's surface is considerably 
greater than that at the surface of the Earth. The ratio 
of the two forces is given by the mass of Sun divided by the 
square of its radius, provided that both quantities are expressed 
in terms of the corresponding quantities for the Earth, or is 
given by 332,000 /(108) 2 . This ratio is about 27. The force 

THE SUN 117 

of gravity at the surface of the Sun is therefore about 27 times 
as great as at the surface of the Earth. 

68. The Rotation of the Sun. If the surface of the Sun 
is observed through a telescope, one or more dark spots will 
usually be observed on its surface. These are termed " Sun- 
spots " and were probably first seen by Galileo. A group 
of spots is shown in Plate IV (a). If these spots are watched 
from day to day, it will be noticed that they appear to move 
across the disc from the east limb to the west. This apparent 
motion is, in the main, due really to the rotation of the Sun. 
It is known that the spots do not, in general, remain fixed on 
the Sun's surface, although their motions are slight and can 
be eliminated on the average when observations of a large 
number of spots have been accumulated. From such obser- 
vations, it is found that the Sun rotates about an axis so 
situated that its equator is inclined at an angle of about 7 
to the ecliptic and its equatorial plane cuts the ecliptic in 
longitudes 75 and 255. The Sun has these longitudes on 
June 6 and December 6, and on these dates, therefore, the 
Sun's equator projects on the Sun's disc as a diameter : on 
the former date it passes west to east from below to above 
the ecliptic ; on the latter date from above to below. At 
the intermediate dates, September 8 and March 8, the equator 
projects as a semi-ellipse, reaching 7 above and below the 
centre of the apparent disc respectively. The spots will 
appear to follow tracks parallel to the equator. When the 
period of rotation is deduced in this way from the motions 
of the spots, but only using spots appearing in a restricted 
range of latitude, it is found that different values are obtained 
for the period of rotation according to the mean latitude of 
the spots utilized. Therefore different parts of the Sun's 
surface do not all rotate at the same rate. The period increases 
from the equator to the poles, and in latitude 60 is greater 
than at the equator by about 20 per cent. It can, moreover, 
be proved that this is not a phenomenon belonging to the 
spots themselves, but does actually belong to the solar surface. 
The mean rotation period, for the surface as a whole, was 
determined by Carrington as 25-38 days. The explanation 
of the unequal surface motion cannot yet be definitely asserted, 


but it is thought that it may be due to the interior of the Sun 
being fluid with an increasing angular velocity of rotation 
inwards. If the inner strata are flattened at the poles rela- 
tively to the outer ones, they will approach nearer the surface 
at the equator than at the poles. Owing to their greater 
velocity, there will be a surface drag tending to increase the 
surface velocity at the equator, so that the period of rotation 
will be shorter there than in higher latitudes. 

69. Spectroscopic Evidence as to Constitution of Sun. 

Our knowledge of the constitution of the Sun is largely derived 
from the evidence afforded by the spectroscope. The spec- 
troscope ( 104) is an instrument which analyses the vibrations 
which are transmitted in a beam of light and separates them 
into their constituent vibrations. Just as a note from a 
piano is complex in nature, consisting of a fundamental tone 
together with certain overtones, so, in general, a beam of 
light is composed of a number of separate light vibrations. 
If a beam of sunlight is passed through a prism it is spread 
out into a coloured band, the colours being in the order red, 
orange, yellow, green, blue, indigo and violet. This coloured 
band is called a spectrum, and to each gradation of colour 
corresponds a definite length of wave and period of vibration 
the red end corresponding to a longer wave-length than the 
blue end. The spectroscope provides a more perfect means 
of analysis than the simple prism, and when a beam of sunlight 
is so analysed it is found that the bright band of light is crossed 
by numerous dark lines, called Fraunhofer lines, after the 
physicist who first mapped and discussed them. 

Without going into the subject in great detail, it may be men- 
tioned that spectra can be classified into three main classes, viz. : 

(1) Bright Line S2^ectra f consisting of a number of definite 
bright lines. These are produced by glowing matter in a 
gaseous condition, e.g. by volatilizing metals in the electric 
arc or by passing an electric discharge through a tube contain- 
ing gas under low pressure. 

(2) Dark Line Spectra. If a mass of glowing vapour is 
giving a bright line spectrum and light from a source at higher 
temperature is passed through it, the spectrum obtained con- 
sists of dark lines which exactly correspond in position with the 



bright lines of the bright line spectrum. The explanation 
of the formation of this type of spectrum depends upon the 
law enunciated by Kirchtroff that a body will absorb radiations 
of the same wave-lengths as those which it emits. Light 
from the source at higher temperature in passing through the 
vapour at lower temperature loses by absorption those portions 
which the latter can itself emit, and passes on deprived of 
them. The lower temperature source is naturally also emit- 
ting vibrations, but in general these are of negligible intensity 
compared with those that are absorbed and appear by con- 
trast to be absent. Such spectra may therefore be called 
absorption or reversal spectra. 

(3) Continuous Coloured Bands. Spectra of this type, 
containing no dark lines,, are emitted by glowing solids or by 
glowing gases, when submitted to great pressure. 

The bright line spectrum of any element is a characteristic 
of that element, and the presence of these lines in any other 
spectrum enables that clement to be identified as existing in 
the source producing the spectrum. 

The fact that the solar spectrum is a dark line spectrum 
indicates therefore that light from the hot interior of the Sun 
passes through a layer of lower temperature at the Sun's 
surface. The elements in this lower temperature layer can bo 
identified from the positions of the lines in the spectrum. 
Rowland made a catalogue and map of most of these lines, 
giving the positions and intensities of about 16,000 lines, which 
has enabled the following elements to be identified in the Sun's 
outer layer or atmosphere, as we may call it : 




Bismuth (?) 








Mercury (?) 




Radium (?) 





The elements are arranged in this table in the order of the 
number of lines identified in each case ; thus, iron heads the list 
with over 2,000 lines. The order does not indicate the relative 
amounts of the various elements present in the Sun. In 
Plate V is shown a portion of the Sun's spectrum. This may be 
compared with the iron arc comparison spectrum, also shown on 
the plate. All the lines in this spectrum may be identified as 
present in the solar spectrum. 

This list does not comprise all the substances contained in 
the Sun's atmosphere. Only about one quarter of the lines 
have been identified. Some belong to compounds such as 
cyanogen and ammonia, both of which contain nitrogen, 
although this clement does not itself appear in the above list, 
no lines of the element nitrogen having yet been identified in 
the solar spectrum. 

The identification of the lines is not, in general, a straight- 
forward matter. Some lines may be blends and such can be 
ascribed to more than one element, for two elements may have 
a line in almost identical positions. Other lines do not corre- 
spond exactly in position with the lines as measured in a 
laboratory, when a terrestrial source is used, on account of 
various disturbing factors. Then again, many lines which 
appear in the solar spectrum originate through absorption in 
the Earth's atmosphere and are not related in any way to the 
Sun. The separation of the terrestrial from the solar lines can 
best be made by the application of what is known as Doppler's 
principle. If there is a relative motion of the source and 
observer towards or away from one another, this principle 
asserts that the wave-lengths of the radiations received will be 
shortened or lengthened respectively, the change being small 
but proportional to the relative velocity. The principle may be 
illustrated by the rise and fall in the pitch of the whistle of a 
train as it approaches and then recedes. During the approach, 
the oncoming waves are crowded together so that the length of 
wave is shortened. If then the spectrum of light from one 
limb of the Sun, at the equator, is compared with light from the 
opposite limb, there will be a relative displacement between 
the solar lines in the two spectra owing to the rotation of the 
Sun carrying one limb away from and the other towards 
the observer. Measurement of this displacement provides a, 

THE SUN 121 

means of determining the velocity of rotation of the Sun. 
Lines which originate from absorption in the Earth's atmo- 
sphere occupy the same position in the spectra of both limbs 
and can therefore at once be distinguished from the true solar 

It is of interest to recall that helium was discovered by 
means of the spectroscope in the Sun before it was found on the 
Earth. Lockyer, in 1868, observed in the solar spectrum a 
prominent line in the yellow close to but not identical to the 
well-known sodium lines. It could not be assigned to any 
known element and was therefore ascribed to a hypothetical 
element helium (f]hoq y the Sun). 

Some years later, Ramsay, on examining the spectrum, 
emitted by an inert gas obtained from the mineral uraninite, 
found the same line and was able to identify the gas with 
Lockyer's helium. 

70. The Surface of the Sun. The surface of the Sun 
appears to us as a disc which is brighter at the centre than at 
the limb. This decrease in brightness from centre to limb is 
more easily seen in a photograph than visually, as the deficiency 
is greater in actinic light. It is due to the absorption 
in the Sun's atmosphere, the light reaching the observer from 
the limb of the Sun having to pass a greater distance through 
the Sun's atmosphere than that reaching him from the centre of 
the disc. In addition, the disc is seen, under suitable magnifi- 
cation to have a mottled or granulated appearance. These 
mottlings may be seen either visually or in a photograph. If 
two photographs are taken in rapid succession the mottlings 
on one cannot, in general, be identified on the other. They 
appear to be in rapid motion, with velocities of from 5 to 20 
miles per second, changing their form continually meanwhile. 
It is not certain what they are ; Langley regarded them as tops 
of columns in which the heated matter from the Sun's interior 
rises to the surface ; others regard them as of the nature of 
clouds. It is not even certain that their velocities represent 
real horizontal movements ; Chevalier compared them with 
the white tops of waves in a choppy sea, which are always in 
motion, but which are composed of different particles of water 
at each instant. 


Sun -Spots. Sun-spots can best be studied by pro- 
jecting with a telescope an enlarged image of the Sun upon a 
screen, or by taking a short-exposure photograph of the Sun's 
surface. Occasionally they are sufficiently large to be seen 
through a dark glass with the naked eye, but this does not 
permit of the detail being studied. 

tThe typical spot consists of an umbra or dark centre sur- 
rounded by a penumbra, in the form of a more or less complete 
ring which is darker than the surroujiding solar surface, but not 
so dark as the umbra/ Plate IV (a) shows an exceptionally fine 
group of spots. On a positive photograph such as this, the 
umbra gives the appearance of a black chasm, but it must be 
remembered that it is dark only in comparison with the 
surrounding surface, for the spot, if it could be removed from 
the Sun, would appear of intense brightness. \ The detail in the 
spot and even its general shape change considerably from day 
to dayj 

The spots are relatively short-lived : some appear and 
disappear in the course of a few days ; others survive for one 
or two revolutions of the Sun, disappearing at one limb and 
reappearing 14 days later at the opposite limb, but they never 
last for more than a few months. With but a few exceptions, 
they occur only within two zones in north and south latitudes, 
extending from the equator to 30. Near the spots may usually 
be seen bright patches on the surface which are called facute, 
though the faculse are not necessarily related to a spot and 
may be observed when no spots are to be seen. A large spot 
usually ends by breaking up into smaller spots which gradually 

The spectra of Sun-spots differ in several respects from the 
general solar spectrum, some lines being weakened and others 
strengthened. There is evidence that lines of compounds 
such as titanium oxide and magnesium hydride are present in 
their spectra : such lines do not occur in the solar spectrum 
and this would appear to indicate that the temperature in the 
spots is lower than that of the rest of the Sun, thus permitting 
the formation of compounds which would dissociate into their 
constituent elements at a higher temperature. On the other 
hand, it might possibly be interpreted as an indication of 
greater pressure in the spot. 

THE SUN 123 

Plate V illustrates the difference between the solar and 
Sun-spot spectra. The relatively greater intensity of certain 
lines in the spot spectrum is at once apparent. The comparison 
spectra o$ the iron arc and iron furnace are shown below. The 
general similarity between the furnace lines and the corre- 
sponding lines in the spot spectrum and between the arc 
spectrum and the corresponding lines in the Sun's spectrum is 
easily seen. The furnace spectrum is characteristic of a lower 
temperature than the arc spectrum. 

It was discovered by Hale, at the Mount Wilson Observatory, 
that an intense magnetic field is associated with Sun-spots. 
This discovery was made by the application of a phenomenon 
predicted by Lorentz and verified by Zeeman that if light is 
passed through a strong magnetic field, each single spectral line 
is turned into a doublet or triplet, a doublet being observed 
when the light is viewed in the direction of the lines of magnetic 
force and a triplet when viewed in the perpendicular direction. 
The differences in the wave-lengths of the separate components 
provides a means of measuring the strength of the magnetic 
fields. If the field is not sufficiently intense, the line is merely 
widened instead of being actually separated. Many of the 
lines in Sim-spot spectra appear widened-on this account, the 
two sides of the line presenting the characteristics of the 
Zeeman resolution. 

j*** ^^^"V/" 

^Hale^lso showed that many Sun-spots are surrounded by 
hydrogen vortices, 1 As will be shown in 72, it is possible 
to photograph the Sun's surface by means of hydrogen or 
calcium light and such photographs give clear evidence of 
the vortical motion. Plate IV (b) shows two spots near to one 
another, associated with vortical motion. /The direction of 
rotation of these vortices is generally counter-clockwise in the 
northern hemisphere and clockwise in the southern, i.e. in the 
same direction as occurs in cyclonic circulation on the Earthi 
The two spots, to be seen in Plate IV (6), show opposite direc- 
tions of rotation. jThe magnetic field associated with a Sun-spot 
is probably produced by the vortical motion of negatively- 
charged particles moving inwards towards the centre. ; 

St. John has found that in the lower regions of Sun-spots the 
general direction of motion is radially outwards, many sub- 
stances participating in the motion. In the upper levels there 



is a corresponding inflow, of which calcium and hydrogen are the 
chief constituents. 

The real cause and nature of Sun-spots must still be regarded 
as unknown. They are evidently violent eruptions of some 
sort and are doubtless indications of deep-seated disturbances. 
But whether they are actually eruptions from the interior, or 
whether they are chasms in the Sun's surface into which gases 
are rushing downwards, the evidence is not at present sufficient 
to decide. 

71. The Periodicity of Sun - Spots . ^If a record is kept 
of the number of spots visible on the Sun each day, or of their 
total area, it will be found that, although these figures are 
subject to irregular variations from day to day, if the averages 
are taken for fairly long periods, say for each year, the numbers 
so obtained oscillate in a well-defined manner j At the Royal 
Observatory, Greenwich, photographs of the Sun are taken on 
every possible day and the areas of all the spots shown on 
these and on other photographs obtained at the Cape of Good 
Hope are measured. The mean daily area of the spots so 
determined, expressed in units of a millionth of the Sun's visible 
hemisphere, are given for a number of years in the following 
table : 

Area Covered 

Area Covered 

Area Covered 


by Spots. 


by Spots. 


by Spots. 

1889 . 

. . 78 

1900 . 

. . 75 


. . . 64 

1890 . 

. . 97 

1901 . 

. . 29 


... 37 

1891 . 

. . 421 

1902 . 

. . 63 


. . . 7 

1892 . 

. . 1,214 

1903 . 

. . 339 


, . . 152 

1893 . 

. . 1,458 

1904 . 

. . 488 


. . . 697 

1894 . 

. . 1,282 

1905 . 

. . 1,191 


. . . 724 

1895 . 

. . 974 

1906 . 

. . 778 


. . . 1,537 

1896 . 

. . 543 

1907 . 

. . 1,082 


. . . 1,118 

1897 . 

. . 514 

1908 . 

. . 697 


. . . 1,052 

1898 . 

. . 376 

1909 , 

. . 692 


. . . 618 

1899 . 

. . Ill 

1910 . 

. . 264 


. . . 350 

It will be noticed that in 1901 there was a pronounced 
minimum in the mean daily spotted area, and that in succeeding 
years the values increased rapidly and attained a maximum in 
1905. After 1905, in spite of a temporary increase in 1907, 
the mean spotted area gradually decreased again and reached 



another minimum in 1913 ; after a lapse of 12 years the cycle 
then repeats itself, the next maximum occurring in 1917. ) Sun- 
spots have been observed and enumerated by various observers 
for about 300 years, and this fluctuation can be traced back 
throughout the records, the mean period of the cycle being 
between 11 and 12 years.) It was first pointed out by Schwabe 
about the year 1843. 

/The distribution in latitude of the spots shows a progressive 
change throughout the cycle. We have already mentioned 
that the spots occur only in two zones extending from about 

7?Uj.^^/J,^^ . 

I I f M i i t i .1 > i M n i i i r ? t i i 

FIG. 45. Distribution of Sun-Spots in Latitude throughout Sun -Spot 


latitudes to 30 on either side of the equator. At the com- 
mencement of a cycle, when the number of Sun-spots is a 
minimum, the spots occur almost entirely in high latitudes ; 
as the number of spots increases, they begin to appear in 
middle latitudes, entirely leaving the higher latitudes. Con- 
tinuing through the cycle, the mean latitude of the spots still 
further decreases and a few years after maximum the spots arc 
found almost entirely in the lower latitudes of the spot zones. 
At or near the time of minimum, spots again commence to 
appear in high latitudes and they then disappear in low 


The periodicity is well shown by Fig. 45, due to Maunder. 
In this diagram, abscissae represent time and ordinates latitudes 
on the Sun. Corresponding to the appearance of any spot, a 
vertical line is drawn in the latitude range covered by the spot 
and with the appropriate abscissa. The figure shows at a 
glance the appearance of the first spots of a cycle in high 
latitudes, the gradual extension in latitude range as the cycle 
develops and the final disappearance of the spots at the end of 
the cycle in low latitudes. The diagram, covering two cycles, 
illustrates the recurrence of these phenomena. 

Not only is the mean period of 1 1 years occupied by the cycle 
an irregular one, but also successive maxima and minima may 
differ very much in intensity. The period may be as short as 
8 years or as long as 17 years ; the maxima may be sharp and 
strongly marked or relatively flat arid weak. Attempts have 
been made to analyse these fluctuations by representing them 
as due to a main period upon which are superposed several 
subordinate periods or harmonics. A period of 33 years has, 
for instance, been strongly suspected. By means of such 
investigations, it is possible if the existence of a sufficient num- 
ber of harmonics is assumed to represent with close accuracy 
past Sun-spot records, but when the analysis is used to predict 
the future course of the Sun-spot activity, it is invariably 
found that the predictions are not verified. No other period 
than the 11-year period has, in fact, been definitely established. 
} There is a remarkable connection between the Sun-spot cycle 
and the occurrence of magnetic storms on the Earth. When 
Sun-spots are numerous, magnetic storms are relatively 
frequent ; when Sun-spots are few in number, the storms are 
rare.) The connection between them was pointed out by 
Maunder, who examined nineteen great magnetic storms 
between the years 1875 and 1903. These storms, in general, 
showed a sudden commencement and in every case there was a 
large spot near the central meridian of the Sun. Further, 
Maunder showed that magnetic storms frequently recur after 
an interval of about 27-3 days, and this is the period of the 
Sun's synodic revolution. \ If a spot is on the central meridian 
at a certain date, it will again be on that meridian after the 
lapse of 27-3 days and will then be in position to cause another 
storm. \ It must be emphasized, however, that the presence of 

THE SUN 127 

a large spot on the Sun is not necessarily an indication that a 
magnetic storm will ensue. The storm is generally held to be 
due to the emission of some form of electrically-charged parti- 
cles from the spot ; these particles are emitted in a restricted 
direction not necessarily normal to the Sun's surface and if, 
in their passage outwards, they come into the Earth's atmos- 
phere, electrical currents are produced in the upper layers which 
cause variable magnetic fields, superimposed upon the general 
magnetic field of the Earth. A magnetic storm is thus pro- 
duced. If, on the other hand, the stream of particles does not 
encounter the Earth's atmosphere, a magnetic storm will not 
follow the passage of the spot across the central meridian of the 
Sun. If this theory is correct, when the stream of particles 
reaches the Earth, the spot should have crossed the Sun's 
central meridian. This is found to be the case, the average 
time between the meridian passage and the commencement of 
the storm being about 30 hours. The theory also explains 
why a storm will tend to be followed by another one after an 
interval of 27-3 days. The motion of a spot relative to the 
Sun's disc is small and therefore if a large spot produces a 
storm and survives another rotation of the Sun, it will remain 
in a position to produce a further storm. 
) The influence of the Sun-spot activity upon the Earth's 
magnetism is also revealed in another way. On normal 
undisturbed days, the several magnetic elements do not remain 
absolutely constant but vary between certain limits during the 
course of a day. It is found that the magnitudes of the 
diurnal ranges of the elements vary throughout the Sun-spot 
cycle. If monthly means of the diurnal ranges are plotted 
against the time and the points joined up in order, an irregular 
curve is obtained ; if the local irregularities are neglected or 
smoothed out, the resulting curve follows closely the curve 
representing the monthly averages of the daily Sun-spot areasj, 
In Fig. 46 are shown curves representing the Sun-spot frequency 
and the mean diurnal ranges of magnetic declination and 
horizontal force at Greenwich, for the period 1841 to 1896. 
The periodic nature of the Sun-spot frequency is clearly shown. 
The fidelity with which its maxima and minima are reproduced 
at the same epochs by the magnetic curves is surprising ; many 
even of the minor fluctuations are reproduced. 



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THE SUN 129 

The actual significance of the Sun-spot cycle which, as we 
shall see, can be traced in other solar phenomenon, is not known. 
It must be the outward evidence of something of a deep-seated 
nature going on in the Sun. There are many stars, or other 
suns, whose light is variable ; in some cases, the variation is 
due to the orbital motion of two bodies of unequal brightness, 
in others it is intrinsic. In some cases, the variation is perfectly 
regular, in others, highly irregular. It appears probable that 
we are justified in regarding the Sun as an irregularly variable 
star, with a very small range of variation. But the cause of the 
variation is not at present known. 

72. The Spectroheliograph. Our knowledge of the Sun's 
surface has been much increased by an instrument called the 
spectroheliograph. When the slit of a spectroscope is pointed 
to a certain portion of the Sun's disc, any line in the spectrum 
obtained indicates the presence and state of a certain element in 
that portion of the disc. The spectroheliograph enables the 
same information to be obtained for the whole disc. If the 
light from a certain line is allowed to pass through a second slit, 
in the focus of the spectroscope, on to the photographic plate, 
and the first slit be moved across the image of the Sun, the 
second slit moving correspondingly, then if the two slits are 
sufficiently long to extend across the image of the Sun, a 
photograph of the Sun will be obtained which is produced by a 
single radiation only. The photograph will therefore give a 
representation of the distribution of that substance over the 
Sun's surface. Alternatively, instead of moving the slits, 
the image of the Sun may be caused to travel slowly and 
uniformly across the first slit, the photographic plate moving 
in unison behind the second slit. This is the principle of 
the spectroheliograph. For most lines, very high dispersion 
is required to prevent the light of the adjacent continuous 
spectrum from blotting out the faint image. It has been 
found, however, that there are certain lines which appear 
" reversed " over certain portions of the Sun's disc, i.e. super- 
imposed on the dark absorption line is a bright line. Typical 
lines showing this effect are the Ha and H/3 hydrogen lines and 
the K calcium line. The dark and bright lines probably 
represent different layers in the sun's atmosphere, the dark 


line being due to absorption in a lower level and the bright line 
to incandescent matter at a higher level, e.g. such as is revealed 
in prominences. In taking photographs with the spectrohelio- 
graph, the bright reversals are utilized as they are easily photo- 
graphed, the Ha line of hydrogen and the K line of calcium being 
generally used. These photographs represent the distribution 
of calcium and hydrogen clouds in the Sun's atmosphere, and 
reveal many interesting phenomena. The faculae surrounding 
a spot are shown much more extensively in the calcium light 
and were called " flocculi " by Hale ; they consist mainly of 
glowing calcium clouds. The prominences which can frequently 
be seen at the edge of the Sun's disc appear in the photographs 
in hydrogen light projected on the disc, whatever their position 
on the surface. The hydrogen vortices surrounding many 
Sun-spots arc also seen in these photographs. 

In Plate VI are reproduced two spectroheliogranis of the Sun, 
photographed on the same day with the K 3 ray of calcium and 
the Ha ray of hydrogen respectively. The correspondence of 
the dark markings in the two photographs may be noted. On 
the limb near the east point is an area of f acute, clearly shown 
in the photograph taken in calcium light. The coarser struc- 
ture of the calcium clouds than of the hydrogen clouds is very 

73. Solar Prominences. -The subject of solar eclipses will 
be dealt with in detail in the next chapter. For the present, it 
is sufficient to remark that from time to time the Moon just 
obscures the Sun's disc over a certain region of the Earth and 
observers in such a region are then able to see the immediate 
surroundings of the Sun's disc. One phenomenon thus revealed 
is the existence of prominences or enormous tongues of flame 
standing out from the Sun's limb and often reaching to very 
great heights. For some time after the discovery of promin- 
ences, it was thought that it was only possible to observe them 
at the time of eclipse. At other times the glare produced by 
the scattering of sunlight by the Earth's atmosphere renders 
them invisible. In 1868, Lockyer and Janssen independently 
discovered that it is possible to observe them at any time, 
without waiting for a total eclipse. \Using a spectroscope with 
a high dispersion, the diffused light is spread out into a band of 

THE SUN 131 

weak intensity which does not obscure the spectrum of the 
prominences, since the latter consists only of a few bright lines. 
If the outskirts of the limb are searched with the spectroscope, 
a prominence will be detected by the hydrogen lines flashing 
out bright and if the slit is then opened, the whole figure of 
the prominence may be seen/ Prominences, which are not 
on the limb, can be detected in spectroheliograms taken in Ha 
light, but it is when they reach the limbs that they can best bo 

Much information in regard to the solar prominences has 
boon gained in recent years. ^As in the case of Sun-spots, their 
distribution is found to vary in the course of the Sun-spot cycle. 
They are found to occur predominantly in two distinct belts, 
both north and south of the equator. The low latitude belts 
are in the same latitudes as the Sun-spot zones and vary in 
activity in a similar manner, drawing in towards the equator and 
gradually dying out as the cycle progresses. The high-latitude 
zones decrease in activity at Sun-spot minimum, but do not 
then disappear and as the solar activity increases, they move 
towards the poles, where they die out at about the time of 
Sun-spot maximum. Some prominences occur in the neigh- 
bourhood of spots, but those in the high-latitude belts cannot 
show any such connection : it is found that the majority of 
those in the low-latitude belts also are not connected with 
spots J 

Prominences vary enormously in shape, size and behaviour. 
Their forms have been studied by Evershed, who has classified 
them into two broad groups. One main group includes small 
prominences in the form of rockets, bright jets and arches and 
metallic prominences (i.e. those whose spectra contain metallic 
lines). Such prominences can usually be definitely connected 
with a spot ; the young and active spots are most frequently 
found associated with prominences, but old spots rarely so. 

- The other broad group includes the large massive forms, long 
groups, pyramids and columns. These types are rarely, if ever, 
associated with spots. They are usually long-lived and may re- 
appear for several rotations, but frequently break up suddenly. 
When they break up in this manner, they are termed eruptive 
prominences. In 1917, a prominence was observed in Kashmir 
and at Kodaikanal, S. India, which in breaking up rose to a 


height of 18' -5 or over half a million miles before fading away. 
The speed of motion was measured and the greatest velocity 
attained was 457 kms. per second. Such eruptions indicate 
the presence of powerful forces which may last for several hours 
and which apparently neutralize gravity, for {he matter is not 
seen to descend again, but fades away at a great height. 

In Plate VII are reproduced two photographs of prominences 
obtained at the total eclipse of 1919, May 29, at Sobral, in 
Brazil and Princes Island respectively. The interval between 
the two photographs was only 2 hours, but the change in the 
form of the prominence during this interval is very marked. 
This prominence was kept under observation on the day after 
the eclipse at the Yerkes Observatory and its break-up Was 

The history of this prominence may be taken as character- 
istic of that of large eruptive prominences in general. It was 
first observed on the east limb on March 22 ; the end was in 
latitude 35 and it extended northwards 13. Successive 
returns were observed, with one exception, until May 27, the 
prominence meanwhile growing gradually in height and 
intensity. On May 27, the crest of the prominence was seen 
just coming over the limb ; it then extended in latitude 
through nearly 40 and its height was l'-5. On the 28th, the 
prominence had come further into view and the height had 
increased to 2'-7. On the 29th, this had increased to 4'-5 and 
the prominence extended in latitude from 42 to 6. The 
north end afterwards broke away from the limb ; this occurred 
between the times at which Plate VII (a) and (b) were taken, and 
in Plate VII (b) the detached end is clearly seen. The streamers 
in the centre of the prominence descended to a spot in latitude 
+ 6-6. At 2 h. 50 m. G.M.T. the south end of the prominence 
began to break away and 20 minutes later had entirely parted 
from the limb. After that the prominence commenced to rise 
rapidly, though it remained connected with the spot by faint 
streamers. At 3h., its height was about 220,000 kms. ; at 
4h., 250,000 kms.; at 5h., 300,000 kms.; at 6h., 360,000 
kms. ; at 7h., 490,000 kms. ; at 8h., 670,000 kms. It was 
last seen at 8 h. 23 m., at a height of 760,000 kms. (17') above 
the surface, a distance exceeding half the diameter of the Sun. 
The two ends of the column were observed at each return until 

THE SUN 133 

August 5, the total life of the prominence therefore exceeding 
4 months. 

o Observation appears to indicate that prominences, even the 
largest, are very tenuous. If this is so, they cannot possess a 
temperature in the ordinary sense, but are luminous on account 
of the absorption of solar radiation. There is not much 
information available as to their speed of rotation, but the 
indications are that they rotate at a faster speed than the 
surface of the Sun, and that the speed of rotation decreases 
with increase of latitude. 

74. The Chromosphere. By bringing the slit of a spectro- 
scope tangential to the Sun's limb, the existence of a layer all 
round the Sun of the same constitution as the prominences can 
be revealed. This layer is called the chromosphere. It is 
immediately above that portion of the Sun where the absorption 
which produces the dark Fraunhofer lines mainly takes place. 
The chromosphere can be most favourably studied at the time 
of a total solar eclipse. As the Moon moves in front of the 
Sun's disc, just before totality commences, the absorbing layer 
is covered and if the limb of the Sun is being examined with a 
spectroscope it is seen that a bright line spectrum appears for 
an instant. This " flash " spectrum, as it is termed, is the 
spectrum of the chromosphere. Immediately afterwards, the 
chromosphere itself is blotted out. Similarly, immediately 
afber totality, the flash spectrum may again be observed. 

The chromosphere contains most of the elements which are 
found in the Sun : its bright line spectrum is not, however, an 
exact replica of the dark line spectrum of the Sun. There are 
differences, due to the physical conditions in the two cases not 
being the same. Thus, for example, the bright lines of helium 
are found in the chromospheric spectrum, whereas dark helium 
lines have not been detected in the Fraunhofer spectrum. 
Lines of some elements such as hydrogen, titanium, chromium, 
etc., though found in the solar spectrum, are relatively stronger 
in the chromospheric spectrum, whilst, on the other hand, the 
lines of such elements as iron, nickel, cobalt, manganese and 
sodium are relatively stronger in the solar than in the chromo- 
spheric spectrum. The differences in intensity are accentuated 
in the case of what are termed enhanced lines. These are lines 


which, in the laboratory, arc found to be more intense in the 
spark spectrum of an element than in its arc spectrum, from 
which they may even be absent, and, for this reason, their 
increased brilliancy in the spark spectrum is generally regarded 
as the effect of the increased temperature. This can hardly be 
the cause which increases their intensity in the chromosphere ; 
it appears more probable that in this case the enhanced lines 
are due to ionised atoms at a relatively great height, so that the 
pressure is much reduced. 

The absorbing layer which produces the dark lines of the 
Fran nhofer spectrum is generally called the " reversing " layer : 
it is immediately beneath the chromosphere but mingles 
gradually with it, so that the two cannot be sharply separated. 
The reversing layer might, in fact, be regarded as the lower 
layer of the chromosphere. By using a slitless spectroscope, 
the lines in the flash spectrum appear as curved arcs of different 
lengths and by measuring the lengths of these arcs, the depth 
of the chromosplieric layer may be calculated. It is found to 
extend up to a height varying from about 6,000 to 14,000 
kilometres above the photosphere. The reversing layer to 
which most of the lines of the flash spectrum are due has a 
depth of from COO to 1,000 kilometres. 

75. The Corona. At the time of a total solar eclipse, 
the instant totality commences a bright aureole surrounding 
the Sun flashes out. This is called the " corona.' 5 Its light 
is relatively so faint that no method has been discovered by 
which it can be observed at any other time than during totality. 
The li^ht of the corona is pearly white and somewhat brighter 
than the light of the full moon A On long exposure photographs, 
the corona is sometimes found to extend from the Sun to a 
distance of two or three solar diameters. The structure of the 
corona is very complex ; it has no definite boundary and is 
usually symmetrical with respect neither to the centre of the 
Sun nor to the Sun's polar axis. Its general shape is found to 
vary with the Sun-spot cycle in a very marked way. At a 
time of Sun-spot maximum, it is compact, without very long 
streamers and more or less uniformly distributed around the 
Sun's disc, so that the direction of the Sun's polar axis is not 
specially indicated. At a time of Sun-spot minimum, on the 

THE SUN 135 

other hand, the two poles are indicated by a number of short 
streamers or tufts issuing from them and suggesting lines of 
force near the two poles of a bar magnet : from the equatorial 
zones stretch curved streamers, reaching to great distances. 
Such a corona is shown in Plate VIII (6), Sun-spot activity being 
a minimum in 1901. At other times the form of the corona is in- 
termediate between these two types. The corona in Plate VIII 
(a) was photographed shortly after Sun-spot maximum and is of 
intermediate type. It will be noticed that the polar tufts are 
much less prominent than in the corona of 1901. So regularly 
do the types recur, that it is possible to predict with consider- 
able accuracy the form of corona which may be expected at a 
future eclipse. It may be mentioned that the existence of the 
polar tufts corresponds with the time when the high-latitude 
prominence zones die away, whilst when the prominences reach 
the poles, the uniform type of corona occurs. 

The structure of the inner corona is very complicated, show- 
ing numerous filaments and curved arches, especially in the 
neighbourhood of prominences or spots which are at or near the 
limb. The corona of 1901 (Plate VIII [b]) shows very interest- 
ing detail in the inner corona. It is not known whether this 
structure persists for any length of time or changes rapidly. 

The light from the corona has been found to be partially 
polarized, i.e. the vibrations which constitute it show a 
predominance for certain directions instead of occurring in all 
directions at random. This is known to bo a characteristic 
of light reflected from small particles. Much of the coronal 
light is, therefore, reflected sunlight. The spectrum of the 
corona consists of a number of bright lines, superposed upon 
a faint continuous background. These bright lines do not 
correspond with the lines of any known element and they have 
therefore been attributed to a hypothetical element which has 
been named coronium. A line in the green part of the spectrum 
is usually very prominent in the coronal spectrum, but in the 
eclipse of 1914, this line was scarcely visible and a previously 
unknown line in the red region was very prominent. 

76. Solar Radiation and Temperature. The determina- 
tion of the temperature of the Sun is closely bound up with 
the determination of what is termed the solar constant. The 


Sun is continually radiating energy into space and of this 
energy only a small part is intercepted by the Earth and 
planets, the remainder passing outwards into space. Of that 
portion which falls on the Earth, a large part is absorbed by 
the Earth's atmosphere. The solar constant ie defined as the 
quantity of heat, measured in calories, which would fall in one 
minute on an area of one square centimetre placed per- 
pendicularly to the radiation at the surface of the Earth, if the 
Earth had no atmosphere and was at its mean distance from 
the Sun. The determination of the constant comprises two 
essentially different problems : first, the determination of the 
amount of energy actually falling on unit area at the Earth's 
surface in one minute ; secondly, the determination of the 
absorption of energy in the Earth's atmosphere. The method 
adopted for determining the actual amount of energy per unit 
area at the Earth's surface consists in allowing the radiation 
to fall on a body which absorbs it and measuring the quantity 
of heat gained by the body during a certain time. To enable 
the measurement to be performed accurately and to avoid 
possibilities of error, a specially designed instrument called 
the pyrheliometer is used. The measurement of the absorption 
in the Earth's atmosphere is a more difficult problem. The 
principle employed consists in measuring the radiation at 
different times of day : the length of the path of the light 
through the Earth's atmosphere decreases from sunrise to 
mid- day and then increases again until sunset. From the 
variations in the amount of radiation passing through with 
change in length of path the total absorption can be estimated. 
It is advantageous to make the observations at a high altitude, 
as then the loss by absorption when the Sun reaches the 
meridian will be relatively small. This method suffers from 
the disadvantage that the light is treated as though homo- 
geneous, whereas in fact the absorption differs in amount for 
the different wave-lengths. Langley devised a method to 
overcome this defect : he employed an instrument called a 
spectrobolometer, which measures the distribution of energy 
amongst the different wave-lengths. If then observations are 
taken with this instrument for various altitudes of the Sun and 
the total energy received is measured with the pyrheliometer, 
the total correction to allow for absorption can be calculated. 

THE SUN 137 

Langley's work has been continued by Abbot and Fowle, 
who find for the mean value of the solar constant 1-93 calories. 
They have shown that this value changes slightly from day 
to day, the changes being confirmed by simultaneous obser- 
vations made at two different stations. The value of the 
constant is greater at the time of Sun-spot maximum than at 
Sun-spot minimum, but superimposed on this variation are 
fluctuations of shorter period, the cause of which are still being 

The value of the solctr constant having been determined, 
the total amount of energy emitted by the Sun can be calcu- 
lated. Imagine a sphere of radius 93 million miles, with the 
Sun at its centre. Then each square centimetre of the surface 
of this sphere receives in one minute 1-93 calories, all of which 
is radiated from the Sun. Since the surface of the Sun is 
1 /46000th of the outer sphere, each square centimetre of the 
Sun's surface must radiate heat at the rate of 1-93 X 46,000 or 
89,000 calories per minute. Every square centimetre thus 
radiates at the rate of a 9 H.P. engine. 

The effective temperature of the Sun can be calculated when 
the solar constant is known. By this term is meant the tem- 
perature of a perfect radiator (the so-called " black " body) of 
the same size as the Sun which is emitting radiation at the 
same rate : such a body would absorb all the radiation falling 
upon it without reflecting (hence the term black) or trans- 
mitting any. The radiation of a black body is proportional 
to the fourth power of its temperature (Stefan's Law), so that 
when the total radiation is known, the effective temperature 
can be determined. It is found to be about 5,600 C. The 
actual temperature of the Sun cannot be less than this value : 
the temperature of the photosphere is probably somewhat, 
though not greatly, higher than 5,600 C. The interior 
temperature, of course, must be very much higher. The 
temperature may be compared with that of the electric arc, 
which is about 3,700 C. 

77. Maintenance of the Sun's Heat. We have seen that 
each square centimetre of the Sun's surface is continuously 
radiating energy at the rate of a 9 H.P. engine : this figure 
corresponds to the enormous total rate of radiation of 


0-58 x 10 24 horse power. If this energy were derived solely 
from the internal store of heat in the Sun, its temperature would 
fall by more than one degree each year. If this were so, the 
future life of the Sun regarded as a source of heat would be 
only a few thousand years. We know, on the other hand, from 
geological considerations that organic life has existed on the 
Earth for many millions of years and dining that period the 
temperature of the Sun cannot have decreased at a rate nearly 
approaching one degree per year. On the contrary, the 
evidence points to the temperature not having greatly altered 
during that period. There must, therefore, be some means by 
which the Sun is able to replenish its store of heat. By what 
process this is achieved has for many years been a matter of 

The theory propounded by Mayer supposed the heat to come 
from the impact of meteors on the Sun. A meteor pulled 
from a great distance into the Sun would acquire a velocity 
of 400 miles per second and its energy would, by the collision, 
be transformed into heat. A quantitative calculation shows 
that this theory is untenable : on any plausible assumption as 
to the quantity of meteors which might be drawn into the Sun, 
the heat so produced would be but an infinitesimal fraction of 
the amount required. Helmholtz proposed an alternative 
theory : the attractive force of the Sun as a whole on a particle 
at its surface will tend to pull it inwards, and as the Sun is 
gaseous, it follows that it will gradually contract under the 
influence of its own gravitation. The effect of this contraction 
will be to generate heat, the process being analogous to the 
generation of heat by the impact of meteors, the meteors now 
being replaced by the outer layers of the Sun which are gradu- 
ally falling in towards the centre. Helmholtz calculated that 
a diminution in the Sun's radius of 75 metres per year would 
liberate sufficient energy to balance that radiated as heat. 
Such a contraction would only produce a decrease in the Sun's 
apparent radius of one second of arc in 29,500 years, and there- 
fore could not be detected at the present time by astronomical 
observation. Supposing this rate of contraction to continue 
unaltered, the Sun would contract until its density was equal 
to that of the Earth in another 17 million years, and we should 
be forced to the conclusion that the Earth would not receive 

THE SUN 139 

sufficient heat to maintain life on its surface for many million 
years longer. 

We can, however, probe backwards and inquire how long 
the Sun can have been radiating heat at its present rate. If 
we suppose that there was a stage when the matter composing 
the Sun was dissipated in the form of a very tenuous nebula, it 
may be calculated that to reach its present state would only 
have required about 22 million years. The Earth is supposed 
to have been formed in the distant past from the Sun, so that 
this figure should give an upper limit for the age of the Earth. 
This period is not nearly sufficient to account for the geological 
processes which have taken place in the Earth, and many lines 
of argument based upon geological considerations combine in 
requiring a much longer period. Other sources of energy must, 
therefore, be looked for : after the discovery of radium and of 
the fact that one gram of radium is continually radiating heat 
at the rate of 138 calories per hour, it was thought that liber- 
ation of heat by radio-active processes in the Sun might account 
for the maintenance of the Sun's radiation. Although the 
existence of radium in the Sun has not been definitely estab- 
lished, one of the transformation products resulting from the 
disintegration of radium, viz. helium, is known to be present 
in the Sun. If each cubic metre of the Sun contained 3-6 grams 
of radium, the present rate of radiation of the Sun could be 
accounted for in this way alone. It is now believed that the 
energy obtained from radio-active processes can be compara- 
tively unimportant. Rutherford has shown that if the Sun 
were made entirely of uranium, only about 5 million years 
would be added to its duration as a heat-giver. Other specu- 
lations have been advanced as to the source of the Sun's heat, 
but no known cause is capable of accounting for the age required 
by geological considerations. It seems probable that it must 
be attributed to a process which is capable of liberating energy 
at the high temperatures (of the order of millions of degrees) 
which prevail in the interior of the Sun, but with which we are 
unacquainted in the laboratory. The mutual destruction of 
electrons by collision, resulting in the transformation of their 
mass into energy, has been suggested as one possible process, 
but at present this can only be regarded as plausible specu- 
lation. Nevertheless, it is certain that to account for the age 


of the Earth required by geologists of the order of 300 million 
y ears n other alternative is open to us but to suppose that 
some such sub-atomic process is occurring, which is conditioned 
by the enormously high temperature existing in the interior of 
the Sun. 


78. Cause of Eclipses. If the orbit of the Moon were 
in the ecliptic, so that the Sun ? Moon and Earth all moved in 
the same plane, then twice in each lunation, the Moon would 
cross the line joining the centres of the Earth and Sun, the 
times of crossing being the times of conjunction (new moon) 
and opposition (full moon). 

Referring to Fig. 47, it will be seen that at opposition, the 
shadow of the Earth E cast by the Sun 8 would then fall on the 

FIG. 47. The Occurrence of Eclipses of Sun and Moon. 

Moon M , and a simple calculation, involving the relative sizes 
and distances of the three bodies, will show that at the distance 
of the Moon, the diameter of the shadow cone is greater than 
the diameter of the Moon. Since the Moon is not self-luminous, 
it would therefore be completely blotted out or eclipsed. The 
eclipse would be visible at all places on the hemisphere of the 
Earth which is turned away from the Sun. 

At conjunction, the shadow of the Moon might or might not 
fall on the Earth according to the distance of the Moon M. 
Should it do so at every place on the Earth within this shadow, 



the Moon would completely obscure the Sun, so that there 
would be a total solar eclipse. It is apparent without calcu- 
lation that the eclipse will not be visible over an entire 
hemisphere of the Earth. In order that the shadow cone of 
the Moon might at the distance of the Earth have a greater 
diameter than that of the Earth, the Moon would obviously 
need to be larger than the Earth. A total solar eclipse can, 
in fact, only be visible within a comparatively small region 
of the Earth's surface. At adjacent points, the Moon will 
only partially obscure the Sun's disc, and at such points there 
is said to be a partial solar eclipse. 

Owing to the variation in the distance of the Moon from the 
Earth, the apex of the Moon's shadow cone occasionally falls 
between the Moon and the Earth. In such cases, the eclipse 
is not total at any point on the Earth's surface, but at all 
points on the Earth within the continuation of the shadow cone 
the Moon will be seen projected upon the Sun's disc, but it will 
be of smaller apparent diameters and its black disc will, there- 
fore, appear surrounded by a narrow, bright ring. Such an 
eclipse is called an annular eclipse. 

If, then, the orbit of the Moon were in the ecliptic there would 
be a total lunar eclipse and a total or annular solar eclipse once 
in each lunation. We know, however, that the orbit of the 
Moon is inclined at an angle of 5 9' to the ecliptic and as the 
angular diameter of the Moon is only about 30', there can be no 
eclipse either at conjunction or at opposition unless the centre 
of the Moon is within an angular distance of about 30' from the 
ecliptic. The Moon will, therefore, in general pass either 
under or over the common tangent cone of the Earth and Sun. 
For an eclipse to be possible the Moon must be sufficiently near 
to one of the nodes of its orbit. 

To ascertain when an eclipse will occur, the times of lunar 
conjunctions and oppositions must be first determined. These 
are the times of new and full moon and occur when the geo- 
centric longitudes of the Sun and Moon (i.e. the longitudes as 
measured by an observer at the centre of the Earth) are equal 
or differ by 180. The positions of the Moon in its orbit at 
these times must be found, and if it is within certain limits of 
angular distance from a node an eclipse is possible. Such 
limits are called eclipse limits. 


79. The Saros. It follows, from the previous section, 
that if for certain positions of the Sun, the Moon and the node 
of the Moon's orbit an eclipse can occur, then another eclipse 
would occur if they returned again to their same relative posi- 
tions. It can easily be shown that they will return in this 
way after a period of 18 years and 11 days. This period is 
called the Saros and was known to the Chaldeans and used by 
them for predicting eclipses. Although they had no accurate 
tables of the Sun and Moon, they were nevertheless able to 
foretell with considerable accuracy the occurrence of an eclipse. 
The period is still used as a rapid means of deciding at which 
conjunctions or oppositions eclipses will occur, the precise 
data of the eclipse being then calculated by modern methods. 

The Raros period can be verified as follows : 

The period of revolution of the Moon relative to the SUD 

= 29-53059 days. 
The period of revolution of the node relative to the Sun 

- 346-62 days. 

For since the daily retrograde motion of the node is 3' 10-64" 
and the mean motion of the sun is 59' 8-33", the relative daily 
motion is 62' 19", requiring 346-62 days for a complete revo- 
lution. Two hundred and twenty-three lunations therefore 
occupy 6,585-32 days, and 19 synodic revolutions of the node 
occupy 6,585-78 days. These periods are very nearly equal, 
so that after 223 lunations or 18 years, 11 clays (or 18 years, 
10 days, if 5 leap years intervene), the Sun, Moon and node 
return to the Scimc relative positions. 

The following table gives the dates of lunar and solar eclipses 
for the years 1914 to 1950 inclusive ; 

Ascending Node. 

Descend ing Node. 







Mar cli 1 1 

Feb. 25 

Sept. 4 

Aug. 21 


Feb. 14 

Aug. 10 


Jan. 20 

(Feb. 3 \ 
(Dec. 241 

July 15 

July 30 


(Jan. 8 ) 
(Dec. 28 j 

( Jan. 23) 
( Dec. 14) 

July 4 

(Juno 19| 
(July 19 / 


Dec. 3 

June 24 

June 8 


Nov. 8 

Nov. 22 


May 29 


Oct. 27 

Nov. 10 

May 3 

May 18 


Oct. 16 

Oct. 1 

Apr. 22 

Apr. 8 


Sept. 21 

March 28 


Aug. 26 

Sept. 10 

March 3 

March 17 


Aug. 14 

(July 31) 

(Aug. 30{ 

Feb. 20 

March 5 


Aug. 4 

July 20 

Feb. 8 

Jan. 24 


July 9 

Jan. 14 


June 15 

June 29 

Dec. 8 

(Jan. 3 ) 

JDcc. 24) 


Juno 3 

(May 19 \ 
(June 17) 

Nov. 27 

Nov. 12 


May 9 

Nov. 1 


April 13 

April 28 

Oct. 7 

Oct. 21 


April 2 

April 18 

Sept. 26 

(Sept. 12 [ 
(Oct. 11 J 


March 22 

March 7 

Sept. 14 

Aug. 31 


Feb. 24 

Aug. 21 


Jan. 30 

Fob. 14 

July 26 

Aug. 10 


Jan. 19 

( Jan. 5 ) 
<|Feb. 3 I 
(Dec. 25 J 

July 16 

/June 30 ) 
(July 30 } 


Jan. 8 

Dec. 13 

July 4 

June 19 


Nov. 18 

Dec. 2 

June 6 


Nov. 7 

Nov. 22 

May 14 

May 29 


Oct. 28 

Oct. 12 

May 3 

April 19 


Oct. 1 

April 7 


Sept. 5 

Sept. 21 

March 13 

March 27 


Aug. 26 

(Aug. 12 } 
(Sept. 10) 

March 3 

March 16 


Aug. 15 

Aug. 1 

Feb. 20 

Feb. 4 


July 20 


Jan. 25 


Juno 25 

July 9 

Dec. 19 

Jan. 14 


June 14 

/May 30) 
(Juno 29 j 

Dec. 8 

(Jan. 3 ) 

(Nov. 23 j" 


Juno 3 

May 20 

Nov. 12 


April 23 

May 9 

Nov. 1 


April 13 

April 28 

Oct. 7 

Oct. 21 


April 2 

March 18 

Sept. 26 

Sept. 12 



The table comprises two complete Saros periods and an 
inspection of it will illustrate the manner in which the eclipses 
repeat themselves after an interval of 18 years, 11 days. 

80. The Number of Eclipses in one Year. In order 
that a lunar eclipse may occur, the distance of the Moon from 
one of its nodes, at the moment of full moon, must not exceed 
12J. This expresses the condition that the latitude of the 
Moon's centre should be equal to the sum of the angular semi- 
diameters of the Moon and of the shadow-cone of the Earth : 
for the latitude of the Moon when 12J from a node is 64', the 
maxmium semi-diameter of the shadow is 47' and the maximum 
semi-diameter of the Moon is 17'. If, then, the Moon is farther 
from the node than 12J, that limb of the Moon which is nearest 
the ecliptic cannot come within the shadow. It can also be 
shown that, if the distance of the Moon from a node be less 
than 9, there must certainly be a lunar eclipse, for in such 
case, the latitude of the Moon's centre will be less than 52' or 
less than the sum of the minimum semi-diameters of the shadow 
(38') and of the Moon (14'). The two distances of the Moon 
from a node at conjunction within the lesser of which an eclipse 
must occur and beyond the greater of which an eclipse is 
impossible are called the lunar ecliptic limits. If the distance 
of the Moon from a node at opposition lies between these limits 
an eclipse may or may not occur, according to the value of the 
diameters of the shadow-cone and of the Moon at the instant. 

Similarly, in order that a solar eclipse may be possible, the 
angular distance of the centre of the Sun from one of the 
Moon's nodes must not exceed 18|,and if the distance is less 
than 13-J there will certainly be an eclipse. These are the 
solar ecliptic limits. 

Using these limits, we can discuss the number of eclipses that 
are possible in one year. 

We have seen that the daily relative motion of the Sun and 
the Moon's node is 62' 19" ; it follows that in 14| days, which is 
the interval between full moon and new moon, the Sun and the 
node will separate by 15|. 

If, then, conjunction occurs exactly at the node there will be 
a solar eclipse ; at the preceding and following oppositions the 
Moon will be 15j from its node ; this distance being outside 



the lunar ecliptic limit, there will not be a lunar eclipse at either 
of these oppositions. Hence near the passage of the Sun 
through the node, there will be one solar, but no lunar eclipses. 

If, on the other hand, opposition occurs exactly at the node 
there will be a lunar eclipse ; at the preceding and following 
conjunctions, the Moon will be within the superior solar 
ecliptic limit, so that at both conjunctions a solar eclipse may 
occur. Corresponding therefore to the passage of the Sun 
through the node in this case, there may be two solar and will 
be one lunar eclipse. 

The same results can easily be shown to hold, if conjunction 
or opposition occur within 2 days on either side of the node. 

Now the Sun takes 173 days to pass from one node to the 
other, and six lunations occupy 177 days. If, therefore, a 
lunar eclipse occurs exactly at one node, there may be two solar 
eclipses near that node, and there will also be another lunar 
eclipse 4 days after the Sun passes through the next node, 
but the Sun is then too far from the node for three eclipses to 
happen near that node, though one solar eclipse may happen at 
the preceding new moon. If, however, a lunar eclipse happens 
2 days before the Sun reaches a node, there will be a lunar 
eclipse at the next node 2 days after the Sun has passed it. 
It is possible, then, for three eclipses to occur at each node. 
The Sun returns again to the first node and opposition will 
occur 6 days after its passage through the node : this will give 
a lunar eclipse and also a solar eclipse at the preceding new 
moon, but a solar eclipse at the subsequent new moon is 

Now the solar eclipse at this new moon will occur exactly 
12 lunations later than the first solar eclipse of the two groups 
of three, which we have mentioned. But 12 lunations occupy 
354 days, so that if the first eclipse occurs early in January, 
it is possible to have seven eclipses within the year. Further, 
12-|- lunations occupy 368f days, so that the eighth eclipse 
cannot come in. If we shift the whole series back so as to 
bring in this lunar eclipse, the first solar eclipse would be dis- 
placed unto the December of the previous year. We may, 
therefore, have either 5 solar and 2 lunar or 4 solar and 3 lunar 
eclipses within a year, but it is not possible to have more than 
7 eclipses in all in any one year. 


By similar considerations, it may be shown that it is possible 
to have a single solar eclipse near one node followed by a single 
solar eclipse near the other node. There cannot be less than 
one eclipse at each node. Therefore, there cannot be fewer 
than two eclipses in any year and if only two occur, both will 
be solar. 

These conclusions are illustrated by the table in 79. This 
table comprises two Saros periods and one year in each has 
7 eclipses, 1917 having 3 lunar and 4 solar eclipses and 1935 
2 lunar and 5 solar eclipses. In no year are there fewer than 
2 eclipses and when there are only two these are invariably 
both solar eclipses. 

Owing to the solar ecliptic limit being larger than the lunar, 
there must be more solar than lunar eclipses. In the Saros 
period there are in fact, on the average, 41 solar eclipses and 
29 lunar eclipses. The number of solar eclipses visible at any 
given spot on the Earth's surface is, nevertheless, fewer than 
the number of lunar eclipses. This is due to the latter being 
visible over one half of the Earth's surface, whilst the former 
are visible only over a small portion. 

The table in 79 shows how the eclipses occur at two seasons 
of the year separated by an interval of nearly 6 months : these 
periods correspond to the passage of the Sun through the nodes 
of the Moon's orbit. Owing to the retrograde motion of the 
nodes, each eclipse season occurs on the average about 19 days 
earlier than in the previous year. 

81. Recurrence of Eclipses. Mention has already been 
made of the Saros period of 18 years, 11 days, after which 
eclipses occur very nearly in the same order. It is of interest 
to note further that in this period the longitude of the Sun 
increases by only 11 and the distance of the Moon from its 
perigee has changed by less than 3 : the recurring eclipses 
are therefore nearly of the same kind, total, annular or partial, 
for a number of returns. 

Considering the recurrence of any particular solar eclipse in 
successive cycles, an eclipse will first occur when the line of 
conjunction makes an angle of about 16 with the line of 
nodes : the Moon will then just touch the shadow-cone and the 
eclipse will therefore be a small partial eclipse, visible necessarily 


only in very high north or south latitudes. After a period of 
18 years, 11 days, the eclipse will recur, but the line of con- 
junction will then be nearly half a degree nearer the line of 
nodes, on account of the half -day difference in length between 
the 223 lunations and the 19 synodic revolutions of the node 
which constitute the Saros cycle. The eclipse will again be 
partial. These partial eclipses will recur until after about 
ten cycles the line of conjunction makes a sufficiently small 
angle with the line of nodes, and the eclipse will then be 
total, but visible only in polar regions. The eclipse in 
successive returns will continue to be total, but will occur 
nearer and nearer the equator until the line of conjunction 
passes near the node : the path of totality will then 
cross in the equatorial regions. This occurs after about 22 
cycles. A further 22 cycles will take the eclipse to the opposite 
pole, the scries of total eclipses then ending and being succeeded 
by about 10 partial eclipses. There will therefore be in all 
about 20 partial eclipses and 44 or 45 total eclipses, the cycle 
comprising altogether about 1,200 years. 

82. Lunar Eclipses. In Fig. 48, 8 and E represent the 
centres of the Sun and Earth respectively and AC, BD the 
common external tangents, AD, BO the internal tangents to 
the sections of the Sun and Moon by the plane of the paper. 
At any point in the cone CPD, the Earth entirely cuts off the 


FIG. 48. To illustrate a Lunar Eclipse. 

light from the Sun : this portion of the shadow is called the 
umbra. Between the umbra and the cone, bounded by the 
internal tangents, is a region at any point of which the light 
from only a portion of the Sun is cut off : this portion of the 
shadow is called the penumbra. 
When the limb of the Moon enters the penumbra at L, there 


is a gradual fading of its light which does not become very 
noticeable to the naked eye until the Moon reaches the umbra 
at R : the limb then rapidly darkens and becomes invisible. 
Near totality, the outline again becomes visible owing to 
illumination of the Moon by light from the Sun which has been 
refracted in the Earth's atmosphere. The absorption of the 
short wave-lengths in the passage of the light through the 
atmosphere, gives the Moon a reddish or coppery colour. The 
brightness of this illumination varies in different eclipses on 
account of the varying conditions under which the light passes 
through our atmosphere, which at some times contains much 
more cloud than at others. 

The duration of the eclipse depends upon the distance of 
the line of opposition from the node ; if the distance is small, 
the Moon will pass almost centrally through the shadow and 
totality may reach 3 hours ; if larger, totality may only just 

The calculation of the details of an eclipse of the Moon cannot 
be described in detail here, but the method may be outlined : 
the condition for an eclipse to occur is that the angular distance 
between the centres of the Moon and shadow as seen from the 
centre of the Earth may be less than 
D's semi-diameter + /^REP 
or <D's semi-diameter + /^CRE - / EPG 
or /J)'s semi-diam. + /_CRE /_AES + /_ EAC. 

But /^CRE is the Moon's parallax, /_AES is the Sun's 
angular semi-diameter and /_EAG is the Sun's parallax. 
Hence the angular distance between the centres of the Moon 
and the shadow must be less than 

})'s semi-diam. + D's parallax + 0's parallax 's semi-diam. 
In evaluating this expression it is customary to increase the 
diameter of the Earth's shadow at the distance of the Moon by 
2 per cent, to allow for the absorption in the Earth's atmo- 
sphere, making the effective diameter of the Earth larger than 
its true value. All the quantities in the above expression are 
known, so that the limiting distance between the centres of 
the Moon and shadow for an eclipse to occur is determined. 

To determine the circumstances of the eclipse, we require 
to know the hourly motions of the Sun and the Moon in longi- 
tude and that of the Moon in latitude. The difference in the 



motions in longitude of the Moon and Sun gives the motion in 
longitude of the Moon relative to the centre of the shadow. 
If in Fig. 49, S is the centre of the shadow, SV the ecliptic, M 
the centre of the Moon at the instant of opposition, MV the 
path of the Moon relative to the centre of the shadow, then 
^....^ MS is proportional to the 
/ \ Moon's hourly motion in 
latitude and SV to its 
motion in longitude relative 
to the shadow. 

If AB are points on MV, 
so that SA and SB equal 
the sum of the semi- 
diameters of the Moon and 
shadow, then the eclipse 
commences when the centre of the Moon reaches A and finishes 
when it reaches B. If SP is the perpendicular from S on MV, 
P is the position of closest approach. If the difference between 
the semi-diameter of the shadow and SP is greater than the 
radius of the Moon, the eclipse will be total, if less it will be 
partial. The distance of the centres at any time t after oppo- 
sition can be readily written down in terms of t : by placing 
this distance equal to SA and solving for t, the times of 
commencement and ending of the eclipse are obtained : by 
placing it equal to the difference of the semi-diameters of the 
Moon and shadow, the times of the beginning and end of 
totality are obtained. 

FIG. 49. Calculation of a Lunar 

FIG. 50. To illustrate a Solar Eclipse. 

83. Solar Eclipses. In Fig. 50, /S, E represent the centres 
of Sun and Earth respectively and CD is the Moon. P (not 
marked in the figure) is the vertex of the cone formed by the 
external tangents to the Sun and Moon, Q the vertex of that 


cone formed by the internal tangents. If P falls within the 
Earth, then at the points on the Earth's surface inside this 
cone the Sun's light is wholly cut off by the Moon and there 
is a total solar eclipse. The relative distances and sizes of the 
Sun and Moon are such that P falls sometimes within and 
sometimes without the Earth's surface, but in the former case, 
the cross section of the cone is never more than a few score 
miles. If P falls outside the Earth's surface, then at points 
within the angle of the cone (produced) an annular eclipse is 
seen, the apparent diameter of the disc of the Moon being 
smaller than that of the Sun. At points outside the zone of 
totality, but within the cone formed by the internal tangents, 
the eclipse is only partial. 

The Sun and Moon being in relative motion, the shadow cone 
in the case of a total eclipse sweeps across the Earth, giving a 
narrow band within which, at different times, the eclipse is 
total. The duration of totality at any one place is never more 
than a few minutes, the maximum possible duration being 
7 m. 30s. On July 5, 2168, will occur an eclipse with almost 
the maximum duration, 7m. 28 s. Owing to the small region 
of the Earth's surface at which a total eclipse is visible, the 
occurrence of a total eclipse at any given place is very infre- 
quent. In the British Isles, there have been total eclipses over 
some part within the last 500 years : on 1424, June 26 ; 1433, 
June 17 ; 1598, March 6 ; 1652, April 8 ; 1715, May 2 ; 1724, 
May 22. There will be other total eclipses on 1927, June 29, 
and 1999, August 11. The former of these will occur soon 
after sunrise, and will be visible in the north of England, with 
a very short duration. The latter will be total near Land's 

The condition that a solar eclipse may take place at or near 
conjunction may be determined as follows, with the aid of 
Fig. 51. For an eclipse to occur the angular distance between 
the centres of the Sun and Moon must be less than the sum of 
the Moon's semi-diameter and the angle SEL, i.e. less than 

D's semi-diam. + /_ELC + /_LVE, 
where V is the point of intersection of AC and BD> 

i.e. /_ D's semi-diam. + D's parallax + /_AES /_EAC, 
or Zl^'s semi-diam. + D's parallax + 0's semi-diam. 
0's parallax. 


The computation of the times of beginning and ending of 
the eclipse generally can be made in the same way as in the 
case of an eclipse of the Moon. To compute the circumstances 
of the phenomenon for any particular place on the Earth's 

surface is naturally 

- ' ~ a much more com- 

plicated problem. 
It comprises two 
parts ; supposing 

FIG. 51,-Thoory of a Solar Eclipse. th Earth fixed > 

the path of the 

shadow across it can be determined when the hourly 
motions in longitude of the Sun and in longitude and latitude 
of the Moon, their parallaxes and diameters are known. The 
modifications produced by the rotation of the Earth must 
then be taken into account. 

The circumstances of a total solar eclipse are represented in 
the Nautical Almanac in a diagram similar to that in Fig. 52, 
which represents the eclipse of 1919, May 28-29. When the 
shadow first meets the Earth, at the point denoted by First 
Contact, the shadow cone will be tangential to the Earth at 
that point ; the Sun and Moon are therefore just below the 
horizon at the point, so that the Sun will be just rising there and 
the eclipse at the same time commencing. At a slightly later 
instant, there will be two points, one in a higher and one in a 
lower latitude, at which the eclipse is just commencing at sun- 
rise. All points at which the commencement of the eclipse 
occurs at sunrise may be connected by a curve. Similarly, 
points at which the eclipse is middle or just ending at sunrise 
may be connected by other curves. There will be two curves 
representing the northern and southern limits of the eclipse, 
at any point of which the eclipse ends at the instant of com- 
mencement. The curves joining points at which the eclipse 
begins, ends or is middle at sunrise meet on these lines. There 
will be similar curves joining points at which the eclipse begins, 
is middle or ends at sunset. The point of Last Contact at 
which the shadow leaves the earth will be on the line " Eclipse 
ends at sunset." The lines representing the northern and 
southern limits are those swept out by the edges of the pen- 
umbra : the umbra itself traces out the narrow path of central 








eclipse : this path must end on the lines at which the middle of 
the eclipse occurs at sunrise and sunset, since totality takes 
place at mid eclipse. It is also customary to give on the 
diagram, curves joining the points at which the eclipse begins 
or ends at certain hours of Greenwich mean time : the 
approximate times of beginning and ending for any place 
within the eclipse region may then be obtained by interpola- 

84. Importance of Solar Eclipses. Total solar eclipses 
are occurrences of considerable astronomical importance, 
which explains why astronomers undertake expeditions to 
great distances and frequently to places difficult of access in 
order to observe a phenomenon lasting at the most but a few 

The solar corona can be observed only during totality, so 
that evidence as to its shape and constitution has had to be 
obtained from the relatively few eclipses which have been 
observed with modern methods : the total amount of time 
which has been available for the spectroscopic study of the 
corona cannot greatly exceed hour. The study of the con- 
stitution of the chromosphere, by means of the flash spectrum, 
can best be made at the instants of commencement and end 
of totality. The eclipse of 1851, July 28, was notable for the 
first attempt to photograph the corona, the method now always 
used to study its structure. The spectroscope was first used at 
an eclipse in 1868 and led to Lockyer and Janssen's discovery 
that the prominences might, with suitable arrangements, be 
seen at any time : previously they had only been observed 
during totality. The first observation of the green line in the 
spectrum of the corona, and the resultant discovery of coro- 
nium, occurred at the eclipse of 1869, August 7. The reversing 
layer was discovered at the eclipse of 1870, 'December 28, and 
the flash spectrum was observed at subsequent eclipses. At 
the eclipse of 1882, May 17, a bright comet was discovered on 
the photographs. At more recent eclipses the problems which 
arise have been further studied. The observations made at 
eclipses provide the best argument against the existence of a 
planet with an orbit nearer the Sun than that of Mercury. If 
such a planet existed, it is improbable that it would have 


escaped observation at all the eclipses which have been well 

A total eclipse provides the only method for testing whether 
rays of light are deflected by a strong gravitational field, as 
predicted by the generalized relativity theory formulated by 
Einstein. The method is to photograph during totality the 
stars in the neighbourhood of the Sun and to compare their 
positions with the corresponding positions on photographs of 
the same region of the sky taken at night a few months previous 
or subsequent to the eclipse. If the light rays are deviated by 
the Sun, the stars will be apparently displaced away from the 
Sun's limb, since the star appears to be in the direction from 
which the ray reaches the observer. The displacement is 
small, only l"-75 for a star at the limb of the Sun and decreases 
outwards, inversely as the distance from the Sun's centre. 
The existence of such a displacement was tested at the eclipse 
of 1919, May 28-29, by two British expeditions and the pre- 
diction of Einstein was confirmed. 

Solar eclipses which happened centuries ago are of great 
importance from the chronological point of view. If any 
event can be connected with the occurrence of a solar eclipse, 
the date of that event can be assigned with great accuracy 
and many disputed points in chronology have in this manner 
been settled. We have also seen that a comparison of ancient 
observations of total eclipses with present tables of the 
Moon enables us to determine the secular acceleration of the 

85. Physical Phenomena associated with Solar 
Eclipses. One of the phenomena which it was customary 
to observe at total solar eclipses some years ago is that of 
shadow bands. These consist of rapid alterations of light and 
shade at the time of commencement and ending of totality. 
If a white sheet is spread out, they appear as rapidly-moving, 
wave -like motions. They are probably due to undulations in 
the atmosphere causing a flickering of the light from the thin 
crescent and are not of great importance. 

The phenomenon known as " Baily's beads " arises from 
the inequalities in the lunar surface : just as totality is 
approaching the last narrow crescent of light breaks up into 


separate portions, giving the appearance of a string of beads. 
They were fully described by Baily, who observed them at 
the annular eclipse of 1836. The disappearance of the last 
bead is generally taken as the commencement of totality, and 
is also the moment for observing the flash spectrum : at this 
instant, also, the corona comes into view. 

86. Occupations. The Moon in its eastward motion 
amongst the stars frequently passes in front of or occults 
a fairly bright star. The circumstances of the occultation 
may be worked out for any point on the Earth's surface in a 
manner generally similar to that adopted in computing a 
solar eclipse ; simplifications are introduced from the star 
having no motion, parallax or semi-diameter. 

In the Nautical Alma,nac is given every year a list of the 
principal occupations visible at Greenwich, with the times of 
disappearance and reappearance and the points of the Moon's 
limb at which they take place. Data are also given for 
enabling the circumstances of the occultation to be calculated 
for any other point of the Earth's surface. As in the case of 
a solar eclipse, the phenomenon is visible only over a portion 
of a hemisphere and at different times at different places. 

On account of the eastward motion of the Moon, the dis- 
appearance of the star always takes place at the eastern limb 
arid the reappearance at the westward limb. Between new 
moon and full moon, the eastern limb is the dark limb ; be- 
tween full moon and new moon the western limb is dark. In 
the first case, the disappearance and in the second case, the 
reappearance therefore occurs at the dark limb. These 
phenomena occur instantaneously ; indicating conclusively 
that the Moon is devoid of an atmosphere, for if it possessed 
an atmosphere, refraction would cause the star to fade away 
or to come into view gradually. The times of disappearance 
or reappearance at the dark limb can therefore be observed 
with great precision ; at the bright limb, on the other hand, 
the star may be lost to view, if faint, before reaching the 
limb. Occultations provide a means of determining the Moon's 
position with a high degree of accuracy, provided the star 
which is occulted is one whose position has been well deter- 
mined by meridian observations and that the latitude and 


longitude of the place of observation are known. If, on the 
other hand, the Moon's position be determined from meridian 
observations and simultaneous observations of a given occul- 
tation are secured at two places on the Earth's surface, the 
difference of longitude of the two places can be determined. 
The method is accurate but suffers from the handicap that 
weather conditions frequently do not permit the observations 
to be secured at both places. 

87. Transits of Mercury and Venus. Another pheno- 
menon which is akin to eclipses is the transit of a planet across 
the Sun's disc. For a transit to occur the planet must pass 
between the Earth and the Sun, and it is therefore only the 
two planets Mercury and Venus which can be observed to 
transit. The planet then appears as a black spot, projected 
upon the Sun's disc. The use of the transits of Venus for the 
determination of the Sun's distance has been described in 
64. The method has now been superseded by more accurate 
methods, but the phenomena are still of importance, as their 
occurrence affords the best opportunity for determining the 
angular diameters of the two planets and they also enable 
accurate determinations of the planets' positions to be 

The inclination of the orbit of Mercury to the ecliptic is 
about 7, and a transit can only occur when the planet is 
very near one of its nodes at the time of inferior conjunction. 
The Earth passes through the nodes about May 7 and 
November 9, and the transits can therefore only occur near 
those dates. The possible transit limit corresponding to the 
mean distance of Mercury is 2 10', but the orbit of Mercury 
is not circular and in May it is nearer to the Earth than its 
mean distance and further away in November. This causes 
the May limit to be smaller than the November limit, so that 
transits are more frequent in November than in May. 

Twenty-two synodic periods of Mercury are approximately 
equal to 7 years, 41 periods are more accurately equal to 
13 years, and as a still better approximation, 145 periods are 
almost exactly equal to 46 years. It follows that a repetition 
of a transit may be looked for at intervals of 7, 13, or 46 years. 
At the May transit, the transit limit is so small that a repetition 



after 7 years is not possible. The dates of the transits during 
the present century are : 

1907, November 12 

1914, November 6 1924, May 7 

1927, November 8 

1940, November 12 

1953, November 13 1957, May 5 

1960, November 6 1970, May 9 

1973, November 9 

1986, November 12 

1999, November 14 

There is a very close approch on 1937, May 10, but no 

In the case of Venus, the inclination of the orbit is about 
3|, and the transit limit is only about 4. The phenomena 
are of very rare occurrence ; 5 synodic periods of Venus are 
nearly equal to 8 years, and as a much better approximation, 
152 synodic periods are nearly equal to 243 years. A transit 
may therefore recur after 8 years, but it is not possible for this 
to happen twice consecutively, and the next transit at the 
same node can only occur after 235 or 243 years. The dates 
of the transits are given in 64. 


88. THE diverse requirements of astronomical observation 
and measurement necessitate the use of many different instru- 
ments. For detailed accounts of the construction and use of 
such instruments reference must be made to treatises dealing 
specially with this branch of the subject. It is only possible 
to give here a very brief description of some of the principal 
instruments used in the observatory. It is assumed that the 
student is acquainted with the elementary laws of optics, 
concerning the reflection and refraction of light and the method 
of formation of images by mirrors and lenses. 

89. Telescopes. Telescopes used for astronomical obser- 
vations are of two kinds, in one of which the image is Jformed 
by a lens and in the other by a mirror. These are called 
respectively refracting and reflecting telescopes. The refract- 
ing telescope was probably invented in 1G08 by Lippershey, 
a spectacle maker of Middleburg, but it was not until Galileo, 
a few years later, made improved models that it was applied 
to astronomical observation. These telescopes were con- 
structed on the principle of the modern opera-glass, and were 
able to give only a very low magnification, and even then 
the images were ill-defined and not free from colour. The 
astronomical telescope with two convex lenses was first 
suggested by Kepler in 1611, but was not constructed until 
many years later. The reflecting telescope was invented by 
Newton about 1670, in order to avoid the chromatic effects 
obtained when a single lens is used for the object-glass in the 
refracting telescope. The achromatic lens for correcting for the 
colour effect was invented by Chest or Moor Hall in 1733, but 
was first applied by Dollond in 1758. 



The Refracting Telescope consists essentially in its simplest 
form of two convergent lenses (Fig. 53), one called the object- 
glass (0), with a long focal length F, and the other the eye- 
piece (E), with a much shorter focal length /. Since the 
objects to be viewed are always at a great distance, the rays 
from any point of the object which fall on the object-glass will 
be parallel. Thus, in Fig. 53, the rays S , S ly S 29 from a 
distant point in the direction OS produced, are parallel and 
must come to a focus at a point s on S Q produced, which is 
in the focal plane Fs of the object-glass : a real image of the 
distant point is therefore formed at s. This image is viewed 
through the eye-piece E. If E is so focussed that its focal 
plane coincides with that of the object-glass, then the bundle 

FIG. 53. The Astronomical Refracting Telescope. 

of rays from s must emerge from E as a parallel bundle and the 
final image is seen at infinity in the direction ES' ' . It will 
be noticed that the telescope is inverting : rays coming from 
a point above the axis OE emerge finally as though coming 
from a point below it. This is immaterial for astronomical 
purposes although inconvenient for terrestrial use. 

If rays from another point Z Q are considered, these arc 
brought to a focus z by the object-glass and the final image 
is seen in the direction EZ'. Suppose the rays S Q } Z Q 
come from two points at the opposite ends of a diameter of 
the Moon : then S OZ is a measure of the Moon's apparent 
diameter as seen by the naked eye. The image of the Moon 
produced by the telescope subtends, however, the larger angle 
sEz. The ratio of the apparent diameters of the image and 
object is a measure of the magnifying power of the telescope. 
But angle sEz \ angle S OZ = angle sEz : angle sOz = 
I// : l/F = F/f. The magnifying power of the telescope is 


therefore the ratio of the focal lengths of the object-glass and 
eye-piece. If the object-glass has a focal length of 20 ft. and 
the eye-piece of in., the magnification will be 480 ; if the 
eye-piece has a focal length of 1 in., the magnification will be 
240. It might be thought that since any desired magnifica- 
tion can be obtained merely by using an eye-piece of sufficient 
power, a large object-glass is not necessary : but a large 
aperture is required for other purposes, viz. for securing 
sufficient brightness of image and resolving power. 

90. Brightness of Image. The objects to be viewed may 
be either extended bright surfaces, such as the Moon, or 
bright points of light, such as stars, which are too far distant 
to give an image showing a disc. In the case of a star, all 
the light falling on the object-glass (except for a certain loss 
by absorption in the object-glass and reflection between its 
surfaces) is collected into a point image : the brightness of 
the image is therefore proportional merely to the area of the 
object-glass or, as is more usually stated, to the square of its 
aperture (A 2 ). Thus a 10-inch object-glass will give an image 
twice as bright as a 7-inch. In order to obtain faint stars, 
a large aperture is therefore essential. 

In the case of an extended object, a small area of the object 
gives rise to an image of finite area in the focal plane of the 
object-glass. For a given aperture, the area of this image is 
proportional to the square of the focal length of the object- 
glass (since any dimension of the image is proportional to F). 
The resultant brightness of the image is therefore proportional 
to A 2 /F 2 , this quantity being proportional to the quantity of 
light falling on the object-glass and inversely proportional to 
the image area over which the light is spread. The important 
consideration is therefore to have a large ratio of aperture to 
focal length : a telescope in which A/F = I : 5 will, for 
instance, give an image of four times the brightness of that 
given by one in which A/F = 1 : 10. It can be shown, 
however, that by no optical arrangement whatsoever can the 
brightness of the image of an extended object be increased 
beyond what it appears to the naked eye. 

91. Resolving Power of a Telescope. We have hitherto 



supposed that the image of a luminous point formed by the 
telescope will also be a point. This result is arrived at on 
the purely geometrical theory of optics which supposes that 
light travels in straight lines ; light is, however, a wave motion 
of very short wave-length, and a slight bending of the waves 
occurs at the edges of an obstacle. When the nature of the 
image of a luminous point produced by a circular aperture is 
investigated by the accurate physical method, it is found to 
consist of a central disc, brightest in the centre and fading 
off gradually towards the edge, which is surrounded by a 
series of bright diffraction rings, the brightness of successive 
rings decreasing rapidly so that generally only the one of 


FIG. 54. To illustrate Resolving Power. 

smallest radius is seen. The diameter of the dark ring between 
the central nucleus and the first bright ring is 1-22 XF /a, X 
being the mean wave-length of the light, a the aperture of 
the object-glass and F its focal length. The larger the aperture 
for a given focal length, the smaller is the diameter of the ring 
Now suppose that two distant bright points, subtending a 
small angle 6 are viewed. If is sufficiently small, the dif- 
fraction rings surrounding the two nuclei may be superposed 
to such an extent that the separate nuclei may not be visible. 
In such a case, the telescope fails to resolve the object into 
its two components. The limiting angle 6 which can be 
resolved is determined as follows : in Fig. 54 are given the 
intensity curves of the images of the two points : o is the 
centre of the nucleus of one image, oc the intensity there, 



a, b the points in the plane of the paper at which the intensity 
vanishes (first dark ring) ; e, f the points in the first bright 
ring. The other point will give a similar intensity curve 
with centre at o'. The resultant luminosity is obtained by 
adding the ordinates of the two curves. If o' coincides with 
6, i.e. the centre of one image coincides with the first minimum 
of the other, the final intensity curve will have two maxima 
(at o and o') with a perceptible dip between : the two nuclei 
will therefore just be seen and the object will be resolved. 
The linear distance apart in the focal plane of the object- 
glass of o and o' will therefore be 1-22 XF/a, corresponding 
to an angular separation =1-22 A/a. If the two luminous 
points subtend a lesser angle than this, the image formed by 
the telescope will be indistinguishable to the eye from that 
of a single point. If a is expressed in inches, this formula 
corresponds for a mean wave-length of 5,500 Angstrom units 
to an angular separation of 5-4"/a. For example, if the 
components of a double star subtend an angle of 1", a telescope 
with an aperture of at least 5 inches will be necessary to reveal 
that the star is double. With a smaller telescope, no increase 
whatever in magnifying power could show this. Large aper- 
tures are necessary, therefore, not only for viewing faint objects, 
but also for revealing fine detail or for separating close double 
stars : or, as this result is generally stated, the resolving power 
of a telescope is proportional to the aperture. 

92. Spherical and Chromatic Aberration. In the pre- 

FIG. 55. Spherical Aberration. 

ceding paragraphs it has been tacitly supposed that the 
object-glass is perfect, i.e. that all parts of the object-glass 
bring the light to a focus at the same point and that all colours 
are focussed together. With a single lens, neither of these 



conditions holds. In the case of a single convergent lens 
upon which parallel light is falling, the rays passing through 
tho outer zones of the lens are focassed nearer to the lens 
than those passing through the central portion (Fig. 55) : in 
the case of a single divergent lens, the converse holds, the 
focus of the central portion being nearest to the lens. This 
defect is known as spherical aberration. For a given focal 
length, it can be reduced by a suitable choice of the radii 
of curvature of the lens surfaces, and by using a compound 
lens the curvatures can be adjusted so that two chosen zones 
bring the light to a common focus and so that the difference 
in focus of any other zones is very small. 

Chromatic aberration arises from the index of refraction of 
glass, in common with other substances, being different for 
different colours. A single convergent lens will bring 
the blue rays to a focus nearer the lens than the red 
rays (Fig. 56). The difference in the refractive powers for 
any two chosen colours varies with the type of glass and it 

FIG. 56. Chromatic Aberration. 

is therefore possible by combining two types of glass to make 
an achromatic lens, i.e. a lens in which any two chosen colours 
are brought to a common focus. By suitable choice of the 
curvatures of the two surfaces, it is possible at the same time 
to correct the spherical aberration also. A compound lens 
is usually made of a convex lens of crown glass, combined 
with a concave lens of flint glass. 

The manner in which the chromatic aberration is corrected 
depends upon whether the telescope is to be used for visual 
or photographic observation. In the latter case, it must be 
corrected for the rays which have the most actinic effect, 
i.e. the blue and violet ; in the former, it must be corrected 
for the yellow and green rays. 

If the lens is required to have a large flat field, it should 


be built up of three single lenses, as the field of an ordinary 
doublet lens is somewhat curved. In fact, the more stringent 
the conditions with which the lens is required to comply, the 
more complicated does its design become. Reference should 
be made to treatises on geometrical optics for detailed accounts 
of the various defects of lens systems and of the methods of 
correcting them. 

93. The Reflecting Telescope. The dispersion of light 
was discovered by Newton, who came to the erroneous con- 
clusion that the dispersive power of different types of glass 
was the same. If this were so, it would not be possible to 
correct the chromatic aberration of a lens without at the same 
time neutralizing its power to bring parallel light to a focus. 
In order to avoid chromatic effects, Newton therefore invented 
the reflecting telescope. In this type of telescope, the light 
falls on to a concave mirror which converges it to a focus 
and so performs the same function as the object glass of a 
refracting telescope. There are several types of reflecting 
telescope, the principal being (i) the Newtonian, (ii) the 
Gregorian, and (iii) the Cassegrain. 

In the Newtonian telescope, the convergent beam of light 
reflected from the mirror is intercepted, just before reaching 
its focus, by a small plane reflecting mirror, situated on the 
axis of the telescope and inclined to it at an angle of 45. 
This mirror reflects the light to a focus at the side of the tube 
where the eye-piece is placed. The Gregorian and Cassegrain 
types are very similar : the large mirror is pierced at its 
centre by a hole and the light coming from it is reflected 
through the hole, in the Gregorian form, by a small concave 
mirror a little outside the focus, on the axis and perpendicular 
to it, and in the Cassegrain form by a small convex mirror, 
placed a little inside the focus. The Cassegrain telescope has 
the advantage of giving a flatter field than the Gregorian, 
which is now but little used. Instead of piercing a hole 
through the large mirror, a small plane mirror may be placed 
in front of the large mirror to reflect the light towards the 
side of the tube. 

With the same large mirror, it is possible by using different 
small mirrors, to convert the same telescope into any one of 



these types, giving different equivalent focal lengths (Fig. 57). 
It is thus possible to use that type of reflector which is best 
suited to the observations required to be made. 

The figure shows the different ways in which the 60-inch 
reflector of the Mount Wilson Observatory is used. The focal 
length of the mirror is 25 feet, and when used as a Newtonian 




FIG. 57. Various Methods of using a Reflecting Telescope to give different 
equivalent Focal Lengths. 

reflector, without secondary magnification, the focal length of 
the telescope is also 25 feet (Fig. 57[a]). In order to use it as 
a Cassegrain reflector, the upper section of the telescope tube 
carrying the plane mirror is removed and replaced by a shorter 
section with a convex (hyperboloiclal) mirror. This returns 
the rays towards the centre of the large mirror, at the same 
time reducing their convergence and so increasing the equiva- 
lent focal length. A small plane mirror is supported at the 


point of intersection of the polar and declination axes and is 
so inclined that it reflects the light down the hollow polar 
axis, where it is brought to a focus on the slit of a powerful 
spectroscope (Fig. 57[6]). The mounting of the plane mirror 
is geared so that as the telescope is rotated about the declina- 
tion axis the light is always reflected down the polar axis. 
This method of using the telescope enables a larger spectro- 
scope to be used than could conveniently be attached to the 
telescope. The equivalent focal length is 150 feet. Fig. 57(c) 
shows the telescope used as a Cassegrain reflector, with an 
equivalent focal length of 100 feet, the light in this case being 
brought to a focus at the side of the tube : in this form the 
instrument is used for large-scale photographs of Moon, 
planets, nebulae, etc. Fig. 57(d) shows a similar Cassegrain 
combination with different focal length (80 feet), used in 
conjunction with a spectroscope. 

94. Relative Advantages of Reflectors and Refractors. 

Each type of instrument has some advantages not possessed 
by the other and they are really complementary to one another 
in their uses. For some types of observation, the reflector, 
and for others, the refractor is preferable. 

In one respect, the reflector has a distinct advantage. The 
construction of a large object-glass is much more difficult 
and expensive than that of a mirror of the same aperture. 
It is essential that the component lenses should be absolutely 
homogeneous throughout and free from striations and other 
defects. The casting and annealing of a large disc of optical 
glass which will meet these requirements is a matter of the 
utmost difficulty. But granting that the discs have been 
successfully obtained, they have then to be ground and 
polished to certain curvatures, obtained by calculation and 
so chosen as to reduce the various aberrations. Finally, local 
polishing by hand must be resorted to in order to obtain the 
best results. The largest object-glass which has been con- 
structed is the 40-inch of the Yerkes Observatory. The disc 
of glass for a reflector need only be reasonably homogeneous 
and well annealed in order to avoid irregular distortion with 
change of temperature. There remains then but one surface 
to be optically worked. To cast a disc sufficiently large for 


a 6-foot or 8-foot mirror is indeed a difficult undertaking ; 
thus the original disc from which the Mount Wilson 100-inch 
mirror was constructed weighed 4| tons. But the combined 
difficulties are much less than in the case of the refractor. 
It must further be remembered that even if it were possible 
to construct very large objectives, their weight would be 
sufficient to distort them to such an extent that they would 
be optically useless : a large mirror, on the other hand, can 
be so supported from behind that its weight is counteracted. 
The employment of a reflector therefore enables a larger 
aperture to be used. Another advantage of jbho reflector is^ 
that a larger angular aperture can be secured than with a 
refractor. The angular aperture is measured by the ratio of 
linear aperture to focal length, and as we have seen in 90, 
a large angular aperture is necessary for securing a bright 
image of a faint extended object. In the case of a refractor, 
there is difficulty in correcting the spherical aberration if the 
aperture ratio is greater than about 1 in 12, whereas with a 
reflector a much greater angular aperture (up to 1 in 4) can 
without difficulty be secured. For all purposes, therefore, for 
which very great light-gathering power is essential, a large 
reflecting telescope must be used. 

The reflector has a further advantage : it is perfectly 
achromatic and if, as is customary, the surface of the mirror 
is worked into the form of a parabola, light parallel to the 
axis falling on every part of the surface will be brought to 
the same focus, so that there is no spherical aberration for 
such rays except that introduced by the reflection at the small 
mirror, which can be reduced if the mirror is made hyperbo- 

Against these advantages of the reflector may be placed 
the following advantages of the refractor. 

reflector with a refractor of the same aperture, the los_s__of 
light by absorption injthe object-glass is less than that by 
reflection at the mirror, except for very large apertures. Hale 
Has estimated that for apertures up to about 32 incKes, re- 
fractors surpass reflectors in light grasp for both visual and 
photographic rays. Between apertures of 32 inches and 50 
inches the refractor gives brighter visual images, but the 
reflector is superior for photographic purposes. For larger 



apertures, the light grasp of the reflector becomes superior for 
both visual and photographic rays. The refractor docs_not 
deteriorate with _age, _whilst_jbhe reflector necds^ resilveiing 
atjret^uent intervals^ owing to the tarnishing of the silvered 
surface. Moreover, thejocus of a refractor is less liable tQ 
change with changes of temperature than is that of a reflector.. 
This is of importance in photographic work of precision in 
which very small displacements of images require to be 
measured. With a refractor it is also possible to obtain a 
larger flat field of good definition than with a reflector. 

95. Eye-Pieces. A simple convex lens gives bad dis- 
tortions of the image and introduces a lot of colour unless 
the object is exactly in the centre of its field. Eye- pieces are 
therefore usually composed of two or more lenses, by which 
means the aberrations may be reduced and good images 


FIG. 58. The Kamsdon Eye- pi ceo. 

obtained over a much larger field. Two of the most common 
forms of eye-piece are the Ramsden and the Huyghenian. 
Each of these eye-pieces is composed of two plano-convex 
lenses made of the same sort of glass ; the one which faces the 
incident light is called the field-lens, the other the eye-lens. 
In the Ramsden eyepiece (Fig. 58), the two curved surfaces 
face towards one another, and in order to secure the greatest 
freedom from colour the focal length of the eye-lens should 
be equal to that of the field-lens and to the distance apart of 
the two lenses, though these distances are varied somewhat 
in practice. This eye-piece gives a very flat field and is 
approximately achromatic for parallel light. It has the 
advantage that the principal focus of the combination is 
outside the field-lens, so that it is possible to place a system 
of spider-webs in the focal plane of the object-glass of the 
telescope, which can be viewed with the image through the 



eye-piece. This is necessary for many purposes in astronomy, 
and an eye-piece which enables this to be done is called a 
positive eye-piece. 

The Huyghenian eye-piece (Fig. 59) consists of two plano- 
convex lenses, the focal length of the eye-lens being one-half 

FIG. 59. The Huyghenian Eyo-pioco. 

that of the field-lens and their distance apart one-half the sum 
of the focal lengths. The curved surfaces of the field and eye 
lenses face the incident light. Since the principal focus of the 
combination is between the two lenses, it follows that when 
focussed on the image formed by the object-glass, the rays 
converging to form this image are intercepted before they 
have come to their focus, and a (virtual) image is formed 
between the lenses of the eye-piece. Such an eye- piece is 
called a negative eye-piece arid cannot be used for any purpose 
in which it is necessary at the same time to focus on a 
graticule system. 

These eye-pieces are not absolutely achromatic. For pur- 
poses not requiring a large field of view a simple compound 
lens may frequently be used with advantage. To obtain the 
most satisfactory results, the colour correction of the object- 
glass should be decided in conjunction with the eye-piece 
intended to be used most frequently with the telescope, and 
the combination made as nearly as possible achromatic. 

It is usual to provide any instrument with several eye-pieces 
of different focal lengths, so enabling the magnification which 
is most suitable for any particular observation to be used. 
It must be emphasized, however, that increase of magnifica- 
tion for an object-glass of given aperture does not entail any 
increase in resolving power and since an increase in magnifi- 
cation also increases the disturbances arising from atmospheric 
irregularities, there is a limit depending upon the definition 



at the time of observation beyond which it is inadvisable 
to increase the magnification. 

96. The Transit Instrument. This instrument, which is 
used for the determination of sidereal time, is necessarily one 
of the fundamental instruments of an observatory. The 
observations consist in determining the times of the transits 
of stars across the meridian. It is known ( 6) that when a 
star is on the meridian, the sidereal time is equal to its right 
ascension ; if, then, the clock time of the transit is determined, 
the error of the clock can be found, provided that the right 
ascension of the star is known. 

The instrument consists essentially of a refracting telescope, 
as illustrated in Fig 60, which is 
supported at the ends of an axis, 
perpendicular to the tube, by 
two trunnions moving in Y-bear- 
ings. The axis is horizontal and 
points east and west, so that as 
the telescope swings on the axis 
it moves in the meridian. It is im- 
portant that the axis should be 
stiff, the telescope tube sufficient- 
ly strong to prevent flexure, and 
the pivots accurately cylindrical, 
equal and coaxial. In the focal 
plane of the object-glass is placed 
a framework carrying a number 
of vertical spider-lines (or " wires" 
as they are usually called), and 

a single horizontal wire across the centre of the field. A 
graduated circle is fixed to the axis and rotates with it, 
enabling the instrument to be set on a star of any required 

The observation of a transit consists in the determination 
of the times that the star image passes across each vertical 
wire ; this may be done by the " eye-and-ear " method, i.e. 
the observer watches the star and listens to the beats of the 
clock, interpolating the times of transit across each wire to 
the nearest tenth of a second. It is now customary, however, 

FIG. 60. Schematic Transit 


and more accurate to record these times electrically, with the 
aid of a chronograph. For this purpose a hand tapper may 
be used, but the most modern method is to use a self-recording 
micrometer. This carries a second frame on whics is mounted 
a single vertical wire ; this frame can be traversed across the 
field of view, immediately behind the frame previously referred 
to. The observer causes the travelling wire to move at such 
a rate that the star image is continuously bisected by it and 
at regular intervals during the transit a record is automatically 
made on the chronograph. By this means, personal errors 
which were inevitable when the hand tapper was used and 
which arose from systematic differences between the methods 
of observing of different observers, are almost entirely 

The advantage of having a number of taps recorded on the 
chronograph is that the accidental error of observation is 
greatly reduced. The intervals between the wires must be 
determined by special observations, and the time of passage 
across each wire can then be reduced to a time of passage 
across the central wire. Provided that the axis of the instru- 
ment is exactly horizontal and points due east and west and 
that the central wire is exactly in the meridian, the time of 
transit is equal to the star's right ascension. In practice none 
of these conditions hold, and the amounts of the three errors, 
level error (axis of the instrument not horizontal), azimuth 
error (axis of the instrument not east and west), and collimation 
error (the optical axis or the line joining centre of object-glass 
with middle wire not perpendicular to the axis of rotation), 
must be accurately determined. If the instrument has been 
carefully set up in the first place and if its support is stable, 
these errors will always be small though somewhat variable, 
and their exact amount must be determined daily by suitable 
observations. A 3-inch transit instrument is shown in Fig. 61. 

97. Adjustments of Transit Instrument. Collimation. 
Small transit instruments can usually be reversed, i.e. the 
axis turned through 180 so that the east and west pivots 
change places. If a distant fixed object is available on which 
the instrument can be pointed, the position of this object 
relative to the centre wire is observed and the telescope is 



then reversed on its axes and again pointed to the same object. 
If the centre wire points in the same direction relatively to 
the distant mark as it did before reversal, there is no collima- 
tion error : if not, the error is given by half the angular 
distance between the two pointings. If a distant fixed mark 

FIG. 61. Three-inch Transit Instrument. 

is not available, a collimating telescope can be used. This is 
an auxiliary telescope, in the focus of which is placed two 
cross wires. The telescope is firmly mounted in the meridian 
with its object-glass towards the transit instrument, so that 
when the latter is horizontal, it is possible to look straight 
through it into the collimator. The cross wires in the focus 
of the latter then serve as a suitable infinitely distant object, 
which can be used as before for determining the collimation. 



If the telescope is not reversible on its axis, two collimatirig 
telescopes must be used, one placed to the north and the other 
to the south of the transit instrument. One collimator must 
first be adjusted to the other and the telescope then set on 
each alternately in order to determine the collimation error. 
Level. In the case of small instruments, the error of level 
is usually determined by means of a sensitive graduated spirit 
level, called a striding level, which is so constructed that, 
when the telescope is horizontal, it can rest on the two pivots. 
The position of each end of the bubble is read, and the level 
is then reversed. The half -difference between the means of 
the readings given by the two ends of the bubble determines 
the amount by which the axis of one pivot is higher than that 
of the other, provided that the pivots 
are of the same size and that the angu- 
lar tilt required to displace the bubble 
of the level through one division is 
known. A striding level cannot be used 
with a large instrument, and a different 
method must then be adopted. Such 
instruments are provided, in addition to 
the fixed framework carrying the wires, 
with a second frame carrying a single 
wire, which can be moved by an accurate 
micrometer screw. The telescope is set 
in a vertical position with the object- 
glass downwards and a bath of mercury 
is placed beneath it. Using a Bohnenberger eye-piece, i.e. a 
common Ramsden eye-piece with a hole in one side and a 
thin glass plate inserted at an angle of 45 (Fig. 62), the 
light from a lamp at the side of the instrument is thrown 
down the tube and the image of the movable wire formed by 
reflection from the mercury surface is observed. A movement 
of the wire produces an equal movement of the image in the 
opposite direction. The micrometer screw is turned until the 
wire and its image coincide. The plane passing through the 
wire and the centre of the object-glass must then be exactly 
perpendicular to the mercury surface and therefore vertical. 
If this direction coincides with that which is perpendicular to 
the axis of the instrument, as determined from the observation 

Fio. 62.-The Bohnon- 
berger Eyo-piece. 



of collimation, the axis is horizontal and there is no level 
error. If, on the other hand, the two directions do not 
coincide, their difference determines the amount of the error. 

Azimuth Error. The amount of this error can only be 
determined from astronomical observations. It is necessary 
first to reduce the level error to a small amount and to know 
approximately the error of the clock. A star near the pole 
is then observed : when the clock time (corrected for error) 
is equal to the R.A. of the star, it is known that the star 
is on the meridian, and the position relative to the tele- 
scope axis of a line which is due north and south is thus 
determined, enabling the 
error of azimuth to be 
deduced. The actual pro- 
cedure is somewhat more 
involved, although the prin- 
ciple is essentially as 
described. It involves 
observation of stars 
different declinations 
which means the error of 
azimuth, which affects 
stars of low altitude to 
the greatest extent, can be separated from level error which 
affects predominantly stars of high altitude. 

It is not possible to adjust the instrument so that these errors 
are exactly eliminated, but with a stable instrument, initially 
well adjusted, they will always remain small in amount. 
Their magnitudes must be determined in the way just described 
and a correction applied to the observed time of transit of a 
star to obtain the true time of transit across the meridian. 
Provided that the errors are small, their effects can be treated 
as independent of one another and the required correction can 
be easily obtained. 

Consider first the error of level. We will assume that the 
axis is perpendicular to the N.S., but that instead of being 
horizontal, it is inclined to the horizon at a small angle 6, the 
east end being the lower. It therefore points to a point E l on 
the celestial sphere (Fig. C3), which is on the prime vertical but 
below E by the amount EE l = b. It is then apparent that 

FIG. 63. Level Error of Transit 



when the instrument is rotated upon its axis, the axis of the 
telescope the collimation error being neglected for the 
present instead of moving in the meridian SZN, moves in the 
great circle SZiJN, whose pole is E L . If a star a is on this circle, 

it will appear to be on the 
meridian, although it actually 
has an easterly hour-angle 
LLi, Li being the point in 
which the hour circle through 

Fro. 64.- 

- Azimuth TCrror of Transit 

a meets the equator. If 
a' is the point at which a 
crosses the true meridian, La' 
= <5, the declination of the 
star, and LZ = PN = <, the 
latitude. Hence a'Z = < d 

and aa' = ZZ cos a'Z = b cos (< 6). Also aa' = 

LL l cos d and so the hour-angle LL V = b cos (< d) sec d, 

and this must be added to the observed time of transit to 

obtain the true time of meridian transit. 

Considering next the effect of a slight error in azimuth of 

amount k, the level and collimation errors now being neglected. 

Suppose the axis is inclined by the amount k to the north of 

east, and so points to a point E v on the horizon such that 

EEi = k (Fig. 64). Then as the instrument is rotated, the 

axis of the telescope moves in a great circle which passes through 

the zenith and through a point S t on the horizon such that 

88 1 = k. A star a there- 

fore appears to be on the 

meridian when its easterly 

hour- angle is aPo'. Then, 

reasoning as before, a a' = 

S8i sin a'Z = k sin (^ d) 

and the hour-angle is 

k sin (<f> - d) sec d, which 

must also be added to the 

observed time of transit to 

obtain the true time. 

Finally, the error in collimation must be considered. Suppose 

that the error in perpendicularity of the axis of the telescope 

and the axis of the instrument is c, so that the end of the axis 

FIG. 65. Collimation Error of 
Transit Instrument. 


describes a small circle S^Z^N^ (Fig. 65) when the instrument is 
rotated, where SS 1 = NN l ~ c and N l , /S x are east of the 
meridian, it being now assumed that level and azimuth errors 
are zero. Then with the same notation as before, era' = c also 
and LL l c sec <5, which must also be added to the observed 
time of transit. 

If all three errors are present, the resulting error in the 
observed time of transit can be represented by 

t = b cos ((/) (5) sec d + k sin (< d) sec d + c sec d. 
If 6, k, c are expressed in time, t will be given in time also. 
Usually, b, k are expressed in angle and c in time. In that case, 
6, k must first be converted into time for substitution in the 
above formula. 

This result is often expressed in the form 

t = m + n tan d + c sec <5, 
where m = b cos $ + k sin <, n 6 sin < & cos <. 

In this form the effect of the declination of the star on the 
time of transit is more readily seen. As d approaches 90, 
i.e. for stars near the poles, the error in the time of transit for 
given instrumental errors increases rapidly. 

98. The Meridian or Transit Circle .The simple transit 
instrument is used only for the observation of times of transits 
of stars. The meridian circle is used to determine in addition 
the declinations or north polar distances of objects. For this 
purpose, a large and accurately graduated circle is attached to 
and concentric with the axis of the instrument and revolves 
with the telescope. When the telescope is set on any object the 
position of the circle may be read by means of four or six read- 
ing microscopes, fixed to the pier supporting the axis. Each 
microscope carries in its focal plane a pair of parallel spider- 
lines which can be moved by a micrometer screw, with a 
graduated head. The reading of each micrometer correspond- 
ing to the position in which the parallel wires are equidistant on 
either side of the nearest graduation of the circle is read. 
Usually one revolution of each micrometer screw corresponds 
to 1' of arc and the micrometer head is divided into 60 parts, 
each being then equal to 1". By estimation, the micrometers 
can be read to a tenth second. The main circle is usually 
graduated every 5' and an index microscope enables the 



position of the circle to be read to the nearest 5' : the reading 
microscopes then give the extra minutes and seconds. The 
purpose of having several microscopes is to obtain increased 
accuracy and also to eliminate errors due to slightly incorrect 
centring of the circle. 

In the meridian circle, the framework carrying the horizontal 
wire and the system of vertical wires is movable in the meridian 
at right angles to the telescope axis by a micrometer screw with 
graduated head. When the star enters the field of view, the 
telescope is clamped in such a position that the star is near the 
horizontal wire. By means of the micrometer screw, the 
horizontal wire is raised or lowered so that it bisects the star 
when it passes across the central vertical wire. From the 
readings of this micrometer and the reading microscopes, the 
exact circle reading for the particular star is obtained. To 
determine the declination or altitude of the star, the reading of 
the circle corresponding to a definite position of the telescope 
must be found : thus, if a close circumpolar star is observed on 
the meridian above pole and 12 hours later below pole, the 
mean of the two readings, corrected for refraction and instru- 
mental errors, gives the reading corresponding to declination 
90, enabling the declination corresponding to any other reading 
to be obtained. 

It is more usual to determine the circle reading corresponding 
to the position in which the telescope is exactly vertical : for 
this purpose, a mercury bath and a Bohnenberger eye-piece are 
used, just as in determining level, but the reading of the 
declination micrometer is obtained corresponding to the position 
in which the horizontal wire and its image coincide. The 
microscopes are also read in this position so that the circle 
reading corresponding to the nadir point is obtained. 

99. The Altazimuth. With the meridian circle, only 
observations in the meridian are possible. Although these 
observations are the simplest and most accurate, there are 
some purposes for which it is necessary to secure extra-meridian 
observations : e.g. just after or just before new moon. When the 
Moon is near the Sun, it is not possible to observe it at meridian 
transit and observations for its position must be secured just 
after sunset or just before sunrise. For this purpose, an 



altazimuth (i.e. altitude and azimuth) instrument is employed. 
This type of instrument is essentially a transit circle which 
can be rotated about a 
vertical axis into any 
desired azimuth. It is 
therefore in principle 
similar to, though much 
larger than, a theodo- 
lite. The azimuth of 
the instrument is deter- 
mined from stellar ob- 
servations, the azimuth 
circle being used only 
to set the instrument 
with sufficient accuracy 
into any desired azi- 
muth. The errors of 
adjustment are deter- 
mined exactly as in the 
case of the transit in- 

100. The Chrono- 
graph. In modern , 
methods of observa- 
tion with the meridian 
circle or altazimuth, 
the times of transit of 
an object across the 
vertical wires are re- 
corded automatically : 
for this purpose an 
instrument called a 
chronograph is em- 
ployed. The most 
common type of chro- 
nograph, such as that 
shown in Fig. 66 (which 

is made by Messrs. T. Cooke & Sons, of York), consists of 
a cylindrical barrel, several inches in diameter and about 15 


inches long, around which is wrapped a sheet of paper. The 
barrel is rotated by clockwork controlled by a governor, to 
secure a uniform rate, usually one revolution in two minutes. 
A pen carried on the armature of a small electromagnet marks 
the paper, and as the barrel rotates, this pen is traversed 
slowly along by means of a screw so that it describes a con- 
tinuous helical trace on the paper. Every two seconds (or 
sometimes every second) a momentary current from the 
sidereal clock passes through the electromagnet, attracting 
the armature and causing the pen to give a slight kick and 
therefore a break in the trace, as illustrated in Fig. 67. A 
longer break every 60 seconds denotes the commencement 
of a minute. 

FIG. 67. Portion of Traco given by Recording Chronograph. 

When the observer at the instrument makes a tap with his 
hand tapper at the instant a star passes one of the vertical 
wires, a current is again sent through the electromagnet, 
causing a corresponding mark on the record. The positions of 
these marks and hence the corresponding times can be read off 
the trace whenever convenient after the evening's observations 
have been completed. Fig. 67 shows a portion of a chrono- 
graph trace, on which the times of the clock taps have been 
indicated. The observer's taps, corresponding to the transits 
of two stars, are shown in the first two lines. The intervals 
between the taps in the first line is greater than in the second, 
showing that the first star had the higher declination. At 
llh. Om. 2s., one of the taps coincides with a clock tap, 
causing a wide break on the record similar to the wider breaks 
at the commencement of each minute. 

101. Clocks. Without an accurate means of measuring 
time, the modern progress in observational astronomy would 
not have been possible. The great improvement on the old 
methods followed the application by Huyghens in 1657 of the 


pendulum as a regulator of the clock mechanism. There is no 
necessity here to enter into the details of clock construction, 
for which information reference should be made to a treatise on 
horology. The astronomical clock does not differ in its essential 
details from an ordinary clock, but it is necessary that it should 
be constructed with very great care, in order that the clock may 
not behave erratically, so that if observations of time are not 
possible for two or three nights, the time given by the clock 
(after correction for its rate) may be nearly correct. The clock 
must therefore have an accurate escapement and must be 
compensated for changes of temperature and of barometric 

If the pendulum were a steel rod, beating seconds, its daily 
rate would change by one-third of a second for each degree 
(Centigrade) change in temperature. This is the direct 
consequence of the change in the length of pendulum &s the 
temperature changes. To correct for this, various compensa- 
tion devices may be used. Thus Graham's mercurial pendulum 
is fitted at the bottom with a vessel containing mercury, the 
amount of which is adjusted so that as the temperature rises, 
the upward expansion of the mercury is exactly sufficient to 
compensate the downward expansion of the steel rod. Another 
common type is based upon the gridiron pendulum of Harrison, 
the inventor of the chronometer : it consists of rods of zinc 
and steel, the upward and downward expansion of which just 
compensate one another. These devices are liable to introduce 
errors through a lag in their adjustment when the temperature 
changes rapidly. The best clocks are therefore now provided 
with pendulums of invar, an alloy discovered by Guillaume 
which, as its name suggests, does not change its length with 
change of temperature. 

Change in barometric pressure causes a change in the resist- 
ance of the air to the swing of the pendulum and therefore alters 
the time of swing. A rise of one inch in the barometric height 
causes an ordinary pendulum clock to lose about one-third of a 
second daily. The simplest method of compensating is to 
enclose the clock in an air-tight, partially exhausted case. 
This is the method adopted by Riefler, of Munich. 
Alternatively, various types of compensation have been 


If a clock is required for use with a chronograph, a toothed 
wheel on the axis of the escapement wheel is usually arranged 
to touch a light spring at alternate seconds, so completing an 
electric circuit and sending a momentary current through the 
coils of the electromagnet to the armature of which the recording 
peri is attached. 

The error of a clock is the amount which must be added to 
the time given by the clock in order to obtain the true time. 
The rate of a clock is the amount of its gain or loss in a day ; 
if the clock is losing, the rate is taken as positive and so increases 
the error when the clock is slow and decreases it when it is 
fast : if the clock is gaining, the rate is negative. A steady 
rate can be determined and allowed for arid is therefore 
immaterial, although it is convenient that the rate should be 
small. The test of the quality of a clock is the absence of 
variations in its rate arising from changes of temperature, 
pressure, or accidental causes. 

102. Equatorial Telescopes. The telescopes which have 
been described in the preceding sections are required only for 
the determination of time or of the positions of celestial objects. 
They are used merely as pointers and, on account of the rotation 
of the Earth, the object under observation remains in the field 
of view for only relatively few seconds. For many purposes, it 
is desirable to retain the object stationary in the field of view. 
For such purposes, an equatorial telescope is used. This type 
of telescope has an axis, supported at its two ends in bear- 
ings so that it is free to rotate. The axis points towards 
the pole of the heavens and is therefore parallel to the 
axis of the Earth. Rigidly fixed at right angles to the polar 
axis, is a second axis called the declination axis, and at the 
end of this axis the telescope is supported with its axis at right 
angles to the declination axis. The telescope can be rotated 
about the declination axis. The polar axis is caused to rotate 
slowly by a clock-work mechanism, at such a rate that the 
rotation of the Earth is exactly counteracted. Owing to the 
great distance of all celestial objects, the combination of the 
equal and opposite rotations about parallel axes results in 
an object remaining stationary in the field of view of the 


The polar axis carries a circle, graduated in hours and minutes, 
called the hour-circle or right-ascension circle. 

The vernier of this circle reads h. when the declination axis 
is horizontal, the telescope being then in the meridian. When 
the telescope is directed to any star, the hour-angle of the star 
can be at once read off, and the right-ascension obtained 
when the sidereal time is known. In some instruments, the 
vernier can be set to read the sidereal time at the instant, and 
then when the telescope is pointed to any object, the reading 
of the hour-circle gives at once its right-ascension. To point 
the instrument to any object of given right-ascension and 
declination it is then only necessary to rotate the instrument 
about the polar axis until the 
reading of the R.A. circle is 
equal to the right-ascension of 
the object and then to turn it 
about the declination axis until 
the reading of the declination 
circle, attached to that axis, is 
equal to the declination of the 

Fig. 68 shows in a schematic 
form a common type of equa- 
torial mounting. The telescope 
is supported at one end of the 

T T j* T , Fio. 68. Tho Schematic Equatorial, 

declination axis and counter- i 

poised by a weight at the other 

end. On all large instruments some typo of mechanical 
counterpoise is introduced so as to reduce the friction of the 
polar axis in its bearings and to enable the instrument to turn 
freely. The form of equatorial mounting shown in Fig. 68 
has some disadvantages in the case of heavy instruments : the 
necessity of counterpoising the telescope necessitates the stand 
of the instrument carrying a weight much greater than that of 
the telescope itself. Some large instruments are therefore 
made with the polar axis in the form of a rectangular frame- 
work of girders, supported at its two ends by independent 
supports. The declination axis is supported by pivots fitted 
to the two sides of this framework. An example of such 
a mounting is shown in Plate X, which represents the 


100-inch Hooker telescope of the Mount Wilson Observatory. 

It is necessary that the telescope should be adjusted so that 
the polar axis points accurately to the pole of the heavens and 
so that the polar and declination axes and also the telescope 
and declination axes are exactly at right angles. The first of 
these is an adjustment of setting and equatorial telescopes are 
generally provided with arrangements for adjusting in azimuth 
and for adjusting the tilt of the polar axis ; the second and third 
adjustments are instrumental, but require to be tested, as in 
general they require small corrections. 

The clock should be controlled by an efficient governor so 
that the rate at which the telescope is turned does not vary. 
Some telescopes are fitted with an automatic electric controlling 
device, governed by a pendulum ; this device, invented by 
Sir Howard Grubb, consists of two parts, one of which, called 
the " detector," detects any irregularity in the clock drive ; 
and the other, called the "corrector," automatically corrects the 
error. The pendulum at the bottom of its swing touches a 
drop of mercury and so completes an electric circuit ; if the 
clock drive does not synchronise with the pendulum, the current 
passes round one or other of two circuits to the corrector ; an 
electromagnet causes an arrest to come into action and by 
a system of sun and planet pinions the rate of rotation is 
either accelerated or retarded until synchronization is ob- 

Equatorial telescopes, whether reflectors or refractors, are 
admirably adapted to photographic observation. It is 
necessary, when so used, that the image should be absolutely 
stationary and therefore it is customary to mount an auxiliary 
visual telescope on the same mounting : during the photo- 
graphic exposure, the observer watches the image of the object 
in this telescope and controls the speed of the clockwork as 
necessary. Sometimes, however, it is simpler to fit an auxiliary 
arrangement to the photographic telescope by means of which 
an image can be seen which is formed by the same telescope 
that is used for the photographic purposes. 

The large 40-inch refractor of the Yerkes Observatory is 
shown in Plate IX. The object-glass is the largest which has yet 
been constructed. The telescope is adapted for visual obser- 
vation. The polar axis is relatively short ; near its lower end 


may be seen the R.A. circle. The telescope is counterpoised 
by weights at the end of the declination axis. The focal length 
of the telescope is very long and it is therefore admirably suited 
to all purposes for which a large scale is essential. The size of 
the instrument may be judged from that of the chairs on the 

Plates X and XI show the 100-inch reflector of the Mount 
Wilson Observatory, California, and the 7 2 -inch reflector of the 
Dominion Astrophysical Observatory, British Columbia, re- 
spectively. These are the two largest reflectors in the world. 
The difference in the methods adopted in the two instruments 
for suspending the telescope tube is of interest. In the 100-inch 
reflector, the polar axis is in the form of a cradle, supported at 
its two ends. The telescope itself is swung in this cradle, the 
declination axis consisting merely of two trunnions which fit 
into bearings in the cradle. No counterpoise weight is therefore 
necessary. In the 72-inch, on the other hand, the telescope is 
supported to one side of the polar axis and counterpoised by 
weights at the other extremity of the declination axis. In 
Plate XI, the telescope is shown with a spectroscope attached 
below the mirror, the light passing through a hole in the mirror. 
In the 100-inch telescope the mirror is not pierced with a 
central hole, the spectroscope being used as shown in Fig. 57 (b). 
The mountings of the large mirrors of these instruments are 
very carefully designed with a complicated counterpoise 
system, to prevent distortions arising from their great weight 
which would spoil their figure. 

Filar Micrometer. If an equatorial telescope 
is used photographically, the photograph can be measured 
at any subsequent time. With a visual instrument, on the 
other hand, any measurements that are desired must be made 
at the telescope. The measurements most commonly required, 
such as the angular diameters of small bodies, or the angular 
separations of double stars, are usually made with a filar 
micrometer, fitted to the eye-end of the telescope. The 
micrometer consists of a rectangular box, containing three 
frameworks which carry spider- wires. One of these frame- 
works is fixed and usually contains two or three close parallel 
wires, running parallel to the length of the box ; the other two 


frameworks are movable in this direction by means of screws 
and micrometer heads, fitted to the two ends of the box. 
These frames each carry a single wire parallel to the short edges 
of the box and therefore perpendicular to the wires of the fixed 
frame. The frames are so constructed that the three sets of 
wires are all nearly in the focal plane of the object-glass and 
therefore can be focussed by the eye-piece together. 

The entire box can be turned around in a plane perpendicular 
to the optical axis of the telescope. A graduated circle is fixed 
behind the micrometer box so that the angle through which 
the box is turned can be read. 

In order to measure the separation of the components of a 
double star, the box is turned until the fixed wires are parallel to 
the line joining the nuclei of the two stars. The two micrometer 
heads are then turned until each wire bisects one of the star 
images, and the readings of the two micrometer heads are taken. 
The movable wires are then crossed over so that each bisects the 
other star and the readings again taken. From these readings, 
double the distance between the stars is at once obtained in 
terms of revolutions of the micrometer screws. The value of 
one revolution in angular measure can be readily obtained by 
observing a close circumpolar star, with the telescope fixed, 
the movement of the star in a certain interval of time being 
measured. The angular reading of the graduated circle 
gives the direction of the line joining the two stars ; to convert 
into " position angle " the angle measured from the north 
point the telescope is stopped and the reading of the circle 
taken when the fixed wires are in such a position that the motion 
of any star is parallel to them. This gives the reading of the 
circle corresponding to a position angle of 90. 

104. The Spectroscope. The purpose of the spectroscope 
is to analyse the light from any source into its constituent 
vibrations. Any spectroscope is composed of three portions : 
(i) The collimator, (ii) the dispersion piece, (iii) the telescope. 
The collimator, Fig. 69, consists of a tube, having at one end 
an achromatic object-glass and at the other a narrow slit, S, 
in its focal plane. The light from the source passes through the 
slit, emerges from the collimator as parallel light and then falls 
on the dispersion piece. This may consist either of a prism, 



or a bundle of prisms or of a diffraction grating. The latter is a 
piece of glass or speculum metal ruled with numerous fine 
equidistant parallel lines, which has the property of analysing 
the light arid reflecting each constituent vibration as a separate 
parallel beam. The light, therefore, emerging from the prism 
or reflected from the grating consists of a series of parallel 
bundles of different wave-lengths travelling in slightly different 
directions. The telescope focusses each of these as a line 
image, parallel to the original slit, and the spectrum produced 
may either be viewed with an eye- piece or photographed by 
placing a sensitive plate in the focal plane. With a fine 
straight slit to the collimator, the lines in the spectrum are 



FIG. GO. Diagram of Single Prism Spectroscope. 

straight and sharply defined. In Fig. 69, B and R represent 
the foci for the blue and red rays respectively. 

A spectroscope may be attached to the end of an equatorial 
telescope in order to obtain the spectra of stars. The image of 
the star produced by the objective must fall exactly on the slit 
of the spectroscope and accurate guiding is necessary in order 
to retain the image exactly in position. 

At the Mount Wilson Observatory, the 60-inch reflector can 
be used for spectroscopic observations as a Cassegrain reflector 
in which the light is brought down the polar axis (Fig. 57 [6]). 
In this way, a massive spectroscope giving high dispersion can 
be utilized and can be easily kept at a constant temperature, 
which is of importance for some types of observation. 

For astronomical purposes, the spectra are usually photo- 



graphed so that their examination and measurement can be 
performed subsequently at leisure. It is customary then to 
give on the same plate a short exposure on either side of the 
stellar spectrum of the spectrum of a terrestrial source, such as 
the iron arc ; this enables the wave-length of many of the lines 
in the spectrum under examination to be assigned with con- 
siderable accuracy and it is then a fairly simple matter to 
deduce those of the other lines. The spectrum is impressed 
on the plate by reflecting the image of the arc on to the spectro- 
scope slit. 

105. The Heliometer. The heliometer is an instrument 
which enables the distances apart of neighbouring celestial 

objects up to a limit of 
about a couple of de- 
grees to be measured 
with a very high order 
of accuracy. As its 
name suggests, it can 
also be used for deter- 
mining the angular dia- 
meter of the Sun. It 
consists essentially of a 
telescope, the object 

5 L j* glass of which is divided 

$1 S% S" tiZ into two along a dia- 

FIG. 70. The Principle of the Heliometer. meter and the two 

parts mounted so that 

they can be moved relatively to one another in a direction 
parallel to this diameter. The image of a point produced 
by a lens always lies on the line (produced if necessary) 
passing through that point and the centre of the lens. The 
effect of moving one half of the object glass is therefore to cause 
the image of a star produced by it to be displaced in a direction 
parallel to that in which the centre is displaced, i.e. parallel to 
the bounding diameter, and since the star is at a very great 
distance, the linear displacement of the image is equal to that 
of the object glass. Suppose, then, that two neighbouring stars 
are under observation and that the two halves of the object 
glass are not relatively displaced: they therefore produce 


coincident images of the two stars Si, S 2 (Fig. 70). If, now, the 
upper half, A, of the object glass is displaced to 0', the images 
produced by it are displaced to Si, S 2 , and if the distance 00' 
is equal to SiS 2 , the image S 2 will coincide with S^ Similarly, 
if displaced to 0", where 00" equals SiS 2 , the images Si" and 
$ 2 will coincide. If then the portion A is displaced by an 
accurate micrometer screw so that first the images S 2 an( l Si 
coincide and then Si" and $ 2 , the total distance through which 
the object glass is moved, d, is twice SiS 2 . The angular distance 
apart of the two stars is therefore d/2f, f being the focal length 
of the objective 

The heliometer was found of great value in the determination 
of the solar parallax by the minor planet method, the heliometer 
being employed to determine accurately the distances of the 
minor planet from several neighbouring stars. It also proved 
invaluable in observations for the determinations of stellar 
parallax, in which minute relative displacements of stars require 
to be measured. The observations which can be made with 
its aid can, however, now be made much more advantageously 
and with a great economy of time by photographic methods. 
With the heliometer, observations were slow and tedious, if 
great accuracy was sought ; and considerable skill on the part 
of the observer was required. The instrument may therefore 
now be regarded as mainly of historical interest, although 
fewer than 50 years ago the results obtained by its aid were 
invaluable for the development of astronomical knowledge. 

106. Zenith Telescopes. -The most accurate and con- 
venient method for the measurement of latitude depends upon 
the determination of the difference in the zenith distances of 
two stars which culminate at nearly equal distances respectively 
north and south of the zenith. For this purpose, a special 
instrument called a zenith telescope is employed. 

The ordinary typo of zenith telescope is generally similar 
to a simple transit instrument, with the addition of an accurate 
declination micrometer to the eyepiece and of a sensitive 
latitude level to the telescope tube, the level being in the plane 
of the meridian. The telescope is set, before an observation, 
on a pair of stars, to the altitude approximately corresponding 
to their mean zenith distance, the setting being made with the 


aid of a graduated circle. The latitude level is then set 
horizontal. As the first star crosses the meridian, its distance 
north or south of the central wire is measured with the aid of 
the micrometer and the reading of the level is taken : the 
instrument is then reversed, the setting of the level remaining 
unaltered and the position of the second star relatively to the 
centre wire determined and the reading of the level again taken. 
If the values in angular measure of one revolution of the micro- 
meter screw and of one graduation of the level arc known, the 
comparison of the two micrometer measures, corrected for the 
difference in level reading, gives the difference of the zenith 
distances of the two stars. 

There are disadvantages attaching to the use of such a 
sensitive level as is required in the zenith telescope. Any 
inequality of temperature, such as warmth from the observer's 
body, is liable to upset the reading. In order to avoid the use 
of levels, a floating telescope was devised by Cookson and is in 
use at the Greenwich Observatory. The telescope floats in an 
annular trough of mercury and is simply rotated through 180 
between the observations of the two stars of a pair. It must 
therefore rotate about an accurately vertical axis and no level 
correction is required. The observations with this instrument 
are made photographically. 


107. The Determination of Time. One of the most 
important and fundamental observations of astronomy is the 
determination of time. The problem reduces to the deter- 
mination of the error of a time-piece and the method almost 
universally adopted in the observatory consists in observing 
with a transit instrument the time as given by the sidereal 
clock of the transit of a star across the meridian. The right 
ascension of a star at the instant of its meridian transit is 
equal to the sidereal time of that instant : therefore, if the 
right ascension of the star be known, the true sidereal time 
corresponding to the time given by the clock is determined and 
thus the error of the clock is obtained. The sidereal time at 
mean noon for the longitude of Greenwich is given for each 
day in the Nautical Almanac, and by interpolation the sidereal 
time at mean noon for any longitude can be obtained. The 
local mean time corresponding to any sidereal time can, there- 
fore, be calculated. 

For the purpose of time determination, a series of bright 
stars are used whose right ascensions have been determined 
from long- continued observations with very great accuracy. 
Such stars are called clock-stars. To obtain a good deter- 
mination of the clock-error, several such stars should be 
observed. It is further necessary to correct the observed time 
of transit for the effect of the instrumental errors of collima- 
tion, level and azimuth as explained in 97. Determinations 
of these errors should be made as nearly as possible to the time 
of the star observations. 

From observations made on two consecutive nights, or during 
the course of a single night, the rate of the clock can be deter- 
mined. Provided the clock is compensated against variations 



of temperature and atmospheric pressure, this rate can be 
carried forward during spells of cloudy weather when stellar 
observations are not possible, without the liability of serious 
error being incurred. 

There is one source of error known as the personal equation 
to which observations, which consist in the determination of 
the instant at which a star crosses a certain wire in the field, are 
liable. The estimation of the exact instant is a subjective 
phenomenon which differs with different observers. One 
observer may make the tap which completes the chronograph 
circuit when he judges the star to be bisected by the wire. 
Owing to the time required for the retinal stimulus to be trans- 
formed into muscular action, his tap will be slightly late : 
another observer may just anticipate the transit, so that he 
actually makes the tap at the exact moment that the star is 
bisected by the wire. Such an observer would record the time 
of transit earlier than the former observer. It is found that 
these personal differences may remain nearly constant for 
skilled observers, for long periods, though they are liable to 
change if the observer gets fatigued after a long spell of observ- 
ing. They may amount to an appreciable fraction of a second 
of time. In the case of all differential observations such errors 
are eliminated, but they enter with full force into the deter- 
mination of time. It is, therefore, desirable that the impersonal 
wire micrometer should be used : with this form of micrometer, 
the observer simply holds the star bisected by the wire by 
traversing the frame carrying the wire across the field of view 
with a steady motion and at the appropriate rate. The con- 
tacts which complete the electric circuit and send signals to 
the chronograph are then made automatically. With this 
type of micrometer, personal equations are reduced to a few 
hundredths of a second of time. 

108. The Determination of Time at Sea. The preceding 
method cannot be used at sea. The most convenient method 
is then to observe with a sextant the altitude of the Sun or 
Moon or of a known star. In the case of the Sim or Moon, 
the altitude observed is that of the lower or upper limb, which 
must be increased or decreased by the semi-diameter to obtain 
the altitude of the centre. The time shown by the chronometer 


at the instant of the observation is noted. In order to deter- 
mine the time, it is necessary that the latitude of the place of 
observation should be known : this must be determined from 
previous observation and, in the case of a ship in motion, 
corrected for the distance made good by the ship in the interval 
between the two observations. 

Referring to Fig. 71, S represents the object observed, P 
the pole, and Z the zenith. ZP is the meridian. Then in the 
spherical triangle ZPS, the 
three sides are known ; ZP is 
the complement of the latitude ; 
ZS is the zenith distance or 
complement of the observed al- 
titude, which must be corrected 
for refraction, for dip of the hori- 
zon, and, except in the case of a 
star, for parallax also. Ptfisthe FlG - 71 - ; ThG Determination of 

, j_ P j i IT j- Tiino at Sea. 

complement of the declination 

of the object, which is known from the Nautical Almanac. 
The angles of the triangle can, therefore, be computed : the 
angle Zl y S, which is the hour angle of the object at the time of 
observation, can thus be determined. This angle (expressed in 
time) added to or subtracted from the right ascension of the 
body, according as the star is west or east of the meridian, 
gives the sidereal time of the observation. 

Instead of a single altitude, a scries of altitudes in quick 
succession should be observed and the mean altitude and mean 
time used for the computation. The method is the more 
accurate the nearer the object is to the prime vertical, for then 
the rate of change of the altitude with the time is most rapid. 
Near the meridian, the method is very insensitive as the change 
in altitude is then so slow. The effect of an error in the assumed 
latitude is also least when the object is exactly east or west. 
The altitude to be observed should not be less than about 10, 
as at lower altitudes, somewhat large errors would be intro- 
duced on account of the uncertainty of the amount of the 
refraction so near the horizon. 

109. The Determination of Latitude. (i) The Talcott 
Method. The most accurate method for the determination of 



latitude is that known as the Talcott method, after Captain 
Talcott of the United States Engineers, who used it in 1845 in a 
boundary survey. The method consists in the determination, 
with the aid of a zenith telescope, of the difference between 
the meridian zenith distances of two stars of known declina- 
tions which culminate at nearly equal distances respectively 
north and south of the zenith. If <f) is the latitude, d s , d n , f C n 
the declinations and meridian zenith distances of the south and 
north stars respectively, then 

* = a. + f , = <* - 
=4(*. + <U+i(C. -O 

The zenith telescope ( 106) provides an accurate determination 
of (, n ) and the declinations of the stars are known. 
Hence the latitude is determined. 

The method has the great advantages of avoiding almost 
entirely errors due to refraction and of not requiring an accur- 
ately graduated circle. A rough setting of the instrument is 
sufficient, the difference of the zenith distances being measured 
by the micrometer screw. 

(ii) By Circumpolars. A simple method of determining 
latitude, which is suitable for observations with a fixed meridian 
circle, is to observe the altitudes of a close circumpolar star at 
its upper and lower culminations, with an interval of 12 hours 
between the observations. Each altitude must be corrected 
for refraction and their mean determines the latitude, the 
altitude of the pole being equal to the latitude. The method 
is not suitable for low latitudes, as refraction then becomes 
very large for circumpolar stars. 

(iii) Suns Maximum Altitude. Observations at sea must 
be made with a sextant, and the method best adapted to this 
purpose is to observe the maximum altitude of the Sun. 
Observations should be commenced a few minutes before local 
apparent noon and a succession of altitudes observed until 
the values begin to decrease. The maximum value obtained 
(after correction for the northward or southward motion of 
ship, for refraction, parallax, dip of the horizon, semi-diameter 
and motion of the sun in declination) gives the latitude by the 
formula <f> = d , the sign being + or according as the 
Sun is south or north of the zenith. 


(iv) By the Use of the Gnomon. This is a method of merely 
historical interest, as it was the only method available to the 
ancients. A vertical stick or column is erected on a horizontal 
piece of ground and the length of its shadow is observed at 
noon each day. This length varies from day to day owing to 
the changing declination of the Sun : it is greatest at winter 
solstice, when the Sun's midday altitude is least, and it is least 
at summer solstice. If AB (Fig. 72) is the gnomon, Si, S 2 
the ends of the noon shadows at the 
winter and summer solstices respec- 
tively, the angles ABSi and ABS 3 
can be calculated : these determine 
the Sun's maximum and minimum 
zenith distances. The mean of 
these angles, therefore, gives the 


distance of the equator from the ^ _ , ~ 

* . fiG. 1Z. Irio (_*nomon. 

zenith, for at winter solstice the 

Sun is as far south of the equator as at summer solstice it is 
north of it. But the distance of the zenith from the equator 
is equal to the distance of the pole from the horizon, i.e. is 
equal to the latitude. 

110. The Determination of Longitude. Whilst the poles 
of the Earth's axis of rotation serve as universal reference 
points for the determination of latitude, there is no corre- 
sponding reference point for longitudes. The determination 
of longitude is, therefore, a more intricate problem. It was 
to the necessity of providing accurate observations of the 
Moon and fixed stars for use in determining longitude at 
sea that the observatories of Greenwich and Paris owe their 

The meridian passing through Greenwich is adopted as the 
arbitrary zero from which longitudes on the Earth are 
measured. The longitude of any other place on the Earth's 
surface is measured by the arc of the equator intercepted 
between the meridians through that place and through Green- 
wich respectively. Longitudes are usually measured in time, 
and the difference of longitude between any two places is then 
the time required for the Earth to turn through an angular 
distance sufficient to bring the meridian of one of the places 


into the position held by the other. The difference of longitude 
is therefore the difference of the local times at the two places. 
We have already explained in 107 how the local mean time 
at any place may be found : the problem therefore reduces 
to finding the corresponding local time at the other place 
without going there. Alternatively, if any common pheno- 
menon can be observed from each of the two places and the 
local time of its occurrence at each place determined, the 
difference of the local times will give the longitude difference. 

(i) By Telegraphic Signals. The most obvious method by 
which to connect the two places is by telegraphic signal, and 
this provides also the most accurate method of determining 
a difference of longitude. Formerly an ordinary telegraph 
cable was employed for the transmission of the signals, but 
the development of wireless telegraphy has superseded the use 
of a cable and has greatly facilitated the determination of 
longitudes. There arc few parts of the Earth which are not 
now within range of one at least of the high-powered wireless 
transmitting stations which send out daily, at certain specified 
times, a series of time signals. These arc emitted at definite 
standard times, and if an observer compares the time at which 
these signals are received with the time by his clock, whose 
error has already been determined from stellar observations, 
he has at once the material for determining the difference 
between his longitude and that of the standard meridian. 

For the accurate determination of the difference of longitudes 
of any two stations, some further precautions must be taken. 
Observers must be stationed at each place and signals from 
some station must be observed and the times of reception 
compared. Special precautions must be adopted to avoid the 
possibility of personal or systematic errors entering into the 
result. It was formerly customary to interchange observers 
in the middle of the series of observations in order to eliminate 
as far as possible personal errors : but the use of impersonal 
micrometers renders this precaution hardly necessary. When 
a telegraph cable was used, it was necessary to compare the 
chronograph at each station with that at the other by means 
of special signals sent in both directions so as to determine the 
time occupied in the transmission of the signals. The velocity 
of wireless waves, on the other hand, is equal to that of light 


and can therefore be neglected unless super-refinement is sought. 

(ii) Longitude at Sea. In order to determine longitude at 
sea every ship carries an accurate time-piece called a chrono- 
meter, whose error and rate must be determined prior to the 
voyage. If this chronometer is set to give Greenwich time and 
the local time is determined by observation of the Sun's altitude 
when near the prime vertical, the longitude can be obtained. 
The accuracy of the result is dependent upon the chronometer 
maintaining its rate without variation, but wireless time 
signals now provide a check upon its behaviour : a comparison 
of the chronometer with the time of reception of signals emitted 
at a definite Greenwich time enables the error of the chrono- 
meter to be determined and the constancy or otherwise of its 
rate verified. 

(iii) Eclipses of the Satellites of Jupiter may be used to 
determine longitude, since they occur at the same instant for 
all observers and therefore provide a common reference signal. 
They also occur with sufficient frequency to be of use. 
Unfortunately, the disappearance of a satellite when eclipsed 
is gradual and not instantaneous as is the case when a star is 
occulted by the Moon ; the accuracy obtainable by this means 
is therefore not very high. 

(iv) Observations of the Moon. One of the oldest methods of 
determining longitudes is based upon the use of the Moon as 
a clock, and although the telegraphic method is now used almost 
exclusively, this niethod is not without interest. The Moon 
changes its place amongst the stars, and therefore also its 
right ascension and declination, much more rapidly than any 
other celestial object. Its position in the sky is given in the 
Nautical Almanac for every hour of Greenwich time throughout 
the year. If then the position of the Moon amongst the stars 
is observed and corrected for parallax, so as to reduce the 
observation to one made by an observer at the centre of the 
Earth, the Greenwich time of the observation can be estimated 
by interpolation from the Nautical Almanac tables, and a 
means provided for the determination of longitude. The dis- 
advantage of the method is that the motion of the Moon 
amongst the stars is relatively slow, so that errors of observation 
enter into the deduced longitude magnified about thirty times. 

The observation of the Moon may consist either (i) in the 



determination of its right ascension at the instant of meridian 
passage with the transit circle, the error of the clock having 
been determined by star observations (which method has the 
advantage that no correction for parallax is necessary) or (ii) 
by observing the distance of the Moon from stars near its path 
which is the method suitable for use at sea with a sextant : 
the distances must be corrected for parallax and compared 
with tables constructed for the purpose ; (iii) alternatively, 
if an occupation of a star by the Moon is observed at two places, 
the longitude difference can be deduced ; the theory in this 
case is similar to that by lunar distances, the distance of the 
star from the centre of the Moon being then equal to the Moon's 

111. The Determination of Position at Sea.- The older 
methods of determining the position of a ship at sea depended 
upon separate observations for the latitude, obtained by 
determining the Sun's meridian altitude, and for the longitude, 
depending upon observations of the Sun's altitude when near 
the prime vertical, and for which a knowledge of the latitude 
was essential. 

These methods have been largely superseded by a more 
convenient and accurate method, first proposed by Captain 
Sumner, of Boston, in 1843. Several modifications of the 

method have been employed, but 
they all depend essentially upon the 
observation of an altitude of the 
Sun, Moon, or star with the corre- 
sponding chronometer time. 

In Fig. 73, SPS' represents a sec- 
tion of the Earth and E its centre. 
Suppose that, at any instant, the 
Sun is in the zenith Z of the point P 
and that at another point S is a ship, 
Z' being in its zenith. Then since 
the angle between the horizons at the 
points S and P is equal to the angle 
SEP, it follows that the zenith dis- 
tance of the Sun observed from 8 will 
also be equal to this angle. If SS' is a small circle with its centre 

FIG. 73. The Sub-Solar 


at P and with radius equal to the angle SEP, at every point 
on this circle the zenith distance of the Sun will be the same. 
If, then, the point P can be determined and the Sun's zenith 
distance obtained from the altitude observation, the circle 
SS r can be drawn on the chart and the position of the ship 
must be somewhere on this circle. Actually, the position is 
approximately known from dead-reckoning observations and 
therefore it is only necessary to draw a small portion of the 
circle in the neighbourhood of the dead-reckoning position. 
This is the basis of Sumner's method of determination of 

The point P is called the sub-solar point. Since the Sun 
(or other body under observation) is in its zenith, it follows 
that the latitude of P must equal the declination of the Sun 
(or other body) at the moment of observation. This can be 
obtained from the Nautical Almanac, if the Greenwich Mean 
Solar Time at the moment of observation is known. This is 
obtained in the usual way from the chronometer corrected for 
error and rate to the moment of observation. The longitude 
of the sub-solar point is the angle between the meridians of 
Greenwich and the Sun, since the Sun is on the meridian of the 
sub-solar point. This angle is the Greenwich Apparent Time of 
the moment of observation, which is equal to the Greenwich 
Mean Time plus the equation of time. In the case of the Moon 
or of a star, it is obtained by adding to the Greenwich Mean 
Time the right ascension of the mean Sun (obtainable from the 
Nautical Almanac) and subtracting the right ascension of the 
Moon or star. The position of the sub-solar point (which is 
fixed by its latitude and longitude) can therefore be obtained 
without difficulty and the corresponding Sumner line drawn. 

As the radius of the Sumner line may be large, it is more 
accurate to use the following method for drawing it on the 
chart : a convenient assumed position for the ship is chosen, 
which must be somewhere near its true position, e.g. the dead- 
reckoning position might be adopted : then, since the latitude 
and the longitude of the sub-solar point are known, the altitude 
of the Sun as observed from this assumed position can be 
readily calculated. If this computed altitude agrees with the 
observed altitude, the assumed position must lie on the Sumner 
line, and a line drawn on the chart through this point in a 


direction perpendicular to the bearing of the Sun will be the 
required line of position. If, as will generally happen, the 
computed altitude is smaller than or greater than the observed 
altitude, then the assumed point is respectively outside or 
inside the Simmer line : if from the assumed point, in the 
direction of the bearing of the Sun, a distance is measured 
towards or away from the sub-solar point equal to the differ- 
ence between the observed and computed altitudes (using the 
relationship 1 nautical mile equals 1 minute of arc), a point on 
the Sumner line is obtained, and the line itself is obtained by 
drawing a line through that point on the chart at right angles 
to the Sun's bearing. 

If a second Sumner line can be drawn for the same instant, 
then since the ship must be on this line also, the intersection 
of the two lines will give its position. To construct the second 
line, the same object may be observed somewhat later : the 
motion of the ship in the meantime is known with sufficient 
accuracy from the compass course and log, and to take the run 
between the two observations into account the first Sumner 
line must be shifted parallel to itself by an amount correspond- 
ing to this run. The intersection of this shifted line with the 
second line determines the position of the ship at the time of the 
second observation. Another method of obtaining the second 
Simmer line is to observe two bodies, say the Sun and the 
Moon, with as short an interval as possible between them. The 
essential thing to secure is that the two lines should intersect 
as nearly as possible at right angles, so that the point of inter- 
section may not be affected too much by errors of observation. 

Instead of actually plotting the two lines on the chart and 
determining their point of intersection, this point may be 
determined by computation ; for the details of which reference 
should be made to a treatise on navigation. 

The advantage of the Sumner method is that both the 
latitude and longitude of the ship are determined from two 
observations at any convenient time of the altitude and bearing 
of a body, with the corresponding chronometer times. With 
the older methods, a determination of latitude was necessary 
by means of a noon-sight (a knowledge of the longitude not 
being necessary), and then a determination of longitude by a 
time-sight, for the reduction of which a knowledge of the 


latitude is necessary. Any error in the determination of 
latitude therefore enters also into the determination of longitude 
and, in addition, some time probably elapses between the 
two observations, so that errors in the dead reckoning also 
enter into the latitude assumed for the time-sight. The 
Sumner method is the best method to use under any circum- 
stances, and even when a noon-sight is taken it is advisable to 
treat it as a Sumner observation and to work out the corre- 
sponding Sumner line. 

112. Determination of Right Ascension and Decli- 
nation. The position of any celestial object is defined by 
its right ascension and decimation. These are best determined 
with the aid of the meridian circle. 

The right ascension of a body is the sidereal time at which 
it crosses the meridian, and therefore all that is necessary for 
its determination is to find first the error and rate of the clock, 
and then to observe the time of meridian passage of the body, 
which must be corrected for errors of collimation, level and 

The declination of the object is obtained from the circle 
reading at the instant of meridian passage, corrected for the 
effects of refraction and, if necessary, of parallax. The zero 
of the circle may be determined from observation of the nadir 
point with a mercury horizon, as previously explained. In 
effect, the zenith distance of the body is observed, and, the 
latitude being known, the declination is deduced. Alterna- 
tively, observations of close circumpolars at an interval of 
12 hours, and corrected for refraction, may be used. The 
circle reading corresponding to the pole is thus obtained, and 
so the north-polar distance, and therefore the declination, are 
directly determined. 

Observations with the meridian circle may be divided into 
two classes : fundamental and differential. Fundamental 
work consists in the absolute determinations of positions of 
certain stars which are then known as fundamental stars ; 
differential work consists in observing the positions of objects 
relatively to one or more of the fundamental stars so that only 
the differences in their right ascensions and declinations are 
actually observed and any instrumental errors enter into the 


final result with much less weight than in an absolute deter- 

We have mentioned that the error of the clock is determined 
from the observation of certain stars whose right ascensions 
are known with great accuracy and that the determination 
of right ascension involves a knowledge of the clock error. 
Although this is the method adopted in practice, it is in reality 
arguing in a circle. The method used to determine the right 
ascensions of the clock stars must be explained. Right ascen- 
sions are measured from the vernal equinox, the imaginary 
point at which the Sun crosses the equator. The absolute 
determination of a right ascension therefore necessarily involves 
a comparison of the star with the Sun. The procedure is to 
observe the clock time of meridian transit of the Sun and its 
declination at that instant on every possible day throughout 
the year. The clock times of transit of the stars chosen as 
clock stars are also obtained throughout the year, during the 
periods when they are visible. The observations of the Sun's 
declination provide a determination of the obliquity of the 
ecliptic : using the value so derived, every observation of the 
Sun's declination gives by trigonometric computation a value 
for its right ascension. Any small error in the adopted value 
of the obliquity will be almost entirely eliminated in the mean 
since it will have opposite effects at winter and summer solstices. 
The sidereal time at the moment of the Sun's transit is equal 
to its right ascension and therefore by comparing the computed 
right ascension with the corresponding observed clock time 
of meridian transit the clock error is deduced. Successive 
observations determine the clock rate, and by interpolation 
the clock error corresponding to the instant of any of the star 
transits can be determined. By correcting the clock time of 
transit for this error the right ascension of the star is obtained. 
By the accumulation of observations, these right ascensions 
become accurately known, and they can then be used as a 
basis for the determination of other right ascensions by the 
methods previously described. 

113. Reduction of Star Places from one Epoch to 
another. Direct observation gives the apparent place of a 
star, i.e. its position as actually seen by an observer on the 


Earth, referred to the actual pole and equinox at that date. 
The mean place is the position at the same instant as it would 
appear to an observer at rest on the Sun. The mean and 
apparent places vary from epoch to epoch on account of the 
fact that the pole and the equinox are not exactly fixed, owing 
to the effects of precession and nutation, and also because the 
stars themselves are in motion ; their distances are, however, 
so great that their apparent angular motions are in general 
small, even over a period of a century. To reduce the apparent 
place to the mean place at a definite epoch corrections must be 
applied for precession, nutation, aberration, annual parallax, 
and for the proper motion of the star. These reductions may 
be expressed in the form : 

Aa - Aa + Bb -\- Cc + Dd + E + ^ a r 

Ad - Aa' + Bb' + Cc' + Dd' + /vr 

In these formula?, A, B, C, D, and E are independent of the 
position of the star, but are functions of the time. At any 
definite time they are therefore the same for all stars, but they 
vary slowly from day to day. They are known as Besselian 
day numbers, after the astronomer Bessel, who first introduced 
them. In the Nautical Almanac they are tabulated for every 
day of the year. The quantities a, a', 6, 6', etc., on the other 
hand, are functions of the place of the star, but are practically 
independent of the time, though they vary slowly over a period 
of years. They are therefore termed the star constants. The 
terms depending on A, B, and E in the above formulae arise 
from the effects of precession and nutation ; those depending 
on C and D from aberration and parallax. The last term in 
each formula represents the effect of proper motion. These 
formulae are used to reduce the observed positions of a star 
to the mean position at the commencement of the year in 
which the observations were made. To reduce from this 
position to the mean position at any other epoch a reduction 
for the effects of precession and proper motion must be applied. 
For a rigorous reduction, trigonometrical methods must be 
employed, for details of which reference should be made to a 
treatise on spherical astronomy. In the case of stars which 
are not too close to the pole and for periods of time which are 
not more than a few score years, an approximate reduction is 
sufficient, using the formulae : 



In these formula) a t , a ; d t , (5 denote respectively the 
right ascensions and declinations at the initial period and at a 
period t years later. The annual variations a and a' include 
precession and proper motion and are usually given in star 
catalogues without the proper-motion component. The 
proper motions, if known with sufficient accuracy, may be 
given separately. The terms b and b' are called the secular 
variations and are also usually tabulated in the catalogues. 

114. Determination of Azimuth. This is a problem of 
importance to surveyors and also to astronomers. The most 
accurate method is to observe with a theodolite, which has 
been carefully adjusted for collimation and levelled, the angle 
between the pole star and either a distant fixed object or the 
cross wires of a suitable rigid collirnator. The time of the 
observation of the pole star must be noted ; its right ascension 
and declination for that instant can then be obtained from the 
Nautical Almanac, and, knowing these, its azimuth can be 
calculated. The azimuth of the collimator or distant object 
may then be obtained, and used as a zero from which the 
azimuth of any other body may be observed. 

In order to compute the azimuth of the pole star, the latitude 
of the place of observation must be known. The advantage 
of using this star is that any slight errors in the assumed latitude 
or in the observed time of observation produce very little 
effect on the deduced azimuth. If the time of observation is 
determined very accurately, the Sun or a star whose altitude 
is not greater than about 30 may be used, but, in general, the 
pole star will be found most suitable and accurate for the 


115. The Planets. The so-called fixed stars retain their 
relative positions on the celestial sphere with such accuracy 
that refined observations are necessary to detect their motion. 
It was known to the ancients that there were a few bodies which 
moved about amongst the other stars, and these were called 
planets or wanderers. Under this term they included Mercury, 
Venus, Mars, Jupiter, and Saturn, as well as the Sun and the 
Moon. The term planet is now restricted to the bodies which 
revolve in definite orbits about the Sun. It includes, in addition 
to Mercury, Verms, Mars, Jupiter, and Saturn, the Earth and the 
two distant bodies, Uranus arid Neptune, which were unknown 
to the ancients, in addition to a very large number of smaller 
bodies, termed minor planets or asteroids, whose orbits lie 
between those of Mars and Jupiter. The Sun, the central 
body of the system, and the Moon, the satellite of our Earth, 
are not now regarded as planets. 

Kepler's Laws. From a study of the extensive 
and long-continued planetary observations of the Danish 
astronomer, Tycho Brahe (1546-1601), Kepler between 1607- 
1620 formulated three empirical laws which he found were 
satisfied by the motions of the planets. These laws are as 
follows : 

^. The orbit of each planet is an ellipse, having the Sun in 
one of its foci. 

V^K. The motion of each planet in its orbit is such that the 
radius vector from the Sun to the planet describes equal areas 
in equal times. 

The squares of the periods in which the planets describe 



their orbits are proportional to the cubes of their mean distances 
from the Sun. 

It will be seen that the first two laws deal with the motion 
of any one planet. The third gives a relationship between the 
periods and distances of the several planets. Thus, if the 
period of any planet be known, its mean distance from the Sun 
in terms of the Earth's mean distance as unity can be deter- 
mined. A determination of any one distance in the solar 
system therefore enables all the other distances to be deter- 
mined, since the periods can easily be obtained by observation. 

The physical meaning of these laws was discovered by 
Newton. He showed that all three laws could be explained 
on the hypothesis that each planet moves under the action 
of an attractive force towards the Sun, proportional to the 
planet's mass and to that of the Sun and inversely proportional 
to the square of its distance from the Sun. The constant of 
proportionality is the same for all the planets and is called the 
constant of gravitation. 

It is desirable to make a distinction between the consequences 
involved in the three laws of Kepler. The second law 
necessarily implies that each planet moves under a force of 
attraction always directed towards the Sun. Moreover, with 
such a force, which in mechanics is called a " central " force, 
whatever the law which it obeys, equal areas must be described 
in equal times. This may be shown from elementary con- 
siderations. In Fig. 74 S represents the attracting centre. 
Suppose first that a body is moving along the line ABO and 
that there is no attracting force ; then its motion must be 
uniform and if AB, BC are lengths described in unit times, 
AB and BC are equal. Hence also the triangles SAB, SBC 
must be equal and the theorem is valid. Now suppose that 
at the moment J5, a velocity of any amount is suddenly applied 
to the body in the direction B8 which in unit time would take 
it to the point 6. Constructing the parallelogram, bBCc, 
it must actually in unit time move to c, and the area described 
by the radius vector is the triangle SBc. But since Gc and SB 
are parallel, the triangles SCB, ScB, being on the same base and 
between the same parallels, must be equal. The area described 
by the radius vector in unit time will therefore be unaltered. 
Now suppose the attracting force to act continuously towards 



S ; whatever its amount, its effect is to produce changes of 
velocity in the direction of the radius vector, and we have just 
seen that these do not affect the rate of description of areas. 
Equal areas will therefore be described in equal times. The 
converse theorem must 
also hold, viz. that if 
equal areas are de- 
scribed in equal times, 
the force must be a 
central one, for if the 
velocity added at the 
point B was not along 
BS, the areas of the 
two triangles SBC, 
SBc would not bo 

Kepler's third law 
involves the univer- 
sality of the constant 
of gravitation. This may be illustrated for the simple case of 
circular orbits. If M is the mass of the central body, m that 
of the attracted body, a the radius of the orbit, w the angular 
velocity of the body, and T its period, then equating the radial 
acceleration of the body to the force of attraction we have 

GMm n 4n 2 

= maw 2 m . -a 

FIG. 74. The Law of Equal Areas. 


OM = 

T 2 

If for another body moving around the same central body 
the radius of the orbit and the period are respectively a', T 1 
and the constant of proportionality (?', 

G'M - 

' 3 


Therefore G/G' =1-^- =l(by Kepler's third law)or G = G'. 

Applying the case of a circular orbit to the motion of the 
Moon around the Earth and denoting by g the force of gravity 
at the Earth's surface, then the Earth's gravitational force 


at the distance of the Moon will be gR 2 /a 2 , R,a being respectively 
the radius of the Earth and the distance of the Moon. But this 
force can also be expressed as GM/a 2 . Hence 

n M R 2 a 

G = = 4jz 2 
a 2 J a 2 T 2 

M denoting here the mass of the Earth, now considered as the 
attracting body. 

Trigonometric measures give a/R 60-27 and the radius of 
the Earth is 6-367 x 10 6 metres. The period of revolution is 
27 d. 7h. 43m. = 39,343 x 60 seconds. Therefore 
g - (60-27) 3 X 4^ 2 x 6-367 x 10 6 /(39343 X 60) 2 = 9-81 metres 

per sec. per sec. 

= 981 cms. per sec. per sec. 

This agrees with the observed value of gravity at the Earth's 
surface. It follows that the gravitational force which holds 
the Moon in its orbit is the same as that which attracts a body 
to the Earth's surface. 

Newton was in this way led to the universality of the law of 
gravitation. It follows that the planets must exert mutual 
gravitational forces upon one another ; the magnitudes of 
these are very much smaller than that of the force due to the 
Sun, on account of the much greater mass of the latter. The 
effect of the combined forces is that the orbits are not accurately 
elliptical, slight deviations occurring when two planets pass 
near one another. It was due to the small deviations of 
Uranus from its predicted position that Adams and Leverrier 
were independently led to the discovery of the then unknown 
planet Neptune. From a mathematical discussion of the 
discordances between prediction and observation, they were 
able to show that these discordances could be accounted for if 
there was a more distant planet whose attraction was disturbing 
the motion of Uranus, and they were able also to assign an 
approximate position to this planet, near which it was dis- 
covered as a direct result of their investigations. 

It should be mentioned that Kepler's third law is not strictly 
accurate, though the discordance is very small. It would be 
accurate provided that the masses of the planets were negligible. 
Actually, they exert an attraction on the Sun, and the attrac- 
tive force per unit mass, relative to the Sun, is therefore 



G(M + m)/a 2 . The accurate form of Kepler's law is thus : 

(M + m)T* : (M + mJT^ - a 3 : a, 3 
m, m 1? M being the masses of the two planets and of the Sun. 

117. Apparent Motions of the Planets. The apparent 
motions of the planets as seen from the Earth are the resultant 
of the actual motion of the planet around the Sun and an 
apparent motion due to the Earth's own orbital movement. 
This combination of two distinct velocities produces certain 
peculiarities in the apparent motion which we shall proceed 
to describe. 

Superior Conjunct, i on 


W. Quadrature 

FIG. 75. Planetary Configurations. 

Certain terms commonly used in connection with planetary 
motions must first be defined. When the planet is in a line 
with the Sun and the Earth, it is said to be in Superior Con- 
function ; when an inferior planet (i.e., a planet whose orbit is 
within the Earth's orbit) is in a line with the Sun and Earth 
and between them, it is said to be in Inferior Conjunction. The 
corresponding position for an outer planet is when the Earth is 
between the Sun and the planet, and the planet is then said to 
be in Opposition. In the case of an outer planet, when the 
direction from the Earth to the planet is at right angles to that 
from the Earth to the Sun, the planet is said to be in quadrature. 



east or west, according as it is east or west of the Sun. The 
angle Planet-Earth-Sun is called the Elongation : for an outer 
planet this angle can have any value from to 180 ; for an 
inner planet it varies between and a maximum, less than 90, 
called Greatest Elongation, whose value depends upon the 
relative sizes of the orbits of the Earth and the planet. The 
positions of greatest east and west elongations are shown in 
Fig. 75, which illustrates also the other configurations, defined 
in this paragraph. 

The apparent motion of a planet can now be described, 
starting from superior conjunction. The planet at first moves 
eastwards amongst the stars, increasing its right ascension. 
After a certain time, the apparent motion becomes less and 
then vanishes, the planet being said to be stationary in this 
position. The elongation of this position depends upon the 
size of the planet's orbit. After reaching the stationary 
position, the planet begins to move westward, with decrease 
of right ascension. It is then said to retrograde. The middle 
of the period of retrogression occurs at inferior conjunction for 
an inferior planet and at opposition for a,n outer planet. The 
retrograde motion is succeeded by a second stationary point and 

then by eastward motion, 
motion bringing the planet 
back to superior conjunction. 
The time spent in the 
direct motion always exceeds 
that spent in the retrograde 

118. Explanation of 
Apparent Motions . We 

can now show how the 
apparent motions can be 
explained by the com- 
bination of the velocities 
of the Earth and the planet. 
In Fig. 76, 8 represents the 

Sun, P and Q any two planets in conjunction on the same 
side of the Sun, P 1 and Q' represent the corresponding posi- 
tions of the two planets after a short interval of time. For 

FIG. 76. Explanation of Planetary 


simplicity, it will be assumed that the orbits are circular ; 
since the eccentricities of the planetary orbits are small, the 
qualitative description of the phenomena will not thereby be 

According to Kepler's third law, the ratio of the periods of P 
and of Q is equal to (SP/SQ) W . Also 

PP'/QQ' = velocity of P/ velocity of Q 

8P X angular rate of motion of P 
SQ X angular rate of motion of Q 

_ ________ SP _____ / ____ SQ _ 

periodic time of P / periodic time of Q 

Therefore since SQ is greater than SP, PP f is greater than 

Let us now suppose Q to be the Earth and P any inferior 
planet. For certain corresponding positions, Pj and Q l9 also 
P 2 , Q 2 of the two bodies, the line joining them is tangential to 
the inner orbit. When the planet moves a short distance to 
P/, which is practically on the line PiQi> the Earth moves to 
Qi and the apparent direction of motion of the planet projected 
on the celestial sphere is evidently the same as the direction in 
which the orbits are described, i.e. direct. At inferior con- 
junction, on the other hand, since PP' is greater than QQ' and 
each is at right angles to PQ, the apparent position of the planet 
in the heavens when the Earth is at Q' is displaced forward as 
compared with its position seen from Q, i.e. in a direction 
opposite to that in which the orbits are described, and the 
apparent motion is therefore retrograde. At the positions 
P 2 Q 2 , the apparent motion is evidently again direct. It 
follows that at some point between Q l and Q, and also between 
Q and Q 2 , the motion changes from direct to retrograde, and 
conversely. These are the stationary points. If P 3 , Q 3 denote 
one of them, the consecutive positions P 3 ', Q 3 ' are such that 
P 3 Q 3 and P 3 'Q 3 ' are parallel. 

For the case of a superior planet, we can suppose P to be the 
Earth and Q the planet. Then, remembering that the apparent 
position of the planet is now given by the line PQ produced 
(not QP as before), a similar line of reasoning proves that when 



the Earth is at P l and P 2 , the apparent motion is direct, at P is 
retrograde, and at P 3 is stationary. 

It follows, from the preceding, that if each planet is seen 
from the other, the apparent motion of each planet will be 
exactly the same at the same time, i.e. both retrograde, both 
stationary, or both direct. 

119. The Ptolemaic System. It is of interest to examine 
in what manner the apparent motions of the planets were 
accounted for by the ancient astronomers who believed the 
Earth to be fixed and the centre of the celestial universe. The 
hypothesis advanced by Ptolemy about A.D. 140 was univer- 
sally accepted for fourteen centuries and continued to receive 
a large measure of assent for some time after Copernicus 
advanced the theory that the Sun was at rest and that the 
Earth in common with the other planets moved round it. 
Ptolemy supposed that to each planet belonged a circular orbit, 
called the planet's deferent. The planet did not itself move 

Fro. 77. The Ptolemaic System. 

upon the deferent, but moved around the circumference of a 
smaller circle called the epicycle, whilst the centre of this circle 
moved round the deferent. Thus the actual motion of a planet 
was compounded of two uniform circular motions, the motion 
of the deferent and that of the planet relative to the deferent. 
In the case of the Sun and the Moon, there was no epicycle, 
these two bodies moving around their deferents. 

The deferents of Mercury and Venus were inside the deferent 
of the Sun and it was supposed that the centres of their epicycles 
revolved around their deferents in a period of one year and in 
such a manner that the line joining them always passed through 


the Earth and through the Sun. The periods relative to a fixed 
direction of the epicyclic motions in the case of these two 
planets were equal to what we now know as the periods of the 
two planets. The revolution of Mercury and Venus in their 
epicycle, evidently on this theory, will make them swing 
backwards and forwards alternatively east and west of the 
Sun, for limited angular distances only, in accordance with the 
observed motions. Also, if the linear velocity of the epicyclic 
motion is greater than that of the epicyclic centre along the 
deferent, the apparent motion will appear retrograde near the 
point where the planet crosses the line joining the Earth and 
the Sun, on the side towards the Earth. The deferents of Mars, 
Jupiter, and Saturn were exterior to that of the Sun, and the 
epicyclic radii at the ends of which the planets were situated 
were supposed always to be parallel to the line joining the 
Earth and the Sun. This ensures that retrograde motion, 
which will only appear near the position of the planet in its 
epicyclic motion in which the radius from the centre of the 
epicycle to the planet passes through the Earth, will only 
occur when the planet is in line with the Sun and Earth. 

Each deferent was supposed to be carried on the surface of a 
perfectly transparent crystal sphere, and all these spheres 
rotated once a day about an axis passing through the poles of 
the heavens. The fixed stars were supposed to be attached to 
an outer crystal sphere which rotated with the others. This 
common rotation, from east to west, gave rise to the diurnal 
phenomena which we now attribute to the rotation of the 

This theory was able to account successfully for the general 
features of the observed motions of the planets ; it explained 
the direct and retrograde motions and the observed periods of 
revolution relatively to the Sun. As observations became more 
accurate, it was found that the theory did not entirely account 
for the actual motions, and it then became necessary to com- 
plicate the theory by adding additional epicycles, i.e. by 
supposing that the planet moved around an epicycle, the 
centre of which moved around a second epicycle, the centre 
of this epicycle moving around the deferent. It was also 
necessary to suppose that the Earth was not exactly in the 
centres of the deferents nor the centres of the epicycles exactly 


on the deferents. The observed irregularities of motion were 
thus explained, but at the expense of making the theory more 
and more artificial. 

Copernicus (1473-1543) was the first to assert that the 
diurnal rotation of the Earth was the true explanation of the 
diurnal motion of the stars and that the planets, including the 
Earth, revolved around the Sun. He supposed their orbits to 
be circular and therefore was obliged to retain some small 
epicycles to account for the principal irregularities. The great 
objection raised against this theory was that if the Earth did 
revolve around the Sun in this way, the fixed stars should 
change their apparent relative positions in the sky. If, for 
instance, there were two stars which were on a line passing 
through the Sun, one more distant than the other, then as the 
Earth rotated in its orbit, the nearer one would appear at one 
time on one side of the more distant star and six months later 
would appear on the other side. The most accurate obser- 
vations at that time failed to reveal any such relative displace- 
ments, and this led Tycho Brahe and other astronomers to 
reject the theory of Copernicus. The explanation of this 
negative result is, of course, to be found in the very great 
distances of the stars ; by modern methods the displacement 
can be observed, and it affords a means of measuring the 
distances of the stars, as we shall see later. 

It was not until the time of Kepler, about 65 years after 
Copernicus, that the planetary orbits were shown to be not 
circular but elliptical, and his work, with the theoretical 
explanations given by Newton, established the theory in the 
form in which we now know it. 

120. Sidereal and Synodic Periods. The sidereal period 
of a planet is the actual period of its revolution around the Sun. 
As seen from the Sun, a planet will again be in the same position 
relatively to the stars after one sidereal period. 

The synodic period is the time between two successive 
conjunctions with the Sun, as seen from the Earth. 

If E, P denote the sidereal periods of the Earth and the 
planet respectively and 8 the planet's synodic period, then, 
since the planet and the Earth move around the Sun in the 
same direction, the angular rate of motion of the planet 



relatively to the Earth is the difference between the angular 
rates of motion of the planet and Earth respectively relatively 
to the Sun. Hence 

1 1 

According as the planet is nearer to or farther from the Sun 
than the Earth. The approximate sidereal and synodic 
periods of the several planets are as follows : 

1 1 1 

- QJ 

S~ P E E 


Earth . 
Jupiter . 
Saturn . 
Uranus . 

Sidereal Period. 

Synodic Period. 

88 clays 

116 days 






12 years 








121. Empirical Laws connecting the Relative Distances 
of Planets from the Sun. A curious empirical relationship 
between the distances of the planets from the Sun was formu- 
lated by Bode in 1772 and is known as Bode's Law. To the 
numbers 0, 3, 6, 12, 24, 48, etc., are added the number 4. The 
resulting series of numbers divided by 10 express approximately 
the mean distances of the planets from the Sun in terms of the 
Earth's distance as unity. The numbers obtained by this 
rule are 

04, 0-7, 1-0, 1-6, 2-8, 5-2, 10-0, 19-6, 38-8. 

The following are the approximate mean distances of the 
planets which were known at the time the law was formu- 
lated : Mercury, 0'39 ; Venus, 0-72 ; Earth, 1-00 ; Mars, 1-52 ; 
Jupiter, 5-20 ; Saturn, 9-54. It will be seen that there was 
a gap at 2-8 and the series ended with Saturn. The gap 
between Mars and Jupiter can be regarded as filled by the 
discovery of the belt of asteroids, with mean distance about 
2-65. The discovery of Uranus, mean distance 19-18, continued 
the series, and it is of interest to note that when Adams and 


Leverrier were computing the position of the hypothetical 
planet which would account for the perturbations of Uranus, 
they provisionally assigned to it the distance required by 
Bode's law. Their investigations led to the discovery of 
Neptune, but later observations showed that this planet 
departs more widely than any other from the law, its mean 
distance being only 30-05. The law is, nevertheless, a con- 
venient aid for remembering the approximate relative distances 
of the planets. 

Bode's law can be represented in the form a -f bc n by putting 
n = oc for Mercury, n = for Venus, n = 1 for the Earth, 
etc., with a 0-4, b =0-3, and c = 2. Other empirical laws 
of this type have been formulated, values for a, 6, c being 
chosen so as to represent some of the distances as closely as 
possible. Thus, Belot adopts a = 0-28, b = 1/214-45, and 
c r^= 1-883. Such laws secure a better general representation 
than Bode's law, but are artificial and have no theoretical 

122. Elements of a Planet's Orbit. In order to define 
the position in space of the orbit of a planet and the position 
of the planet in its orbit, seven quantities are necessary. 
These quantities, with their usual designations, are as 
follows : 

1. The semi-major axis of the orbit, a. 

2. The eccentricity of the orbit, e. 

3. The inclination of the plane of the orbit to the ecliptic, i. 

4. The longitude of the ascending node, a. * 

5. The longitude of perihelion, a>. 

6. The epoch, T. 

7. The period, P, or mean motion, n. 

Of these quantities, the first and second define the size and 
shape of the orbit, the third and fourth define the plane of the 
orbit, the fifth defines the direction of the major axis in the 
plane, and the sixth and seventh are used to define the position 
of the planet in its orbit at any time. 

The seven elements are represented in Fig. 78. S represents 
the position of the Sun, ASA' the major axis of the orbit, A 
being perihelion and A' aphelion. ENN'E' is the plane of the 
ecliptic and PNN'P' that of the planet's orbit. NN' is 



therefore the line of nodes. T, =^=, points in the ecliptic plane, 
represent the position of the vernal and autumnal equinoxes. 
If C is the midpoint of A A', then CA = CA' = a, the semi- 
major axis of the orbit which is usually expressed in terms of 
the mean distance of the Earth from the Sun as a unit 
(astronomical unit). 
CS/OA is equal to the 
eccentricity of the 
orbit. The inclination, 
i, is given by the angle 
PNE, the angle be- 
tween the plane of the 
orbit and the ecliptic. 
The longitude of the 
ascending node, a, is 
the angle wSN, the 
direction of motion of 
the planet in its orbit 
being in the direction 
of the arrow-head. 
The so-called longi- 
tude of perihelion, o>, is 
the sum of two angles, 
one & measured in 
the plane of the ecliptic, and the other, a) or NSA (taken 
in the sense shown) measured in the plane of the orbit ; it is 
not, strictly speaking, a longitude. The mean motion or period, 
together with the epoch, i.e. the position of the planet at some 
specified time, are sufficient to determine its position in the orbit 
at any subsequent time, the shape and size of the orbit being 
given. It will be seen that a defines the line of nodes ; i then 
defines the plane of the orbit ; a) defines the position of the 
axis major ; a and e then give the shape and size of the orbit. 

123. Stability of the Solar System. The elements of 
any planetary orbit would be absolutely constant if the 
Sun, assumed spherical, and the planet alone constituted 
the solar system, except for the slight shifting of the 
perihelion required by Einstein's theory of gravitation 
(p. 264). The mutual attraction of the planets, however, 

Fia. 78. The Elements of a Planet's 


introduce small disturbing forces which produce slight changes 
in their orbits. Is it possible that these slight changes may in 
the course of time so add up that the orbits of the planets may 
be gradually modified to such an extent that their physical 
conditions may be entirely altered or the system itself even 
destroyed ? That branch of astronomy which seeks to deter- 
mine the motions of the planets under the forces of gravitation 
and to answer this question is called " celestial mechanics." 
Although the problem of determining the subsequent motion 
of three bodies started in any manner under the action of their 
mutual gravitation is not capable of solution in general, it is 
possible to give an answer to the above question. This 
possibility is due to the preponderating mass of the Sun in the 
solar system, which ensures that the planetary orbits must be 
very nearly ellipses. The mathematical investigations of 
Laplace, Lagrange, and others have shown that the major axes 
of the orbits can undergo only slight changes and that these are 
of a periodic nature, so that the average values taken over 
a sufficiently long period of time will show no change. It 
follows, from Kepler's third law, that the periods can show only 
small periodic changes. The eccentricities and inclinations of 
the orbits relative to a fixed plane may show greater variations 
in the course of thousands of years, but these variations cannot 
exceed certain definite limits. As the fixed plane of reference 
for the inclinations, the ecliptic at a certain epoch may be 
chosen. This is not, however, a natural plane of reference, 
being connected with the orbit of the Earth arid not being 
therefore absolutely fixed. Laplace showed that there was a 
certain plane the position of which remains absolutely un- 
changed by any mutual action between the planets ; this 
plane is called the invariable plane and is defined by the follow- 
ing condition : If the radius vector from the Sun to each 
planet is projected upon this plane and each planet's mass 
multiplied by the area described in unit time by this projected 
radius vector, then the sum of the products so obtained will 
be a maximum. The inclination of the ecliptic to the invariable 
plane is about 2. The limits between which the eccentricities 
and inclinations of the planetary orbits must lie are given in 
the subjoined table : 




Inclination relative 
to invariable Plane. 






9 11 
3 16 

Mercury .... 






i / 
4 44 








3 6 

5 56 
1 1 
1 7 

Jupiter .... 

Uranus .... 

Neptune .... 






Since the major axes and periods in the mean remain constant 
and since the eccentricities and inclinations vary only within 
narrow limits, it follows that the solar system is stable in so 
far as the effect of the mutual attractions of its component 
parts is concerned. 

124. The Determination of a Planetary Orbit. A 

knowledge of the elements of a planetary orbit and cf the 
manner in which they vary with time enables the position of the 
planet at any future date to be predicted. For this purpose, 
long-continued observations are necessary so that the theory 
can be worked out with a high degree of approximation. For 
some purposes it is necessary quickly to determine an approxi- 
mate orbit ; for instance, a minor planet may be discovered, 
and after a few observations have been secured may be lost in 
the Sun's rays at conjunction. If these observations suffice to 
determine the orbit, it becomes possible to identify the planet 
again when it emerges from the Sun's rays. For such purposes 
the assumption is made that the orbit is accurately an ellipse, 
with the Sun in one of the foci, and the calculations follow a 
process invented by Gauss, or a modification of his method. 
There are six elements to determine, since of the seven elements 
which are necessary to define an orbit in space, the mean 
motion or period and the mean distance are connected by 
Kepler's third law. It is therefore necessary to have six 
observational data at known times in order to derive the orbit. 


For instance, if for a given instant of time, the three co-ordinates 
and velocity components of the body relative to three fixed 
planes through the Sun were known, the orbit could be deter- 
mined ; or, again, if the three co-ordinates were known for 
two given instants. Actual observations provide positions 
relative to the Earth, and the most convenient form in which 
the necessary data can therefore be supplied is to give the 
values of the right ascensions and declinations of the body at 
three instants. Gauss's method is based on these data ; by a 
mathematical process the geocentric positions at the three 
times are first derived from the observed right ascensions and 
declinations. The heliocentric positions are then deduced, 
after which the actual determination of the elements is straight- 
forward. For the details of the process, reference should be 
made to a treatise on celestial mechanics. Using Gauss's 
method, three observations of the modern degree of accuracy, 
separated only by a week or two, will give an orbit sufficiently 
accurate for the body to be found again after the lapse of a 
considerable time. Such a preliminary orbit having been 
found, it can subsequently be corrected, if necessary, by 
differential methods based upon further observations. For 
predicting future positions, allowance must be made for the 
perturbing actions of other planets. 

125. Determination Of Diameter s of Planets. There are 
two methods of determining the angular diameter of a planet. 

(i) A filar micrometer may be used, with which may be 
measured the actual linear dimensions of the image produced in 
the focal plane of the telescope ; the value so obtained, divided 
by the focal length of the instrument, gives the angular diameter 
in circular measure. The wire micrometer is generally used, 
the wires being placed tangentially to the two limbs of the 
planet and then crossed over and the observation repeated. 
The observation is subject to an error due to irradiation, 
which is physiological in nature. A bright object appears to 
the eye somewhat larger than it actually is ; although the error 
may be reduced by employing a large instrument and a bright 
field of view, it is very difficult to ensure that it is entirely 
eliminated. It is therefore better to employ a double-image 
micrometer, which forms two images of the planet whose 


distance apart can be varied. The two images are adjusted 
so that they are tangential to one another, and the irradiation 
error in making this observation is less than in setting a dark 
wire tangential to a bright limb. 

(ii) A more accurate method is that devised by Michelson, 
which is entirely free from irradiation errors. If two parallel 
narrow slits are placed in front of the object glass of the tele- 
scope, which is set to view the planet, then the image produced 
in the focal plane consists, in general, of a series of short parallel 
alternately light and dark interference fringes, extending in a 
direction at right angles to the length of the slits. There is 
thus a gradation of light in the field. If the distance apart of 
the slits is varied, this gradation changes, and if the amount of 
light thrown into the bright fringes is increased and that into 
the dark fringes decreased their visibility becomes plainer. 
There are, however, certain distances apart of the slits for 
which the gradation entirely vanishes, the light and dark 
fringes then becoming of equal brightness and therefore ceasing 
to be visible. The distances apart of the slits for which this 
happens are given by : 

d = (1-22, 2-24, . . . ) A/a 

where A is the mean wave-length of the light (which may be 
taken as 5,500 angstrom units or 5-5 x 10" 5 cms.) and a is the 
angular diameter of the object viewed. The determination of 
the least distance apart of the slits for which the visibility of 
the fringes vanishes enables the angular diameter of the body 
to be determined from the relationship a = l-22A/d. This 
observation can be made very accurately and has the advan- 
tages not only of being free from irradiation error but also of 
being relatively independent of atmospheric definition. 

The angular diameter may be converted into linear diameter 
on multiplying by the distance of the planet from the Earth. 
The distances of the planets can all be deduced by Kepler's 
third law when the distance of the Earth from the Sun has been 
determined and the planet's period has been measured. The 
methods by which this distance can be found have already been 
described in 63. 
Synodic period, i.e. the interval between two successive opposi- 

126. Determination of the Period of a Planet. The most 
accurate method of determining a planet's period is to find its 


tions or conjunctions of the planet. In practice, of course, 
the times of the oppositions, i.e. the moments when the longi- 
tudes of the Sun and planet differ by 180, must be observed. 
At opposition, a planet will cross the meridian near midnight. 
The procedure involves the determination of the right ascension 
and declination of the planet at meridian transit for several 
days before and after opposition, the Sun also being observed 
at apparent noon. By interpolation from the latter obser- 
vations the longitudes of the Sun corresponding to the times of 
the planetary observations can be obtained. The planetary 
observations give the longitudes of the planet at the same 
instant. The differences of longitude between Sun and planet 
are tabulated with the corresponding times, and, by another 
interpolation, the exact time of opposition, corresponding to a 
longitude difference of 180, can be derived. 

The planetary orbits not being exactly circular, the mean 
synodic period is not thus obtained. By extending the 
observations over a sufficient number of oppositions, however, 
the mean period can be obtained with any desired degree of 
accuracy. Once the synodic period is known, the true sidereal 
period is obtained from the relationship, l/P = l/E l/S 
(see 120). 

127. Determination of the Mass of a Planet.- If a 
planet has a satellite, its mass can readily be determined as 
follows : If M is the mass of the planet, m that of the satellite, 
a, a' the radii of the orbits of the planet and its satellite respec- 
tively, T, T' their periods, S the mass of the Sun, and G the 
gravitational constant, the accelerating force acting on the 
satellite is given by O(M + m)/a' 2 . But the acceleration in a 
circular orbit is given by the square of the angular velocity 
multiplied by the radius or (2n/T') z X a'. Hence 

a' 2 T' 2 ~ 

Similarly, considering the motion of the planet around the Sun, 


M + m __ t 



The relative distances and the periods must therefore first be 
determined. In general, the mass of the satellite can be 
neglected compared with that of the planet and the mass of 
the planet can be neglected compared with that of the Sun, 
so that we have simply, 

M /a'y /T\ 2 

s " w \r) 

This determines the mass of the planet in terms of that of 
the Sun, and we have previously explained how the masses 
of the Earth and Sun can be determined. Therefore, the 
mass of any planet possessing a satellite can be found. 

In the case of those planets which do not possess a satellite, 
the determination of the mass is more indirect and difficult. 
It must be based upon the magnitude of the perturbation 
produced by the planet on a neighbouring planet when the 
two planets are near their distance of closest appioach : 
knowing the paths of the two planets, it is possible to calculate 
the mass which would produce the observed deviations from 
elliptic motion. Thus Venus perturbs the Earth, and from 
the magnitude of the perturbation the mass of Venus may be 
deduced. So also Mercury perturbs Venus, and this pertur- 
bation enables the mass of Mercury to be deduced. 

The methods by which the masses and linear diameters of 
the planets may be determined have now been detailed. By 
dividing the mass by the volume, the actual density of the 
planet may be obtained. Or, if mass and radius are expressed 
in terms of those of the Earth, the density in terms of the 
density of the Earth as unity can be obtained from the simple 
formula : 

d = M/R* 

For instance, Jupiter's mass derived from satellite observa- 
tions is about 316 times that of the Earth, and its radius is 
about 11 times the Earth's radius. Hence its density is 
3 16/1 1 3 , or about 0-24 of that of the Earth. Assuming for 
the value of the Earth's mean density 5-53, the density of 
Jupiter is found to be about 1-33 times that of water. 

The value of the gravitational attraction at the surface of 
a planet compared with that at the surface of the Earth is 
of importance in forming a conception of the physical con- 


ditions on the planet's surface. Expressing the mass and 
radius in terms of those of the Earth, the surface gravity is 
M/K* or M/R* multiplied by R, i.e. equal to the planet's 
density multiplied by its radius, both quantities being expressed 
in terms of the corresponding quantities for the Earth. At 
the surface of Jupiter, the force of gravity would therefore 
be 11 X 0-24, or 2-64. A body of given mass would therefore 
weigh 2-64 times as much at the surface of Jupiter as at the 
surface of the Earth. 

128. Motion in a Resisting Medium. It is of interest 
to consider in what way the motion of a body moving under 
gravitational attraction would be affected by the presence of 
a resisting medium. It will suffice for an explanation of the 
principles to consider only the case of a circular orbit. 

If v, w are respectively the linear and angular velocities 
when the radius of the orbit is r , and M denotes the mass of 
the attracting body, 

, OM 
rw 2 = 

and v = rw = VGM/r 

so that, for equilibrium, the linear velocity must increase as 
the radius of the orbit decreases. 

The resistance of the medium may be supposed small, and 
proportional to the square of the velocity, say kv 2 . Then, in 
one revolution, the work done by the body against the resis- 
tance is Znrkv 2 = 2nkGM. This must be performed at the 
expense of its kinetic and potential energies. 

The attracting force acting on the body is GMm/r 2 , and if 
the body moves outwards a distance Ar, the decrease in 
potential energy is consequently (GMm/r 2 ) Ar. Also since 
the kinetic energy is %mv 2 , when the velocity increases by Av, 
the diminution in the kinetic energy is mv . Av. Hence we 
must have, by the principle of conservation of energy, 

rt T !.* GMm A . 

znfcGM = Ar mvAv 


But since v* = GM/r, vAv = %GM/r 2 . Ar 
so that Ar = nkr 2 /m 
and vAv = + 2nkGM/m. 


It follows therefore that the effect of the resisting medium 
is to decrease the radius of the orbit and to increase the linear 
velocity and consequently to decrease the period. The increase 
in the velocity appears at first sight to be a paradoxical result, 
but it is in reality a consequence of the decrease in the radius 
of the orbit. 

129. Velocity at any point under Gravitational Attrac- 
tion. It can be shown by dynamics that when any body is 
moving under the action of an attractive central force of 
amount /^/r 2 , its orbit must always be a " conic section/' 
i.e. a curve which may be obtained by cutting a right circular 
cone. Such curves are the circle, ellipse, parabola, and hyper- 
bola, with, as a special case, a straight line. The planets 
afford examples of the elliptic motion ; certain comets, 
examples of the parabolic and possibly of the hyperbolic 

If a is the semi-axis major of the orbit, it can be shown that 
the velocity at any distance r is given by 

The velocity is greatest when r is least, i.e. at perihelion, and 
least when r is greatest, i.e. at aphelion. In the case of an 

ellipse is positive, for a parabola it is zero, and for a hyper- 

bola it is negative. 

If a body is moving in a straight line towards the attracting 
force, the velocity which it acquires in moving from rest at 
a distance s to a distance r is given by v 2 = 2f,i(l/r l/s) ; 
if, therefore, it starts from rest at an infinite distance, the 
velocity acquired in falling to a distance r under the action 
of the attracting force will be v = Vfy/r. If, on the other 
hand, the body is moving in a parabola, its velocity at distance 

r will also be \/2^/r (putting = 0). Hence this velocity is 


called the " velocity from infinity," or the " parabolic velocity." 
The parabolic velocity due to the attraction of the Sun is, 
at the mean distance of the Earth from the Sun, equal to 
26-2 miles per sec. At this distance, a body projected in any 



direction with this velocity would describe a parabolic orbit : 
if projected with a greater velocity, the orbit would be hyper- 
bolic, and if with a lesser velocity, it would be elliptic. 

FIG. 79. Velocity in an Elliptic Orbit. 

From the above considerations, the following simple method 
of representing the velocity of a body at any point in an 
elliptic orbit may be derived. If about 8 (Fig. 79) a circle 
be described of radius equal to the major axis of the elliptic 
orbit (2a), then the velocity of the planet at any point P is 
equal to that which it would have acquired by falling from 
rest at the point p, in which SP produced meets the circle, 
to the point P. For the velocity so acquired would be 
given by 


which is the actual velocity at the point P of the orbit. 

The parabolic velocity at the surface of the Earth, due to 
the Earth's attraction, is 6-94 miles per second. A body pro- 
jected from the Earth with a velocity equal to or greater than 
this would describe a parabolic or hyperbolic orbit (neglecting 
the resistance of the Earth's atmosphere) and would not return. 
Other parabolic velocities in miles per second are : for the 
Sun, 38-3 ; Moon, 1-5 ; Mercury, 2-2 ; Venus, 6-6 ; Mars, 1-5 
Jupiter, 37 ;. Saturn, 22 ; Uranus, 13 ; Neptune, 14. 



130. Statistics of Planets. The following table gives, for 
reference purposes, the following statistics for planets : the 
period, semi-axis major (in terms of that of the Earth as unity), 
eccentricity, inclination to ecliptic, mean daily motion and 
mass in terms of that of the Sun. 









ation to 


of Mass 


o / 













3 24 













1 51 







1 18 



Saturn . 




2 30 














1 47 




131. Mercury. As we have seen in the preceding chapter, 
the angular distances from the Sun of the two planets, Mercury 
and Venus, whose orbits lie within that of the Earth, can 
never exceed a certain value. This angle is attained when 
the planet reaches greatest elongation. Owing to the eccen- 
tricity of their orbits, the angle of greatest elongation is not 
constant, but in the mean it equals 23 for Mercury and 46 
for Venus. These planets can therefore be observed by eye 
only in the early evening after sunset or in the morning shortly 
before sunrise, as they rise and set within a comparatively 
short period of the Sun's rising and setting. Owing to their 
great brightness, however, it is sometimes possible to observe 
them with a telescope in broad daylight. The popular desig- 
nation, Evening Star or Morning Star, is used to denote which- 
ever of these planets is visible in the western sky shortly 
after sunset or in the eastern sky shortly before sunrise. 

Mercury is relatively infrequently seen with the naked eye 
on account of its small angle of greatest elongation. In high 
latitudes it is more difficult to observe than at places nearer 
the equator, as its maximum altitude for places in high 
latitudes is smaller owing to the smaller angle of inclination 
of the ecliptic to the horizon. Under favourable conditions, 
it is possible to observe the planet for about two weeks at 
each elongation ; in the northern hemisphere it is best seen 
in the evening at eastern elongations in March or April. Not- 
withstanding the difficulty of observation, Mercury has been 
known from very early ages and no record of its discovery 
exists. By the ancients, it was given different names accord- 
ing as it appeared as a morning or as an evening star, so that 



for some time it was not recognized as the same body in the 
two cases. Thus, the Greeks called it Mercury when seen 
as an evening star and Apollo when seen as a morning star. 

The mean distance of Mercury from the Sun is 36 million 
miles. The eccentricity of its orbit is larger than that of 
any other planet (apart from certain asteroids), having the 
value 0-2056. Its actual distance from the Sun therefore ranges 
from 28| to 43-J million miles, with a corresponding range in 
orbital velocity from 35 miles per second at perihelion to 
23 miles per second at aphelion. The inclination of the orbit 
to the ecliptic is about 7. 

The sidereal period of Mercury (the planet's " year ") is 
equal to 88 days. The synodic period is 116 days. Greatest 
elongation occurs about 22 days before and after inferior 
conjunction, and therefore about 36 days before and after 
superior conjunction. 

The apparent diameter of Mercury, as obtained by micro- 
metric observations, varies from about 5" to 13" according 
to its distance. The most reliable measures correspond to a 
linear diameter of about 2,950 miles, only slightly greater than 
one-third of the Earth's diameter. There is no reliable evidence 
of any flattening at the poles. Owing to the great brightness 
of the planet, and its small angular distance from the Sun, 
the diameter is not easy to measure ; the most reliable observa- 
tions are those obtained during a transit across the Sun's 
disc. The surface area of Mercury is only about one-seventh 
that of the Earth and its volume about one-nineteenth part. 

Mercury has no satellite, and this makes its mass difficult 
to determine. The method of perturbations is the only one 
available, but as the planet's mass is small and it is near the 
Sun, its disturbing effects on the other planets are not large. 
The uncertainty attaching to its mass determination is there- 
fore large. The most probable value is 1/9,700,000 of the 
Sun's mass or 1/29 of the Earth's. This value corresponds to 
a density of rather more than 6/10ths that of the Earth and 
a surface gravity of about 0-24. 

132. Telescopic Appearance and Rotation Period. 

Mercury, seen in the telescope, shows phases similar to those 
of the Moon. At inferior conjunction, when the planet is 


nearest to the Earth, the dark side is towards us. Between 
inferior conjunction and greatest elongations, it shows a 
crescent phase. At greatest elongations, it appeals practically 
like a half-moon. Between greatest elongations and superior 
conjunction, it is gibbous (i.e. more than half-phase), whilst 
at superior conjunction the illuminated surface is towards us, 
but the apparent diameter is then least. 

There are no well-defined markings on the surface of Mer- 
cury. Such markings as can be perceived are of interest 
mainly for the information which their apparent motion may 
give about the period of rotation of the planet. In this way 
Schroter, a contemporary of Herschel, announced that the 
rotation period was 24 hours 5 minutes. Later, Schiaparelli 
contradicted this result : he stated that the surface markings 
showed 110 apparent motion in the course of several hours, 
so that the period must be much longer than found by Schroter. 
Schiaparelli concluded that the period was 88 days, in other 
words, that the planet in its orbital motion round the Sun 
always turns the same face towards it, and so behaves to the 
Sun as the Moon does to the Earth. This value for the rota- 
tion period seems to be more probable than the shorter period, 
but it has remained up to the present unconfirmed. Such 
surface markings as are seen on Mercury are very faint, 
diffuse, and ill-defined. Their position cannot be accurately 
determined and it is doubtful whether the markings can be 
regarded as in any sense permanent. The true value of the 
rotation period of Mercury must therefore be regarded as an 
open question. 

133. Physical Nature and Atmosphere. If Mercury 
does turn the same surface always towards the Sun, its physical 
conditions might with some plausibility be expected to be not 
dissimilar to those existing on the Moon, which is characterized 
by the absence of air and water and by a rough, irregular 
surface. Some information on this point is given by the 
planet's albedo, i.e. the fraction of the incident sunlight which 
is reflected back by the body. The mean value of the albedo 
for the Moon is about 0-13, but varies with the phase : near 
new Moon the amount of reflected light is less than the 
theoretical value for a smooth sphere, this being due to the 


roughness of the Moon's surface. The mean albedo found for 
Mercury is about 0-14 and shows the same variation with 
phase as that of the Moon : this supports the hypothesis that 
the surface conditions of the two bodies are very similar. 

The low value of the albedo is strong evidence that the 
planet is not cloud-covered, and it is plausible to assume that, 
if it has an atmosphere, the density is very much less than 
that of the Earth's. This assumption is supported by the 
appearance of Mercury when it enters the limb of the Sun 
at a time of transit : in the case of Venus, a bright ring is then 
seen round the planet due to refraction in its atmosphere, 
but with Mercury no such ring is seen. Spectroscopic obser- 
vations of Mercury also support the same view : there is no 
marked difference when examined in the spectroscope between 
the light reaching us directly from the Sun and that reaching 
us after reflection from Mercury, making the presence of a 
dense atmosphere very improbable. 

Such scanty observations as are available, supported by 
various lines of indirect reasoning, lead therefore to the con- 
clusion that Mercury is probably similar as regards physical 
conditions to the Moon, with a rough surface and little or no 
atmosphere. The one side is turned always to the Sun, the 
other side always away from it. Its density is relatively 
high and not greatly different from that of the Moon. 

134. Venus. The next planet in order from the Sun is 
Venus, the brightest of all the planets. Although so bright 
and easily observable, our knowledge of the conditions on the 
planet are hardly more complete than in the case of Mercury. 
The great brightness of the planet is, in fact, in some ways 
a hindrance to observation. 

The mean distance of Venus from the Sun is about 67 
million miles, and as the eccentricity of the orbit is only 0-007, 
the least in the solar system, the greatest and least distances 
from the Sun do not differ by as much as 1 million miles. 
The sidereal period of Venus is 225 days, and the synodic 
period is 584 days. Its orbital velocity is 22 miles per second. 
Greatest elongation occurs about 71 or 72 days before or after 
inferior conjunction. The inclination of its orbit is only 3 24'. 

The distance of Venus from the Earth varies from 26 million 



miles at inferior conjunction to 160 million at superior con- 
junction. Its apparent angular diameter correspondingly 
varies from 67" to 1 \" . Its real diameter is 7,700 miles, and the 
size of Venus is not therefore greatly different from that of 
the Earth. 

As Venus has no satellite, the mass must be found by the 
method of perturbations. This method gives a more reliable 
result in the case of Venus than in that of Mercury. The 
most probable value of the mass is 1/403,490 that of the Sun, 
or about 0-826 of that of the Earth. The density and super- 
ficial gravity in terms of those of the Earth are respectively 
0-94 and 0-90. The mass, density, and surface gravity of 
Venus are therefore comparable with those of the Earth. 

""O * 

FIG. 80. The Telescopic Appearances of Venus. 

135. Phases and Brightness of Venus. Venus exhibits 
phases similar to those of the Moon and Mercury. They are 
more easily observed than in the case of Mercury on account 
of the larger angular diameter of Venus ; a telescope of very 
moderate power will reveal them easily. When showing the 
crescent phase, the planet appears much larger than when 
seen full at superior conjunction, on account of the great 
difference in the distance from the Earth in the two cases. 
The phases and relative sizes of Venus in different positions 
are shown in Fig. 80. 


It is of interest to note that according to the theory of 
Ptolemy, Venus could never be seen larger than the half- 
moon shape. In Fig. 81, S, V, E, represent the relative 
positions of the Sun, Venus, and the Earth according to 
Ptolemy. The centre of 
the epicycle of Venus is 
on the deferent of Venus 
and also on the line join- 
ing the Sun to the Earth. 
It is clear from the 

diagram that the angle .X^ \Venus 

SVE could never be so .X deferent 

small as a right angle 

Since the radius of the tf IG . 81. Ptolemy's Theory of Vomis. 

epicycle of Venus is small 

compared with the distances VS and VE. But it is only 
when this angle becomes less than a right angle that the 
planet can appear more than half illuminated. The dis- 
covery of the gibbous phase of Venus by Galileo was one 
of the early fruits of his application of the telescope to 
astronomical observation and provided a strong argument 
for the theory of Copernicus and against that of Ptolemy. 
The variations of brightness of Venus are due partly to the 
changes in the phase of the planet and partly to the changes 
in the distance from the Earth. Since the full-moon phase 
occurs when the planet is at its greatest distance from the 
Earth, the two effects tend to compensate one another, and 
Venus does not show the wide range in brightness which the 
changes in its distance alone would require. It can be shown 
by a simple mathematical investigation that the greatest 
brightness occurs about 36 days before or after inferior con- 
junction. The phase then corresponds to that of the Moon 
when about 5 days old. Venus is then six or seven times as 
bright as the brightest fixed star, Sirius, and can be easily seen 
with the naked eye in broad daylight, if one knows where to 
look for it. 

136. Telescopic Appearance and Rotation Period. 

Owing to its great brightness, Venus can best be observed 
telescopically in the twilight, just after sunset or just before 


dawn. She does not show any conspicuous or well-defined 
surface markings. When in the crescent phase, ill-defined 
darkish shadings can be seen near the terminator. These 
markings possess no distinct outline and may be mere atmo- 
spheric objects and not true surface markings. Lowell claimed 
to have seen more definite markings, but the observations are 
so difficult that it is doubtful whether they can be substantiated. 

The absence of well-defined markings makes the determina- 
tion of the rotation period of Venus difficult, and the actual 
period is still a subject of dispute. Cassini found a period 
of 23 h. 15 m., and Schrotcr found 23 h. 21 m. Other investi- 
gators have asserted that the period is much longer than 
this. Schiaparelli concluded that it was 225 days, in which 
case the planet would always turn the same face towards the 
Sun. Although several series of observations seem to support 
the shorter period found by Cassini, yet these same series 
give for the inclination of the axis of Verms to its orbital 
plane values which differ by more than 20. It does not seem 
possible that visual observations will settle the question : an 
observer with great acuity of vision, possessed of excellent 
judgment, and making observations under the most favourable 
atmospheric conditions, would be necessary, but these are con- 
ditions which it is difficult to combine, and even if obtainable 
success could not be guaranteed. 

The most promising method of attacking the problem is the 
spectroscopic method, using Doppler's principle, as explained 
in connection with the determination of the period of rotation 
of the Sun. As applied in the case of the Sun, the method 
consists in photographing the spectra of the light from the 
eastern and western limbs of near the equator and measur- 
ing the relative displacement of the lines. As a result 
of rotation one limb moves towards the observer and the 
other away from him as compared with the centre. In spite 
of the much smaller relative displacement to be expected in 
the case of Venus than in that of the Sun, the method would 
probably give accurate results, if it could be applied in this 
way. When Venus is near the Earth, however, one limb is 
always in darkness, and when near superior conjunction the 
image is small and errors due to irregular guiding become 
important. It is not surprising, therefore, that the results 


furnished by the spectroscopic method are discordant, as it 
cannot be used differentially. Belopolsky found a rotation 
period of 12 hours ; Lowell and Slipher 30 days, the spectro- 
scopic method being used in each case. The true period 
remains unknown, and the problem is one of the most difficult 
astronomical problems awaiting solution. 

137. Physical Conditions. The albedo of Venus has the 
high value of 0-76, which is about equal to the reflecting power 
of freshly fallen snow. As few, if any, rocks or soils have so 
high a reflecting power, the value would seem to indicate that 
the planet is mostly or entirely cloud-covered. 

This conclusion is supported from other considerations. 
When Venus is entering the Sun's disc at a transit, its black 
disc is seen surrounded by a bright ring of light which must 
be due to refraction by the atmosphere of Venus. From 
observations made at transits, it has been concluded that the 
depth of the atmosphere must be at least 55 miles. When 
Venus is seen in its early crescent phase, the horns of the 
crescent extend appreciably beyond their geometrical position 
and sometimes a thin line of light, completing the whole 
circumference of the planet, may be observed. This also is 
an effect of refraction. 

Such definite knowledge as we possess of the nature of the 
atmosphere of Venus is negative in character. St. John has 
photographed the spectrum of Venus when its velocity rela- 
tive to the Earth is a maximum. If water- vapour or oxygen 
are present in the planet's atmosphere, there should be a 
double series of absorption lines in its spectrum, due to the 
absorption in the planet's atmosphere and the Earth's atmo- 
sphere respectively, the separation being the relative Doppler 
displacement. No lines due to either substance in the 
atmosphere of Venus were found, however. It therefore 
seems probable that oxygen and water-vapour are not present, 
at any rate in the outer layers of the atmosphere of Venus. 
This suggests the question : of what is its atmosphere 
composed ? 

It has been asserted by many observers, and denied by 
many others, that at times a faint illumination of the dark 
portion of the planet's surface may be seen, akin to the 


phenomenon of the old Moon in the arms of the new. Since 
Venus has no satellite, such illumination if not a subjective 
phenomenon must either originate on the planet's surface or 
be due to reflection of light by the Earth. The study of Earth- 
shine on the Moon shows that the Earth's surface has a high 
albedo, so that, seen from Venus, the Earth under favourable 
conditions would appear several times as bright as Venus at 
its brightest appears to us. It is, nevertheless, doubtful 
whether the reflected light from the Earth is capable of 
explaining the phenomenon, more particularly as it is stated 
to have been observed even in daylight. If not explicable 
in this way, the phenomenon may be of an electrical nature 
and possibly comparable with the aurora. 

138. Mars. The planets whose orbits lie outside that of 
the Earth are much more suitably situated for observation 
than Venus and Mercury. They are seen fully illuminated 
by the Sun when at their nearest to the Earth, instead of when 
at their greatest distance. They may be observed at certain 
seasons throughout the night, since their elongations may have 
all values from to 180. Their phase changes are also much 
less important than for the two inner planets. 

The nearest to the Earth of the outer planets is Mars, 
which has been known from remote antiquity. Its mean 
distance from the Sun is 141-5 million miles, and the eccen- 
tricity of its orbit is 0-0933, which, after Mercury, is the largest 
value for any of the major planets. In consequence of this 
eccentricity, its distance from the Sun varies by about 26 
million miles. The inclination of its orbit to the ecliptic is 
small, 1 51'. The sidereal period is 687 days and the synodic 
period is 780 days. The latter is the longest in the solar 
system. The planet retrogrades during 70 days of the&e 780, 
through an arc of about 18. 

The average distance of Mars from the Earth at opposition 
is 48-5 million miles. The actual distance depends upon 
whether the opposition occurs near the planet's perihelion or 
aphelion. In the former case it is only 35 million miles ; 
in the latter it is 61 million. At conjunction, the average 
distance from the Earth is 234-5 million miles. 

The apparent diameter of the planet varies between 3"-6 


and 25"-0, the latter value being obtained at a favourable 
opposition. Its true diameter is about 4,200 miles, so that 
its surface is rather more than one-quarter and its volume 
about one-seventh those of the Earth. Its mass can be deter- 
mined with accuracy, as it possesses satellites. Compared 
with the Sun it is 1/3,093,500, or 0-108 of that of the Earth. 
This figure gives for its density 0-72 and surface gravity 0-38 
in terms of the corresponding quantities for the Earth. 

Since the orbit of Mars is outside that of the Earth, the 
planet cannot come between the Sun and the Earth and 
therefore does not show any crescent phases. Both at opposi- 
tion and superior conjunction, the whole of the illuminated 
hemisphere is turned towards the Earth, but at quadrature 
a distinct gibbous phase may be seen, which corresponds to 
the appearance of the Moon when about three days from full. 

It follows that the variation in brilliancy of the planet is 
much greater than is the case with Venus. At conjunction, 
Mars is about as bright as the pole star, but at opposition, 
owing to its relative nearness to the Earth, it is on the average 
about twenty -three times as bright. At a favourable opposi- 
tion, it may be sixty times brighter than at conjunction. 
The difference in apparent brightness between favourable and 
unfavourable oppositions exceeds four to one. The favourable 
oppositions always occur in the latter part of August, as the 
Earth then passes through the line of apses of the orbit of 
Mars. The interval between consecutive favourable opposi- 
tions is 15 or 17 years. Fifteen years is somewhat longer 
than 7 synodic periods of Mars and 17 years is somewhat 
less than 8 synodic periods. The last favourable opposition 
was in 1907 and the next will occur in 1924. 

139. Telescopic Appearance and Rotation Period. The 

early telescopic observations of Mars in the seventeenth cen- 
tury revealed certain markings on the planet which altered 
their position from hour to hour. From a study of these 
markings, Cassini found for the period of rotation 24 h. 40 m. 
Later observations have enabled the period to be determined 
with very great accuracy : by comparing modern observations 
with old ones, an approximate knowledge of the period suffices 
to determine the exact number of revolutions in the interval 


between the observations, and thence an accurate value of 
the period may be deduced. In this way a value of 24 h. 
37 m. 22-6 s. has been determined. A comparison of the two 
drawings in Plate XLI (a), the interval between which was 
23 h. 15 m., illustrates the rotation of the planet. 

The inclination of the planet's equator to its orbital plane 
is about 24-5. This inclination may be deduced from obser- 
vations of the surface markings, or, in particular, of the polar 
caps. It was noticed by the early observers, Huyghens, 
Cassini, and others, that around each pole of Mars was to be 
seen, at certain times, a white cap, which they compared with 
the regions of ice and snow at the two poles of the Earth. 
The size of these caps was found to vary and also the times 
when they could be observed. If opposition occurred near 
perihelion, the south-polar cap was turned towards the Earth ; 
if near aphelion, the north-polar cap. Herschel first pointed 
out that the period of variation of the size of the polar caps 
is equal to the sidereal period of Mars and suggested that tho 
decrease in size of, say, the northern cap, was due to the 
melting of ice and snow by the heat of the Sun in the planet's 
northern summer, and that when winter returned the cap 
increased in size as the water froze again. The polar caps are 
shown in the drawings of Mars (Plate XII (a) ) and in the 
photographs (Plate XL [I (a) ). 

Besides these polar caps, whose interpretation as formed of 
ice and snow can hardly be doubted, there are other markings 
visible on Mars whose nature is more controversial. The most 
noticeable features are patches of a bluish-grey or greenish 
shade which cover usually about three-eighths of the planet's 
surface and are found mainly in the southern hemisphere 
near the equator ; there are also extensive regions of brownish 
or orange shades which occur mainly in the northern hemi- 
sphere and cover more than half the surface. There is a 
tendency to interpret these markings in terms of the physical 
conditions existing on our Earth and to consider the greyish 
regions as sheets of water and the brown regions as land, 
probably deserts of sand or rock. The names of various 
markings on Mars to be found on maps of the planet, such 
as Mare Tyrrhenum, Mare Sirenum, etc., must, however, not 
be interpreted literally, any more than similar names applied 


to portions of the surface of the Moon. It is not improbable 
that future investigation will show that the names are in reality 
unsuitable. Most of the names by which the various forma- 
tions on Mars are known were given by the Italian observer 
Schiaparelli, who from 1877 onwards made numerous obser- 
vations of Mars. Favoured with exceptional eyesight and a 
good telescope, he added greatly to the existing knowledge 
of the various formations : the smallest markings were observed 
and measured micrometrically, enabling accurate maps of the 
Martian surface to be constructed. 

In 1877 Schiaparelli discovered that the so-called continents 
were intersected by numerous straight greyish lines, which he 
interpreted as a network of channels for water intersecting 
the land ; these he designated by the Italian word canali. 
The nature of the canals has given rise to much speculation 
and controversy. For nine years, until 1886, only Schiaparelli 
could see them, but they were observed in that year by Perrotin 
and Tholloii at the Nice Observatory and subsequently by 
many other observers, in particular by Lowell and Sliphcr at 
the Flagstaff Observatory. The careful observations of these 
observers, made under favourable atmospheric conditions, led 
them to assert that at times some of the canals became double ; 
that the so-called seas were also intersected by canals and are 
therefore probably not of an aqueous nature at all ; that at 
the intersections of the canals are small round dark spots 
which have been variously called lakes and oases ; and that 
as the polar caps melt the canals darken. On the other hand, 
other careful observers, such as Barnard, observing under 
favourable conditions, have failed to detect the canals, and 
photographs of Mars have not revealed the thin, sharp lines 
delineated by Lowell (see Plate XIII (a) ). It is possible, there- 
fore, that the canals are really subjective phenomena arising 
from the tendency of the eye to connect by straight lines 
faint markings which are visible only with difficulty. When 
observing at the limit of resolution of an optical instrument, 
it is well known that the observed details may not correspond 
with fact ; thus, for example, with a microscope, totally 
different structures of diatoms may be observed with objectives 
of differing perfection and resolving power. Whilst it would 
be unwise to make too definite an assertion, the balance of 


probability seems to be in favour of the supposition that the 
canals are subjective. The appearance of Mars in the tele- 
scope may be judged from Plate XII (a), reproducing drawings 
by a skilled observer under good conditions. 

Lowell built up a speculative theory of the canals which 
has not received general acceptance. He supposed that the 
polar caps are composed of ice and snow, which melt in summer, 
the water flowing towards the equator, through the canals, 
which he considers are artificial water-channels constructed 
by intelligent beings for irrigation purposes. On his theory, 
the dark regions formerly considered as seas are land covered 
with vegetation, whilst the ruddy portions are deserts. As 
the water flows along the channels, vegetation springs up along 
them, arid these we observe as canals. Where the canals cross, 
oases are formed. This theory involves many difficulties : 
if intelligent beings are at work in the way suggested, it would 
be expected that they would construct the canals to follow 
the contours of the planet's suifaoe instead of making them 
absolutely straight for thousands of miles. Also it is difficult 
to imagine that canals could be constructed to carry water 
from the melting north-polar cap well down into the southern 
hemisphere and from the south cap well into the northern 
hemisphere. There is the further difficulty that the rate of 
disappearance of the polar caps is such that it is difficult to 
believe that they can be thick masses of ice and snow : more 
probably they are thin deposits of snow or hoar-frost. The 
only portion of the theory which receives fairly general 
acceptance is the existence of seasonal changes on Mars which 
could reasonably be attributed as due to changes in vegetation. 

140. Atmosphere and Temperature of Mars. It is 

very probable that Mars possesses an atmosphere, though much 
less dense than that of our Earth. At times thin, whitish 
veils of cloud have been observed which appear to admit of 
no alternative explanation. The deposition and dissipation 
of the polar caps also point to the presence of an atmosphere. 
From the planet's low surface gravity it would be anticipated 
that the density of the atmosphere would be low. Spectro- 
scopic observations confirm this : Campbell, in 1909, photo- 
graphed the spectrum of Mars from the summit of Mount 


Whitney (15,000 ft.), so reducing the effect of the absorption 
of the Earth's atmosphere. To estimate the residual amount 
of absorption, the spectrum of the Moon, was obtained for 
comparison. These results enabled Campbell to conclude that 
at the surface of Mars the density of the atmosphere is not 
more than one-half that of the density of the Earth's atmo- 
sphere at the summit of Mount Everest, and that, in particular, 
the Martian atmosphere contains very little, if any, water- 
vapour. Incidentally, this observation provides a strong 
argument against the hypothesis that there is much water 
on Mars, and that the polar caps are composed of thick masses 
of ice or snow. The albedo of Mars is 0-22, which is higher 
than those of the Moon and Mercury, but much smaller than 
that of Venus : it is consistent with the existence of a rare 

On the assumption that the heat received by Mars from 
the Sun is just equal to the amount which it radiates into 
space, it is possible to form some idea of the temperature of 
Mars : some uncertainty is introduced by our ignorance as 
to the effect of the atmosphere on Mars in regulating the day 
and night temperatures. The problem was carefully investi- 
gated by Poynting on the supposition that the planet rotates 
about an axis perpendicular to the plane of its orbit. Assuming, 
firstly, that the effect of the atmosphere would keep the tem- 
perature in any given latitude the same, day and night, ho 
found that the equatorial temperature of Mars would be 
20 C. and its average temperature 38 C. If, on the 
other hand, like the Moon, it has no atmosphere, the temperature 
would still be considerably below the freezing-point of water. 
The only escape from this conclusion is to be found by 
assuming that an appreciable amount of heat is issuing from 
beneath the surface. It is evident, however, from a comparison 
of the polar and equatorial temperatures on the Earth that 
the internal heat of the Earth has very little effect on its 
surface temperature, and it is therefore reasonable to assume 
that the same is true for Mars. This probable low temperature 
is a further argument against Lowell's theory of the canals. 

The method of determining the temperature of Mars can 
be applied also to Mercury and Venus. The values deduced 
for their temperatures depend, however, upon whether they 


rotate on their axes in a short period or so as always to turn 
the same face towards the Sun. In the former case, they are 
about 170 C. and 55 C. respectively hotter than the Earth ; 
in the latter case, the hemispheres facing the Sun must be at 
much higher temperatures still, whilst those away from the 
Sun must be at very low temperatures. 

141. Satellites of Mars. Mars possesses two tiny satellites 
which were discovered in 1877 by Asaph Hall at Washington. 
They are both very small, the larger one having a diameter 
of less than 40 miles and the smaller of only 8 or 10 miles. 
Their smallness, combined with their nearness to Mars itself, 
renders them difficult objects of observation. The outer and 
smaller one, Deimos, is only 14,600 miles from the centre of 
Mars ; the inner one, Phobos, only 5,800 miles. Their periods 
of revolution arc correspondingly short, viz. 30 h. 18 m. and 
7 h. 39 m. respectively. Thus the month of Phobos is less 
than one-third that of the Martian day. Although both 
planets revolve about Mars in the same direction as Mars 
revolves around the Sun, Phobos would appear to an observer 
on Mars to rise in the west and to set in the east after an 
interval of 4] hours, since its rate of rotation is so much more 
rapid than that of Mars on its axis and is, in fact, the shortest 
in the solar system. 

The period or month of Deimos is nearly equal to the rotation 
period of Mars. Its orbital motion eastward amongst the stars 
is therefore nearly equal to its diurnal motion westward. As a 
result, it rises in the east at intervals of 132 hours, equal to 
more than four of its months, so that in the interval between two 
successive risings, it goes through all its phases four times. 

The orbits of the two satellites are almost exactly circular 
and in the equatorial plane of the planet. Mars is sensibly 
flattened at the poles, the polar compression being about 
1/200 ; the equatorial bulge tends to keep the satellites in the 
plane of the equator. 

It is of interest to note that in Gulliver's Travels Swift relates 
that the astronomers of Laputa " have discovered two lesser 
stars, or satellites, which revolve about Mars, whereof the 
innermost is distant from the centre of the primary planet 
exactly three of his diameters, and the outermost five ; the 


former revolves in the space of ten hours and the latter in 
twenty-one and a half." If Swift had actually observed the 
satellites, these figures would have been creditably near the 
truth. As a conjecture, they are a remarkable coincidence. 

As givers of moonlight to an observer of Mars the satellites 
would be of very little importance, but Phobos, with a motion 
relative to the stars ninety times as rapid as that of our Moon, 
would provide an excellent object for use in longitude deter- 
minations on Mars. 

142. The Minor Planets. The minor planets or asteroids, 
as they were named by Sir William llerschel, are a numerous 
group of very small planets circulating in the space between 
Mars and Jupiter, with a mean distance closely corresponding 
to that given by the vacant place in Bode's law. The total 
number discovered up to the present is approaching one 
thousand. The first asteroid to be discovered was Ceres ; an 
extended search was being carried out for a planet assumed 
on account of the gap in Bode's law to exist between Mars 
and Jupiter. On January 1, 1801, Piazzi at Palermo, in the 
course of observations for his star-catalogue, observed a 
seventh-magnitude star which the next evening had perceptibly 
moved. Thus was Ceres accidentally discovered. Shortly 
afterwards it was lost in the rays of the Sun, but Gauss was 
able to compute an ephemeris, by employing his recently- 
discovered method, which enabled the asteroid to be found 
again exactly one year after Piazzi first observed it. 

Pallas was discovered by Olbers in 1802 ; Juno by Harding 
in 1804; and Vesta, the brightest of all the asteroids, in 1807. 
The fifth, Astrsca, was not discovered until 1845, but since that 
date fresh discoveries have been made continually, and the list 
is still growing, though any members of the group not yet 
discovered must be small and faint bodies. They are usually 
discovered on photographs of regions near the ecliptic, taken 
with an exposure of two or three hours. The telescope follows 
the stars during the exposure, and the duration is sufficiently 
long for the motion of the asteroid relative to the stars to be 
perceptible, so that its image on the plate will not be round 
but an elongated trail. One of the group having been found, a 
comparison of its position with the positions of previously 


discovered asteroids known to be in the same region of the sky 
is made in order to ascertain whether or not it is a new member. 
The largest numbers of discoveries have been made by Palisa at 
Vienna, Charlois at Nice, and Wolf at Heidelberg. 

Plate XIV (a) is a reproduction of a photograph obtained at 
Heidelberg, on which three minor planet trails were discovered. 
The positions of these trails may be readily found by means 
of the arrow-heads which point to them. Attention may be 
drawn to the difference in the directions of the three trails and 
to the difference in the brightness of the three asteroids. 

143. Size, Mass, and Brightness of Asteroids. The 

angular diameter of the four largest members of the system 
have been measured by Barnard with the large refractors at the 
Lick and Yerkes Observatories. These values, reduced to true 
diameters, give for Ceres a diameter of 480 miles, for Pallas 
306 miles, for Vesta 241 miles, and for Juno 121 miles. These 
diameters are exceptionally large ; those of the majority of the 
asteroids must be considerably less than 50 miles. 

Photometric measures have shown that the albedos of the 
asteroids are small, falling for the most part between the 
albedos of Mercury and Mars, i.e. between 0-14 and 0-22. If 
the albedo is assumed to have a value of, say, 0-20, then from a 
knowledge of the orbit of the asteroid and its apparent bright- 
ness at opposition it is possible to compute its diameter. The 
percentage error in the resulting value may be considerable, 
but the order of magnitude obtained will be correct. 

From the smallness of the majority of the group, it follows 
that their total mass cannot be large. It is known that it 
must be less than that of Mars, otherwise noticeable per- 
turbations in the orbit of Mars would be detected. The total 
mass of the asteroids which have actually been discovered prob- 
ably does not exceed one-thousandth of the mass of the Earth. 

Most of these bodies are fainter than the tenth magnitude, 
but the brightness varies with the distance from the Earth 
and the phase of the illumination. After allowing for the 
variations due to these two causes, it is found that some show 
small residual fluctuations in brightness. It is possible that 
these are due to axial rotation and a variation in the reflecting 
powers of different portions of the surface. 


None of the asteroids appears to possess an atmosphere. The 
low values of their albedoes tends to confirm this conclusion. 

144. Asteroid Orbits. The asteroid zone extends from 
Mars to Jupiter. Eros has a semi-axis of 1-46 astronomical 
units, which is smaller than that of Mars (1-52), whilst that of 
Hector (5-28) exceeds that of Jupiter (5-20). Eros is of great 
importance for the determination of the solar parallax ; its 
orbit has a high eccentricity (0-22) and small inclination to the 
ecliptic, so that it can approach nearer to the Earth than any 
other known planet. When its perihelion passage coincides 
with the time of opposition its distance from the Earth is only 
about 14 million miles. It is therefore more suitable as an 
object of observation than Mars, and in addition, the obser- 
vations can be made with greater accuracy since it does not 
possess an appreciable disc. A favourable opportunity for the 
determination of the solar parallax under these conditions will 
occur in the year 1931. 

The six planets, Achilles, Hector, Patroclus, Nestor, Priam us 
and Agamemnon, constitute the Jupiter group. Their orbits 
are all near that of Jupiter and they provide an interesting 
illustration of a particular case of the problem of three bodies. 
If a body were at the third corner of an equilateral triangle at 
whose other two corners were the Sun and Jupiter, it can be 
shown that it would always remain in the same relative position 
if the initial velocity were suitably chosen. The above four 
planets satisfy the conditions approximately and oscillate 
about the positions of equilibrium which they would fill if 
they exactly satisfied them. 

There are gaps in the asteroid orbits corresponding to 
periods of revolution which are simple fractions of the period of 
Jupiter. It has been supposed that these gaps have been 
caused by the perturbing effect of the large mass of Jupiter, 
which under these circumstances might be expected to bo 
cumulative and gradually to have pulled such asteroids out 
of their orbits. This conclusion is not, however, supported by 
theoretical considerations. 

145. Origin of Asteroids. It has been suggested that the 
asteroids are the relics of a larger planet which has broken up. 
It is not possible definitely to state whether or not this is so. 


The present assemblage of orbits, some of which lie entirely 
within others, offers no resemblance to a series of orbits passing 
through one point which should result from a sudden explosion. 
It must be remembered, on the other hand, that there may 
have been several successive explosions and that Jupiter and 
other planets have perturbed the orbits by greater or less 

Another suggestion is that the matter forming the asteroids 
was originally uniformly distributed in a ring about the Sun, 
analogous to the rings of Saturn, and that perturbations by 
the planet Jupiter broke it up into many fragments which 
ultimately formed the aggregates which we now know as the 
asteroids. But whether either of these theories is near the 
truth, it is not possible to tell. 

146. Jupiter. The next planet in distance from the Sun 
is Jupiter, the largest and most massive of the planets. In 
apparent brightness, it is exceeded only by Venus. Although 
Venus is only one-seventh of the distance of Jupiter from the 
Sun, its average brightness exceeds that of Jupiter by only 
about one magnitude. Jupiter is, on the average, about five 
times brighter than Sirius, the brightest of the stars, and when 
near opposition is a very brilliant object. 

The moan distance of Jupiter from the Sun is 483 million 
miles. The eccentricity of its orbit is 0-0483, so that its greatest 
and least distances differ by 42 million miles, being 504 and 
462 million miles respectively. At opposition, its averagp 
distance from the Earth is 390 million miles, and at conjunction 
it is 576 million. When opposition occurs at Jupiter's 
perihelion, i.e. early in October, its distance is only 369 million 
miles. At a perihelion opposition it is about 50 per cent, 
brighter than at an aphelion opposition, and nearly three times 
as bright as at conjunction. Its orbital velocity is about eight 
miles per second. The inclination of its orbit to the ecliptic is 
1 19'. 

The sidereal period of Jupiter is 4332-6 days or 11-86 years 
and its synodic period is 399 days, so that in the course of one 
year it practically moves through one sign of the Zodiac. 

Its apparent diameter varies from about 50" at a favourable 
opposition to 32" at conjunction. Even a small telescope will 


show that it is perceptibly oblate, the polar diameter being 
about one-seventeenth part smaller than the equatorial. 
Its equatorial and polar diameters are respectively 88,700 and 
82,800 miles ; its mean diameter is therefore about eleven 
times that of the Earth. Its surface is 119 times and its 
volume 1,300 times that of the Earth. So that not only is 
Jupiter the largest of the planets, but it is also larger than all 
the others combined. 

The mass of Jupiter has been more accurately determined 
than that of any other planet. The determinations have been 
based both on the motions of its satellites and on the perturba- 
tions which it causes in the orbits of Saturn, certain asteroids 
and periodic comets. On account of its large mass, these per- 
turbations may be very considerable. In terms of the mass of 
the Sun, Jupiter's mass is 1/1,047*355, which is about 318 times 
that of the Earth. This corresponds to a density of about 
one-quarter of that of the Earth. Its surface gravity is 2-64 
times that of the Earth. 

147. Telescopic Appearance and Rotation Period. 

When seen in a telescope of moderate power, Jupiter is an 
object of great interest, much detail being visible which can 
be more easily observed and drawn than the detail visible on 
Mars. It is at once apparent that the principal markings 
run in belts or zones across the disc at right angles to the polar 
axis of the planet. Since the axis of Jupiter is very nearly 
perpendicular to its orbital plane, the boundaries of the several 
zones appear practically as straight lines and the complications 
which arise, as in the case of Mars, when the axis of the planet is 
inclined at a considerable angle to its orbital plane, are therefore 
absent (Plates Xil (b) and Xlil (b) ). 

The equatorial zone is usually very bright and conspicuous, 
although it occasionally takes on a tawny hue, the sequence of 
changes in its appearance occurring with some regularity. 
It is bordered to the north and south by two darker brownish 
zones (see Plate XIII (b) ). Numerous small well-defined bright 
and dark spots are frequently observed in these zones, which 
frequently last for several months, and may be utilized to 
determine the period of rotation of the planet. Such obser- 
vations show that, as in the case of the Sun, the period of 


rotation is different in different latitudes, the rotation in the 
equatorial zone being more rapid than that of most of the 
remaining portion of the surface. At the equator, the period is 
about 9h. 50 in. 25 s., whilst at the poles it is about 9h. 55m. 40s., 
the transition between the two being practically instantaneous. 

The increase in rotation period from equator to poles is not 
uniform, for Denning and Williams have shown that between 
latitudes 24 and 28 north is a zone with a rotation period 
somewhat shorter than 9 h. 49 m. It is evident that the 
observations cannot relate to markings on a solid body, but 
rather to markings of an atmospheric nature. The zones on 
Jupiter have been compared with the trade-wind belts on the 
Earth and it has been suggested that the higher velocity of 
certain portions of the surface are due to downward atmospheric 
currents which communicate a greater velocity to the lower 
layers and continually hasten them onwards. It is doubtful 
whether there is any solid surface beneath the clouds, as the 
variation in rotation period can be much more readily reconciled 
with an entirely gaseous constitution. 

There is one marking on the planet which is of a semi- 
permanent nature. This is known as the Red Spot. It is 
clearly seen on Plate XIII (b). It was first observed in 1878 and 
was then a pale, pinkish oval spot ; it rapidly attracted the 
attention of observers as it developed and attained a brick-red 
colour and a length about one-third the diameter of the planet. 
It faded considerably in subsequent years, but remained a 
conspicuous object on the planet until about 1919, when it 
gradually faded away ; in 1921 it was only very faintly visible. 
The period of rotation of the spot is slightly variable and rather 
longer than that of the atmospheric belt surrounding it. The 
cause of the phenomenon is uncertain, but it is generally 
supposed that in 1878 there was an eruption of some sort on the 
planet and that the gases poured out over the highest cloud 
zone and remained in a practically stationary position relatively 
to it. 

148. Physical Constitution of Jupiter. The nature of 
the surface markings observed on Jupiter and the variations 
in the periods of rotation of different zones suggest a dense 
atmosphere. The high value of the albedo, 0-62, supports 
this, and other evidence is also confirmatory. The brightness 


of the planet is not uniform across its disc, but decreases 
towards the limbs ; the limb turned away from the Sun appears 
the darker of the two, but this is merely an effect of phase. 
The decrease at the other limb is due to the greater absorption 
which the light from the Sun undergoes due to its longer 
passage through Jupiter's atmosphere at the limb. The 
spectrum of Jupiter also shows an appreciable strengthening 
of the telluric lines, i.e. those lines which are due to absorption 
in the Earth's atmosphere, and of certain other lines ; this can 
only be due to additional absorption in the atmosphere of 
Jupiter itself. 

The rapidity of the changes occurring on the surface of 
Jupiter would seem to indicate that the temperature of the 
planet must be high. Its low density further suggests that it is 
largely, if not entirely, in a gaseous state except in so far as the 
interior may be liquefied by the pressure of the outer layers. 
The variable rotation period also leads to the same conclusion, 
and the flattened shape of the planet is in harmony with it. 
On the other hand, the planet, if self-luminous, can be only 
feebly so. When one of the satellites of Jupiter passes between 
the Sun and the planet, its shadow is thrown on to the planet's 
surface ; the portion of the surface in the shadow then appears 
dark, so that in contrast with the illumination of Jupiter by 
sunlight its own luminosity is negligible. 

Jupiter is therefore a gaseous or semi-gaseous body at a high 
temperature and in rapid rotation. The rotation gives rise 
in all probability to winds blowing parallel to its equator which 
throw the clouds in its atmosphere into belts parallel to its 
equator, so giving rise to the well-known appearance of Jupiter 
in the telescope. 

149. Satellites of Jupiter. Jupiter is known to have nine 
satellites. Four of these were discovered by Galileo in 1610, 
one of the first results obtained with his telescope. He was 
able to satisfy himself as to their true character, in spite of 
much opposition and vituperation from the Churchmen, and 
to determine their periods of revolution with a very fair degree 
of accuracy. These four are usually called the first, second, etc. 
satellite, in the order of their distance from Jupiter, though they 
have the names lo, Europa, Ganymede, and Callisto. They are 



comparatively bright and are the largest satellites in the solar 
system. With a pair of field glasses they may easily be seen. 
The remaining five satellites arc named the fifth, sixth, etc., in 
the order of their discovery. The fifth, which was discovered 
by Barnard in 1892 is the nearest of the satellites to Jupiter. 
These five are all faint and have only been discovered within 
recent years, large telescopes and, in the case of the faintest, 
tho employment of photography, having made their discovery 
possible. The particulars as to their distances, brightness, 
periods, and size arc given in the following table : 



terms of 

Distance in 





Year of 

d. li. rn. 





1 18 28 







3 13 14 







7 3 43 







16 16 32 







11 57 



























The four major satellites, when viewed with a large telescope, 
show sensible discs on which faint markings may be seen 
under favourable atmospheric conditions. These markings 
have been studied with a view to determining the periods of 
rotation of the satellites. From such observations, combined 
with observations of slight variations in brightness of the 
satellites, it is believed that their period of axial rotation is 
equal to the period of their rotation about Jupiter, so that they 
always turn the same face towards the planet. Their orbits 
are almost circular and lie very nearly in the plane of Jupiter's 
equator. With the exception of the fourth, they pass through 
the shadow of the planet and therefore suffer eclipse at every 
revolution ; they also transit across the disc of the planet, and 
the shadows which they cast may easily be observed as black 
dots upon the planet's surface. It is difficult to observe the 



satellites themselves during transit. The fourth satellite 
usually suffers eclipse, but its orbital plane is sufficiently 
inclined to the plane of Jupiter's equator for it to pass 
at certain times either above or below the shadow of 
Jupiter. Exactly at opposition or conjunction, the eclipses 
cannot be observed, as the shadow of the planet then 
lies straight behind it and therefore out of sight. At quad- 
rature, on the other hand, the shadow projects out obliquely 
to the line of sight and the whole eclipse of the second, 
third, and fourth satellites takes place clear of the planet's 

The eclipses of the major satellites were used by the Danish 
astronomer, Roemer, in 1675, to determine the velocity of 
light. His announcement that light was propagated with a 
finite velocity was received at the time with ridicule. The 
periods of the satellites being known with high precision, the 
times of the eclipses could be accurately calculated and should 
recur at regular intervals, lloemer found that this was not the 
case : during half the year they occurred earlier than his 
calculated times and during the other half they occurred later. 
He noticed that when they came early the Earth was nearer 
Jupiter and when late it was farther from Jupiter than the aver- 
age. He correctly explained 
tliis discrepancy as clue to the 
decrease or increase of the dis- 
tance which the light had to 
travel before reaching the Earth. 
Thus, in Fig. 82, if S, J are 
respectively the positions of the J\ 
Sun and Jupiter, when the 
Earth is at E^ the light has to 
travel the distance JE t before 
an eclipse becomes visible 
to an observer on the Earth ; 
when at E 2 , it has to travel 
a distance JE 2 . These are the extreme distances, so that the 
greatest difference between the observed and calculated times 
for the accelerated and retarded eclipses corresponds to the 
time taken by light to travel the distance E^^ i.e. the dia- 
meter of the Earth's orbit. lloemer found for this difference 

FIG. 82. lioemer'H Discovery. 


22 minutes ; more accurate modern observations give 16 min. 
38 sec. 

The observations of the times of the eclipses provide an 
easy method of determining longitudes approximately. In the 
Nautical Almanac are given the Greenwich times of the eclipses ; 
if the local time at the instant of eclipse be observed and 
compared with the calculated Greenwich time, the difference 
gives the longitude of the place of observation, in time, 
measured from Greenwich. The method is only approximate 
as, on account of the appreciable discs which the satellites 
show, the satellites are eclipsed gradually, not instantaneously. 
The determination with as much accuracy as possible of the 
times of eclipse is also important for the accurate determination 
of the orbits of the satellites ; they enable the positions of the 
satellites in space to be determined. 

The satellites slightly disturb each other's motions by their 
mutual attractions, and a study of these perturbations enables 
tho masses of the major satellites to be determined. The 
masses are respectively 1/22,230, 1/39,440, 1/12,500, and 
1/22,200 of Jupiter's mass. Comparing these figures with the 
diameters given in the table above, it will be seen that these 
four satellites differ considerably in density, the density of the 
third, for instance, being about double that of the fourth. 
The mass of the largest of the satellites, Ganymede, is about 
double that of the Moon. 

The outer satellites fall into two pairs ; the sixth and seventh 
are at approximately the same distance and have nearly the 
same period and their orbits are interlocked the one with the 
other and somewhat highly inclined to Jupiter's equator. 
The eighth and ninth are also at about the same distance and 
their orbits also are interlocked. They are of particular interest 
because the orbits are described in the " retrograde " direction. 
With but few exceptions, the various members of the solar 
system describe their orbits in the same direction in space and 
also rotate about their axes in the same direction. Such 
motion is termed " direct," whilst motion in the contrary 
direction is termed " retrograde." When the eighth satellite 
was discovered it was at first uncertain whether it was an 
asteroid or a new satellite ; its apparent motion was compli- 
cated by the motion of the Earth relative to Jupiter, which 


caused large apparent displacements, on account of its great 
distance from Jupiter, and some asteroids are known whose 
orbits extend beyond the orbit of Jupiter. 

150. Saturn. The most distant of the planets known to the 
ancients, Saturn is unique amongst the heavenly bodies with it-* 
system of rings and its ten satellites. By many it is considered 
the most beautiful object to be seen in a telescope. 

The mean distance of Saturn from the Sun is 886 million 
miles. The eccentricity of its orbit, 0-0559, is somewhat 
greater than that of Jupiter, so that its actual distance from the 
Sun can vary by about 50 million miles. When opposition 
occurs near Saturn's perihelion, i.e. towards the end of 
December, it is at its smallest possible distance from the 
Earth, about 744 million miles. Its greatest distance, at an 
aphelion conjunction, is about 1,028 million miles. The varia- 
tion in distance is therefore less than in the case of the nearer 
planets, so that its changes of brightness are not so extreme. 
The inclination of its orbit to the ecliptic is 2 30'. 

The sidereal period of Saturn is 10,759*2 days or about 29| 
years, and its synodic period is 378 days. As the sidereal 
period of an outer planet increases, the synodic period naturally 
decreases, and for an infinite sidereal period (such as a fixed 
star), the synodic period naturally becomes equal to the length 
of the year. 

The apparent mean diameter of Saturn varies from 20" to 
14". The planet is much more oblate than Jupiter ; it is in 
fact more flattened at the poles than any of the other planets, 
the equatorial diameter being about 75,100 miles and the 
polar diameter about 67,200. Its mean diameter is therefore 
rather more than nine times that of the Earth, its surface eighty- 
two times and its volume 760 times that of the Earth. 

Its mass can be determined with accuracy since it has 
numerous satellites ; it is found to be 1/3,501-6 of the mass of 
the Sun, or 95-2 times that of the Earth. This value corre- 
sponds to a mean density of one-eighth that of the Earth or 
about 0-69 that of water. It is much the least dense of all the 
planets. Its surface gravity is only 1-14 times that of the 
Earth. The inclination of its equator to the orbital plane has 
the high value of 27. 


151. Telescopic Appearance and Rotation Period. The 

albedo of Saturn has the very high value of 0-72, which suggests 
a densely clouded atmosphere, and it is probable, for the same 
reasons as in the case of Jupiter, that Saturn is entirely gaseous 
and at a high temperature. Spectroscopic observations con- 
firm the presence of an atmosphere, the lines due to atmospheric 
absorption being even stronger than in the case of Jupiter. 

The characteristic feature of the planet's disc when seen in 
the telescope is again a series of belts running parallel to its 
equator, which, however, are much less clearly defined than are 
those of Jupiter. They are shown in Plates XI I (c) and XIII (c). 
Well-defined spots can rarely be seen within them, and such as 
are seen are usually only short-lived. The rotation period can 
therefore not be determined with the accuracy with which 
Jupiter's is known. The observation of a number of spots 
appearing near the equator gave a rotation period of 10 h. 14 m., 
with an uncertainly of about one minute. In 1903, a large 
white spot appeared in the planet's north latitude 35, and this 
had a period of rotation of 10 h. 38 in. Different zones of the 
planet, therefore, rotate with different periods, and as in the 
case of Jupiter it seems that on the whole the higher the 
latitude the longer is the period. 

It is the ring-system of Saturn, however, which makes it so 
striking an object when viewed in the telescope. The aspect 
under winch the rings are seen varies with the relative positions 
of the Earth and Saturn, on account of the high inclination of its 
equator to its orbital plane. The rings are parallel to the planet's 
equator, and their nodes are in longitudes 168 and 348. 
The plane of the ring passes through the Earth twice during 
each revolution of Saturn, i.e. when the Earth passes through 
the nodes of the ring system, and the rings are then seen 
edgewise. On account of their small thickness, they then 
become invisible for a short time. This last occurred early in 
1921. Midway between the nodes, the apparent width of the 
rings is almost half their length. The general appearance of the 
rings may be seen from Plates XII (c) and XI II (c). 

The changing appearance of Saturn, depending upon the 
angle at which the rings are viewed, was a great puzzle to 
astronomers from the time of Galileo onwards. Their small 
telescopes, of poor defining power, were not adequate to reveal 


the true nature of the rings. It was Huyghens, in 1655, who 
first succeeded in perceiving the ring form. Cassini, in 1675, 
first discovered that the ring was double, consisting of two 
concentric portions with a narrow black division between them. 
The outer ring is called ring A ; the inner, ring J3. In 1837, 
Encke observed a fine shading on ring A, which is visible only 
with difficulty. It is not yet known whether this is an actual 
division, similar to the Cassini division. The brightness of 
ring B falls off strongly towards the planet, and in 1850 Bond, 
in America, and almost simultaneously Dawes, in England, 
independently discovered a third inner ring 0, called the Crepe 
ring, from its dusky appearance. It does not appear to be 
separated from ring B by an actual division, but there is a 
strong contrast in brightness at the common boundary. This 
ring is semi-transparent and the edge of the planet can be seen 
through it. 

152. Dimensions of Ring-System. The width of ring 
A is about 11,000 miles ; that of the Cassini division is 2,200 
miles ; the ring B is about 18,000 miles wide, and ring G is 
just a little narrower than ring A. Between the edge of the 
planet and the inner edge of ring is a space of several thousand 

The thickness of the rings is extremely small in comparison 
with their width. This is proved by the vanishing of the rings 
when the Earth passes through their plane. It may be con- 
cluded from this that their thickness cannot exceed about 
60 or 70 miles, though it may well be less than this amount. 

Structure of the Rings. It was suggested originally 
by Cassini in 1715 that the rings were composed of a 
swarm of small satellites. Telescopes of increasing power 
failed to resolve the rings into separate particles, and this 
suggestion was not generally accepted. Bond revived it on 
the discovery of the Crepe ring, through which the disc of the 
planet was visible. There was, as yet, no direct experimental 
evidence to prove conclusively that the rings were so con- 
stituted. The theory became firmly established, however, 
when Clerk Maxwell showed by a mathematical investigation 
that if the rings were either SJiid or liquid they would form 



what is termed in mechanics an unstable system, i.e. the 
smallest disturbing force would cause them to break up. He 
also showed that, if they consisted of a swarm of separate 
bodies, moving in orbits nearly circular and in one plane, they 
would form a stable system. 

In 1895, Keeler obtained direct observational evidence in 
support of this conclusion. If the rings were solid and rotating 
about the planet as a rigid body, the linear velocity of a point 
on the outside of a ring would evidently be greater than that 
of a point on the inside of the ring. If, on the other hand, 
they form a system of satellites, then the period of revolution 
would increase and the linear velocity would decrease with 
increasing distance from the planet. We have previously seen 
how the motion of a source of light towards or away from 
the observer causes a slight displacement of the spectral 
lines in the directions of shorter or longer wave-lengths 
respectively, the measurement of the displacement enabling 
the velocity of motion to be determined. Keeler placed the 
slit of his spectroscope across the image of the planet as shown 
in Fig. 83 and found the shape of the spectral lines to be as 

FIG. 83. Spectroscopic Observation of Saturn's Kings. 

shown above the planet. The portion AB of a line is due 
to the planet ; the end 13, corresponding to the limb which 
is receding from the observer, is displaced relatively to the 
centre towards the red, and the end A, corresponding to the 
limb which is approaching the observer, is displaced towards 



the blue. The portions ab, cd are due to portions of the ring 
which are moving towards and away from the observer arid are 
displaced towards blue and red respectively. The displace- 
ments of the outer extremities a, d are, however, less than 
those of the inner points 6 and c, proving that the inner edges 
of the ring rotate with a greater linear velocity than the outer 

The same conclusion as to the nature of the rings was 
obtained by Seeliger, based upon careful photometric obser- 
vations of the brightness of the rings and its change with 
phase ; and also by Barnard, who observed one of the outer 
satellites, Japetus, pass through the shadow of one of the rings, 
proving that the rings cannot be an opaque solid mass. 

153. The Satellites of Saturn. Saturn is known to 
possess ten satellites, the brightest of which (Titan) was 
discovered by Huygliens in 1655, at a period when the rings 
were invisible. Four more were discovered by Cassini between 
1670 and 1685, also when the rings became invisible. Two 
more were discovered by Sir William Herschel and another by 
Bond. The two faintest are recent discoveries, which photo- 
graphy and the construction of large telescopes have facilitated. 

T!K details of the satellite system are given in the following 
table : 


Distance in 
terms of 

Distance in 






















Tethys. . . . 







Dione .... 







Bhea .... 







Titan .... 














Hyperion . 














Phoebe . . . 








The satellites are of considerable theoretical interest. It 
will be noticed that the rotation period of Tethys is almost 
exactly double that of Mimas, and that that of Dione is almost 
exactly double that of Enceladus. Also the periods of Mimas 
and Enceladus and those of Tethys and Dione are very nearly 
in the ratio of 2 : 3. These four satellites, together with llhea, 
have almost exactly the same orbital plane. The result is a 
kind of resonance effect, large mutual perturbations being 
set up which cause considerable variations in the orbital 

Titan, as its name suggests, is much larger than any of the 
other satellites and has a diameter of several thousand miles. 
It causes large perturbations in the orbits of Themis and 
Hyperion. The latter provided the first known instance of a 
special case of the problem of three bodies, its line of apses 
always keeping in conjunction with Titan. 

The variations in brightness, noticed in the case of Jupiter's 
satellites, arc still more marked in the case of those of Saturn. 
Of particular interest are the variations of Japetus, noticed by 
Cassini in the seventeenth century and amounting to 1-7 mag- 
nitudes : its greatest brightness occurs at western elongation 
and its least at eastern elongation. The western side of the 
planet must be at least twice as bright as its eastern side 
and, like the Moon, it must rotate on its axis in a period 
equal to that in which it completes one revolution about 

The most distant satellite, Phoebe, provides another example 
of the retrograde motion previously noticed in the case of the 
two outer satellites of Jupiter. It can be shown theoretically 
that greater stability is obtained by the retrograde motion for 
such a distant satellite than would be obtained by direct 

154. Uranus. The two outer planets, Uranus and Neptune, 
are of much less interest from the point of view of telescopic 
observation than are Jupiter and Saturn. Their appearance is 
very similar to the appearance which we might suppose that 
Saturn would have if it had no ring and were removed to their 
distances. Uranus was the first planet to be " discovered," 
the other planets known at the time of its discovery having 


been known from prehistoric times. It was discovered by 
William Herschel on March 13, 1781, in the course of a systema- 
tic sweep of the heavens upon which he was engaged with a 
7-inch reflector. The discovery was at first announced as a 
comet, but the computations of Laplace, five months later, 
showed it to be a new planet at a greater distance than Saturn. 
The discovery ranks as one of the most important astronomical 
discoveries of the eighteenth century and at the time caused 
great excitement. Herschel named it the Georgium Sidus, in 
honour of the king, who knighted him, gave him a pension and 
funds for the construction of his great 4-foot reflector with 
which he afterwards discovered two of the satellites of Saturn. 
The name of Uranus was suggested by Bode. It has since been 
found that Uranus had been observed at least twenty times 
previous to its discovery by Herschel, but it had been thought 
to be a star. These observations date back to one by Flam- 
steed in the year 1690. 

The mean distance of Uranus from the Sun is about 1,800 
million miles. The eccentricity of its orbit is 0-0471, slightly 
less than that of the orbit of Jupiter ; the inclination of its 
plane to the ecliptic is only 46'. The sidereal period is 
22,869 days or about 84 years, the synodic period being slightly 
less than 370 days. The orbital velocity of the planet is rather 
more than 4 miles per second. 

The apparent mean diameter is about 4" and varies by 
very little. This angular diameter corresponds to a real 
diameter of about 31,000 miles. There is a relatively large 
uncertainty attaching to this value, due to the difficulty of 
measuring with accuracy the small disc of the planet. The 
width of the spider lines in the micrometer would, at the 
distance of Uranus, correspond to two or three thousand miles. 
Adopting the above figure, however, the surface and volume 
are respectively about fifteen and sixty times greater than those 
of the Earth. Micrometric measures support the view that 
the planet is flattened at the poles, but the amount of the 
flattening cannot be stated with accuracy. Analogy with 
Jupiter and Saturn would confirm the probability of a flat- 

Uranus possesses four satellites, so that its mass can be 
determined with accuracy. It is 1/22869 of that of the Sun, 


or about 14-6 times that of the Earth. Its density is therefore 
about 0-25 times the Earth's, or about the same as that of 
Jupiter. Its surface gravity is 0-96 that of the Earth. Uranus 
can be seen with the naked eye, appearing as a faint star of the 
sixth magnitude. 

155, Telescopic Appearance and Period of Rotation. 

The albedo of Uranus is 0-60, indicating a high reflecting power 
and a cloud-laden atmosphere. It is therefore improbable 
that much detail could be seen on the disc even under the most 
favourable conditions. There are sometimes visible faint 
bands or belts, which arc analogous to the appearance Saturn 
would give if at the distance of Uranus. There arc no markings 
sufficiently definite to enable the rotation period to be deter- 
mined. As Uranus is so distant from the Earth, the influence of 
phase is negligible and cannot be detected even by the most 
accurate photometric observations. It is, therefore, possible to 
apply the spcctroscopic method, particularly as the position of 
the planet's axis is known with accuracy. The satellites have a 
retrograde motion and the planes of their orbits are inclined 
at over 80 to the plane of the planet's orbit : it can be shown 
that the plane of their orbits must very nearly coincide 
witli the plane of the planet's equator, and the position in 
which the spectroscope slit should be placed in order to detect 
the rotation is therefore determined. The observations of 
Lowell in 1911 indicated a period of rotation of about lOf 

156. Satellites of Uranus. Uranus is known to have 
four satellites. Two of these, Titania and Oberon, which are 
the brightest and most distant from the planet, were discovered 
by Sir William Hcrschel, shortly after his discovery of Uranus. 
The other two, Ariel and Umbriel, were discovered by Lassell 
in 1851. They are small bodies and difficult to observe in the 
telescope. Particulars of their distances and periods are as 
follows : 




Distance in 
terms of tho 
Radius of 

in Miles. 




h. m. 





12 29 

Umbriel .... 




3 28 





16 56 





11 7 

The orbits of the four satellites are nearly circular and 
co-planar, their plane being inclined at 82 to the orbit of 
Uranus and being therefore almost perpendicular to it. 

The orbits are all described in the retrograde direction. 
When the Earth passes through the line of nodes of their orbit 
plane, the orbits appear edgewise as straight lines, providing a 
favourable opportunity for determining the inclination of the 
orbits. The next occasion on which this will happen is 1924. 
Twenty-one years after passing through a node, the plane of the 
orbits is seen almost perpendicularly. It is believed that the 
periods of axial rotations of the satellites agree with their 
respective revolution periods. 

Their probable diameters arc of the order of 500 miles and 
are therefore much larger than the satellites of Mars, though 
much smaller than the Moon. 

157. Neptune. As mentioned in 154, it was found after 
the discovery of Uranus that it had frequently been observed 
before and taken for a fixed star. Although the older 
observations were not of very great accuracy, they proved 
sufficiently accurate to be of value for the calculation of the 
orbit of the planet, covering as they did a complete revolution. 
It was found by Bouvard that it was not possible to satisfy 
both the old and the new observations, and he therefore based 
his computed orbit entirely on recent observations. After a 
short time, it was found that Uranus was not in the position 
computed by Bouvard's tables, and Bessel, who examined the 
matter thoroughly, concluded that the discrepancies between 
the computed position and both the new and the old observed 
positions must be due to a real physical cause and suggested 


the possibility of the existence of an unknown planet beyond 
Uranus, which was invisible to the naked eye, but which by 
its attraction on Uranus was causing the observed discrepancies. 
The French astronomer Leverrier and the English astronomer 
Adams adopted this suggestion and independently computed 
the position of the hypothetical planet. Neptune was dis- 
covered very close to the positions which they assigned to it 

( 121). 

The mean distance of Neptune from the Sun is about 2,800 
million miles, or rather more than thirty times the mean 
distance of the Earth. The eccentricity of its orbit is only 
0-0085, which, with the exception of Venus, is much smaller 
than that of any other planet. Owing to the large size of the 
orbit, this small eccentricity corresponds to a difference of over 
50 million miles between the greatest and least distances of the 
planet from the Sun. The inclination of the orbit to the ecliptic 
is 1 47'. The sidereal period of the planet is GO, 181 days, or 
about 105 years, nearly double the period of Uranus. Relatively 
to the fixed stars, the planet therefore moves only about 2 per 
year ; this amount, though apparently small, corresponds to 
a motion of about 20" per day. Relative to the Earth, the 
daily motion varies between 101" at opposition and 133" at 
conjunction. This motion enabled the planetary nature of 
Neptune to be established within twenty-four hours of its 
discovery. The orbital velocity is about 3^ miles per second. 

The apparent mean diameter is about 2-G" and is subject 
to little variation. This angular diameter corresponds to a 
linear diameter of about 33,000 miles. Neptune is, therefore, 
somewhat Larger than Uranus ; its volume is about eighty-five 
times that of the Earth. It has one satellite, which enables the 
mass to be determined ; it is found to be 1/19,314 that of the 
Sun, or about seventeen times that of the Earth. Its density 
is therefore only about 0-24 of that of the Earth, or about 1-30 
of that of water. 

158. Telescopic Appearance. In the telescope, Neptune 
appears as a small greenish disc upon which 110 markings are 
visible. Nothing is known as to its period of rotation. Its 
physical constitution is probably similar to those of Jupiter, 
Saturn, and Uranus. Everything points to the existence of a 


dense atmosphere. The albedo, though not so high as that of 
Jupiter, has the relatively high value of 0-52. The spectrum is 
similar to those of Jupiter and Uranus, with a number of dark 
absorption lines, due to the absorption in the planet's atmo- 
sphere. There is in particular one prominent band in the 
spectra of all three planets which has not been identified in the 
spectra of any known terrestrial substance. 

vX t59. The Satellite of Neptune. Neptune possesses one 
satellite, which was discovered by Lassell within a month after 
the discovery of the planet. It is distant from Neptune about 
15 of the planet's radii, or 284,000 miles. Its period of 
revolution is 5 d. 21 h. 2m. It is a faint object Neptune 
itself is only of the eighth or ninth magnitude which is best 
observed photographically. It is of about the same brightness 
as Oberon, the outer satellite of Uranus. Its size has been 
estimated as about equal to that of the Moon. The inclination 
of its orbit to the ecliptic is about 35 and the orbit is described 
in the retrograde direction. 

160. On the Possibility of an Intr a -Mercurial or Trans- 
Neptunian Planet. -The possibility of the existence of a 
planet between Mercury and the Sun, or beyond Neptune, has 
received much discussion. A planet between the Sun and 
Mercury was actually announced as having been discovered 
by a Dr. Lescarbault in the year 1859, and accepted as genuine. 
The name Vulcan was given to the supposed planet. Little 
doubt exists to-day that the planet does not exist. The transit 
of such a planet across the Sun's disc could scarcely have 
escaped observation and, if it existed, it should be a conspicuous 
object at the time of a total eclipse, unless hidden behind the 
Sun's disc. But such a planet has neither been seen nor 
photographed, although photography can reveal stars as faint 
as the eighth magnitude which are only a few minutes of arc 
distant from the Sun. 

Leverrier discovered that the perihelion of the planet 
Mercury had a motion greater than it was possible to account 
for theoretically. The attractions of other planets cause a 
slight progressive motion of the perihelia of their orbits : in 
the case of all the planets except Mercury, the observed advance 
in the perihelia agreed with their calculated values within the 


errors of observation. In the case of Mercury, there was an 
unexplained residual of about 43" per century. One suggestion 
put forward to account for this motion was that there were 
one or more planets within the orbit of Mercury. This 
explanation, however, raises other difficulties, and the motion 
of the perihelion can now be otherwise accounted for by the 
relativity theory of gravitation formulated by Einstein, for 
the details of which the reader should refer to one of the 
many books which deal with this theory. There is therefore 
at present no reason to suppose that an intra-Mercurial 
planet exists. 

The question of a possible trans -Neptunian planet is still un- 
decided. There are slight discordances between the calculated 
and observed positions of Uranus and Neptune which have been 
tentatively attributed to the attraction of an unknown planet. 
But as their amount is only about 2 seconds of arc, and not 
much greater than the possible error of a single observation 
of the position of the planets, it is evident that the position 
assigned to this hypothetical planet from a discussion of these 
discordances must be liable to considerable error. As the 
planet, if it exists, will be found to be of only about the 
thirteenth magnitude and its motion relative to the stars 
will be very slow, it must prove a difficult object to detect. 
It is even possible that the discordances between the observed 
and calculated position of Uranus and Neptune may be due to 
other causes. At present, therefore, it seems that the prob- 
ability of the discovery of an ultra -Neptunian planet is small. 


161. Comets, or stellce comatce, i.e. hairy stars, as they were 
formerly called, arc bodies which, moving under the influence 
of the Sun's gravitation, appear in the heavens at irregular 
intervals. They gradually increase in brightness for a while, 
after which they grow fainter and fainter until they can no 
longer be observed. Only a small proportion, of the comets 
which are discovered become sufficiently bright to be visible 
to the naked eye. Those which do so appear as a hazy cloud 
with a brighter nucleus from which a fainter tail extends in 
the direction away from the Sun, visible sometimes to a great 
distance. In physical constitution they are very different 
from planets, although they are to be considered, even if only 
temporarily, as members of the solar system. 

162. Number of Comets. It has been estimated that, on 
the average, only about one comet out of five discovered 
becomes visible to the naked eye. The discoveries of comets 
since the invention of the telescope have therefore greatly 
increased in number. There are records extant of about 400 
comets previous to the year 1600, which must all have been 
bright objects : this number includes the different returns of 
periodic comets. The appearance of a bright comet was 
formerly regarded with great alarm and considered as an 
omen of impending disaster, and it is, perhaps, for this reason 
that the records of their appearances are numerous. The 
employment of " comet-seekers," telescopes with a large field 
of view and a low-power eyepiece, has greatly increased the 
numbers of discoveries of comets : Pons, for instance, dis- 
covered 27 comets between 1800 and 1827. Newcomb 
estimated that between 1500 and 1800 there were 79 comets 



visible to the naked eye. It is probable that some of the less 
conspicuous naked-eye comets were cither not observed or else 
that no records were kept of their appearance, for Denning 
finds that, between 1850 and 1915, 78 comets were discovered 
which became visible to the naked eye. Between 1800 and 
1850, there appear to have been at least 30 naked-eye comets. 
It would therefore seem that, on the average, there is at least 
one naked-eye comet per year. Some years are particularly 
favoured ; thus, in 1911, there were four naked-eye comets. 
In other years there may be none. The average number of 
comets now discovered per year is probably about five or 

163. Old Views on Comets. Aristotle and his followers 
believed that comets were exhalations from the Earth which 
had become inflamed in the upper regions of the atmosphere. 
This theory was doubtless based to a certain extent upon 
observation : comets attain their greatest brightness when 
nearest the Sun and would then be visible shortly after sun- 
set or before sunrise ; the tail of a comet points always away 
from the Sun, so that the comets would then be observed 
with their tails pointing upwards and appearing like a rising 
flame. The views of Aristotle were generally accepted until 
about the seventeenth century. Tycho Brahe first showed 
that comets must be more distant than this theory required : by 
comparing observations of the position of the comet of 1577 
amongst the stars, made at different places in Europe, he con- 
cluded that its distance must be much greater than that of the 
Moon. He found the same results for the comets of 1582, 1585, 
1590, 1593, and 1596. Tycho supposed their paths to be 
circular ; Kepler, on the other hand, supposed them to move in 
straight lines. That their orbits were parabolas was first sug- 
gested by Hevelius, who noticed that the path of a comet was 
curved near perihelion. It was proved by Doerfel for the par- 
ticular case of the comet of 1681 that the path was a parabola 
with the Sun in the focus. A method for determining the 
parabolic orbit of any comet was first given by Newton, and, 
using this method, Halley was able to determine the paths 
of 24 bright comets which appeared in the years 1337 to 1698. 
From the similarity of the paths of the comets of 1531, 1607, 


and 1682, he concluded that the three comets were identical 
the famous comet bearing his name : the records of this 
comet go back without a break to the year 989 and with a 
few gaps back to the second century B.C. Halley therefore 
concluded that the orbit, though very nearly parabolic, was 
in reality elliptic ; and predicted the comet's return in 1759. 
It was observed in that year, after Halley's death. Its most 
recent appearance occurred in 1910. A photograph of the 
comet obtained in 1910 at the Union Observatory, Johannes- 
burg, with Venus in the same field of view, is reproduced in 
Plate XIV (6). 

164. Orbits of Comets. The orbits of about three hundred 
comets have been determined with some certainty. The 
majority of these orbits 
are parabolic. If a come- 
tary orbit were accurately 
parabolic or hyperbolic, 
it would imply that the 
comet had entered the 
sphere of attraction of the 
Sun from outer space and 
as a result had been de- 
viated from its path al- 
though not captured, the 
comet again passing away FIG. 84. Comotary Orbits, 

out of the solar system. 

But supposing the path of a comet were elliptic and its period 
several hundreds or thousands of years, the accuracy of 
the observations might not be sufficient to distinguish between 
an elliptic or a parabolic orbit. Comets are visible only in 
that portion of their path near to the Sun, and in that portion 
the differences between an ellipse, parabola, and hyperbola are 
small (Fig. 84) : also a comet, being diffuse, cannot be observed 
with the same accuracy as a planet. When the orbit has been 
determined as a parabola, the logical inference is, therefore, 
that its eccentricity is so nearly unity that a period cannot 
be assigned. In a few cases, hyperbolic orbits have been 
determined. A careful examination of these has been made 
by Stromgren, who concluded that either the observations 

This portion 

of orbit 

is observed 


were not sufficiently accurate or consistent to justify the 
assumption of a hyperbolic orbit, or that the hyperbolic orbit 
had been caused by planetary perturbations. There thus 
seems to be no conclusive evidence of any comet having 
entered the solar system from outside. This is further con- 
firmed by the absence of any preference shown by the direction 
of approach of comets for the direction in space in which the 
solar system is moving as compared with the opposite direction. 
If comets entered the system from outside, more would be 
overtaken in the direction in which the system is moving 
than would overtake the system in the opposite direction. 
There is no evidence for such a tendency. 

For about fifty comets elliptical elements have been deter- 
mined, but only about twenty of these can be properly 
described as " periodic " comets, i.e. comets which have been 
observed at more than one return. The return of a comet 
with a definitely elliptical path may be missed, either because 
of an inaccuracy in the determined period or because the orbit 
may have been changed by planetary perturbations : in other 
cases, the return may not be observed on account of the comet 
having broken up. 

The inclinations of the comets' orbits to the ecliptic have 
all values from to 90, but with a few exceptions, of which 
Halley's Comet is the most notable, the direction of motion 
is the same as that of the planets. 

165. Origin of Comets. We have already pointed out 
that there is no evidence of any comet having entered the 
solar system from outside. From the fact that the axes of 
cometary orbits do not show any preferential direction, it 
has been conjectured that comets originated in a system of 
cosmical matter, at a very great distance from the Sun and 
moving with it. In the absence of disturbing forces due to 
planetary attractions, portions of matter from this system 
attracted towards the Sun would describe parabolic orbits. 
The perturbations due to planetary attractions would change 
these orbits into either an elliptical or hyperbolic shape. In 
the latter case, the comets would pass out of the system and 
not return. The nearer the comet passed to a planet, the 
greater would be the perturbations. They would also be 


greater in the case of comets with orbits inclined at only a 
small angle to the ecliptic and moving in the direct sense than 
in the case of comets with orbits inclined at large angles to 
the ecliptic or moving in the retrograde direction in orbits 
of small inclination. For, in the first case, the comet would 
remain for a longer time in the sphere of influence of the 
planet. It would therefore be anticipated that the comets 
with the most elliptical paths would be those which move in the 
direct sense in orbits inclined to the ecliptic at a small angle. 
Experience tends to confirm this conclusion : of 20 periodic 
comets, Halley's and Temple's Comets are the only two with a 
retrograde motion, and 16 have inclinations of less than 30. 

166. Families of Comets. Of the comets for which 
elliptical orbits have been determined, it is noticeable that 
for the large majority the aphelion is at about the same distance 
from the Sun as the orbit of Jupiter. The periods of this 
group of comets, comprising some thirty or more members, 
all lie between three and eight years. They, therefore, pass 
at some point in their paths very near to Jupiter's orbit 
and arc spoken of as Jupiter's "family" of comets. The 
orbits of the various members of the family have not neces- 
sarily any special resemblance to one another except in period 
and aphelion distance. Other planets also have families of 
comets, though not so numerous as that of Jupiter. Uranus 
has two, Neptune has six (of which Halley's Comet is one), 
and Saturn has two. The large number belonging to Jupiter 
is to be attributed to its large mass, its great perturbing force 
having enabled it to capture more comets than the other 
planets. It is possible that some of the members of Jupiter's 
family may have originated in parent comets which w r ere split 
up under the attracting forces of the Sun and Jupiter which 
differ for different parts of the extended comet. 

There are two comets, 1862 III (i.e. the third comet to pass 
perihelion in 1862) and 1889 III, whose aphelion distances are 
respectively 47-6 and 49-8 astronomical units ; or about 50 
per cent, greater than the distance of Neptune. It has been 
suggested that these may be members of a family of comets 
belonging to an undiscovered planet beyond Neptune. 

The capture of a comet by a planet may be illustrated by 


the case of Comet Brooks, 1889 V. This comet was found 
to have a period of about 7 years, and Chandler, by com- 
puting back from the observed positions, found that, in 1886, 
the comet and Jupiter had come within a small distance of 
one another, with the result that the comet's previous orbit 
had been entirely altered : he also computed that the previous 
orbit was much larger and that the period was then 27 years. 
In 1886, the comet probably passed so near to Jupiter that 
it was within the orbit of its first satellite ; in 1889, after 
its discovery, Barnard observed that the comet was double, 
and that the two parts were slowly separating at such a rate 
that the disruption could be traced back to the time of its 
close passage to Jupiter. 

A comet belonging to one of the planet families may not 
necessarily remain permanently a member of that family. At 
some other part of its orbit it may suffer perturbations by 
another planet and its orbit again be considerably modified. 
The orbit may even be changed into a hyperbola so that the 
comet never returns. The masses of comets are so small that 
they produce no perceptible perturbation on the planets. 

167. Groups of Comets. Comets which have orbits whose 
elements are so similar that it may be concluded that they 
had a common origin are said to form a group. Several such 
groups are known. The most remarkable group is one com- 
posed of the great comets 1843 I, 1880 I, and 1882 II. Of 
these 1880 I had a tail extending over 40 and 1882 II had 
one nearly as long. These comets all had retrograde motion, 
very small perihelion distances, and long periods. A bright 
comet which appeared in 1668 is probably either identical 
with one of the three comets mentioned or is a further member 
of the group. Other instances which may be mentioned are 
the comets 1742-1907 II ; 1812-1884 I ; 1884 III-1892 V. 

From the close similarity of the elements of the orbits of 
two comets it is not to be assumed, however, that they have 
necessarily had a common origin, although there is an a priori 
probability. For conclusive proof, it would be necessary to 
trace back the previous history of each comet, allowing for 
the effects of planetary perturbations. If the paths approached 
close together and the comets reached the adjacent points at 



the same time, the probability of a common origin would be 
greatly strengthened. On the other hand, the failure of the 
comets to converge in this manner would only disprove the 
possibility of a common origin if it were certain that at no 
point in their previous history had either comet disrupted, 
with consequent alteration of path. 

The actual disruption of a comet has been witnessed more 
than once. The case of Comet Brooks was mentioned in the 
previous section, the disruption 
having been due in this case 
to the attraction of Jupiter. 
Biela's Comet provides another 
example ; this comet was dis- 
covered in 1826 and was found 
to have a short period, 6-6 
years. It was observed at 
return in 1832, missed in 1839 
on account of unfavourable 
position, and observed again 
in 1846. JYhen first observed 
in that year it had the normal 
appearance of a comet, but 
shortly afterwards the nucleus 
divided into two parts which 
gradually separated. At the 
next return in 1852, the two 
twin comets were again ob- 
served, but their separation had 
greatly increased. They did 
not appear at subsequent re- 
turns, but in the year 1872, 

as the Earth passed the track 

Of the lost COmet, there was 
a fine meteor display. This 

was repeated at subsequent 

returns, proving that the comet had in the interval completely 

disrupted into fragments. 

Fig. 85 illustrates an early stage in the disruption of a comet. 
On June 4, 1910, a tail was detached from the head of Halley's 
Comet and receded rapidly from the nucleus. The figure 

- Radius Sector 

June 8-740 Mt. Hamilton 

8-475 Cordoba 

7-836 Christchurch 
7-735 Mt. Hamilton 

7-505 Cordoba 

7-286 Beirut 

7-057 Dar/en 
6-997 Tokyo 
6-832 Honolulu 
6-737 Mt. Hamilton 
6-655 Yerkes 
6-494- Cordoba 

6-064 Damn 
4-783 Mt.Hamitton 

-^ OK & T> * * *i 

FIG. 85. Successive Portions of the 

Inner End of a Detached Tail of 
Halley's Comet, June 4-8, 1910. 


shows the position of the inner end of the tail as observed 
at various observatories within the few succeeding days. The 
detached tail appears to recede with an accelerated motion, 
finally becoming completely separated from the parent comet. 

168. Appearance of Comets. A comet, when first dis- 
covered, usually appears in the telescope as a faint nebulous 
or hazy cloud, in which may sometimes be distinguished a 
central condensation. As the comet approaches the Sun, the 
appearance greatly alters : the typical comet at this stage 
consists of a triple structure a head, a nucleus, and a tail 
(Plates XIV, XV, XVI). The head is the cloud of nebulous 
matter which was seen when the comet was much fainter : it 
now has a more clearly-defined outline which is, however, never 
sharp. In shape it is usually round or elliptical. The nucleus 
is a bright point near the centre of the head ; it is more or 
less star-like and is the most suitable portion of the comet 
to point on in determining its position. From the head there 
streams out a nebulous tail, with a cylindrical outline, whose 
axis lies in the plane of the comet's orbit and is directed away 
from the Sun. The brightness of the tail increases towards 
the nucleus, from which it seems to spring. Particularly in 
the neighbourhood of the nucleus, much structure may be 
seen in the tail : this is best studied by the employment of 
photographic methods, which have in a comparatively short 
while provided more information than was obtainable from 
all the earlier visual observations of comets. Comet More- 
house (Plate XVI) provided a good example : the tail of this 
comet was continually and with great rapidity changing its 
shape The structure shown in Plate XVI (a) is entirely 
different from that in Plate XVI (b). 

As the comet approaches the Sun, the tail follows it, but 
when it has passed perihelion, the tail precedes it, pointing 
always away from the Sun. It does not therefore consist of 
matter left behind by the nucleus. In some comets, the 
structure of the tail exhibits rapid variations from night to 
night, the course of which may be traced by photographs 
obtained at sufficiently short intervals. 

169. Size and Mass of Comets. The dimensions of 


comets vary greatly, some being comparatively small and 
others almost inconceivably large. The diameter of the 
nucleus can only be approximately assigned, but may attain 
to several thousands of miles. The head itself may be very 
much larger ; that of the celebrated comet of 1811 has been 
estimated to have had a head whose volume was about 350 
times that of Jupiter. On approaching the Sun the head 
appears to contract, though this may possibly be only an 
optical phenomenon, as it is not easy to give a physical 
explanation. The length of the tail may be a few million 
miles only, or, in some cases, may exceed the distance of the 
Sun from the Earth, with a volume some thousands of times 
that of the Sun. 

In view of these enormous dimensions, it is surprising that 
the masses of the comets are insignificant. Although in no 
case has the mass of a comet been determined, several lines 
of evidence confirm this assertion. In the first place, although 
comets frequently pass so near to the Earth or to one of the 
other planets that their orbits are completely changed, it has 
never been possible to detect any perturbing effect produced 
by the comet on the planet. Further, the comet of 1770 
and comet 1889 V passed through the satellite system of 
Jupiter without producing any perturbations. Certain optical 
phenomena presented by comets also tend to confirm the 
smallness both of their mass and of their density. Thus the 
bright daylight comet of 1882 became quite invisible when 
it passed in front of the Sun. 

The only conclusion which can be drawn as to a comet's 
mass is, therefore, that it is very much smaller than that of 
the Earth, probably not exceeding 1/100,000 of the Earth's 
mass and possibly being much smaller still. Even so, the 
mass may amount to many millions of tons. 

The density must therefore also be extremely small. Small 
stars may be seen through a comet's head, quite near the 
nucleus, without suffering any perceptible diminution in 
brightness (see Plates XV and XVI). The average density of 
the head is probably of the order of the density of the residual 
air in a chamber exhausted by a good air-pump : that of the 
tail must be much smaller even than this. From the fact that 
many comets have broken up and subsequently produced 



showers of meteors, it does not seem unreasonable to suppose 
that the head of a comet consists of meteoric stones, widely 
separated from one another. The fact that the heads of 
comets show no phase effects is consistent with this supposition. 

170. Spectra of Comets. The first spectroscopic obser- 
vation of a comet was made by Donati in the year 1864, and 
revealed certain bright lines superposed upon a faint con- 
tinuous background. This is typical of comet spectra. It 
was shown by Huggins in 18G8 that the bright lines usually 
observed are identical with those given by the blue flame of 
a Bunsen burner and indicate, therefore, the presence in the 
comet of gaseous carbon compounds (such as cyanogen). In 
some cases bright metallic lines of sodium, magnesium, and 
iron may be observed. These observations settled the ques- 
tion, which had previously been much discussed, whether 
comets shone by their own light or merely by reflected 
sunlight. Attempts to determine to what extent the light 
from comets is polarized have yielded negative results, the 
faintncss of most comets making the observations very difficult. 
The presence of bright lines in the spectra can only be due, 
however, to a self-luminous body. The continuous back- 
ground, on the other hand, is doubtless due to reflected 

With the application of photography, it has become possible 
to study the spectrum in the ultra-violet and also the spectrum 
of the tail. This has revealed differences from the normal 
hydrocarbon spectra, which are explained as being due partly 
to a mixture of carbon monoxide and cyanogen and partly 
to the reduced pressure. The electrical phenomena obtained 
by discharge through a Geissler's vacuum tube enable the 
assertion to be made with a high degree of probability that 
a comet's self-luminosity is due not to an actual combustion, 
but to an electrical phenomenon. This is in accordance with 
what might be anticipated from the low density of the comet. 

171. The Nature of a Comet's Tail. Many theories have 
been advanced to account for the apparent repulsion of a 
comet's tail from the Sun. Zolhler suggested that it was due 
to an electrostatic repulsion of matter ejected from the nucleus. 



Bredichin developed a more complete theory based upon 
this suggestion. The repulsion was supposed to be inversely 
proportional to the molecular weight of the ejected gas. The 
repulsive force would therefore be greatest if the ejected gas 
were hydrogen. He divided the tails into three types: (1) 
Long straight rays, the cross-section being very small compared 
with the length. Comet Morehouse (Plate XVI) had a tail of 
this type. The repulsive force due to the Sun's electric 
field was supposed to be eight or more times as great as 
the gravitational attraction : this would cause particles 
to leave the nucleus with a relative velo- 
city which would rapidly increase. Such 
tails were supposed to consist mainly of 
hydrogen. (2) The second type is shorter, 
more curved, and with a relatively larger 
cross-section (Plate XV (6) ). In this case, the 
repulsive force is supposed to be about double 
the gravitational attraction, and the tail to 
consist mainly of carbon compounds. (3) The 
third type is short, stubby, and greatly 
curved, and due to matter for which the 
repulsive force is only a fraction of the gravi- 
tational force. This type is supposed to be 
composed of metallic gases. In the case of 
bright comets, the three types merge and may 
all be detected more or less clearly in the 
compound tail. Compound tails are to be seen 
in Plates XV and XVL. The three types are 
illustrated in Fig. 86. 

More modern views are in favour of a different theory. 
It was shown theoretically by Maxwell, and demonstrated 
experimentally by Lebedeff and by Nichols and Hull, that 
when light falls upon a body it exerts a pressure upon it. 
This pressure is very small and proportional to the area upon 
which the radiation impinges. Suppose it falls upon a small 
sphere near the Sun : then the gravitational force of attraction 
on the sphere due to the Sun is proportional to the cube of 
its radius, and therefore if the radius becomes sufficiently 
small it may be anticipated that the gravitational attraction 
will ultimately become less than the radiation pressure, which 

Fro. 86. Typos 
of Comets' Tails. 


is proportional to the square of the radius. Theoretical 
investigation shows that this would be the ease if the diameter 
of the particle were between 1-5 // and 0-07 ^ where 1 // = 
1/1,000 mm. A sphere of diameter equal to the smaller of 
these limits would still be large compared with the size of a 
gaseous molecule : if, therefore, this theory is correct, the 
tail of a comet is not truly gaseous in nature, but is composed 
of a cloud of very small discrete particles. Bredichin's three 
types of tails may be accounted for by supposing there are 
particles of three different densities. 

The tail of a comet accordingly appears to be composed of 
small particles of matter, ejected from the nucleus and repelled 
by radiation pressure from the Run. The total matter so 
repelled is probably small in amount and presumably is 
dissipated into space. 

172. Effect of Possible Collision with the Earth. In 

June, 1921, the Earth narrowly escaped collision with Poiis- 
Wimiccke's comet, which passed its perihelion a few days 
too early for a collision to occur. It has been estimated that 
such an occurrence should happen on the average once in 
every 15 million years. Our present knowledge of the con- 
stitution of a comet does not render it probable that such a 
collision, should it occur, would be a serious matter. There 
is only one piece of evidence of such a collision having occurred 
in the past, the existence of a cup-shaped crater in Arizona 
which bears a general resemblance to the lunar craters. This 
is about three-quarters of a mile in diameter and several 
hundred feet deep. Many small iron meteorites have been 
found in its vicinity. It is estimated that the outer crust of 
the Earth is 100 million years old, and this is the only instance 
of a supposed collision which is known. It has also been 
thought that the gaseous tail of the comet would have poisonous 
effects, but the density of the gas is so low that the passage 
of the Earth through the tail would pass entirely unnoticed. 

173. Meteors. Closely connected with comets are meteors 
or shooting stars. It is only within comparatively recent 
years that the nature of these bodies has been definitely 
settled. Shooting stars may be seen on any clear moonless 


night, though on some nights they are much more frequent 
than on others. The brightness of the majority no ted is about 
equal to that of the naked-eye stars, though a few are as bright 
as or brighter than Jupiter and Venus. The brig] it er ones leave 
trains which may persist as long as two or three minutes. 
Occasionally, very bright meteors are observed which from 
their appearance are called fire-balls. These may be dissipated 
in an explosion of considerable violence, accompanied by a 
loud report. No sound, however, accompanies the dissipation 
of the ordinary shooting star. 

The heights and velocities of meteors may be determined 
by observing their positions relatively to the stars from two 
stations some miles apart and noting their time of flight. 
It is thus found that the mean height at which they are first 
observed is about 80 miles, and that at which they disappear 
is about 50 miles. The length of visible path may be any 
distance up to several hundred miles. The average velocity 
with which they enter the Earth's atmosphere is probably 
about 26 miles per second ; this is the velocity which a body 
moving from rest at a great distance under the action of the 
Sun's attraction would attain when it had attained the dis- 
tance of the Earth. The larger meteors or fire-balls first 
become visible at greater heights, up to 100 miles, and penetrate 
more deeply into the Earth's atmosphere, sometimes to a 
height of only 5 or 10 miles. Their velocity decreases rapidly 
during their flight, owing to the resistance of the Earth's 
atmosphere to the motion. 

The flight of a fire-ball is accompanied by a succession of 
explosions, by which fragments are torn off from the principal 
body. These are accompanied by variations in brightness. 
The path of the fire-ball is not, in general, straight, but more 
or less irregular, probably due to the resistance of the air on 
a body of irregular shape. In the case of shooting stars, 
these variations in brightness and direction of motion are 
not apparent to the naked eye. Occasionally, however, they 
are photographed by accident, and an examination of their 
trails on the photographic plates reveals the irregular changes 
in brightness and slight deviations from rectilinear motion. 

The numbers of shooting stars which may be observed is 
very large. The average hourly number of visible shooting 


stars which may be seen by a single observer varies from about 
six or seven on some nights to as many as sixty or seventy 
on others. Tf the whole sky could be watched, the average 
number visible in one hour would probably not be less than 
thirty to sixty. These, however, would be only the meteors 
which enter the Earth's atmosphere within a few hundred 
miles' distance from the observer : it has been computed that 
the total daily number of meteors which enter the Earth's 
atmosphere cannot be fewer than several millions. Very few 
of these ever reach the Earth's surface, the large majority 
being completely burned up before they reach the surface 
by the heat generated by friction in the Earth's atmosphere. 
Such meteors as do reach the surface are generally called 
aerolites or meteorites. They are probably essentially the 
same as the normal meteor or shooting star, the distinction 
being one of size only. 

The majority of the aerolites which are found or are observed 
to fall consist of masses of stone limestone, magnesia, or 
siliceous stone, generally mixed with grains or globules of 
iron. A small percentage consist of nearly pure iron, usually 
alloyed with a relatively small amount of nickel. Some con- 
tain iron and stone in nearly equal proportions. No chemical 
element has been found amongst them which is not known 
on the Earth. The mass of an aerolite may vary from a few 
ounces to several tons. The masses of the shooting stars 
are not known with certainty, but they are believed to be 
very small, in general not exceeding a fraction of an ounce. 
The entrance of meteors into the Earth's atmosphere is con- 
tinually adding to its mass, but the rate of growth is relatively 
exceedingly slow. Even though each meteor had a mass of 
a quarter of an ounce (an estimate probably much in excess 
of the truth), the daily addition to the Earth's mass would 
only be about 100 tons, and at this rate of increase it would 
take 1,000 million years to add sufficient matter to increase 
the radius of the Earth by 1 inch. The effect of the increase 
in mass would be a shortening of the length of the year ; but 
the total effect would be negligible, amounting to less than 
1/1,000 of a second in 1 million years. 

174. Meteor Radiants .If the paths of all the meteors 


visible on any one night are noted and then plotted on a 
celestial globe, it will be found that they are not distributed 
at random over the heavens, but that many of them when 
produced pass, within the accidental errors of observation, 
through a single point. Such a point is called a meteor 
radiant. The right ascension and declination of the radiant 
is the same for all observers. This indicates that the meteors, 
whose paths pass through the radiant, were, before they 
encountered the Earth's atmosphere, moving through space 
in parallel paths. For since the position of the radiant amongst 
the stars is independent of the position of the observer, it 
must be considered as infinitely distant, and two parallel 
lines in space, if seen projected on the celestial sphere, would 
appear to pass through the same point, viz. the point in which 
a line from the centre of the sphere parallel to this direction 
meets the sphere. 

All meteors having the same radiant are said to constitute 
a meteor swarm. If the path, before encountering the Earth's 
atmosphere, of one of the meteors of a swarm passes through 
the position of the observer, that meteor would apparently 
be stationary in the radiant point. If the actual lengths of 
path through the Earth's atmosphere are about the same 
for all the meteors of a given swarm, then the apparent length 
of path will be greater, the greater the distance from the 
radiant point at which the meteor appears : for the angle 
between the direction of the meteor's motion and the direction 
to the point of appearance will be greater. 

Besides those meteors which belong to definite swarms, 
there are a large number of sporadic meteors. The frequency 
of appearance of these has a daily and a yearly variation : 
the hourly number observed during any one night increases 
from the early evening until about three or four hours after 
midnight : the mean hourly frequency for a single night has 
a minimum in the spring, a maximum in the autumn, and 
intermediate values at the solstices. The directions of motion 
are also not uniform : more than 50 per cent, come from the 
east, and about equal numbers from the north, south, and west. 
These phenomena can be simply accounted for by the motion 
of the Earth around the Sun. The direction of motion of 
the Earth at any instant is towards the point in the ecliptic 


at which the Sun was three months previously : this point 
is called the apex of the Earth's motion. The combination 
of the motion of the Earth and that of a meteor swarm causes 
an apparent displacement of the radiant point towards the 
apex, a phenomenon comparable to the displacement of the 
position of a star on account of aberration. The greater the 
altitude of the apex, the fewer the number of radiants which 
could not be observed. It may be deduced from the relative 
velocities of meteor swarms and of the Earth, that if the apex 
was in the observer's zenith, about live-sixths of all radiants 
would be seen ; if on the horizon, about one-half ; and if in 
the nadir, only one-sixth. The diurnal variation in the hourly 
frequency of meteors follows from this, for the altitude of the 
apex is least at 6 o'clock in the evening and greatest at 6 
o'clock in the morning : the maximum frequency would, 
however, be observed somewhat earlier than the latter time, 
as with the on-coming of dawn the fainter meteors would be 
missed. The yearly variation is due to the change in the 
declination of the apex from 23 to -|- 23, following the 
declination of the Sun. The apex is on this account highest 
(in the northern hemisphere) at the autumnal equinox and 
lowest at the vernal equinox. The preponderance of meteors 
from the east is due to the apex being east of the meridian 
during the greater portion of each night. 

175. Connection between Comets and Meteor Swarms. 

The existence of meteor radiants points, as we have seen, 
to the occurrence of swarms of meteors moving in parallel 
paths : the fact that their velocity on entering our atmosphere 
is the parabolic velocity suggests that these swarms are moving 
in parabolic (or nearly parabolic) orbits around the Sun. The 
similarity between the orbits of several periodic comets and 
those of certain meteor swarms suggested a connection which 
was verified when, after Biela's periodic comet had been lost, 
a magnificent meteoric shower was observed in 1872 as 
the Earth passed through the old track of the lost comet 
( 167). 

The following meteor showers and periodic comets are known 
definitely to be related to one another : 



Kadi ant in 

Meteor Shower. 

Date of Shower. 





April 20-21 

1861 I 

415 years 



August 10-11 

1802 III 




November 14-15 

1866 I 




November 23 



The meteors in each of these cases are extended more or less 
uniformly in an elliptical ring about the Sim, one of the points 
of intersection of which with the ecliptic lies near the Earth's 
orbit. In the case of the Leonids and Bielids, the meteor 
swarm does not occur every year, showing that the meteors 
are stretched out through a relatively small portion of the 
orbit : thus, in the case of the Leonids, the swarm takes 
about three years to pass any given point. The Lyrids and 
Perseids, on the other hand, occur with about the same 
frequency each year, indicating that in these instances the 
swarm is fairly uniformly distributed along the orbit. There 
seems little doubt that the above-mentioned meteor swarms 
and comets are definitely associated, but it is not so certain 
whether in all cases the meteor swarm is the result of the dis- 
ruption, partial or complete, of the comet : for instance, the 
Leonid meteor shower has been traced back to the year 902, 
and in this case it would appear as though both the comet 
and the swarm might be constituent parts of a cosmic cloud. 

The extension of the swarm along the orbit is due to the 
attraction of the Sun and planets arid is probably to some 
extent a criterion of the age of the swarm. On this view, 
the age of the Lyrid and Perscid showers would be much 
greater than that of the Leonids, which themselves are at 
least several hundred years old. 

176. The Zodiacal Light. The zodiacal light is a faint 
hazy band of light extending from the Sun along the ecliptic. 
The brightness decreases with increase of distance from the 
Sun to a distance of over 170, after which it increases to a 
patch of light a few degrees in diameter at a point exactly 
opposite to the Sun, called the counter-glow. The zodiacal 
light is best seen in the evening in February, March, and April, 


because the portion of the ecliptic east of the Sun's position 
is then most nearly perpendicular to the western horizon. 
In the early morning, it is best seen in the autumn. For the 
same reason, it can be better observed in the tropics than in 
more northern latitudes : it may then be seen extending 
entirely across the sky, forming a complete ring. The portion 
near the Sun is relatively bright, but the more distant portions 
are so faint that clear air, free from smoke and dust and the 
glare from artificial illumination in cities, is necessary for it 
to be observed. 

The spectrum of the zodiacal light shows no bright lines 
but is mainly a continuous spectrum in which some of the more 
prominent Frauiihofcr lines have been detected. This probably 
indicates that the light is mainly or entirely reflected sun- 
light. The most plausible explanation of the zodiacal light 
is, therefore, that there exists in the neighbourhood of the 
Sun, and extending beyond the orbit of the Earth, a thin 
flat sheet of rarefied matter, lens-shaped and symmetrical 
with respect to the ecliptic. The light from the Sun reflected 
or scattered by this matter gives the appearance of the 
zodiacal light. If the particles have a rough surface, the 
phenomenon of the counter-glow can be theoretically accounted 
for. If the total mass of this ring of matter is appreciable, 
it will have an influence upon the motion of the planets. 
This question has been examined by Seeliger, who has shown 
that on the assumption that the density of the matter decreases 
with increasing distance from the Sun, it is possible to account 
for the acceleration of the motion of the perihelion of Mercury 
without introducing any appreciable discordances into the 
motions of the other planets. The amount of matter so re- 
quired, as computed by Seeliger, has a mass about one -tenth 
of that of the Earth. The plausibility of this explanation is 
discounted by Cromnielm's study of the motion of comets 
with small perihelion distance. The five comets considered 
emerged from the region near the Sun with no appreciable 
loss of velocity : from the manner in which meteors become 
incandescent on entering the Earth's atmosphere at a height 
of 100 miles, it follows that the comets could never get past 
the Sun if the density of the matter responsible for the 
zodiacal light at all approached that of the Earth's atmosphere 


at a height of 100 miles. The actual density of the matter 
must be but a very small fraction of this amount. It there- 
fore appears as though this matter cannot be held to give a 
satisfactory explanation of the acceleration of the perihelion 
of Mercury. 


177. WE have hitherto been dealing with the various 
members of the solar system. We have now to consider the 
bodies which to the ancients were known as the " fixed stars." 
They were so called because they apparently did not alter their 
positions with respect to one another, in contrast to the 
wandering stars or planets. We now know that the smallness 
of their apparent motions is due solely to their great distances, 
the distance of the nearest star at present known being about 
260,000 times the distance of the Sun from the Earth. But the 
ancients had no knowledge of stellar distances, nor was there 
then any means by which they could determine them. In 
the sixteenth century, Copernicus was able to infer that their 
distances must be very great because they did not reflect the 
annual motion of the Earth about the Sun ; this was, however, 
used by his opponents as an argument against the theory rather 
than as a proof of the great distance of the stars. That the 
so-called fixed stars were, at least in some instances, not actually 
fixed was first proved by Halley, who, in 1718, showed that the 
bright stars Sirius, Proeyon, and Arcturus were gradually 
changing their positions with reference to the neighbouring 
stars. The apparent relative displacements are so small even 
in the course of centuries that the appearance of the con- 
stellations of bright stars does not appreciably alter in the 
course of thousands of years ; the constellation of Orion has the 
same configuration to-day that it had several thousand years 
ago to the writer of the Book of Job. 

178. Stellar Constellations and Names. The stars 
were divided by the ancients into groups or constellations, to 
which were given names of persons or objects famous in olden 



mythology. In a few cases a fanciful resemblance may be seen 
between the outline of a constellation and the object from 
which it derives its name, but in general no resemblance can 
be seen nor can any reason be assigned for the name. The 
division of the sky into constellations in this way was 
probably made for reasons of convenience ; at a time when 
there were no instruments by which accurate positions might 
be assigned, the division facilitated the description of the 
sky and was an aid to remembering the number and arrange- 
ments of the stars. It is therefore not surprising that several 
peoples, including the Babylonians, Chinese, and Egyptians, 
divided the sky into constellations. 

The oldest document in which is to be found a description of 
many of the constellations as known to-day is one by Eudoxus 
(409 to 326 B.C.) in which each figure is described together 
with the positions of the principal stars. Ptolemy's star 
catalogue divided the stars into forty-eight constellations, 
twelve in the zodiac, twenty-one to the north, and fifteen to the 
south. Some of these constellations have since been modified 
and others have been added from time to time, particularly 
in the neighbourhood of the south pole. The total number of 
constellations now generally recognized is eighty-eight. 

A knowledge of these constellations and of the names and 
positions of the brighter stars in them is valuable. On several 
occasions, the appearance of a bright new star has been detected 
by persons thoroughly familiar with the aspect of the constella- 
tions who have at once noticed the changed appearance due 
to the outburst of the new star. For acquiring this familiarity , 
the study of the sky at different seasons of the year with a good 
star atlas is essential. 

Individual stars are designated in various ways. Most of the 
brighter stars have names of their own of Greek, Latin, or 
Arabic origin, but the names of only some fifty or sixty stars 
are in common use, many of the fainter naked-eye stars having 
Arabic names which are practically obsolete. Bayer, in 1603, 
in the star maps of his Uranometria, was the first to adopt the 
plan of designating stars by the name of the constellation, 
prefixed by the letters of the Greek alphabet, usually assigned 
in the order of magnitude. Thus Polaris, the brightest star in 
the Little Bear, is a Ursoe Minoris ; Arcturus, the brightest 


star in Bootes, is a Bootis ; Aldebaran, the brightest star in 
Taurus, is a Tauri, etc. The next brightest star would have the 
prefix ft and so on. When the letters of the Greek alphabet 
have all been assigned, the letters of the Roman alphabet or 
the numbers assigned by Flamsteed are used, so that every 
naked-eye star has some letter or number in its constellation 
by which it may be identified. 

In the ease of the fainter stars it is convenient to have an 
easy means of identification and it is usual to refer to such a 
star by its number in a well-known star catalogue, usually the 
first important catalogue in which its place is given. Thus, a 
star might have the designation Lalande 45,585, indicating that 
it is No. 45,585 in Lalaiide's star catalogue (1790), or Groom- 
bridge 990, indicating that it is No. 990 in Groombridge's 
catalogue (1810). The majority of the stars down to a limit 
between the ninth and tenth magnitudes can thus be 

If a star cannot be designated in this way, it is necessary for 
its identification to give its place (i.e. its right ascension and 
declination) at a definite epoch. The purpose of a star cata- 
logue is to give the places of a number of stars for a definite 
epoch. The observations for these catalogues arc made with 
the meridian circle. Certain brighter stars for which accurate 
positions have first been determined are adopted as funda- 
mental stars and the positions of the other stars are based upon 
these. The star catalogues usually give also for each star the 
values of the precession in right ascension and declination and 
of its secular variations, so that the position of the star at any 
desired epoch may readily be computed. 

179. Stellar Magnitudes. The relative brightness of 
different stars is indicated by a number, termed the star's 
magnitude. The conception of the magnitude of a star dates 
back to the time of Hipparchus, although it is only within 
recent years that precision has been given to the notion. 
Hipparchus selected about twenty of the brightest stars which 
he could see and called them first-magnitude stars. All the 
stars that he could just see he called sixth-magnitude stars. 
Stars of intermediate luminosity (by which we refer to apparent 
and not to intrinsic brightness) he placed in intermediate 


classes, thus obtaining a somewhat rough and purely arbitrary 
classification. Ptolemy carried this classification a stage 
further, recognizing gradations in brightness between adjacent 
classes ; these he recognized by attaching the words ^LEI^MV 
(greater) and iXdaacov (less) to the magnitudes, to denote that 
it was somewhat brighter or fainter than that magnitude. 
He therefore practically divided each class into three. The 
decimal division of the magnitude intervals was first used by 
Argelander and Schoiifeld in the preparation of the extensive 
survey of the sky known as the Bonn Durchmusterung or B.D. 
Thus, a star whose magnitude was assigned as 8-3 was inter- 
mediate between magnitudes 8 and 9, but judged to be only 
three-tenths of the interval fainter than 8. This method of 
denoting magnitudes has been adopted and extended in the 
modern more precise magnitude determinations. 

The question arises as to what this arbitrary magnitude 
classification corresponds to in terms of apparent brightness. It 
was not until the time of Sir John Herschel that attention was 
given to this question ; Herschel concluded that a decrease of 
light in geometrical progression corresponded to an increase of 
magnitude in arithmetical progression and estimated that the 
actual ratio of the light of a star of the first magnitude to one 
of the sixth is at least 100 to 1. 

Hcrschel's conclusion is in accordance with a physio- 
physical law enunciated by Fechner in 1859 that as a stimulus 
increases in geometrical progression, its resulting sensation 
increases in arithmetical progression. If then I lt I 2 denote the 
brightness of two stars whose magnitudes are m t and m 2 , a 
relationship must hold of the type 

l 2 m ' 

where k is a constant quantity denoting the ratio in the bright- 
ness of stars of consecutive magnitudes. Adopting Herschel's 
estimate that if m 2 m l is 5 magnitudes, I l /I 2 = 100, we have 
100 = k 5 or log k = 04, so that k - 2' 512 ... Now the 
magnitudes assigned in the Bonn Durchmusterung and other 
early catalogues, which for the naked-eye stars fit in closely with 
previous estimates of magnitude extending back to the time of 
Hipparchus, agree very closely with this value of the " light- 


ratio, " as the quantity Ic is termed. Pogson tlicref ore suggested 
that k should be definitely adopted as the quantity 2-512 . . . 
whose logarithm is 0-4 ; a star of one magnitude is then about 
2-5 times as bright as one of the next lower magnitude and a 
difference of five magnitudes corresponds exactly to a ratio in 
brightness of 100 : 1. 

The adoption of this value for k gives logical precision to the 
conception of magnitude and is sufficiently in accordance with 
old estimates to avoid serious discontinuity. Modern deter- 
minations of visual magnitudes are based upon this ratio and 
the zero of the scale is adjusted so that the mean magnitude 
of stars near the sixth magnitude agrees with the mean value 
of the magnitudes assigned to these stars in the Bonn Durch- 
rnusterung. In this way, extensive determinations of visual 
magnitudes have been m^tdc at Harvard and Potsdam which 
are available for fixing the zero of the scale in future deter- 

Logically, the scale of magnitudes can be continued without 
limit in both directions. Thus stars which are one magnitude 
brighter than stars of the first magnitude are said to be of 
magnitude and still brighter stars have a negative m$!gnitudc. 
Thus, the magnitude of Sinus, the brightest star, is about 
~ 1-4, whilst on the same scale the magnitude of the Sun is 
- 26-7. 

180. Photographic Magnitudes.- With the development 
of photographic methods in astronomy, it was natural to 
attempt the determination of stellar magnitudes by photo- 
graphic methods. Stars of different brightness produce 
images of different sizes on the photographic plate and the 
size of the image may be used as the basis of magnitude 
determination. Photography has the advantage of economy 
of time at the telescope, since when the plates have been 
obtained they may be measured at leisure. It also enables 
fainter magnitudes to be reached, since, by lengthening the 
exposure, the plate continues to respond to the stimulus of 
the incident light and gives an integrated effect, which the eye 
is not able to do. 

But since, as compared with the eye, the photographic plate 
is relatively more sensitive to the blue end of the spectrum and 


less sensitive to the red end, it follows that if two stars, one of 
which is blue and the other red, are of equal visual magnitude, 
their photographic magnitudes will not be equal, but the blue 
star will appear photographically the brighter. The difference 
photographic minus visual magnitude therefore provides a 
criterion as to the colour of the star and is called its " colour 
index." The zero of the photographic scale of magnitudes is 
adjusted so that for stars of the sixth magnitude of the Harvard 
type AO (see 189), i.e. for bluish stars in whose spectrum the 
hydrogen series Ha, Hft, etc., reaches its greatest intensity, 
the photographic and visual magnitudes are equal. The same 
light-ratio is adopted for photographic as for visual magnitudes, 
so that for any star of type AO the photographic and visual 
magnitudes are equal. Of recent years, by employing 
isochromatic plates in conjunction with a yellow filter, 
magnitudes have been determined photographically which 
correspond very closely with visual magnitudes, the sensitivity 
curve of the isochromatic plate under these circumstances 
being similar to that of the eye. Magnitudes so determined 
are termed photo- visual. By proceeding in this way, the scale 
of visual magnitudes can be extended to much fainter stars 
and the magnitudes can be determined with less labour. 

As an illustration of the difference between photographic 
and visual magnitudes, reference may be made to Plate XXII I. 
The two large star images are those of a and ft Crucis, two bright 
stars in the Southern Cross. Both these stars arc blue stars. 
On a level with a Crucis, the star nearer the top of the plate and 
near the left-hand edge of the plate is a much smaller star 
image. This is the image of y Crucis, a very red star, which 
visually is of the same brightness as ft Crucis. 

181. Determination of Visual Magnitudes. Visual 
magnitudes are determined with an instrument called a 
photometer, of which there are several different types. The 
two best types are perhaps the Zollner and meridian photo- 
meters, which were used for the extensive series of magnitude 
determinations at Potsdam and Harvard respectively. A brief 
description of these two instruments will suffice to illustrate the 
general principles of visual magnitude determination. 

In the Zollner type of photometer, the star under observation 



is compared with the image of an artificial star whose brightness 
can be varied at will until equality between the two images is 
obtained. An arm is attached to the telescope tube at right 
angles to its axis (Fig. 87) and at the end of this arm is a small 
pinhole diaphragm o, through which passes light from a 
petroleum or other type of standard lamp giving constant 
illumination. The size of the aperture o can be varied to 

FIG. 87. The Zollner Photometer. 

simulate stars of different magnitudes. The divergence of the 
rays is increased by a double concave lens, m, and they then 
fall upon a Nicol prism, k, which polarizes the light, i.e. permits 
only the vibrations in one definite plane to pass through. On 
emergence from the Nicol prism, the light passes through a 
crystal of quartz, Z, cut perpendicularly to its axis and a second 
Nicol i, so that by rotating the first Nicol relatively to them, the 


colour of the artificial star can be varied so as to produce 
approximate equality with that of the star under observation. 
The light then falls upon a third Nicol prism, A, which acts 
as an analyser. By rotating the system containing the Nicols 
k, i and the plate I, the Nicol h remaining fixed, the light 
emerging from the latter is varied in intensity, its colour 
meanwhile remaining the same ; the intensity of the emergent 
light is proportional to the square of the sine of the angle 
between the principal sections of the two prisms i and h. The 
light then falls on a double convex lens, /, which brings it to a 
focus in the focal plane of the telescope, after reflection from a 
plane unsilvered glass plate ee*. Two images g, g of the artificial 
star are formed by reflection at the front and back surfaces 
respectively of the mirror. To find the magnitude difference 
between two stars, therefore, the image of the artificial star is 
brought into equality with the two star images respectively, 
and if 1 ,(? 2 are the angles through which the Nicol i is turned 
in the two cases, the ratio of the brightness of the two stars is 
sin 2 0i/sin 2 2 an d therefore their difference in magnitude is 
2-5 {log sm 2 ^! log sin 2 2 }. There are four positions in which 
equality of light is produced, the reading corresponding to each 
being taken and the mean value used. The principal dis- 
advantage of this type of photometer is the use of an artificial 
star, whose image is not absolutely comparable under conditions 
of average atmospheric definition to that of a real star ; this 
causes a liability to personal errors in observation. 

The meridian photometer of Pickering consists of a tele- 
scope with two similar object glasses side by side (.4 and B) and 
of the same focal length (Fig. 88). It is placed in a horizontal 
position, and in front of each object glass is a, right-angled prism 
(C, D) which serves as a mirror to reflect into the tube the light 
from stars on or near the meridian. One of these mirrors, D, 
is capable of slight adjustments by means of the rods E and F, 
so as always to send the light from the pole star intp the lens, 
whilst the other, C, can be turned about an axis so as to reflect 
light from any star on or near the meridian into the second 
lens. The position of the mirror, (7, is given by the graduated 
circle 0. The beams of light from the two objectives fall upon a 
double-image prism, K, of Iceland spar, compensated by glass ; 
each beam is split up into two beams, polarized at right angles 







FIG. 88. The Meridian Photometor. 


to one another. A diaphragm is so placed as to cut off the 
extraordinary beam from the one objective and the ordinary 
beam from the other, allowing the other two beams which are 
polarized at right angles to unite and to pass through the 
eyepiece, L. The beam then falls on to a Nicol prism, M, 
which can be rotated, and then through the eycstop, N, to the 
observer's eye. By rotating the Nicol the relative intensities 
of the two beams can be varied, causing a corresponding 
variation in the two images in the focal plane of the eyepiece. 
If is the angle through which the Nicol is turned from the 
position in which the image of Polaris disappears to that in 
which equality between the two images is obtained, and if 
7 , / are the brightnesses of Polaris and the other star respec- 
tively, then when the images are equal 

7 cos 2 - 7 siii 2 

and the difference in magnitude of the two stars is 2-5 log tan 2 
or 5 log tan 0. As in the case of the Zollner photometer, there 
are four settings for which equality can be obtained, and the 
position of each is observed. This photometer has the advan- 
tage that two star images are directly compared, but has 
the disadvantage that observations can only be obtained on or 
near the meridian and that the images are produced by different 
optical trains. It does not provide any means of compensating 
for colour, and personal and subjective errors of considerable 
magnitude may enter when the brightness of images of different 
colours are compared ; this is due to a phenomenon called the 
Purkinje effect if two sources of light, one red and the other 
green, appear of equal brightness and the intensity of each is 
increased on the same ratio, the red light will appear the 
brighter ; if the intensity of each is decreased in. the same ratio, 
the green will appear the brighter. Also the relative sensitive- 
ness of the eyes of different persons to light of different colours 
are not the same, so that it is not surprising that series of 
observations by different observers, even with the same 
instrument, show at times somewhat large discordances 
depending upon colour and magnitude. The subjective effect 
of the eye cannot be entirely eliminated and it is best to take 
the mean values obtained by several observers. 

It will have been noticed that the photometer determines only 
differences of magnitude and therefore the zero of the magnitude 


scale must be fixed before the magnitudes can be made absolute. 
As previously explained, the zero is fixed by adjusting the 
magnitudes so that in the mean they agree with those in the 
B.D. at about the sixth magnitude. 

There are other types of photometer, such as the wedge 
photometer. In this type, a wedge of dark neutral-tinted 
glass is used, through which the star is viewed. The position 
of the wedge is found for which the star just disappears. This 
type of observation, besides being extremely fatiguing to the 
eye, depends upon the retinal sensitiveness at the moment of 
observation, and the wedge photometer has not therefore been 
greatly used. 

With any type of photometer, the magnitudes determined 
must be corrected for the absorption of tight in the Earth's 
atmosphere. The lower the altitude of the star at the moment 
of observation, the longer the path of the light from it in our 
atmosphere and the greater the absorption. By observing the 
same star at different altitudes or otherwise it is possible to 
deduce a correction depending upon altitude, so that the 
magnitude can be corrected to the value it would have if the 
star were observed in the zenith. All magnitude observations 
should be corrected in this way. 

182. Determination of Photographic Magnitudes. 

The determination of magnitudes photographically proceeds 
upon rather different lines. It is somewhat complicated by 
the laws of photographic action. The intensity of the image 
produced on a photographic plate by a given stimulus is not 
proportional to the time of exposure but follows a more complex 
law. All comparisons must therefore be made via exposures of 
the same length. It is advisable also that only images on the 
same plate should be compared, so as to avoid possible errors 
arising from slight differences in the sensitivities of the plates 
or in their treatment during development. The principle of the 
method by which photographic magnitudes are determined is 
to give one exposure on the field of stars under investigation 
and then a second exposure in which the intensities of the light 
falling on the plate from all the stars have been reduced in the 
same ratio and therefore by a definite magnitude interval. In 
the case of a reflecting telescope, this reduction can be most 


easily effected by reducing the aperture, the light falling on the 
plate being then reduced proportionally to the area of the 
aperture ; with a refractor, this method would introduce errors, 
as it would modify the proportion of light absorbed in the object 
glass, the central portion being thicker than the edge. With 
either type of telescope, the reduction may be effected by means 
of a fine wire-mesh screen whose absorption must be determined 
in the laboratory. Another method is to place over the objec- 
tive a grating composed of a number of parallel equidistant 
wires. Such a grating, in use at the Royal Observatory, 
Greenwich, is shown in Plate XVII (a). Each star image then 
consists of a central image, with a series of diffraction images 
on either side of it extending along a direction perpendicular to 
the direction of the wires, as illustrated by Plate XVII (6). By 
a suitable choice of the diameter of the wire and of the intervals 
between adjacent wires, the first diffracted image on either side 
of the central image will be round and will differ in magnitude 
from the central image by a desired amount, which should 
not exceed four magnitudes, and which can be calculated from 
the dimensions of the grating. This method has the great 
advantage that the two images of any one star, differing by a 
known magnitude, are obtained with one exposure, so that 
errors which might arise from any variation in the transparency 
of the sky during the exposures for any one plate are avoided. 

Having the plate with the two series of images differing by a 
known magnitude, suppose an image of one star in the first 
series is equal to that of another star in the second series ; then 
the magnitude difference of these stars is obviously equal to 
the constant difference between the two series. In practice, 
both series of images are compared with a set of images of 
a single star, photographed with the same telescope with 
exposures so adjusted as to give approximately equal difference 
in magnitude between successive images. The magnitude of 
each image can be estimated on tliis arbitrary scale by com- 
paring them in a micrometer and the scale can then be 
standardized from the known magnitude differences of the 
two images of each star. 

In this way, differences of magnitude only are determined 
and the zero must be adjusted so that for stars of type AO the 
photographic magnitudes agree with the visual magnitudes. 



183. Numbers of Stars of Different Magnitudes. The 

numbers of stars in the whole sky down to various limits of 
magnitude from the second to the tenth are given in the 
following table, for both visual and photographic magnitudes. 
The data for the visual magnitudes were obtained at Harvard, 
and those for the photographic at Greenwich : 

Number of Stars. 

Magnitude Limit. 






























The limit of magnitude visible to the average eye is somewhere 
between 6 and 7, so that approximately about 10,000 stars in 
the whole sky are within the reach of the naked eye. Not 
more than about one -third of these can probably be seen at 
any one time. This number is much smaller than is generally 
popularly supposed. 

184. Total Light of the Stars. The equivalent light of the 
stars of different magnitudes in terms of first-magnitude stars 
can easily be calculated. On the photographic-magnitude 
scale, the three brightest stars, Sirius, a Carina), and a Centauri, 
are equal respectively to 11, 6, and 2 first- magnitude stars. 
The eight stars between magnitudes and 1 are equal to 14 
first-magnitude stars and so on. The greatest light from stars 
within one magnitude interval is between magnitudes 9-0 and 
10-0, the total light from the 174,000 stars within this range 
being equal to that of 69 first-magnitude stars. For fainter 
stars, the increase in total number is not sufficient to com- 
pensate for the decrease in brightness, and the total equivalent 


light from stars with a range of one magnitude continuously 
decreases. The equivalent light of all the stars is equal to 
that of about 700 first-magnitude stars (photographic) or of 
about 900 to 1,000 first-magnitude stars (visual). The photo- 
graphic magnitude of the full Moon has been found to be 11-2, 
and from this it follows that the full Moon gives 100 times as 
much light as all the stars together. Another way of expressing 
the results is that the total star light is equal in photographic 
intensity to that of an ordinary 16-candlc-power lamp at 
47 yards distance. 

185. Proper Motions of Stars. That the stars have 
motions of their own was first shown by Halley. The real 
motion of any star may be along any direction in space, but it is 
only that component of 
the motion of any star 
which is at right angles 
to the line of sight which ^ 
will cause an apparent 
displacement of the star 
on the celestial sphere 
relative to other stars. 
This component of the 
motion, expressed in an- FJG. 89. Proper Motion, 

gular movement per year 

or per century, is called the proper motion of the star. 
It is actually a combination of the angular displacements 
resulting both from the actual motion of the star and from the 
actual motion of the Sun. 

Thus, in Fig. 89, if E is the position of the Earth, and A and B 
represent the positions relatively to the Earth of the star at the 
beginning and end of a year, then AB represents the real relative 
motion of the star in a year, AB being very small compared 
with the distance AE. AB may be split into the two com- 
ponents, CB along the line of sight and AC at right angles to it. 
The angle AEB is the angular displacement of the star upon the 
celestial sphere due to the component AC of its proper motion, 
i.e. it is the annual proper motion of the star. 

For a given real velocity, the angular velocity will be greater 
the nearer the star. A large proper motion is generally to be 



ascribed to the star being relatively near rather than to an 
intrinsically large velocity. The brighter stars are on the 
average nearer than the fainter ones and therefore have on the 
average larger proper motions. The largest proper motion 
known at present is that of a faint star of magnitude 94 m, 
known as Munich 15,040. This star, which is known also as 
Barnard's proper-motion star, has an annual proper motion of 
10" -29. The next largest motion known is that of a star of 
magnitude 8-0, known as Cordoba Zones, 5 hours, No. 243, 
which has an annual proper motion of 8"-75. Other large 
proper motions are : 

Mag. P.M. 

Mag. P.M. 

Groombridge 1,830 
Lacaille 9,352 . . 

. 6-4 7-03 
. 8-7 6-89 

Cordoba 32,416 . 

. 8-3 6-11 

GlCygni . . . 
Lalando 21,185 
e Indi .... 

. 5-4 5-24 
. 7-3 4-74 
5-2 4-67 

Lalanrlo 21,258 . 
2 -Eridani 
ft Cassiopeia) . 
Argolander 11702 . 
a Centauri 
Lacaillo 8,760 



A proper motion of 10" annually would carry a star through 
360 in about 130,000 years. The number of stars with proper 
motions known to exceed 1" per year is about a couple of 

Proper motions of stars can be deduced by comparing the 
positions given in star catalogues whose epochs differ 
preferably by at least fifty years. In making the comparison, 
the effect of precession on the star's right ascension and 
declination between the two observations must be allowed 
for. For most of the brighter stars, positions may be 
found in several catalogues. In the case of faint stars, for 
which early observations are not available, proper motions 
can be obtained with a fair degree of accuracy by com- 
paring photographs obtained at an interval of about twenty 

If the distance of the star is known (AE in Fig. 89) the proper 
motion can be converted into linear motion. 

186. Line-of-Sight Velocity The component of the 
velocity of the star in the line of sight (BC in Fig. 89) can be 
determined by measuring the displacement of the lines in the 



spectrum of the star produced by the motion. This method 
was first used visually by Sir William Huggins in 3867. The 
principles involved have already been explained in 69. 
Determinations of line-of-sight velocity are now always made 
photographically. The spectrum of the star is photographed 
with the aid of a spectrograph attached to the eye end of the 
telescope, and a suitable comparison spectrum of a terrestrial 
source is photographed with the same spectrograph on the same 
plate, with the aid of which it is possible to determine directly 
the displacements of many of the lines. The displacement is 
measured accurately in a micrometer. Special precautions 
must be taken in the observations to avoid temperature changes 
by insulating the spectrograph. 

The line-of-sight velocity is determined directly in miles 
or kilometres per second. The accuracy of modern observation 
is very great ; provided the spectral lines are sharp, a probable 
error of under J mile per second can be obtained. Velocities 
greater than 50 miles per second are relatively few. The 
greatest yet observed is that of Lalande 1,966, with a velocity 
of 203 miles per second. The star Cordoba Zones 5 h. 243, 
which has the large proper motion of 8"-75, has also a largo 
line-of-sight velocity of 151 miles per second. 

187. Solar Motion. The observed motions of the stars 
are their motions relative to the Earth, the motion of which is 
in turn a combination of 
its orbital motion about 
the Sun and the motion 
of the Sun itself. The / \\ 
apparent displacements of 
stars on the celestial 
sphere due to the orbital 
motion of the Earth are 
small and oscillatory, 
whilst the influence of the 

FlG - 90- Effect of Solar Motion on 
Proper Motion of Stars. 


orbital velocity on the line-of-sight measurements can be 
allowed for and the observations reduced to an observer on the 
Sun. We need therefore consider only the combination of the 
real motions of the star and the Sun. 
Suppose first that the stars have no real motions and that 


the Sun is in motion directly towards a point A on the celestial 
sphere and away from the diametrically opposite point B 
(Fig. 90). Then, in small regions of the sky around A and 13 
respectively, the stars will show no proper motions, since the 
relative velocity is along the line of sight in each case. Any 
star in a direction at right angles to A B has a relative motion, 
on the other hand, which is entirely at right angles to the line of 
sight, and the motion will appear as a proper motion of the star 
along a great circle passing through A and B and in the 
direction towards B. The stars near A, therefore, appear to be 
opening out, those near B to be closing in. The proper motion 
will be greater the smaller the distance of the star from the 
Sun, and the nearer the angular distance of tLe star from A 
approaches 90. If V is any star, at a distance 11 from the 
Sun in a direction SV making an angle with 8 A or SJi, and 
if D is the distance through which the Sun moves in one vear 
in the direction 8 A, the relative displacement of the star will 
be Vv = D parallel to 813 and its angular displacement on the 
celestial sphere, obtained by dividing the projection of Vv by 
the distance SV, will be D sin 0/J2. This will be the observed 
proper motion of the star. 

But this simple result is complicated by the intrinsic motions 
of the stars and by their varying distances. If, however, a 
sufficiently small area of the celestial sphere be considered, and 
the proper motions of the stars in this area be determined, it 
may be assumed that since the real motions occur in all direc- 
tions at random they will average zero when the mean is taken. 
If stars of a limited range of magnitude are used, their distances 
will cluster about a certain mean value and the mean observed 
proper motion for the group will be D sin 0/jR, where E is the 
mean distance of the group. 

Sir William Herschel was the first to notice that the stars in 
one region of the sky were apparently separating from one 
another and that those in the opposite region were closing in, 
as shown in Fig. 90. He interpreted this as due to solar motion 
and used the result to determine the direction of the Sun's 
motion. That this interpretation is correct is confirmed by 
the radial- velocity observations. The mean radial velocity 
of stars near A should be towards 8 and that of those near B 
should be away from S } the mean values being equal. At G, 


on the other hand, the solar motion does not affect the line-of- 
sight velocity, and the mean value of a sufficiently large group 
of stars should be zero. At an intermediate point, the mean 
value should be proportional to the cosine of the angle between 
the direction to the star and AB. These results are confirmed 
by the observations, which prove also that the Sun has a 
velocity of about 13 miles per second towards a point with 
right ascension approximately 270 (18 hours) and declination 
+ 30, in the constellation of Hercules. This point is called the 
Solar Apex. The point from which the Sun is moving, i.e. 
the point on the celestial sphere diametrically opposite to the 
solar apex, is called the Solar anlapex. 

Using the value so obtained for the velocity of the solar 
motion, D the distance described by the solar system in a year 
can be determined and then from the mean value of the proper 
motions for a group of stars can be deduced the mean value of 
1 111 or, alternatively, the mean parallax of the group of stars. 
This method is frequently used for the determination of the 
mean distances of groups of stars. 

188. Star Streams. Tn discussing the solar motion in the 
preceding section, it was assumed that the real motions of the 
stars in a sufficiently small area were in random directions. 
Tn 1904 it was shown by Kapteyn that this is not exactly so, but 
that there is a peculiarity in 
the stellar motions which causes 
the stars to move in two 
favoured directions. If we con- 
sider the stars in a limited area 
and count the number of stars 
observed to be moving in dif- 
ferent directions, say from to 

10 10 to 20 20 to ^0 oto Fia ' 91. Distribution O f Proper 
1U , 1U to ZO , ZU DO 3V , CtC., Motions due to single Drift Motion. 

the angle being measured from 

the north through east, the results can be plotted on a polar 
diagram, the radius vector in a given direction being pro- 
portional to the number of stars moving in that direction. If 
the motions were at random, the curve would be a circle if 
there were no solar motion. The effect of the latter is to 
superpose on the real motions an apparent motion in the 



opposite direction ; there will therefore be a maximum number 
of stars apparently moving in the direction opposite to that 
of the motion of the Sun and a minimum number with it. 
The curve representing the observed distribution would 
therefore be an oval (Fig. 91), symmetrical about the great 
circle through the solar apex and antapex and with its greatest 
elongation towards the antapex. The curves actually obtained 
are not of this simple nature but are of a more complex type. 
They show, in general, two favoured directions of motion 
instead of a single one, and in every case it is found that they 

Fia. 92. Tho Combination of two Single Drift Motions. 

can be represented within the limits of error of observation by 
the superposition of two simple diagrams of the type discussed. 
This is represented in Fig. 92. The directions OP, OQ, are the 
two favoured directions of motion, and the combination of the 
effects produced by two simple drifts gives a distribution of 
proper motions represented by the curve on the right-hand side 
of the diagram. If similar curves are constructed for different 
regions of the sky, using the observed proper motions, and the 
directions OP, OQ thus determined are continued across the 
celestial sphere as great circles, it is found that these circles 
intersect one another, within the limits of error, in two distinct 
points ; this indicates that the two favoured directions are for 


every region of the sky towards the same two points on the 
celestial sphere and represent real drifts of the stars. One of 
these points is at R.A. 90, Dec. 15, and the other at R.A. 
285, Dec. 64. It is further found from the mathematical 
analysis that the first of these drifts contains about 60 per cent, 
of the stars and the second about 40 per cent, and that the 
speed of the first drift is about double that of the latter. 

These motions are, of course, measured relative to the Sun. 
In Fig. 93 suppose that SA and SB represent the drift velocities; 
then if AB is divided 
at G so that AC : CB 
2:3, i.e. in the 
proportion of stars in 
the drifts SB and SA 
respectively, SC will 
represent the motion 
of the centroid of the 
Stars with reference Fia 93. Star-streaming and Solar Motion. 

to the Sun. (7$ must 

therefore represent the solar motion and point towards the 
solar apex whilst AB represents the velocity of the one drift 
relatively to the other. The points on the celestial sphere 
towards which the line AB is directed are called the vertices ; 
their positions are approximately R.A. 95, Dec. + 12, and 
R.A. 275, Dec. 12. These points fall exactly in the 
plane of the Milky Way. 

Viewed in this way, it is evident that the inclination of the 
directions of the two stream motions, illustrated by Fig. 92, is 
due to the fact that the observed motions are a combination of 
the real motions entangled with the solar motion. Actually 
the directions of motion of the streams are opposite to one 
another in space. It is evident, in fact, that if there exist two 
streams of stars moving relatively to one another in space, all 
that can possibly be learnt about their motions is the direction 
and magnitude of their relative motion. From this point of 
view, it is clear that the phenomenon of star-streaming indicates 
the existence of a direction which is of fundamental importance 
in connection with stellar motions and about which the motions 
are symmetrical. 

Schwarzschild emphasized this aspect of the matter and 


pointed out that the observed phenomenon can be explained 
by supposing that the stars have a greater freedom of movement 
in a certain direction (the line joining the vertices) than in 
perpendicular directions. From this point of view the separa- 
tion of the stars into two separate streams is not essential to 
the explanation of the observed distribution of the proper 
motions, although, of course, this method of representation 
does not deny and is not in variance with the tendency of the 
proper motions in any region to favour two definite directions. 
Observations of liiic-of-sight velocities are also in accord with 
the phenomenon of star-streaming, the radial velocities after 
eliminating the component due to the solar motion showing a 
distinct tendency to be greater near the vertices than elsewhere. 

180. Spectral Types of Stars. The spectrum of a star 
consists in general of a continuous emission spectrum upon 
which is superposed a discontinuous absorption or emission 
spectrum. Different stars give spectra which differ very 
considerably inter se, but Seech i, who was the first to examine 
stellar spectia on a considerable scale, found that the spectra 
could be classified into four broad classes, and that this classifi- 
cation was practically a classification according to the colour 
of the star. Sccchi's classification has now been practically 
superseded by that of the Draper Catalogue of the Harvard 
Observatory ; about a quarter of a million stellar spectra have 
been classified at Harvard. This classification forms a single 
continuous linear sequence, successive classes being denoted 
by the letters, 0, B, A, F, G, K, M, N, R, the order in which 
the letters are here given being connected with the successive 
stages of the evolution of a star. At present it is sufficient to 
state that it is believed that a star in the course of its evolution 
passes through successive stages, which are characterized by 
these spectra. Each class is further subdivided, these sub- 
divisions being indicated by letters in the case of types and M, 
thus Ob, Ma, etc., and by figures (representing a decimal 
subdivision) in the case of types B, A, F, G, K, thus BO, AO, 
F8, G5, K2, etc. The following is a brief description of the 
types : 

Type O. The spectrum consists of a faint continuous 
background upon which arc superposed bright bands. NO dark 


bands occur except in the subdivisions Od, Oe, and are then due 
chiefly to hydrogen and helium. In the subdivisions Oa, Ob, 
and Oc the hydrogen and helium lines are bright. Bright bands 
at wave-lengths 4633 and 4686 are present in all subdivisions 
except Oe5. Stars of type with bright bands are often 
called Wolf -Ray et stars. 

Type JB. Stars of this type are sometimes called Orion or 
helium stars. The spectra of this type contain only dark lines, 
of which the helium lines are the most prominent ; they reach 
their maximum intensity in the subdivision B2 and in later 
subdivisions become gradually loss prominent, finally practi- 
cally disappearing in B9. At the same time the intensity of 
the hydrogen lines gradually increases. In B8, some lines 
which occur in the solar spectrum appear for the first time. 
The H and K lines of calcium are present but not prominent. 

Type A. The most prominent feature in the spectra of this 
type is the great intensity of the hydrogen lines, which reach a 
maximum in A2. Helium is absent. The H and K lines of 
calcium increase in prominence throughout the type. The 
magnesium line of wave-length 4481 is prominent, and other 
metallic lines begin to appear in the early subdivisions and 
grow progressively stronger, becoming the chief feature in 
class A5. 

Type F. In this type, the intensity of the hydrogen lines 
diminishes and the metallic lines increase in prominence. The 
H and K lines are very prominent. The spectrum is gradually 
approaching that of the Sun. 

Type G. The solar spectrum is a typical example of a type G 
spectrum. Numerous metallic lines are present and are as 
conspicuous as the hydrogen lines. The H and K lines of 
calcium are the most prominent feature. The increased 
absorption at the blue end of the spectrum gives the stars a 
yellow colour. 

Type K. In this type, the hydrogen lines are much less 
prominent and the relative intensity of the blue end of the 
spectrum becomes appreciably less. Bands due to hydro- 
carbons appear for the first time. 

Type M. This type is characterized by broad absorption 
bands which have their greatest intensity towards the violet end. 
The solar lines decrease in number and intensity. Bands due 




to titanium oxide are very prominent. Sub -type Md is a 
special subdivision, in which bright hydrogen lines reappear. 
All stars of this sub-type are long-period variables. 

Type N. Stars of this type have broad absorption lines in 
their spectra which are most intense towards the red. These 
lines are mostly due to carbon monoxide and to cyanogen. A 
few bright lines may be present. All the stars of this type are 
very red. 

Type R. This is a small class of stars which are not so red 
as stars of types M and N, although the most prominent 
absorption bands of type N are present. 

The characteristics of stars vary so much with spectral type 
that a knowledge of the type is of fundamental importance in 
all discussions connected with the physical state of a star. 

190. Spectral Type and Colour-Index. We have seen 
in the preceding section that the recognized classification of 
stars according to the nature of their spectra is also a classifi- 
cation according to colour, the 0, B, A or early-type stars being 
blue, the stars of intermediate type, E, G, K, yellow, and the late, 
type stars, M, N, etc., red. The colour of a star was defined in 
180 by means of its " colour-index " or the difference between 
photographic and visual magnitudes. A comparison between 
colour-index and spectral type is therefore suggested. In the 
following table the average colour-index for stars of different 
spectral types, as determined by King and Parkhurst, is given: 

Colour-index according to 












+ 0-32 




+ 0-86 


+ 1-17 

+ 1-32 


-f 1-68 

+ 1-77 

It will be seen from this table that the increase in colour- 
index from type to type is in the mean practically uniform, so 


that a knowledge of the colour -index of a particular star is 
sufficient to determine its spectral type with a fair degree of 
accuracy, and vice versa. 

191. Effective Temperatures of Stars. The effective 
temperature of a star is defined as the temperature of a perfect 
radiator, or black body as the physicist calls it, which has the 
same distribution of energy amongst different wave -lengths as 
the star. For a perfect radiator, there is a definite relationship 
between the energy radiated of any specified wave-length, the 
wave-length, and the temperature. For any given temperature, 
the curve connecting the spectral intensity with the wave- 
length is a smooth curve which has a single maximum of 
intensity for a certain wave-length, the intensity falling away 
for wave-lengths longer or shorter than this value. The wave- 
length corresponding to maximum spectral emission is inversely 
proportional to the temperature. By measuring the dis- 
tribution of intensity throughout the spectrum of a star and 
comparing the smoothed intensity curve with the black-body 
curves, the effective temperature may be determined. The 
temperature so obtained may be higher or lower than the 
true value at the surface of the star, since the stars are not 
perfect radiators ; but it is probable that the values so found 
are not greatly wide of the mark and there is no doubt that they 
express correctly the relative temperatures of different stars. 
The temperatures show a strong dependence upon spectral type, 
as is indicated by the following table in which the mean effective 
temperatures determined at Potsdam for stars of different type 
are given : 

Stellar Type. Effective Temperature. 


Oa . 23,000 

BO 20,000 

B5 14,000 

AO 11,000 

A5 9,000 

FO 7,500 

GO 5,000 

KO 4,200 

Ma 3,100 

N 2,950 

B 2,300 


These figures show that the temperatures of the blue stars 
are the highest and those of the red stars the lowest. The rate 
of change of temperature with spectral type and therefore also 
with colour is much more rapid at the beginning of the series 
than at the end. 

The effective temperature of the Sun, which is a G-type star, 
is between 6,000 and 7,000. The mean value determined at 
Potsdam for this type is 5,000. This value and the subsequent 
values in the table above are almost certainly somewhat too 

192. lonization in Stellar Atmospheres. A physical 
explanation of this clearly marked relationship between 
effective temperature and the type of spectrum was given in 
1921 by Saha. According to his theory, in fact, the temperature 
of the stellar atmosphere is the factor which, more than any 
other, fixes its spectrum. 

If a gas, say a chemical compound, is gradually heated, a 
stage will arrive at which the molecules of the compound will 
be dissociated into the molecules of simpler compounds or of 
the constituent elements themselves. On still further heating, 
the molecules will in turn be dissociated or decomposed into 
atoms. The progress of these phenomena can be treated 
mathematically, using the principles of thermodynamics. If 
the gaseous mass consisting of atoms only be heated, Salia 
supposes that the outer electrons in the atom are gradually 
torn off : the energy of this ionization can be calculated by 
physical principles, and it then becomes possible to apply 
the reasoning of thermodynamics to the process. At any 
given temperature, a steady state will result if the temperature 
does not vary, in which there will be a definite degree of 
ionization and in which, on the average, as many electrons 
rccombine with atoms as are dissociated from them. The 
extent of the ionization at any particular temperature for any 
given element can thus be calculated. 

Now, spectroscopically, an ionized element can be distin- 
guished from a unionized element, for the former shows lines 
in its spectrum which are called enhanced lines whose intensity 
is much greater than in the spectrum of the unionized element. 
It therefore becomes possible to correlate the intensities of 


various enhanced lines of different elements with the tempera- 
ture and pressure required to produce the necessary degree 
of ionization. This is the basis of the method developed by 
Saha ; from a study of certain lines in spectra of different 
types he has found it possible to derive the temperatures in 
the stellar atmospheres. The temperatures so deduced agree 
very closely, for types 0, B, A, with those determined at 
Potsdam and given in the preceding paragraph. For types 
FO, GO, KO, and Ma, values of 9,000, 7,000, 6,000, and 
5,000 respectively are obtained. These values are somewhat 
higher than those determined at Potsdam for the same types, 
but, as has previously been mentioned, the Potsdam values 
for the red stars are probably too low. 

It may incidentally be mentioned that the theory also gives 
a satisfactory explanation of the occurrence of lines due to 
certain elements in the spectra of the Sun and stars and of 
the absence of lines due to other elements. It has for long 
been a puzzle why the lines of some elements should be so 
prominent, whilst those of other elements, which must almost 
certainly be present in the stars in large quantities, are either 
absent or relatively inconspicuous. It is, for instance, impos- 
sible to believe that stars of type consist solely of hydrogen 
and helium. If a star passes through successive types in the 
course of its evolution, the elements whose lines appear in 
the later stages must also have been present in the early stages, 
although no lines which characterize their spectra were evident 
in the spectrum of the star at that time. The presence or 
absence of the lines due to any individual element is almost 
solely a question of the relative extents of the ionization of 
that element at the temperature prevailing in the stellar 
atmosphere. If the ionization is complete the ordinary lines 
will be absent, and the lines due to the ionized atoms may be 
far away in the ultra-violet. If, on the other hand, the ioni- 
zation has not commenced, the enhanced lines will be absent. 
As an example, the yellow D lines of sodium are much more 
prominent in the spectra of Sun-spots than in the spectrum 
of the Sun's chromosphere. This is because the Sun-spots 
are at a lower temperature, so that there is less ionization, 
and the D lines, being due to non-ionized atoms, are therefore 
more prominent. The theory also gives a satisfactory expla- 


nation of the relative heights at which different elements 
appear in the flash spectrum. 

193. Measurement of the Distances of Stars. The 

distances of stars are so great that it is customary, for con- 
venience, to refer to the parallax of a star rather than to its 
actual distance. In speaking of a stellar parallax it is not the 
diurnal parallax that is referred to, but the annual parallax. 
The former is the angular semi-diameter of the Earth as seen 
from the body in question : this semi-diameter forms the 
natural base-line for the measurement of the distances of the 
members of the solar system. For the stars, on the other 
hand, the natural base-line is the semi-diameter of the Earth's 
orbit, and the parallax of a star is the angle which this distance 
subtends at the distance of the star. For no star is it known 
to be as great as 1 second of arc. 

The unit of distance used for the convenient expression 
of the linear distance of a star is termed a parsec, and is 
that distance which corresponds to a parallax of 1 second of 
arc. Parallaxes of 0"-1, 0"-0l, 0"-001, etc., therefore correspond 
to distances of 10, 100, 1,000 . . . parsecs respectively. 
Another unit which is frequently used in popular language 
is the light-year, i.e. the distance which light (whose speed is 
186,000 miles per second) travels in one year. The light-year 
is about 63,000 times the distance of the Earth from the 
Sun. The scientific unit, the parsec, is equal to 3-26 light- 

It is owing to the smallness of stellar parallaxes that attempts 
to measure them were for so long unsuccessful. It was not 
until 1838 that the first stellar parallax was determined, and 
then, strangely enough, the problem was solved simultaneously 
for three separate stars by three different astronomers. 
Although the accuracy of these determinations was much 
inferior to that of modern determinations, they marked a 
great step forward in astronomy. The results obtained, 
together with the values derived from modern observations, 
are given in the table. The distances are given in terms cf 
the mean distance of the Earth from the Sun as unit. 





Morlorri Observations. 




Centauri (Henderson) 
Cygni (Bossel) . 
Lyrae (Struve) . 







Three different methods were used, which well illustrate 
the principles of parallax determination. Henderson deduced 
his parallax from observations of the zenith distance of a 
Centauri at different seasons of the year ; this was the method 
which Bradley had used in his attempt to determine a stellar 
parallax which, though failing in its object, had led him to 
the discovery of the aberration of light. Struve used a clock- 
driven equatorial telescope and measured with a position 
micrometer the distance of a Lyne from a faint star at an 
angular distance of about 40" : the distance of the faint star 
was assumed to be much greater than that of a Lyrse, so that 
its parallactic displacement could be neglected in comparison 
and the relative parallactic displacement determined by the 
observations was attributed solely to the bright star. Bessel's 
method was analogous to that of Struve, but the distance 
apart of the two stars was determined with the aid of a 
hcliometcr ( 105). Until the application of photographic 
methods, the heliometer provided the most accurate means 
of measuring small parallactic displacements. It suffered, 
however, from the disadvantages that the observations were 
slow and called for a high degree of skill in the observer if 
systematic errors were to be avoided. Great care is required 
in the adjustments and in the accurate determination of the 
scale value at different temperatures. 

The only method now used for the direct determination of 
stellar parallaxes is the photographic one. It has the advan- 
tage of being not only more rapid, because economical of observ- 
ing time, but it is also superior in accuracy even to the best 
heliometer determinations. The principle of the method 
adopted is to obtain a photograph of the star whose parallax 
is desired and then a second photograph about six months 


later, when the parallax displacement is in the opposite 
direction. The position of the star is compared with the 
positions of several fainter stars in its neighbourhood by 
measuring the plate in an accurate micrometer. The variation 
in the relative distances between the two exposures will be 
due partly to the relative parallactic displacement and partly 
to the relative proper motions of the stars. By taking a 
further plate, at another interval of six months, when the 
parallactic displacement will have its former value but the 
proper-motion displacement will have doubled, it is possible 
to disentangle the two effects and so determine the parallax. 
Several refinements are necessary in order to secure a high 
degree of accuracy. Three or four plates should be obtained 
at each of three six-monthly epochs. The comparison stars 
must be as symmetrically placed as possible about the central 
star. The magnitude of the central star must be reduced by 
some means to that of the comparison stars, as, for example, 
by exposing it behind a rotating shutter with a sector of 
suitable size cut out the object being to avoid spurious 
displacements due to slight errors in the motion of the tele- 
scope, these displacements varying with the magnitude of the 
image. The photographs should also be taken on or very 
close to the meridian. 

When due care is taken the probable error of a determina- 
tion by photographic methods should not exceed 0"-01. The 
parallax derived in this way is relative to that of the faint 
comparison stars ; the correction required to reduce it to an 
absolute value depends upon their magnitude, but can be 
determined from statistical considerations. It averages in 
general only about 0"-004 or 0"-005. The number of parallaxes 
determined with this accuracy is now somewhere about two 

Particulars of the stars with largest known parallaxes are 
given in the following table ; 






in Light- 

Sim = l. 


Proxima Centauri 





a Contauri .... 






Munich 15,040 . . . 






Lalando 21,185 . . . 












Cordoba, V h. 243 . . 






i Coti 






e Eridani .... 












61 Cygni 






It will be noticed that all the stars in this table have large 
proper motions, illustrating the fact that large proper motion 
may be taken as a criterion of nearness. The values of the 
luminosities show that the stars in question are all dwarfs 
( 199). It will also be noticed that they are mainly of late 
spectral types. 

194. Absolute Magnitude of a Star. If the apparent 
magnitude of a star be known arid also its parallax, it is possible 
to deduce the magnitude which the star would have when 
moved to a standard distance. This magnitude is termed an 
absolute magnitude, because it gives a relative measure of the 
intrinsic as contrasted with the apparent magnitude of the 
star. The standard distance most commonly used is 10 
parsecs, corresponding to a parallax of 0"-1. The absolute 
magnitude, M, is then expressed in terms of the apparent 
magnitude, ra, and parallax co (expressed in seconds of arc) 
by the relationship 

M = m + 5 + 5 log co. 

For in bringing a star from a parallax o>" to one of D'-l its 
distance is increased in the ratio 10 co : 1, and its apparent 
brightness is therefore decreased in the ratio 1 : (10 co) 2 , and 
the consequent increase in magnitude (M m) equals 


2-5 log (10 o>) 2 , leading to the above formula. If the standard 
distance adopted is that corresponding to a parallax of 1", the 
relationship becomes M = m + 5 log co. 

Although the latter is not the method generally adopted for 
measuring absolute magnitude, it is of interest from the 
following fact : the apparent magnitude of the Sun, according 
to the most reliable determinations, is 26m- 72. The 
parallax of the Sun to be used in the above formula being 
1 radian or 206, 2G5", it follows that the absolute magnitude 
of the Sun is Om- 15. Thus, if the standard distance is 
taken as 1 parsec, the absolute magnitude of the Sun, within 
the limits of observational error, is zero, so that a star with 
an absolute magnitude M has a luminosity L, which, expressed 
in terms of that of the Sun as a unit, is given by Iog 10 L 
- - 0-4M or L = l<r' 41f . If M - - 5, the star has a 
luminosity 100 times that of the Sun. If the unit of distance 
is taken as 10 parsecs, the corresponding formula will be 

L =- 10 - () '*(^~5) ^ 10() x 1Q 0-4^ and a gtar of zer() magni _ 

tude has a luminosity about 100 times that of the Sun. 

195. Determination of Absolute Magnitudes. The 

absolute magnitude of a star can be determined if its apparent 
magnitude and its parallax are known. The uncertainty 
attaching to the determination of both these quantities 
therefore enters into the value of the absolute magnitude. 

Recently a method has been developed for determining 
absolute magnitudes which does not involve directly a prior 
knowledge of the parallax. The method is based upon the 
discovery that amongst the stars of a given spectral type there 
are slight differences in their spectra which depend upon their 
absolute magnitudes. It is known that spectral type is no 
criterion as to the physical nature of a star, for stars of the 
same type may and do show a very great range in density. 
It is at first surprising, therefore, that their spectra are so 
similar : we have seen, however, in 192 that this is because 
the temperatures of their atmospheres are the same. Careful 
investigation has nevertheless revealed distinct differences in 
the intensities of certain definite lines, and an investigation 
of these differences for stars whose parallaxes, and therefore 
whose absolute magnitudes, are known, shows that the absolute 


magnitudes and the spectral intensities can be directly corre- 
lated. By using all available material, the correlation has 
been placed upon a firm basis. A determination of the relative 
intensities of the crucial lines in the spectrum of a star therefore 
enables its absolute magnitude to be deduced with an accuracy 
much greater than that possible by calculation from its 
parallax, unless the parallax is relatively large. 

Having determined in this way the absolute magnitude, 
the parallax of the star can be deduced if its apparent magni- 
tude is known. Parallaxes so determined are termed spectro- 
scopic parallaxes to distinguish them from the directly 
determined or trigonometric parallaxes. 

196. The Angular Diameters of Stars. Owing to the 
great distances of the stars their angular diameters are in all 
instances very small, probably in no case exceeding 0"-05. 
The direct determination of their angular diameters is, there- 
fore, a difficult matter, for even in the most powerful telescope 
no star shows a perceptible diameter. 

The determination of a few angular diameters has recently 
been accomplished at the Mount Wilson Observatory by 
means of a method suggested first by Fizeau in 1868, but 
developed by Michclson. If two narrow parallel slits are 
placed over the object glass of a telescope (or anywhere in 
the converging beam of light) and the telescope is set upon a 
star, the light reaching the focal plane of the instrument will 
consist of two pencils which have passed through the two 
slits, and these pencils will be in a condition to produce 
interference. In the focal plane of the eyepiece a series of 
interference fringes, parallel to the direction of the slits, will 
in general be seen. For a certain distance apart of the slits 
these interference fringes will disappear, and mathematical 
investigation shows that this distance (d) is connected with 
the angular diameter (a) of the object by the relationship 
d = 1-22 A/a,* where A is the mean wave-length of the light 

* In deriving this formula it is assumed that the star disc is of uniform 
brightness. If the brightness falls off towards the limb, as in the case of 
the Sun, the numerical constant in the formula requires to bo increased. 
For the law of falling off obtained for the Sun, the constant would be 



from the star. If the slits are placed in the converging cone 
of light their distance apart as projected conically on the 
object glass of the instrument must be used. If d is expressed 
in inches and a in seconds of arc, this relationship beccmes 
approximately a = 5"/d. By this method, an angular diameter 
of 0"-05 might just be measured with a telescope of 100 inches 
aperture, and this is the aperture of the largest existing 

In order to measure still smaller diameters it is necessary 

Fia. 94. Stellar Interferometer as used with the Mount Wilson 
100-inch Reflector. 

to increase the aperture of the telescope. This is effectively 
done, in the case of the Mount Wilson 100-inch reflector, by 
placing a steel girder, 20 feet in length, across the upper end 
of the telescope tube. Attached to this girder are two plane 
mirrors, M 1} Jf 4 , at equal distances from the axis of the 
telescope and inclined at an angle of 45, so that the light from 
a star is reflected by them along the girder to two other plane 
mirrors, M 2 , M 3 , near its centre, 4 feet apart, which in turn 
reflect the light down the tube of the telescope, as shown in 
Fig. 94. The two beams, after reflection from the parabolic 
mirror a, the convex mirror b, and the flat mirror c, unite at 


d and produce interference bands. The distance apart of the 
outer mirrors determines the separation of the two interfering 
beams. To vary the distance, these mirrors can be moved 
along the girder, remaining always at equal distances from 
the axis. The fringes were found to vanish in the case of 
Betelgeuse when the separation of the mirrors was 10 feet, 
corresponding to an angular diameter of 0"-046, and in the 
case of Arcturus, when their separation was 19 feet, corre- 
sponding to an angular diameter of 0"-024. Antares was 
found to have an angular diameter of 0"-040. 

The angular diameters can be calculated approximately 
from theory. The apparent surface brightness corresponding 
to each spectral type is known with fair accuracy from the 
determination of energy distribution in stellar spectra, by 
means of which estimates of effective temperature and surface 
brightness can be made, on the assumption that the stars 
radiate as black bodies. If the total apparent brightness is 
divided by this surface brightness the result is the angular 
area subtended by the star. Since the surface brightness is 
independent of the distance, the value obtained is not depen- 
dent upon a knowledge of the distance of the star. Thus, if 
the surface brightness J is estimated in terms of that of the 
Sun as unit and if D denotes the apparent angular diameter 
of the star, m its visual magnitude, then, since the mean 
angular diameter of the Sun is 32', and the Sun's apparent 
magnitude is 26-7, we have 

Light from Sun : light from star - (2-512) m + 26 ' 7 


D = 1920 J-*(2-512)- 13 ' 35 - " 

= 0"-0088(0-631) J-* approximately. 

Hence D can be determined when m and J are known. The 
angular diameter so determined for Betelgeuse is "-05 1 and 
for Arcturus is "-020, in close agreement with the observed 
values. This agreement lends support to the following table, 
given by Eddington, showing the probable angular diameters 
for giant stars of various types and visual magnitudes. 



Vis. Mag. 






























197. Phenomena associated with Spectral Type of 
Stars. (i) Velocities. For the complete specification of the 
velocity and direction of motion in space of a star, it is necessary 
to know its radial velocity, proper motion, and distance. As 
the distances of relatively few stars are known with sufficient 
accuracy, the relationship between velocity and type can best 
be discussed through the radial velocities, because these are 
not dependent upon the distances and because the effect of 
the solar motion can easily be eliminated. For a single star, 
the radial velocity may be small although the actual velocity 
of the star is very large, but if the mean of a group is taken, 
the mean radial velocity (corrected for solar motion) will be 
proportional to the mean total velocity and so will furnish a 
criterion of relative speeds. 

When this is done, it is found that the average velocity 
increases continually as we pass through the series of spectra 
from the earliest to the latest types. This was first shown 
conclusively by Campbell, from the extensive radial- velocity 
determinations made at the Lick Observatory. His figures 
are as follows : 

Type of Spectrum. 

Radial Velocity, 
km. per sec. 

No. of Stars. 

B . . . 















The velocities given in this table are not corrected for the star 



stream motions, but when the effect of these systematic motions 
is allowed for, the progression in velocity is still shown. 

The proper motions of the stars can be used to confirm the 
reality of this phenomenon. If the proper-motion of each 
star is resolved into two components, one towards the solar 
antapex and the other at right angles to it, the mean parallax 
of a group of stars of any type can be determined from the 
mean motion towards the antapex, as explained in 187. 
Using this mean parallax, the component of proper motion 
at right angles to the direction to the antapex (the cross motion) 
can be converted into linear measure. These velocities should 
be comparable with the radial velocities. In this way, Boss 
determined the following values : 


Cross Linear Motion, 
km. per sec. 

No. of Stars. 
















These values, though naturally somewhat more uncertain 
than those derived from the radial velocities, are in sufficient 
agreement with them to confirm the gradual progression in 

198. (ii) Distances. The mean distance of a group of stars 
of given spectral type depends upon the mean magnitude of 
the group, for, on the average, the fainter a star the more 
distant it will be. In comparing the relative distances of 
stars of different spectral types, it is necessary, therefore, to 
take groups of stars of the same apparent brightness. Then, 
as has just been explained, the mean parallax of the group 
can be calculated from the mean component of the proper 
motion in the direction of the solar antapex. Such investiga- 
tions have been made by Boss, Campbell, Jones, and others ; 
in all cases the mean parallax is found to increase from type 



B A F G K M N 

+ 2 











. -;- 

: : '" 


r> \ 



\ i 


) . 



+ 9 







*X x 





\ *: 

.v . 
\ , 

\ , 

V i 

: : \ 

: .\ 


v :; 



x \ 





: K 

: . \ 


1 Q 




V * 





+ c> 



t \ 



4 a 



FIG, 95. Giant and Dwarf Stars (Russell), 


B to type P and then to decrease gradually to type M, whose 
mean parallax is somewhat greater than that of the early 
B-type stars. On the average, therefore, the stars of type F 
are the nearest to the Sun, those of type B the most remote. 
These results are of importance in connection with the 
general question of stellar evolution. 

199. (iii) Luminosities. When the absolute magnitudes of 
stars are arranged according to their spectral type a remarkable 
relationship is found. If the unit of distance to which the 
absolute magnitudes are referred is 10 parsecs, it is found that 
for stars of type B, practically all the stars fall within a 
magnitude range of 3 m. to + 1 m. (1,600 to 40 times the 
luminosity of the Sun). For succeeding types, the range of 
magnitude gradually increases, the maximum brightness 
remaining about the same, but the minimum brightness 
decreasing. In the case of type M, the range is from about 

2 m. to + 12 m., but the stars are clustered in the neighbour- 
hood of the two limits, there being few if any stars of this 
type with absolute magnitudes between + 2 m. and + 6 m. This 
separation into two classes can be seen, though less distinctly, 
in type K. It was pointed out by Russell that if absolute 
magnitudes as ordinatcs are plotted against spectral types as 
abscissso, the general configuration of the plotted points is 
along two lines, as shown in Fig. 95. There thus appear to 
be two series of stars : the members of one of these are very 
bright with an average luminosity several hundred times that 
of the Sun, and independent of the type of spectrum, whilst 
the luminosities of the members of the other series diminish 
rapidly in brightness with advancing type. These two series 
of stars were called by Russell "giant " and "dwarf " stars 

Other lines of argument have confirmed the reality of the 
distribution of stars into these two classes, and the separation 
is now of fundamental importance in astronomy. 

200. The Velocities of Stars and their Brightness. 

Recent work has shown that in addition to the gradual pro- 
gression with spectral type of the average velocities of the 
stars there is also a progressive increase of velocity with 



diminishing absolute brightness. This result has been indicated 
from discussions of both radial velocities and proper motions, 
and is not connected with the separation into giants and 
dwarfs, since it is found to hold for the giant stars of a given 

The relationship may be illustrated by the stars in the 
immediate neighbourhood of the Sun. It is probable that 
most of the stars within a radius of 5 parsecs are known, and 
the distances of these stars have been measured. That the 
stars of least luminosity have the greatest velocity is shown 
by the following results : 

Luminosity Mean Cross Velocity 
(Sun = 1). (km. per sec.)- 

9 brightest stars .... 48-0 to 0-25 29 

10 faintest stars .... 0-10 to 0-004 68 

The radial motions of these faint stars have also been found 
to be more than twice that of the brighter stars of their type. 

It has been suggested that the progressions of speed with 
type and with velocity may be in reality a mass phenomenon. 
If the stellar universe is in a steady state, the average kinetic 
energy of each star would be the same, as in the case of the 
molecules of a gas in thermal equilibrium. The stars of largest 
mass would therefore have the smallest velocity, and vice 
versa. It is known that the average mass of the B-type stars 
is several times that of the stars as a whole, and from this 
view-point their small velocity is to be expected. To account 
for the progression with luminosity, it is only necessary to 
assume that the stars of small mass do not attain to as great 
a luminosity as those of large mass. 

Although it is not probable that the Universe is in what 
is termed, in the kinetic theory of gases, a steady state 
of statistical equilibrium, it is probable that it is sufficiently 
near to such a state for the law of equipartition of energy 
according to which the average kinetic energy of translation 
of each star should be the same to hold approximately. The 
progression of speed with type and with mass then follow 
as a rational consequence. 


201. Double Stars. Many stars which to the naked eye 
appear single are found when examined in a telescope to be 
double ; as well-known instances of double stars may be 
mentioned Castor and 61 Cygni. The existence of such stars 
has been known since the seventeenth century, but the first 
systematic search for doubles was carried out by Sir William 
Herschel commencing in 1779. To-day many thousands of 
such stars are known. In fact, down to about the ninth 
magnitude, about one star in eighteen is found to be a double. 

Stars may appear double from two causes : (i) the two 
stars may be at very different distances but nearly on the 
same line of sight, in which case there is no physical connection 
between them : such pairs of stars are termed optical doubles, 
(ii) They may be at the same distance and physically con- 
nected, revolving about their common centre of gravity under 
the action of their mutual attraction. Such pairs are termed 
binary systems and are the ones of interest to the astronomer. 

Herschel had supposed that the double stars discovered by 
him were only optically double, and his object in observing 
them was to use the relative parallactic displacements of the 
two components to determine the solar motion. It was shown 
by Michell, in 1784, from considerations of probability, that 
some at least of HerscheFs pairs must be physically connected. 
If the stars visible to a good eye were distributed over the 
celestial sphere at random, the probability against two or 
three pairs being so nearly in the same direction that they 
would appear single to the naked eye would be very great, 
and the chance that any of these would be as close as some 
of Herschel's closer pairs would be so small as to be negligible, 



In any particular instance, only by observations extending 
over a period of time is it possible definitely to decide whether 
a double star is a binary system or merely an optical double, 
although, in many cases, the physical connection of the two 
components can be regarded as probable. The nearer the 
two stars and the brighter they are, the greater is the prob- 
ability that they constitute a binary system. Of the 20,000 
or so double stars which have been catalogued, it is not possible 
at present to state how many are binary systems, but in general 
it may be assumed that if the pair is wide and shows appreciable 
relative motion it is probably optical, whilst if the relative 
motion is much smaller than the proper motion of the pair, 
or if the stars are less than 5" apart and as bright as the ninth 
magnitude, they are probably physically connected. 

In the case of stars which are physically connected, the 
period over which observations must be extended in order 
that the relative orbital motion may be detected will be greater 
the larger the angular separation of the two components. If 
the separation is greater than 2", relative motion will probably 
not be detected with certainty in a century. 

202. Measurements of Double Stars. The two quantities 
which it is necessary to measure in connection with a double 
star are the angular separation of the components and the 
position angle, which determines the direction of the great 
circle on the celestial sphere passing through the two stars. 
Both these quantities can be determined with the lilar micro- 
meter. For accurate observations, good atmospheric seeing 
and acuity of vision are essential. Since the resolving power 
of a telescope of " d " inches aperture is approximately 5/d 
seconds of arc, it follows that for the measurement of pairs 
with separations of 1", 0"-5, 0"-2, telescopes of apertures of 
at least 5, 10, and 25 inches respectively are required. 

The interference method used for the determination of the 
angular diameters of stars ( 196) can be applied with advantage 
to the observation of close double stars. If the stars are of 
the same brightness, the interference fringes produced in the 
focal plane by the double slit disappear for an appropriate 
separation, Z, of the slits provided that the line joining the 
stars is at right angles to the slits. The distance apart of the 


slits for which the fringes disappear is connected with the 
angular separation, a, by the relationship a -J-A//, A being 
the mean wave-length of the light. The method consists, 
therefore, in determining the orientation and separation of 
the slits for which the fringes disappear. The former quantity 
determines the position angle of the pair, the latter the angular 
separation. If the two stars are not equal in brightness, the 
fringes do not completely disappear for any orientation of the 
slits ; there are then positions of maximum and of minimum 
visibility, and it is the latter which must be observed. This 
method possesses the advantages that it is practically inde- 
pendent of the quality of the atmospheric definition at the 
time of observation, and that it is greatly superior in accuracy 
to the filar micrometer method. It will also be noticed that 
for a telescope of aperture d, the smallest angular separation 
which can be measured is f A/VZ, whereas by ordinary methods 
the least angle measurable (corresponding to the limit of 
resolution of the telescope) is 1-22 A/cZ. The resolving power 
is in effect increased by this method of observation in the 
ratio of about 2-44 to 1. With a 25-inch telescope, stars with 
separation down to 0"-08 are measurable as compared with a 
limit of 0"-2 with the filar micrometer. 

203. Orbits of Binary Stars. According to the law of 
gravitation, each component of a binary system must describe 
an elliptical orbit about the common centre of gravity as one 
focus. The two ellipses, and also the elliptical orbit of the 
one star relative to the other, are precisely similar, differing 
only in linear dimensions. The orbit described by the smaller 
star is larger than that described by the other in the inverse 
ratio of their masses. The major axis of the relative orbit 
is equal to the sum of the major axes of the two real orbits. 

In general, the plane of the relative orbit is inclined to the 
line of sight so that observation gives only the projection of 
the orbit upon the celestial sphere. The projected relative 
orbit in such a case will still be an ellipse, but the larger star 
will no longer be at its focus ; the projections of the major 
and minor axes of the real orbit will not be at right angles 
and will not be the major and minor axes of the projected 
orbit. Since, however, equal areas project as equal areas, 


the smaller star will still describe equal areas about the larger 
star in equal times in the projected orbit. 

Using the knowledge that the larger star is in the focus of 
the real relative orbit, it is possible to determine theoretically 
the shape of the orbit, its angular dimensions, and its inclina- 
tion to the line of sight as well as the period in which the 
orbit is described. Three observations, if free from error, would 
suffice to determine the orbit. In practice, a greater number 
must be used in order to eliminate observational errors. 
Generally speaking, the position angles in the case of close 
pairs are more valuable than the separation, and the values 
of the angles at various epochs suffice to determine the shape 
of the orbit. 

The orbits of about eighty double stars are known with a 
fair degree of accuracy. 

204. Masses of Double Stars. If raj, m 2 are the masses 
of the two components of a double star in terms of the Sun's 
mass, P the period in years, a the semi-axis major in seconds 
of arc, co the parallax in seconds of arc, then Kepler's laws 

(m l + m 2 )P 2 co 3 = a 3 . 

If, therefore, the parallax of a double star whose orbit has 
been computed can be determined, it is possible to deduce 
the mass of the system. It is only for a small number of 
systems that all the required data are known. For fourteen 
such systems, the masses are found to vary from 0-45 to 3-3 
times the mass of the Sun, with an average of 1-8. This result 
is in accordance with other lines of evidence which indicate 
that the range in mass of the stars is not very great. 

If, then, it be assumed that the total mass of the system 
is twice that of the Sun, the above relationship becomes 
2P 2 co 3 = 3 , and the parallax can be deduced when the 
elements of the orbit are known. It should be noted that in 
determining o>, the mass enters as a cube root, so that if the 
mass of the system were sixteen times instead of twice that 
of the Sun, the value deduced from the formula 2P 2 co 3 = a 3 
would be only double the actual value. This method is very 
useful for determining theoretical or' 'hypothetical "parallaxes 
of double-star systems. 



On the assumption that the combined mass of the system 
is double that of the Sun, the following table has been 
compiled giving the approximate periods of binary stars 
for different angular semi-axes ma j ores and parallaxes : 

Revolution Period for Parallax. 








2J years 

8 years 

90 years 

250 years 





















The method just explained for determining the hypothetical 
parallax of a double-star system whose orbit is known can be 
extended to the case of any double star which shows appreciable 
relative angular motion, but which has not completed an arc 
sufficiently large to enable the orbit to be computed. 

If co is the parallax, d the mean separation, and w the mean 
angular motion in degrees per year, then 

o> = k . dw 2 /* 

where k is a constant whose value depends upon the precise 
assumptions made, but which may be taken as 0-022 without 
serious error. 

Thus, in the case of the star No. 4,972 in Burnham's General 
Catalogue, the position angles and separations at two epochs 

1830 47-5 20-4 

1914 68-5 18-9 

The angular motion is 21 in 84 years, so that w = . Also 
d 19-65". Hence co = Tc X 19-65/ 3 y'16 and the hypotheti- 
cal parallax is 0"-17. The trigonometrically determined value 
is 0"-15. In this way, reliable parallaxes of many double stars 
may be determined. 

205. Spectroscopic Binaries. Some double stars are so 
close that it is not possible to separate them visually with 


any existing telescope. The duplicity of many such stars 
can be detected with the spectroscope. Suppose the two com- 
ponents are of the same spectral type and that the orbital 
plane is edgewise to the observer. Then, when the line joining 
the stars is perpendicular to the line of sight, one of the stars 
will be moving towards and the other away from the observer. 
The lines of the spectra of the two component stars will be 
displaced, according to Doppler's principle, in opposite direc- 
tions, and the lines of the resultant spectrum which is obtained 
by superposing the two component spectra will therefore 
appear double. On the other hand, when the two components 
are in the line of sight, the lines will appear single. 

If the two components of the star are not of the same spectral 
type, the spectrum will be more complex, but the displacements 
of the two sets of lines can still be detected. Double stars 
whose components are so close that they cannot be separated 
visually, but whose duplicity can be detected spectroscopically, 
are termed spectroscopic binaries. 

The first spectroscopic binary to be discovered was the 
brighter component of the double star Mizar in the Great 
Bear. Pickering in 1889 found that the dark lines in its 
spectrum appeared double at regular intervals corresponding 
to a period of 20| days. 

The number of spectroscopic binaries which have been 
discovered amounts to many hundreds and is rapidly increasing. 
They have been found principally amongst the blighter stars 
because these are the stars whose spectra are most easily 
obtained. Those of faint stars, on a scale suitable for radial- 
velocity determination, require large instruments and long 
exposures. Of the stars examined, Campbell states that at 
least one in live is found to be a spectroscopic binary, and there 
is no reason why this should not hold for the fainter as well 
as for the brighter stars. For some spectral types, the pro- 
portion seems to be even greater ; thus, for class-B stars, 
about two stars in five are spectroscopic binaries. 

206. Orbits of Spectroscopic Binaries. If a photograph 
of the spectrum of a spectroscopic binary is obtained on a 
sufficiently open scale along with a suitable comparison 
spectrum, the displacements of the lines can be measured 


directly and the velocity in the line of sight of one or, in some 
cases, of both components determined. 

Theoretically, five such observations at different points of 
the orbit would suffice to determine the orbital elements, but 
practically the number of observations must be considerably 
increased to allow of the elimination of the accidental errors of 
observation. It may then happen that the observations have 
extended over more than one revolution of the components, 
since the periods of spectroscopic binaries are much shorter 
than those of visual doubles. The observations enable a 
provisional value of the period to be determined, by means 
of which they may all be reduced to a single revolution. A 
curve can then be drawn to represent the velocity at any 
epoch during the revolution. 

Since the radial velocity is a maximum or minimum at the 
two nodes of the orbit, i.e. at the points of intersection of the 
orbit with a plane perpendicular to the line of sight, the positions 
on the velocity curve of these points can be assigned. By a 
simple mathematical procedure it is further possible to deduce 
the positions of apastron and periastron, relative to the node 
of the eccentricity, and of the quantity a sin i ; a is the semi- 
axis major of the orbit expressed in kilometres and i is the 
angle between the line of sight and the normal to the orbital 
plane. Neither a nor i can be separately determined from the 
observations of radial velocity. Observations of visual doubles, 
on the other hand, determine both a and i, a being found, 
however, in angular and not in linear measure and the sign of i 
remaining undetermined, i.e. the observations cannot dis- 
tinguish between the two planes, which make an angle i with 
the plane perpendicular to the line of sight. The longitude 
of the node cannot be determined for spectroscopic binaries. 

It is possible in the case of a few systems to obtain both 
visual and spectroscopic observations. In such cases the 
sign of i can be determined and also the actual linear value of a. 
Since a is then known in both angular and linear measure, the 
parallax of the system can be deduced. If, in addition, the 
velocity of both components can be observed spectrographically 
the ratio of their masses can be determined, the masses being 
inversely proportional to the amplitudes of the two velocity 
curves. From visual observations, the total mass can be 


determined when the parallax is known and the values of 
the separate masses can therefore be obtained. A complete 
knowledge of such systems can therefore be obtained. 

In this way W. H. Wright determined the parallax of a 
Centauri, which has been observed both visually and spectro- 
scopicaliy as a double, and obtained a value 0"-76 in exact 
agreement with the heliometer determination by Gill. The 
masses of the two components are each nearly the same as that 
of the Sun. The components are separated at periastron by 
eleven astronomical units and at apastron by thirty-five units. 

207. Triple and Multiple Stars. Many stars which were 
at first thought to be double are now known, to be more complex 
in nature, and the number of triple and multiple systems is 
steadily growing. In some cases the additional components 
can be detected spectroscopically ; in others their existence is 
inferred indirectly. A good example of a multiple system is 
the well-known visual double, Castor. Each component 
is now known to be a spectroscopic binary ; the period of one 
is nearly three days and of the other rather more than nine days. 
The period of the one binary about the other is somewhat 
uncertain, but is of the order of 300 years. 

The North Star, Polaris, provides an interesting example of a 
triple system. It was first found to have a variable radial 
velocity and then proved to be a spectroscopic binary with a 
period of nearly four days. The velocity of the centre of mass 
of the visible system is, however, subject to a slow variation, 
indicating that the binary is attracted by and is moving around 
a third star which is invisible to us. The orbit is eccentric and 
the period of this motion exceeds twenty years. 

Sometimes it is the smaller or fainter star of a pair which is 
a close binary. As an example may be mentioned the star 
40 Eridani. This star consists of a bright component of 
magnitude 4-5 with a faint component of magnitude 9-2, 
separated from it by a distance of 82". The faint component is 
itself a visual double with a period of 180 years and the smallest 
eccentricity of any known visual double. Both bright and 
faint stars have a large proper motion, which indicates a 
physical connection between them, but they show very little 
relative motion, as might be anticipated from the wide 


separation. The period of revolution of the binary about the 
primary is of the order of 7,000 years. The orbit of the faint 
components is nearly as large as Neptune's, whilst that of the 
faint about the bright star is about 470 astronomical units. 
The system is therefore somewhat similar to but on a much 
greater scale than the Earth-Moon-Sun system. 

Many other instances of such complex systems are known. 
They are of interest from the information which they enable us 
to obtain as to masses, luminosities, etc., of various stars ; 
information which is of value on account of its bearing upon the 
general question of stellar evolution. 

208. Variable Stars. Another class of stars of great 
interest and importance are the stars whose light is variable. 
The number of known variable stars is several thousands and is 
being added to continually. Many of these stars vary in a 
very irregular manner ; others, on the other hand, exhibit a 
remarkable constancy in the period of their variation, so that 
the maxima and minima of brightness can be predicted with 
certainty beforehand. The periods of the light variations 
range from a few hours to several hundred days ; on the average 
the greatest variations in brightness occur with the long-period 

There are so many different types of variation and so many 
different features present themselves from one star to another 
that it is necessary, in order to obtain a broad view of the 
problems presented by variable stars, to divide them into 
several main classes. Various systems of classification have 
been adopted, but that suggested by Pickering is perhaps the 
simplest for our present purpose. He divided variable stars 
into the following classes : I, Novae or temporary stars ; 
II, long-period variables ; III, variables of small range or 
irregular variation ; IV, short-period variables ; V, eclipsing 

The characteristics of these several classes will be considered 
briefly in the following sections. 

209. Novae or New Stars. A star to which the name 
of nova or new star is applied is one which experiences one 
sudden and usually considerable increase in brightness, after 
which its light diminishes at first somewhat rapidly, and then 


more slowly, to a more or less steady value. No instance is 
known of the same star experiencing two such outbursts. The 
characteristic features of the light changes of a nova are the 
rapid and large increase of the brightness to a maximum, and 
the more gradual falling away, accompanied by numerous 
small and irregular oscillations. These features may be 
illustrated by the two most recent novse, Nova Aquike III, 
1918, and Nova Cygni III, 1920. 

Nova Aquilse III before its outburst was a faint star of 
magnitude between 10 and 11, which showed irregular varia- 
tions in light with an amplitude of about one magnitude. It 
is not known whether this is a characteristic feature of the 
early history of iiovao, for, in general, such history can only be 
investigated after the outburst has taken place by finding 
earlier photographs on which the nova is shown. A photo- 
graph of the region around Nova Aquilso was obtained at 
Heidelberg on 1918, June 5, and the star was then of magnitude 
10-5. On a photograph obtained at Harvard on June 7, it 
appears of the sixth magnitude. On the following evening, 
when it was discovered by several observers independently, it 
reached almost the first magnitude. The next evening (June 9), 
it was with the exception of Sirius and Canopus the brightest 
star in the sky ( 0-5 m.). Its increase in brightness in a 
period not exceeding four days was therefore about 2500 : 1. 
The light then commenced to decrease ; by June 17, its 
magnitude had fallen to about 2 ; by June 22, to 3. 
At the end of June, the irregular oscillations in brightness 
commenced, these being superposed upon a progressive fall in 
mean brightness. By the end of the year the brightness had 
decreased to about 6 m. It is now fainter than 10 m. 

Nova Cygni III was discovered by Denning on 1920, August 
20, its magnitude being then about 3-5 m. Little is known of 
its history before the outburst, but the star does not appear on 
earlier photographs reaching to the fifteenth magnitude, and 
it must therefore have been extremely faint. Photographs of 
the region around it were obtained at Harvard on August 9, 
and on August 20. On the earlier photograph, going down to 
9-5 m., the star does not appear. On the later, taken the night 
before its discovery, it was of magnitude 4-8 m. It was 
photographed in Sweden on August 16, and was then of 



magnitude 7-0 m. On August 22, it had attained a magnitude 
of 2-8 m., and on August 24 it reached a maximum of about 
2 m. Thus, although its maximum brightness was inferior to 
that of Nova Aquilae, the increase in brightness was consider- 

10 15 SO 25 30 5 10 15 20 25 30 S 10 15 

Aug.1920 Sept. Oct 

Fie. 9(>. The Light Curve of Nova Cygni (1920). 

ably greater. The decrease in brightness was much more 
rapid. By the end of August, the magnitude had fallen to 4 m. 
and oscillations had started, and by the end of October it was 
fainter than 9 m. The light curve is shown in Fig. 96. 

Whether the final magnitude of a nova after its outburst 
becomes in general equal to its mean magnitude beforehand is 
uncertain, though it is known that in some cases this is not so. 
Thus, Nova Corome, 1866, before its outburst was of magnitude 
9-5 m. and is now of magnitude about 11*5 m. 

Although these two examples illustrate the usual course of 
light changes in novae, there are exceptions. Thus, the star 
P Cygni, when discovered by Jansonin 1600, was of the third 
magnitude. In 1602 the star was observed by Kepler and was 
still bright, but in 1621 it had become invisible to the naked eye. 
In 1655 it again attained the third magnitude, but vanished 
in 1660, and in 1665 was again visible, though fainter. Since 
1677 its brightness has remained constant at about 5 m. 
Somewhat similar changes occurred in the case of the Nova II 
Vulpeculse, discovered by Anselm in 1670. 

The brightest novae on record are Nova Cassiopeiae, dis- 


covered by Tycho Brahe in 1572, which reached 4 m., 
brighter even than Venus ; and Nova Ophiuclii, discovered by 
Kepler in 1604, which reached 2m. Both these stars are 
now very faint and cannot be identified. The next brightest 
novae appeared in recent years, Nova Persei, 1901 (0-0 m.) and 
Nova Aquil0e, 1918 ( 0-5 m.). The majority of the novae 
which have been observed have been situated in or near the 
Milky Way. 

210. Spectral Changes of New Stars. The changes in 
brightness of a new star are accompanied by remarkable 
changes in the spectrum of an exceedingly complex nature, the 
explanation of which is not at present fully understood. The 
spectrum at an early stage of the outburst is characterized 
by absorption lines of hydrogen and calcium, which are strongly 
displaced towards the violet ; this displacement is similar in 
nature to a Doppler displacement, but if interpreted in this 
manner, the velocities involved are very great and variable 
from day to day. In the case of Nova Aquiloe III, the dis- 
placements corresponded to velocities of about 4,500 km. per 
second. Most of those absorption lines are accompanied by 
faint broad emission lines which are displaced but little from 
their normal positions. The absorption lines gradually become 
more numerous and prominent and are due mainly to enhanced 
spark lines of titanium and iron. A few days later the 
absorption lines become double, the displacements of the two 
components corresponding in the case of Nova Aquilae to 
velocities of about 1,400 and 2,100 km. per second ; the 
continuous background of the spectrum at the same time 
becomes weaker and the bright bands broaden. The spectrum 
is at this stage very complex and is somewhat similar to a 
superposition of spectra of types B and A, displaced relatively 
to one another. The B-type spectrum gradually becomes more 
prominent and the A type less prominent, the latter finally 
disappearing. Nebular lines now begin to appear and the 
hydrogen lines to become less prominent, the spectrum being a 
bright band spectrum, due mainly to hydrogen, helium, and 
nebulium. The hydrogen lines continue to become fainter 
and the spectrum reaches a stage when it is practically nebular. 
At a later stage, the hydrogen and helium lines become stronger 


again and the spectrum assumes the typical features of the 
spectra of Wolf-Rayet stars. 

The details vary considerably from star to star and the 
changes are so rapid and complex that it is a difficult problem 
to elucidate their meaning. Nevertheless rapid progress has 
been made in the case of recent novae which should in the 
future throw much light upon the physical causes responsible 
for the outburst. 

211. Theories of New Stars. Many theories have been 
put forward to account for the various phenomena presented by 
the novae. Whilst none of these theories can at present be 
regarded as satisfactorily proven, the balance of probability 
is strongly against some of them, which we can therefore 
disregard here. There are really two problems requiring 
solution ; the cause of the original outburst and the nature of 
the subsequent occurrences which are responsible for the 
complex changes evidenced by the spectra. 

The doubling of many of the lines in the spectra of novoa 
naturally suggested the theory that two stars were concerned 
in the production of the outburst. The two stars were supposed 
on this theory either to have collided directly or to have 
approached so near to one another that enormous disturbances 
of a tidal nature were set up ; such a theory could account for 
the suddenness of the outburst and its more gradual decay. It 
must be supposed that one of the stars has an absorption 
spectrum and the other a bright lino spectrum ; also that the 
former star is moving towards the Earth and the latter away. 
The frequent occurrence of novae in the Milky Way would then 
be attributed to the greater density of stars there. On this 
theory we must suppose that in all the novae which have been 
spectroscopically studied, one of the stars involved possessed 
a bright line spectrum, and was receding from the Earth with a 
small velocity, and that the other possessed an absorption 
spectrum and was moving towards the Earth with a large 
velocity. It is extremely improbable that chance encounters 
of stars could have produced results obeying such a clearly 
marked law. In addition, the density of stars in space is so 
small that it is improbable that collisions between two stars 
would occur as frequently as novae are observed. 


A more plausible theory attributes the outburst to the 
passage of a dark or feebly -luminous star through a mass of 
nebulous matter. The star is heated by frictional resistance 
to incandescence, on entering the nebula, and the composite 
absorption and bright line spectrum would be anticipated. 
The displaced absorption lines are on this theory diie to 
expanding and cooling gases moving out from the centre of the 
disturbance, and the displacement would naturally correspond 
to a motion towards the observer. If successive streams of 
matter are emitted, the double, or in some cases the triple 
structure of the lines, could be accounted for. The alternations 
between bright bands and strong absolutions would be expected, 
the absorption appearing when the stream was pointed towards 
the observer. If the star were a binary system certain 
additional peculiarities of some novtc would be accounted for. 
This theory is to some extent directly confirmed by observation ; 
both in the case of Nova Persei IF and Nova Aquiioe III, the 
star was found some time after the outburst to be surrounded 
by a disc of nebulous matter of appreciable angular diameter. 
This may be the glowing envelope formed by the ejected 
matter. When this envelope becomes the chief seat of the 
radiation, the spectrum would change to a bright line spectrum. 
We have also seen that the final stage of a nova spectrum is 
similar to that of a Wolf-Rayet star. It is significant that 
these stars are usually found in association with nebulosity 
and that they occur (apart from the Magellanic Clouds) only in 
the Milky Way. Collisions between stars and nebulic would 
naturally be much more frequent than between two stars and 
would occur mainly in the Milky Way. It may, therefore, be 
not improbable that all the Wolf-Rayet stars known are merely 
novse in their late stages. In the case of Nova Persei II, a 
nebula was certainly associated with the outburst, for bright 
nebulosity was seen to be moving out from it and was inter- 
preted as the brightening of a dark nebula by light spreading 
out from the original outburst ; the distance of the nova calcu- 
lated on this assumption agreed with that directly determined 
by the ordinary methods. 

212. Long-period Variables. When variable stars are 
classified according to their period, it is found that there are 


a large number with periods of less than 11 days and a large 
number with periods between 150 and 450 days, but that only 
a relatively small number have periods between 11 and 150 
days. A fairly definite subdivision into two classes according 
to period is therefore possible with some overlapping for 
periods between, say, 50 and 150 days. Long-period variables 
are those with periods exceeding 150 days. 

There are several well-pronounced characteristics of long- 
period variation. The range of variation is large, usually 
from three to eight magnitudes ; the period and the magnitude 
at maximum are usually somewhat irregular ; the stars are 
reddish in colour, and the redder the tint, the longer is the 

The best known long-period variable is Mira or o Ceti, 
discovered by Fabricius in 1596. This star has a mean period 
of 333 days, which is subject to large variations. Its brightness 
varies from the second to the ninth magnitude. The magnitude 
at minimum varies from about 8-5 m. to 9-6 in., but the value 
at maximum is still more irregular, as the following table 
indicates : 

1868 . . . .5-2 mag. 

1869 .... 3-9 
1875 .... 2-5 
1879 .... 4-2 

1885 .... 2-8 

1886 .... 5-0 mag. 

1896 .... 4-0 

1897 .... 3-2 

1898 .... 2-4 
1900 .... 3-4 

The rise to maximum brilliance is more rapid than the 
decline. The spectrum is of type Md, with bright hydrogen 
emission lines. The radial velocity observed near maximum 
does not vary, indicating that the star is not a spectroscopic 
binary, but the value given by the dark lines is 62 km. per 
second away from the observer, whereas that given by the 
hydrogen lines is only 48 km. per second. This suggests 
that the increase in brightness is due to outbursts of hydrogen 
gas occurring with approximate regularity. 

The spectra of long-period variables are all of classes Ma, 
Mb, Me, Md, or N ^ no star with spectrum of type Md is known 
which is not a long-period variable. 

The periodic outbursts occurring in the long -period variables 
have been compared with the outbursts of activity on the Sun, 




evidenced by the Sun-spot cycle and allied phenomena. It has 
been suggested that the Sun should be regarded as a long-period 
variable, with a very long period and a small range in brightness; 
the rise to maximum spot activity in the case of the Sun is 
more rapid than the subsequent decline, so that in tliis respect 
the analogy holds. On the other hand, no long-period variable 
is known with so long a period, so small a range of brightness, 
and a spectrum of such early type. The comparison seems 
therefore to be somewhat misleading. 

213. Special Long-period Variables. There are a few 
long-period variables whose behaviour is somewhat different 
from that described in the preceding section. These were 
assigned by Pickering to sub-classes 111) and lie. The first of 
these includes U Geminorum, SS Cygni, SS Aurigyo. As a typical 

Interval varies 
from about 40 to 


Day sO 

4 Q 12 16 20 4 8 12 

FIG. 97. The Light Curve of U Geminorum. 

example U Geminorum, discovered by Hind in 1855, may 
be considered (Fig. 97). The normal state of the star is one of 
constant (minimum) brightness, about 13 m. ; from time to 
time its light increases very suddenly to a maximum of 9-5 m., 
which does not last for any regular interval of time and is then 
followed by a more gradual fall to its constant minimum value. 
Two distinct types of maximum are known and occur alter- 
nately ; they are called the long and the short, the star remain- 
ing above minimum brightness for about twelve and twenty 
days respectively. Successive outbursts occur at irregular 
intervals which may vary from 60 to 152 days. In the case 
of SS Cygni, there is even a third type of maximum. 

Owing to the faintness of these stars, little is known about 
their spectra or radial velocities and no plausible theory has 
been advanced to account for the phenomena. 


The sub-class lie includes the stars R Coronae Borealis, 
RY Sagittarii and SU Tauri. R Coronac Borealis is normally 
of about the sixth magnitude. At irregular intervals, which 
may be months or years, its light decreases, passes through a 
minimum value, and finally attains again its normal brightness. 
The variation may range from as much as 9 m. to as small as 
1 m., and its duration from a few years to several months. 
The decrease in brightness is usually more rapid than the 
subsequent increase. These changes suggest that the light 
from the star is from time to time obscured by an absorbing 
medium passing between the star and the observer, which may 
have some physical connection with the star. 

214. Irregular Variables. Class III in the classification 
is a large one comprising stars whose variation is so irregular 
that no period can be assigned to them. The variation in 
brightness of stars of this class is usually small, averaging less 
than 2 m. It comprises stars of all spectral types from G to N, 
including many stars having peculiar spectra. The cause of the 
variation is not at present known, though it is not impossible 
that in these red stars there may be a crust forming over the 
surface. As the crust forms the light diminishes. But a 
certain stage is reached at which the pressure of the imprisoned 
gases becomes so great that they break through the crust with 
more or less violence, and with corresponding increase in 
brightness. Such occurrences would naturally take place at 
somewhat irregular intervals, but, averaged over a sufficiently 
long period, the mean interval may be expected to remain 
fairly constant. 

215. Short-period Variables. Cepheid Variables. The 
stars of Classes IV and V are all short-period variables and 
are characterized by a small range of variation and a perfectly 
regular period. These variables can be divided into two 
main groups stars which are known to be binary systems and 
to be variable on account of one component periodically 
eclipsing the other (Class V), and stars whose variation cannot 
definitely be attributed to eclipses (Class IV). The latter will 
be considered first. 

The short-period variables belonging to Class IV are generally 



called Cepheid variables from the typical star <5 Cephei (Fig. 98). 
The range of variation is small, generally less than 1 m., and the 
periods range from less than one day to one or two months. 
d Cephei has a range of about 0-7 m. and a period of nearly five 





Days r 2 3 4 5 6 7 

FIG. 98.The Light Curve of 5 Cephei. 

and a half days. The rise from minimum to maximum occurs 
in about one and a half days, and is, therefore, much more 
rapid than the decline to minimum which occupies the remain- 
ing four days of the period. The decline is not uniform, but is 
accompanied by secondary oscillations. Some stars belonging 
to this group, such as f Geminorum, have a practically sym- 
metrical light curve. 

The Cepheid variables have spectra of all types, though types 
F and G are most common. This seems to indicate that the 
cause responsible for the light variation is independent of the 
physical state of the star. From the positions of the spectral 
lines, it is found that all these stars have variable radial velo- 
cities, the period of this variation being equal to that of the 
light variation. The two phenomena are therefore intimately 
connected. Supposing for the moment that these stars are 
binary systems, the maximum light is found to occur always 
approximately at the time when the brighter component (whose 
spectrum alone can be observed) is approaching us most 
rapidly, and the minimum when it is receding most rapidly. 
The constancy of the period of a Cepheid variable and the 



variation of the radial velocity with the same period as the 
light variation suggest that such stars are double, consisting 
of two components whose orbital plane is inclined to the line 
of sight at a sufficiently small angle for eclipses to occur. If 
the two components are of equal brightness, a symmetrical light 
curve of the f Geminorum type would be obtained (Fig. 99) ; 




DaysO Z 4 6 8 10 

FIG. 99. Tho Light Curve of Gominorum. 

if the orbit is markedly eccentric an unsymmetrical light curve 
would bo obtained, the minimum not occurring midway 
between the two maxima, as in the normal d Cephei type. On 
such a theory it would be expected that the light minimum 
should occur at conjunction when one star is obscured by the 
other and not at quadrature, and further, the light curve should 
show straight stretches, since at quadrature there would be 
no light variation. This forms a difficulty in the interpretation 
of the phenomena as due to eclipses which has not been success- 
fully overcome ; one explanation has been advanced which is 
based upon the hypothesis that a resisting medium exists 
around the two stars ; this would brush back the atmosphere 
on the advancing side of the bright star and make it appear 
brightest when approaching most rapidly ; another explanation 
supposes large tidal effects at periastron. But there is a still 
more serious difficulty in the way of the binary interpretation ; 
the Cepheids have been found to be stars of great absolute 
brightness, several hundred times that of the Sun. The 
average Cepheid has therefore a volume between fifteen and 
twenty thousand times as great as that of the Sun, since the 


surface brightness cannot be very different from that of the Sun. 
Now, adopting the binary hypothesis, it is possible, as in the case 
of a spectroscopic binary, to deduce the value of a sin i for the 
orbit ; the average value of this quantity found for 15 Cepheids 
is 1,116,000 km., with a maximum of 2,000,000 km. The 
value of i cannot be very small or eclipses would not be observed : 
it follows that, interpreted as binaries, the radii of the orbits 
are less than one -tenth the radii of the stars themselves. A 
further difficulty is that the spectral type shows a continuous 
change throughout the variation ; thus in the case of RS Bootis, 
period about nine hours, the type changes from FO at minimum 
to B8 at maximum. The interpretation of this on a binary 
hypothesis is not clear. 

It appears therefore that an alternative explanation must be 
sought. The most plausible explanation yet advanced is that 
the variations of radial velocity are due to periodic pulsations 
of the star and that these pulsations cause changes in the rate 
of emission of light. The greatest light would occur when the 
rush of hot gases from the interior was greatest and would 
therefore coincide, at least approximately, with the maximum 
velocity of approach. The change in the temperature of the 
star's atmosphere would coincide with a change of spectral 
type, and would take place when the star was hottest. At 
maximum brightness the spectrum would be of an earlier type 
than at minimum, when the star would be coolest. This is 
in accordance with observation. The theory also accounts 
naturally for the absence of a second spectrum. 

This theory therefore provides an adequate explanation of 
the observed phenomena. The question arises as to how the 
pulsations are maintained from whence comes the energy 
necessary to maintain pulsations with a total amplitude of the 
order of 2,000,000 km. This is a question which has not yet 
been satisfactorily answered ; until such an answer is forth- 
coming the pulsation theory will not command universal 
acceptance, even though no equally plausible theory has yet 
been advanced. 

216. The Luminosity-period Relation for Cepheids. 

A remarkable relationship was discovered by Miss Leavitt 
between the absolute magnitude and length of period for 



Cepheid variables. In the Small Magellanic Cloud many of 
these stars have been found. The distance of the Cloud is so 
large compared with its linear dimensions that all the stars in it 
may be assumed to be at the same distance without appreciable 
error. Their apparent magnitudes therefore differ from their 
absolute magnitudes merely by a constant, which depends upon 
the distance of the Cloud. Miss Leavitt found that for the 
Cepheid-type variables in the Cloud there was a definite 
relationship between the period of the light variation and the 
apparent magnitude of the star from which the period of any 
Cepheid variable in the Cloud could be deduced if its apparent 
magnitude had been determined. 

In many stellar clusters occur variable stars which are 
usually termed cluster variables ; the variation of these stars is 
essentially of the Cepheid type, though they include stars with 
periods much longer than are found amongst the group of stars 
to which the name Cepheid variable was originally applied. 

FIG. 100. 

+ 1-8 +1-4 +1-0 +0-6 +0-2 -0-2 ~0'6 
Log Period* 

The Luminosity-period Relationship for Cepheid Variables. 

Shapley found that such variables in any one cluster con- 
formed to the same type of relationship connecting period and 
magnitude as the variables in the Magellanic Cloud, and it is 
logical, therefore, to assume that it is a universal characteristic 
of Cepheids. From a study of the near Cepheids, Shapley was 
able to assign an absolute magnitude to a definite period and 


hence to fix a point on the curve enabling all the material 
available to be reduced to a common basis and a definitive curve 
connecting the absolute magnitude and the period to be con- 
structed. This curve is shown in Fig. 100. 

This relationship is of great importance. If the period of any 
Cepheid variable is observed, its absolute magnitude can be 
deduced. It is then only necessary to determine its apparent 
magnitude in order to obtain its distance. The accuracy of 
the distances so obtained far surpasses that of direct trigono- 
metrical determinations and enables the distances of the 
Magellanic Cloud and of various stellar clusters to be 
derived with certainty. 

217. Eclipsing Variables. The last class of variable 
stars to be considered comprises the large number of stars 
which are definitely known to be binary systems, the variation 
in light being due to one component eclipsing the other. The 
extent of the eclipse and the corresponding variations in bright- 
ness depend upon the brightness of the two components, their 
relative size, and the inclination of their orbital plane to the 
line of sight. If this inclination is large, eclipses will not be 
visible to an observer on the Earth. The circumstances of the 
light variation, in fact, enable very definite conclusions to be 
drawn as to the nature of the system. 

The light curves may be divided into four main classes : 

(i) There may be a series of equal minima, occurring at equal 
intervals. Such a curve could be produced by a system 
containing one dark body and one bright body, or two bright 
bodies of equal size and luminosity. If the orbit is not circular 
but elliptical, the major axis of the ellipse must coincide with 
the line of sight or the eclipses would not occur at equal 
intervals. The spectrum will indicate whether two bright 
bodies are concerned and the radial- velocity curve will decide 
whether the orbit is circular or elliptical. 

(ii) There may be a series of equal minima, occurring 
alternately at two different intervals. Such a curve can only 
be due to two bodies equal in size and brightness, with an 
elliptical orbit whose major axis is not in the line of sight. 

(iii) There may be a series of minima which are unequal 
but occur at equal intervals, the alternate minima being equal, 


Such a curve must be produced by two unequally bright stars 
moving in a circular orbit or in an elliptical orbit whose axis 
is in the line of sight. 

(iv) The minima and intervals may both be unequal but 
alternate ones equal. Such a curve would be given by un- 
equally bright stars moving in an elliptical orbit whose major 
axis is not in the line of sight. 

If the eclipses are partial the duration of the minima will be 
very short, whereas if they are total or annular the minima may 
remain constant for some time. Between the eclipses, the 
brightness will remain constant if the two stars are spherical 
and have uniform surface brightness, but if one or both com- 
ponents is elliptical or has a non-uniform surface brightness, 
there will be small variations in brightness between the eclipses. 
If the two components are not spherical, but are tidally 
distorted, there will be a continuous change in brightness 
from minimum to maximum and back to minimum, the light 
curve then being a representation of two superposed effects, 
the light variations due to the eclipses and those due to the 
rotations of the non-spherical bodies. 

A general conception of the nature of the system can therefore 
usually be obtained from the light curve. By the application 
of mathematical methods it is possible, with a few simple 
assumptions, to deduce the ratio of the major axes of the two 
components, their ellipticities, the ellipticity of the orbit, the 
ratios of the axes of the two components, to the major axis 
of the orbit, the brightness of the two components in terms of 
that of the Sun, the ratio of their surface brightness, the 
inclination of the orbital plane to the line of sight, the mean 
densities of the two components, and the time of passage 
through periastron. If, in addition, the velocity curve is 
known from spectroscopic observations, the actual dimensions 
and densities of the components can be obtained. A knowledge 
of the parallax of the system further enables the absolute 
magnitudes and surface brightnesses of the two components 
to be deduced. 

218. Examples of Eclipsing Variables. (a) Algol. The 
regular variability of Algol (/? Persei) was discovered by John 
Goodricke (1764-1786), although the name (the " demon " 
star) suggests that its variability was known to the Arabs long 



before. It was certainly known to Montanari a century before 
Goodricke. The light curve is shown in Fig. 101. The magni- 
tude is about 2-3 m. at maximum and remains practically at 
this value for several hours ; it then decreases rapidly and 
falls by about 1-2 in. to 3-5 m. in live hours. On reaching 
the minimum, the magnitude immediately commences to 





t tours 10 20 dO 40 50 60 
FIG. 101. The Light Curve of Algol. 

increase again and reaches its original value after a further 
period of five hours. About twenty-live hours later, there is a 
secondary drop in magnitude of about 0-05 m. The whole 
period is 2-807301 days, or nearly 09 hours. 

From an inspection of the light curve the following infor- 
mation may be deduced. In each period there are two minima 
which occur at equal intervals ; the two eclipsing bodies 
therefore move in a circular orbit or in an elliptic orbit whose 
axis is in the line of sight. The light does not remain constant 
for any length of time at the minima ; the eclipses are therefore 
partial, for if they were total the light would remain appreciably 
constant during the passage of the one star in front of the other. 
Since one minimum is much deeper than the other, one of the 
components is bright and the other faint. The brightness 
between the two minima varies slightly ; this suggests that 
the components are elliptical or are of non-uniform surface 




3 6 

brightness. If the former, the magnitude would be a maximum 
at quadrature, midway between primary and secondary 
minima ; this is not the case, and the curve is not symmetrical 
about this point, so that both effects are concerned. 

The mathematical discussion shows that the radii of the 
bright and faint bodies in terms of that of the orbit as unit are 
respectively 0-207 and 0-244, the faint body being therefore the 
larger. The light of the bright body (in terms of the maximum 
light of the system) is -925 ; that of the faint body, which is of 
non-uniform brightness, varies between 0-045 and 0-075. The 
angle between the normal to the orbit and the line of sight is 
about 82. Radial-velocity observations confirm that the orbit 
is circular. Both 
bodies are slightly 
elliptical. The 
mean density of 
the system in 
terms of that of 
the Sun as unity 
is 0-07. The par- 
allax of the system 
is estimated as 
0"-032 ; this would 
make the total 
light of Algol 
about 200 times 
that of the Sun, 
whilst the darker DaysO 
body has a surface 
intensity ten times 
that of the Sun. 

(6) /? Lyr&. The light curve of this star, whose variability 
was also discovered by Goodricke, in 1784, is shown in Fig. 102. 
It has two unequal minima separated by two equal maxima and 
is characterized by a continuous variation in light, there being 
no period of steady luminosity either at maximum or minimum. 
The period is 12-916 days ; the maximum magnitude is 
3-4 m. and the ranges at primary and secondary minima are 
respectively 0-97 m. and 0-45 m. The continuous variation 
of light is due to the large eccentricity of figure of the two coni- 





FIG. 102. The Light Curve of /3 Lyrae. 


ponents, so that, even though the eclipses are total, the light 
does not remain constant at the minima. The orbit is very 
nearly circular, the minima being nearly equidistant. The 
spectral lines due to both components can be observed in the 
spectrum of the system, the two components being of types 
B5 and B8. The mathematical investigation shows that the 
components are strongly ellipsoidal, the ratio of the other two 
axes to the major axis being 0-76 and 0-69 respectively. The 
inclination of the orbit is about G2. The radii of the brighter 
and fainter components in terms of that of the orbit are 
respectively 0-27 and 0-68. The ratio of the surface intensity of 
the brighter to that of the fainter body is about 9-4 to 1. Both 

bodies have very small density. 

^" --^ This system therefore consists 

/'' s \ of two stars of early type and 

/ \ low density revolving about 

\ B Qne ano th er nearly or possibly 
actually in contact. Under 
their mutual attracting influ- 
ence they are tidally distorted, 
/ and to their elliptical shape 

the peculiarities of the light 

v ^~ -"" curve are mainly due. A hypo- 

FIG. 103. The System of p Lyrse. thetical representation of the 

system is shown in Fig. 103, in 

which A,B are the fainter and brighter components respec- 

219. Remarks on Eclipsing Variables. Whilst the 
long-period variables are chiefly to be found amongst the red, 
late-type stars, the eclipsing variables occur most frequently 
amongst stars of early type, and particularly amongst types 
B and A. Out of 93 stars examined, 18 were of type B, 54 of 
type A, 12 of type F, 8 of type G, and only one of type K. 
The mean densities of the eclipsing variables are small ; of 
these 93 stars, only one had a density exceeding that of the Sun. 
The mean density for the B-type stars was 0-12 in terms of the 
Sun's density as unit, and that of the A- type stars was 0-21. 
The extreme densities, both high and low, are to be found 
amongst types F and G. 



The range of variation in magnitude at principal minimum 
is connected with the relative sizes of the two components. If 
the range exceeds two magnitudes, the faint star is almost 
certain to bo the larger ; if the range exceeds one magnitude, 
the faint star is probably the larger ; but if it is less than 0-7 m. 
the faint star is either equal to or smaller than the bright star. 
On the other hand, in the majority of systems examined, the 
brighter component has a somewhat greater mass than the 
fainter, and in no case has it a less mass. Therefore in general 
the fainter component is larger, but less massive, and therefore 
has a smaller density ; on the average the density of the 
fainter star is about one-third that of the brighter. The fainter 
components are usually of a later spectral type than their 
primaries, so that the eclipsing variables provides instances of 
bodies of late type with low densities. 

The eclipsing variables are not uniformly distributed in 
space, but show a marked condensation towards the plane of the 
Milky Way, though not so marked as in the case of the Cepheid 

If the inclination of the orbital plane of an eclipsing variable 
is large, the eclipses are not visible, but the binary nature is 
detected by the spectroscopic observations and the system is 
then termed a spectroscopic binary. It is interesting to note 
that for spectroscopic binaries and visual double stars there is a 
progressive increase of eccentricity with period. This is 
indicated in the following table, which represents the mean of a 
large number of systems : 




3 days 


Spectroscopic Binaries . . J 




1J years 





Visual Binaries . . , . J 





220. The Origin of Binary Systems. The principal 
phenomena to be taken into account in formulating a theory 


of the origin of binary systems are (i) the great relative number 
of such systems, (ii) the correlation between eccentricity of orbit 
and length of period to which we have just referred, (iii) the 
correlation between length of period and spectral type, the 
short-period spectroscopic binaries being most numerous 
amongst early -type stars and the long-period visual binaries 
amongst late-type stars. 

One theory which has received much support supposes that 
the two stars of a binary system were originally independent, but 
that having at some time approached one another sufficiently 
closely, they were captured by their mutual gravitation. The 
theory has been worked out in great detail, but the large number 
of binary systems seems a fatal argument against it. From the 
density distribution of stars in the neighbourhood of the Sun 
as determined by parallax observations, it has been estimated 
that two stars will only approach one another within 200 
astronomical units, on the average, once in 10,000 million years. 
It would on this theory be even more difficult to account for 
triple systems, many of which are known. 

Another theory attributes the origin of binary systems to 
the fission of an original parent star the genesis being thus 
somewhat similar to that of the Earth-Moon system. The 
parent body, in its primitive state, is supposed to have been in 
rotation and gradually to have contracted. The rate of rota- 
tion would have increased as the body contracted. A certain 
stage was at length reached when the body became first pear- 
shaped and then dumbbell-shaped and subsequently divided 
into two detached masses. These were at first nearly in con- 
tact, as in the case of the two components of /3 Lyroe. Tidal 
friction would then cause the distance apart of the stars 
gradually to increase. 

The objections to this theory are that the critical density at 
which fission takes place exceeds that of many eclipsing 
binaries and that tidal friction is not competent (when the 
masses of the components are equal) to increase the initial 
period more than a few-fold, whereas we find in nature periods 
ranging from a few days to thousands of years. 

At present, therefore, there is no theory which can be held 
to explain at all satisfactorily the various phenomena presented 
by binary systems. (See also 244.) 


221. The Milky Way or Galaxy. The Milky Way or 
Galaxy is the name given to the luminous belt of stars which 
encircles the heavens, nearly in a great circle. The Galactic 
Plane is the plane passing as nearly as possible through the 
centre of this belt. The Milky Way passes in the northern 
sky within about 30 of the North Pole, runs through the 
constellations of Cassiopeia, Perseus, and Auriga to the horns 
of Taurus, where it crosses the ecliptic near the solstice at an 
angle of about 60. Thence it passes between Orion and 
Gemini, through Monoceros to Argo, Crux, and the feet of the 
Centaur. Here it divides into two branches, the brighter of 
which passes through Ara, Scorpio, the bow of Sagittarius, and 
Aquila to Cygnus, where it rejoins the other branch. Both 
the width and the brightness of the Milky Way vary greatly 
from one point to another. In. its widest part, between Orion 
and Canis Minor, it has a widtli of about 45 ; in other places 
the width is as small as 3 or 4. The densest part of the 
Milky Way is in the star clouds in Sagittarius (Plate XVIII). 
In the denser parts of this region the plate shows a con- 
tinuous background of stars, the images of which are too close 
to separate. 

The northern galactic pole is situated in R.A. 190, Dec. 
+ 28, the southern galactic pole being therefore in R.A. 10, 
Dec. 28. The position of a star may be defined with 
reference to the galactic plane. The galactic latitude of a body 
is defined as the angular distance of the body north or south 
of this plane. The galactic longitude is defined as the angular 
distance of the body measured along the galactic plane from 
the point of intersection of the galaxy with the equator in 
R.A. 280, Dec. 0, These galactic co-ordinates are of import- 




ance in many stellar investigations, for the galactic plane is the 
fundamental plane of reference for sidereal astronomy and 
corresponds in this respect to the plane of the ecliptic of 
planetary astronomy. A further advantage of galactic co- 
ordinates is that they are not affected by the precession in 
the equinoxes, so that changes in the galactic co-ordinates of 
a star determine at once the star's proper motion. 

222. The Galactic Condensation of the Stars. The 

fundamental importance of the galactic plane in sidereal 
astronomy is due to the fact that it forms a plane of symmetry 
for Hie stellar universe. The stars, both bright and faint, 
crowd towards it. This was first pointed out in the case of 
the brighter stars by Sir William Herschel, and Schiaparelli, 
Sceliger, and others have extended the relationship. The 
following table gives the mean densities of the stars per square 
degree for galactic zones of 20 width, for stars down to magni- 
tude limits of G m., 9 in., and 14 m. respectively : 

Galactic Latitude Zones. 

Densities of Stars down to various Limits of 

6'0 m. 

9-0 m. 

14-0 m. 

+ 90 

to +70 




+ 70 

, +50 




+ 50 

, +30 




+ 30 

, +10 




+ 10 

, -10 




- 10 

, -30 





, -50 





, -70 





, -90 





The values in the last column are derived from HerscheFs 
"star gauges," or counts of stars visible in a definite area in 
the field of his 18-inch reflector. 

From this table it will be seen that the naked-eye stars 
exhibit a slight but perfectly definite preference for the galaxy, 
apparently indicating that a proportion of the bright stars 
which are visible in the Milky Way are either situated in it 



and form part of it or are in some way definitely connected 
with it. Down to a limit of the ninth magnitude, the stars 
are nearly three times as dense in the Milky Way as at the 
galactic poles, whilst for the stars of Herschel's gauges the 
ratio is nearly twenty to one, the large ratio being due to the 
great number of faint stars in the Milky Way. The stellar 
system, considered as a whole, is therefore very markedly 
flattened towards the Milky Way. 

A comparison of the densities in the above table for corre- 
sponding zones of north and south galactic latitude shows that 
in each case the density in the southern zone is the larger. 
The explanation of this result is that our solar system is 
situated somewhat to the north of the galactic plane. A close 
study of the course of the Milky Way in the heavens confirms 
this view. Struvo found that it docs not actually trace out 
a great circle on the celestial sphere, but a small circle at a 
distance of about 88 from the south galactic pole. 

The galactic condensation just considered was a mean value 
for stars of all spectral types. The condensations for the 
individual types show very considerable variations. In the 
following table are given the numbers of the bright stars of 
different types, down to a visual limit 6- 5 m., in eight zones of 
galactic latitude, so chosen as to have approximately equal 
areas. The table indicates therefore the distribution of the 
stars nearest the Sun. 

Mean (jalaetio 
Latitude of Zone. 
































































+ 9-2 

It will be seen that the stars of type B exhibit a strongly 
marked crowding towards the galaxy but that with successive 
types the distribution in the various zones becomes more 

A A 


uniform until with type M there is very little evidence of 
galactic condensation. If the pole of the plane of concentra- 
tion of the B-type stars is determined, the value R.A. 182, 
Dec. + 28 is obtained, which is very near the position of the 
galactic pole. 

The most remarkable concentration is shown by the small 
class of stars known as Wolf-Rayet stars, which we have seen 
may be a state through which all novse must pass. With the 
exception of some Wolf-Rayet stars in the Magellanic Clouds, 
the remainder are all in the Milky Way and the pole of their 
plane of condensation coincides exactly with the galactic pole. 

Other classes of objects which show strong galactic concen- 
tration are eclipsing variables, Cepheid variables, and gaseous 

223. The Distance of the Milky Way. The distances of 
the individual stars composing the Milky Way cannot be 
determined by direct trigonometrical methods. Some idea of 
the distance of the Milky Way can nevertheless be obtained 
indirectly. If we assume that a condensation of any class of 
stars towards the Milky Way indicates that some stars of such 
class belong to the Milky Way, then it is certain that stars as 
bright as 9'0 m. belong to the Milky Way ; a star whose 
luminosity is 10,000 times that of the Sun would appear of 
the ninth magnitude at a distance of 5,000 parsccs. It may 
therefore be concluded that the nearest parts of the Milky 
Way are not more distant than this limit. If, as we have seen 
is probable, some of the sixth-magnitude stars belong to the 
Milky Way, the distance is reduced to 1,200 parsecs. 

We can get some more definite information from the Copheid 
variables, whose marked galactic concentration would appear 
to indicate that they are connected with the Milky Way. In 
216 it was explained how the distance of any Cepheid variable 
can be determined with accuracy when its period is known. 
Using this method, Shapley has determined the distances of 
the known Cepheids. The distances so found range from 
about 100 parsecs to 6,000 parsecs, but inasmuch as there are 
undoubtedly many Cepheid variables which still await dis- 
covery, and as the nearest are most likely to be discovered 
first, it is not improbable that there are Cepheids at distances 


much greater than 6,000 parsecs. But the important point is 
that the known Cepheids are fairly uniformly distributed in 
distance between the two limits. 

Some investigators have supposed that the Sun occupies a 
nearly central position in a fairly uniformly distributed and 
spherical local system of stars outside which lies the Milky 
Way, with a form somewhat like that of an anchor ring. 
Recent investigation has modified this conception, as we shall 
see when dealing with stellar clusters. The distribution of the 
Cepheid variables would seem to indicate rather that our stellar 
system is strongly oblate or lens-shaped and that the great 
star-density in the Milky Way is due at least partly to the 
greater distance to which the system extends in that direction. 
This conception is not opposed to the possibility of the exist- 
ence in the Milky Way of numerous " star clouds " of great 
star-density and at very great distances. That such star 
clouds exist is rendered probable by the example of tho 
Magellanic Clouds (Plate XIX). Although the latter may be 
isolated portions of the Milky Way, it is possible that, in view 
of their high galactic latitude, they may not be related in any 
way to it. They contain many Cepheid variables, and this 
fact has enabled their distance to be determined as about 
10,000 parsecs. There are many similar aggregations in the 
Milky Way ; Easton has discussed the star-densities in some 
of these regions and in adjacent regions just outside and found 
that at the fourteenth magnitude an overwhelming proportion 
of the stars were associated with the local aggregations. 
Recently, Cepheid variables and other stars of high luminosity 
have been found amongst the fifteenth-magnitude stars of the 
galactic clouds, indicating that these clouds are at a distance 
of at least 15,000 parsecs. Nevertheless, there is no reason to 
suppose that these clouds are more than local aggregations in 
the general oblate system, and it is very probable that the 
Cepheid variables extend to distances as great as or greater than 
many of these clouds. If this view is correct, we ought not 
to speak of the distance of the Milky Way, but rather of the 
depth or extent of the sidereal system in the galactic plane. 

224. Stellar Clusters. In addition to the large-scale 
aggregations of stars in the Milky Way which we have been con- 


sidering and of which the star clouds in Sagittarius (Plate XVIII) 
may bo cited as an illustration, there are also known many 
aggregations of a different nature, frequently showing a regular 
and definite configuration. These are called star clusters. 
The typical star cluster, such as the cluster in Hercules (Plate 
XXI, b) is spherical in shape with a density decreasing from the 
centre outwards, at first rapidly and then more gradually ; the 
boundary of the cluster is not sharply defined in general, the 
density falling gradually to zero ; superposed upon the cluster 
are the field stars (i.e. the stars which would still be seen in 
the same region of the sky if the cluster were removed), so that 
it is only possible to define approximately the boundary by 
careful counts for the determination of the relative density. 
Clusters of this type are termed globular clusters. But re- 
garding a true cluster as any group of stars which have a 
physical relationship with one another, there are two other 
types which must be included under the general designation 
of star-cluster. One of these types is illustrated by the 
Proesepe cluster ; it consists of a group of stars with a some- 
what irregular boundary but less condensed towards the 
centre and with fewer stars than the typical globular cluster. 
Such clusters are termed open clusters. The other class con- 
sists of groups of isolated stars, each member of which has a 
velocity along the same direction in space and of the same 
magnitude. The Pleiades stars form one example of such a 
group of stars having a common motion ; there is undoubtedly 
a physical connection between them (Plate XX). Another 
example is given by the Taurus cluster, which comprises some 
of the stars in the Hyades and other neighbouring stars ; these 
stars are in the immediate neighbourhood of the Sun, and their 
relationship can only be established by the identity of their 
motions. Thirty-nine members of the Taurus cluster are 
known ; all of these are stars much brighter than the Sun, and 
it is therefore probable that many fainter stars belong to it 
which have not yet been identified. These last two classes 
are probably analogous and differ only in distance from us. 
The first class possesses somewhat different properties and has 
been more extensively studied. 

225. Globular Cluster s. The globular clusters, of which 


the Hercules cluster (Plato XXI, 6) is the typical example, form 
a limited class of objects, the number known being about ninety- 
five. Their distribution shows some important features ; in 
galactic longitude, they mostly occur between longitudes 235 
and 5, in the constellations of Ophiuchus and Sagittarius, 
whilst between longitudes 41 and 195 there is not a single 
one. The distribution is approximately symmetrical with 
respect to the line 145 to 325. In galactic latitude the dis- 
tribution is also irregular ; whilst it is approximately sym- 
metrical with respect to the galaxy, there is an almost complete 
absence of clusters from a zone extending from + 10 to 10 
galactic latitude. In this respect they diifer from the other 
two classes of clusters, which are concentrated towards the 
galactic plane and are most numerous in the central zone. 
The globular clusters are most numerous just outside this 
central zone. The physical meaning of this zone of avoidance 
is not known, but it has been conjectured that if a globular 
cluster should enter the central zone in the neighbourhood of 
the large masses of matter concentrated in the star clouds of 
the Milky Way, the effect of gravitational attraction would 
be to destroy the cluster, which would ultimately appear as 
a loose aggregation of stars the typical open cluster. 

The distances of the globular clusters can be estimated with 
great accuracy by indirect means. The investigations of 
Shapley at the Mount Wilson Observatory have resulted in 
several mutually concordant methods, with the aid of which 
the distances of all the clusters have been determined. The 
foundation upon which these methods are built is the 
luminosity-period relation which holds for Cepheid variables. 
In many of the clusters variable stars occur whose light curves 
indicate that they are typical Cepheids. A determination of 
their periods and apparent magnitudes enables the distance of 
the cluster to be at once deduced. For such clusters, it is then 
found that there is a definite correlation between apparent 
diameter and distance. This implies that the globular clusters 
do not differ greatly in actual linear dimensions, and it there- 
fore becomes possible to determine with some accuracy the 
distance of any globular cluster by measuring its apparent 
diameter. The distance so determined can be checked in 
another way. Shapley jjnds that the mean absolute magni- 


tude of the brightest stars in the cluster (say the twenty-five 
brightest) has practically a constant value for different clusters, 
and it is not unreasonable to suppose that this result holds for 
all clusters. Such stars are typical giant stars, and it is not 
probable that the absolute magnitudes of the giant stars differ 
greatly wherever they occur in the universe. For any cluster, 
therefore, in which no variables have been discovered it is only 
necessary to determine the apparent magnitudes of the twenty- 
five brightest stars in order to be able to deduce the distance, 
the corresponding absolute magnitude of these stars being 
known. By a combination of these three methods the dis- 
tances of the globular clusters have been determined with a 
relatively small uncertainty. 

The nearest clusters are 47 Toucanse and a> Centauri, which 
have parallaxes of 0"-000148 and 0"-000153 respectively, 
corresponding to distances of about 7,000 parsecs or 22,000 
light-years. N.G.C. 7,006, on the other hand, has a parallax 
of 0"-000014, corresponding to a distance of 67,000 parsecs or 
220,000 light-years. In this cluster, a star of the brightness 
of Sirius would appear to us as only of the seventeenth magni- 
tude. The diameters of the clusters are of the order of several 
hundred light-years. 

That the bright stars in the globular clusters are in fact 
giants and not dwarfs has been established by photographing 
the spectra on special photographic plates, sensitive in the 
blue and the red but insensitive in the green-yellow region. 
In this way spectra divided in the middle are obtained and the 
relative intensities of the blue and red portions can be esti- 
mated. Giants and dwarfs of the same spectral type show a 
markedly different intensity ratio : it is found that many 
giants are present amongst the cluster stars. 

226. The System of the Globular Clusters. The deter- 
mination of the distances of the globular clusters enables their 
actual positions in space to be assigned. Taken together, 
they form a system whose centre lies in the Milky Way in the 
direction of Sagittarius and at a distance of about 20,000 
parsecs or 65,000 light-years. The greatest diameter of the 
system outlined by the clusters is about 300,000 light-years. 
Our solar system therefore occupies a markedly eccentric 


position in this larger system. This accounts for the unequal 
distribution of the clusters in galactic longitude. 

These investigations have necessitated some revision of the 
views which were held only a few years ago with regard to the 
structure of the universe. The concentration of the clusters 
towards the Milky Way tnd their symmetrical arrangement 
with respect to it indicate a definite relationship. We must 
therefore conclude that our stellar universe has a longest 
diameter of at least 300,000 light-years. Our solar system 
lies somewhat to the north of the central plane of this system 
and about 60,000 light-years from its centre. It seems prob- 
able that the Sun is near the centre of a large local cluster 
situated eccentrically in this larger system. Investigations of 
star density in the neighbourhood of the Sun have shown that 
the density falls off with distance in all directions. This was 
formerly taken as an indication that the Sun was near the 
centre of the Universe. In the light of recent discoveries this 
view must now be revised. The evidence in support of the 
hypothesis that such a local cluster exists is strong and is 
summarized in the next section. 

227. The Local Cluster. We start from the hypothesis 
that the stars in the neighbourhood of the Sun (i.e. mainly 
those which are contained in the various catalogues of star 
positions and proper motions) fall into two classes : (1) mem- 
bers of a moving local cluster of limited extent ; (2) members 
of the general galactic system. The distribution of the B-type 
stars in the neighbourhood of the Sun (within about 500 
parsccs) has been studied by Chaiiier. On the above hypothesis 
it must be assumed that the B-type stars belong mainly to the 
local group, for stars of this type do not increase in number 
with decreasing apparent magnitude as rapidly as do other 
types and they are very infrequent beyond a distance corre- 
sponding to apparent magnitude 7-5. The system outlined 
by these stars has its centre in the constellation Carina, in a 
direction which is nearly at right angles to the direction of the 
centre of the general galactic system, and at a distance of 
about 88 parsecs from the Sun. The Sun is a few parsecs 
above its central plane of symmetry. The pole of the central 
plane of the system of the B-type stars was found by Charlier 


to be inR.A. 184-3, Dec. + 28-7, whilst the pole defined by 
the Milky Way clouds is in R.A. 191-2, Dec. -- 27-4. This 
difference is a real one and indicates an inclination of the 
central plane of the local system to the plane of the galaxy. 
The reality of the difference is established by plotting down in 
projection the positions of the B-type stars : the brighter stars 
are found to be concentrated along the projection of the central 
plane of the local cluster, whilst the faint ones are concentrated 
along that of the galactic plane. The pole of the true Central 
plane of the local cluster is in this way found to be in R.A. 
178, Dec. + 31-2, corresponding to an inclination of the 
central plane to the galactic plane of about 12. It is signifi- 
cant that Gould's belt of bright stars, defined by the naked-eye 
stars, coincides closely with this central plane, whilst fainter 
stars, to the ninth magnitude, give the same pole for the Milky 
Way as is given by the galactic clouds. In support of this 
hypothesis of a local cluster, it may be mentioned that in 
many typical globular clusters the existence of what may be 
termed a " galactic plane " has been found. 

It appears from Shapley's researches that the cluster stars 
comprise most of the B-type stars, a majority of the A-type 
stars brighter than about the seventh magnitude, and a number 
of the redder stars in the neighbourhood of the Sun. The 
stars of the general galactic system which interpenetrate it 
include a few stars of types B and A, a majority of the redder 
stars, Cepheid variables, etc. The greater degree of concen- 
tration towards the galactic plane of the early than of the late 
type stars is thus explained, for a greater proportion of the 
early-type stars belong to the local cluster. The motion of the 
local cluster through the surrounding field may possibly give 
an explanation of the phenomenon of star-streaming. 

228. The Absorption of Light in Space. For many 
astronomical observations, it is necessary to know whether 
there is any appreciable absorption or scattering of light in 
space. Is there cosmic dust distributed through space in 
sufficient density to produce an appreciable amount of absorp- 
tion when light travels through it for a great distance ? If 
such absorption does occur, it will be greater for light of short 
wave-length than for light of long wave-length, and the effect 


will be to make distant stars appear relatively redder than 
near stars and to increase their mean colour-index. An 
extremely small density of matter would be required to cause 
an increase in colour-index of 0-00015 m. for every parsec 
distance travelled by the light. In a distance of 10,000 
parsccs this would result in an increase in colour-index of 
1-5 m., so that, even with this small amount of absorption, a 
star at such a distance could not have a negative colour-index. 
This indicates the argument by which it is possible to conclude 
from the known distances of stellar clusters that absorption in 
space is cither non-existent or is so small in amount that it is 
of no importance to the astronomer. In clusters whose dis- 
tance is greater than 10,000 parsecs, stars with negative colour- 
indices occur, and in as large a proportion as in the nearer 
clusters. Since there is no reason to suppose that uniformity 
of conditions and of stellar phenomena does not prevail through- 
out the galactic system, it is natural to assume that the stars 
in the clusters are similar to those adjacent to the Sun. The 
similarity in the range and proportions of the colour-indices 
therefore indicates that the absorption of light in space is quite 
negligible. This conclusion must not be taken, however, to 
prove the non-existence of absorption by isolated nebulous 
clouds of matter. Conclusive evidence of such absorption is 
in fact known and is illustrated by Plates XXIII and XXIV. 

Plate XXIV (frontispiece) shows a dense star region in Ophiu- 
chus. In the lower left-hand portion of the photograph is seen a 
dark lane almost entirely devoid of stars. At its boundaries the 
star density falls off with remarkable suddenness. This lane 
is connected with the nebulosity near the centre of the photo- 
graph. There is no doubt that between the Earth and the 
stellar background is an absorbing cloud of gas, which is non- 
luminous and which entirely cuts off the light from the back- 
ground of stars. This dark nebula, as it may be called, is 
directly connected with the luminous nebula at the centre. 
The few stars appearing in the dark lane are stars nearer to the 
Earth than the absorbing matter. 

In Plate XXIII is represented a portion of the Southern Cross 
and the "hole" in the Milky Way known as the Coal- 
Sack. This is undoubtedly another instance in which absorb- 
ing matter is present, although its boundaries are not so 


sharply defined as in the case of the dark lane illustrated in 
Plate XXIV. 

229. Open Clusters.- The open clusters, in which class 
may be included the group ; of stars having a common motion, 
differ in several respects from the globular clusters. Their 
shape is more irregular and they are frequently associated 
with nebulosity, whilst the globular clusters never are. They 
are strongly condensed towards the galactic plane, but in 
galactic longitude their distribution is far from uniform ; they 
are most numerous at the opposite point in the galactic plane 
to that in which most of the globular clusters are found. Many 
of their distances have been determined and vary between 
400 and 16,000 parsecs, with a mean value of about 6,000. 
The number of stars in them is far fewer than in the globular 
clusters and their angular size is in many instances greater. 

230. Nebulae. Nebulae may be divided into three main 
classes, which must be considered separately. 

(1) Irregular Nebulce. Of these, the Great Nebula in Orion 
is the most conspicuous example. This class comprises 
nebulae of many varied shapes, whose names are often given 
from a more or less striking resemblance to some terrestrial 
object, such as the Dumbbell Nebula, the Crab Nebula, the 
North America Nebula, the Keyhole Nebula, etc. In the 
same category may be placed the nebulous backgrounds, 
obviously associated with stars, such as the nebulosity around 
the Pleiades (Plate XX) and in the constellation of Taurus and 
also the dark nebuloe ( 228 and Plate XXIV). These irregular 
nebulae occur mainly in the neighbourhood of the galaxy and 
in many cases show undeniable connection with certain stars. 

(2) Planetary or Gaseous Nebulce. These were first classified 
as such by Sir William Herschel, though he did not originally 
recognize their nebulous nature. " We can hardly suppose 
them to be nebulae," he says ; " their light is so uniform as 
well as so vivid, their diameters so small and well defined, as 
to make it almost improbable that they should belong to that 
species of bodies." He considered that they might be planets 
attached to distant tuns, but later recognized that this supposi- 
tion was untenable. Their spectra present many analogies to 


the spectra of the Wolf-Rayet stars and, like the latter, they 
occur almost exclusively in the Milky Way. 

(3) Spiral Nelulce. These constitute by far the most numer- 
ous class. Their discovery was the one striking achievement 
of the great Parsonstown reflector, constructed by Lord Rosse. 
Curtis estimates that the number of them which it is possible 
to photograph with the 36-inch Crossley reflector of the Lick 
Observatory lies between 700,000 and 1,000,000. By far the 
most conspicuous object of this class is the great nebula in 
Andromeda, easily visible to the naked eye as a small blurred 
patch, very different from a star in appearance (Plate XXI, a). 
It was the only nebula discovered before the invention of the 
telescope. Photographs show it to consist of a bright central 
nucleus, with long, spiral, nebulous arms wreathing round it. 
The spiral nebula) that we know are placed at all inclinations 
to the line of sight. Many are seen perpendicular to the plane 
of the spiral arms ; in this case, and in others, the two arms 
are clearly seen starting out from opposite edges of the central 
nucleus. Two examples of spiral nebulae seen broadside on 
are shown in Plate XXII . Some, again, are viewed edge on, and 
in these the spiral arms are seen as a narrow line, evidence that 
they lie in one plane. This line is seen dark where it crosses 
in front of the central nucleus, owing to the absorption which 
the light from the latter undergoes in passing through the 

231. Irregular Nebulae. In this class must be included 
not only the brighter objects, such as the Orion (Plate XX, 6), 
Omega, and Keyhole nebulae, but also the extended nebulous 
backgrounds which have been photographed by Barnard and 
others, in the star clouds of the Milky Way and particularly 
in the constellations of Taurus, Scorpius, and Sagittarius. Of 
the same nature must be many of the dark spaces in the Milky 
Way (Plates XXIII and XXIV). There are many regions in the 
Milky Way where the star-density as shown on photographs 
is very small, whilst in adjacent regions it is very great, the 
boundary between the regions of large and small density being 
usually sharply defined. At first it was thought that the 
regions of small density were holes in the Milky Way, but it is 
now certain that they are due to the presence of dark nebulae 


lying between us and the Milky Way star clouds and cutting 
off the light from the latter. Only the nearer stars are seen in 
projection on them. The nebulosity is often apparent at the 
edges of these dark regions, and they are always found in 
portions of the sky where nebulosity abounds. Some of these 
dark nebula) give the appearance of curved lanes carving their 
way through dense starry regions, the nebulosity becoming 
visible at the end (Plate XXIV). Sir William Herschel was the 
first to point out the tendency of gaseous nebula) to sweep 
clear a space amongst the stars, this phenomenon being the 
same as that under consideration. 

These irregular nebula) occur almost exchisively in the 
Milky Way. In many cases, it seems probable from the 
evidence of photographs that they are actually associated 
with the distant galactic star clouds ; in others, as in the case 
of the Orion nebula and the Pleiades nebulosity, the stars are 
definitely in the midst of the nebula (Plate XX). The great 
brightness of some of these stars appears to indicate that then- 
distance is not very great ; such parallax determinations as 
are available support this statement the parallax of the 
Orion nebula, for instance, having been determined as 0"-005. 

The spectra of nebulas of this class consist of a line emission 
spectrum superposed upon a weak continuous spectrum. The 
typical spectrum contains lines of hydrogen and helium to- 
gether with certain characteristic lines which do not occur in 
the spectra of known terrestrial elements : these lines are 
attributed to a hypothetical element called "ncbulium." It 
is possible, however, that two elements are concerned in their 
production, for the relative intensities of the lines denoted NI 
and N 2 is the same in all nebulae, whilst the relative intensities 
of NI and N 2 to the line N 3 vary with different nebulae and 
sometimes even within the same nebula). 

The most important irregular nebula is that in Orion (Plate 
XX, 6), which long-exposure photographs show extends over an 
area of about 45 in declination and of two hours in right 
ascension. Owing to its large size and brightness it has been 
extensively studied. Spectroscopic observations, as well as 
photographs taken with and without colour filters, have estab- 
lished that the distribution of the various gases responsible 
for the emission lines in the spectrum is not uniform through- 


out the nebula : the relative intensities of different parts of 
the nebula vary according to the nature of the colour filter 
used. The radial velocities at different points of the nebula 
have been determined. The velocity varies from point to 
point. After correcting the observed values for the solar 
motion it is found that although the nebula as a whole has no 
appreciable velocity in the line of sight, yet there are irregu- 
larities of speed in certain portions of the nebula amounting 
to as much as 10 km. per sec. There are also great collective 
movements : with respect to the mean velocity, the south- 
west region is approaching at a speed of about 5 km. per second 
and the north-east region is receding at about the same rate. 
Such movements may indicate a rotatory motion about a line 
south-east to north-west. The stars in the nebula have the 
same motion as the nebula. 

By measuring the breadth of the emission lines of the nebula 
and applying the laws of the kinetic theory of gases, it has been 
estimated that the bright double line with wave-length 3,727 
belongs to an element with an atomic weight about 3 and the 
nebular line 5,007 to another with a smaller atomic weight. 
These considerations also indicate a temperature of about 

The small radial velocity of the Orion nebula appears to be 
a general characteristic of the irregular nebulae. All whose 
radial velocities have been determined have been found to have 
only a small velocity. 

232. Planetary Nebulae. This class includes small 
nebula) of regular shape and outline which are not spirals. In 
the telescope, such a nebula in general appears as an object 
with a sensible disc without a central condensation and similar 
therefore to one of the outer planets ; hence the name. But 
many nebulae of this class show a pronounced central con- 
densation. Their apparent diameters vary between 15' and 
a few seconds. In this class should also be placed the ring 
nebula) and nebulous stars. All these objects show a strong 
concentration towards the Milky Way, though a few may be 
found in high galactic latitudes. The smallest of them, and 
therefore probably on the average the most distant, are found 
in the Milky Way. 


Many planetary nebulse have a nucleus of sufficient sharp- 
ness to enable the parallax to be determined by the ordinary 
methods. The parallaxes of several nebulcB have already 
been obtained ; these are naturally comparatively near objects, 
their distances being of the order of 50 parsecs. The parallax 
being known, the absolute magnitudes of the central nuclei 
and the linear diameters of the objects can be deduced. The 
central nuclei are thus found to have absolute magnitudes 
ranging from about 7 m. to 10 m. and are therefore dwarf 
stars. The diameters of the nebulyo are of the order of about 
6,000 astronomical units (about 0-09 light-years), the smallest 
being 1,350 units. For comparison, it may be noted that the 
diameter of the orbit of Neptune is about 60 units. 

The number of planetary nebulas at present known is about 
150. Their velocities have been measured in many cases and 
average about 40 km. per second, a value which, though much 
smaller than that found for spiral nebuloe, is greater than that 
of the irregular nebula). There are a few individual velocities 
which greatly exceed this amount ; thus N.G.C. 6,644 has 
a velocity of 202 km. and N.G.C. 4,732 of 136 km. The 
velocities do not show a preference for any particular direction. 
In many cases there is evidence of internal motion or of 
rotation : particularly in those nebulae whose boundary is 
elliptical is there strong evidence for rotation about the shorter 
axis. The rotational velocity is greatest at the centre and 
decreases outwards. Curtis has used these results to obtain 
an estimate of the masses of several nebula? and obtains values 
ranging from 4 to 150 times the mass of the Sun. 

The spectra of the planetary nebulae are closely related to 
those of the Wolf-Rayet stars. The relationship may be illus- 
trated by a comparison of the two objects N.G.C. 6,572 and 
B.D. + 30 3,639, investigated by Wright. The former of these 
is a planetary nebula with a nebulous nucleus ; the latter is 
a Wolf-Rayet star surrounded by a weak, nebulous shell of 
apparent diameter 1". The spectrum of this shell is identical 
with that of the planetary nebula, whilst the nucleus of the 
latter has a typical Wolf-Rayet spectrum. The principal 
nebular lines are absent from the spectrum of the nucleus, 
whilst on the other hand the nebula itself does not show the 
usual bands associated with Wolf-Rayet stars. This relation- 


ship is typical, and a gradual transition in spectra is indicated 
through successive stages, from the typical planetary nebula 
without a distinct nucleus to the nebula with a nebulous 
nucleus, thus to the star with a nebulous shell, and finally to 
the typical Wolf-Rayet star with no indication of nebulosity. 
As in the case of the irregular nebuloo, it is frequently found 
that the various elements which together produce the observed 
spectrum of a planetary nebula are not uniformly mixed 
throughout the nebula, in the spectra of some portions certain 
lines predominating and other lines in those of other portions. 
Iri view of the relationship between planetary nebulae and 
Wolf-Rayet stars indicated by their spectra, the difference in 
the average velocities of the two classes of objects is somewhat 
surprising. The latter have a low velocity, smaller than that 
of the B-type stars, averaging about 6 km. per second, whilst we 
have seen that the former have velocities five or six times this 
value. Ludendorff has shown that in the case of the B-type 
stars those of largest mass possess on the average the largest 
velocities. The masses of the planetary nebulae are un- 
doubtedly greater than the average star masses, so that a 
similar result would appear to hold for these objects, and it 
would then seem that their occurrence in the scheme of stellar 
evolution is in some manner conditioned by their large mass. 

233. Spiral Nebulae. As we have already remarked, the 
spiral nebulae are by far the most numerous class of nebulae. 
Their distribution with respect to the galactic plane is remark- 
able and not similar to that of any other class of object. They 
are never found in or near the galaxy ; there are many more 
in north than in south galactic latitudes, and they are most 
numerous in the neighbourhood of the north galactic pole. 
In galactic longitude, the distribution is also far from uniform ; 
they are mainly clustered in that portion of the sky which is 
opposite to the region in which the globular clusters are most 

The radial velocities of many spiral nebulae have been 
determined, and with few exceptions these velocities are large, 
in some cases surprisingly large. Several have been found to 
have velocities exceeding 1,000 km. per second. No other class 
of celestial object has velocities which at all approach these 


figures. What is even more surprising is that these large 
velocities are in every case velocities of recession. This result 
must be of fundamental significance, but the precise interpre- 
tation is at present obscure. On one theory which considers 
the spirals as other universes comparable with our own galactic 
system it is explained by assuming that these systems our 
own included have velocities of several hundred km. per 
second, and the observed velocities are velocities relative to our 
system. On the theory which considers them as definitely 
associated with and part of our universe, it is supposed that 
they are in some way repelled from the galactic plane. 

In several spirals evidence of internal motion has been 
obtained. Many of the nuclei on the spiral arms are suffi- 
ciently sharp and well defined io enable their relative positions 
on photographs taken at an interval of twenty or more years 
to be determined. In this way, angular motions have been 
detected which can be represented as the combination of a 
rotation of the nebula as a whole with an outward motion of 
the nuclei along the spiral arms. The rotational motion 
corresponds to a period of about the order of one or two hun- 
dred thousand years, though the uncertainty attaching to this 
figure is naturally great. In the case of spirals which are seen 
nearly edgewise on, a rotational motion can bo detected by 
means of radial- velocity observations. Such observations have 
established for several nebuljB a rotation about an axis per- 
pendicular to the plane of the spiral arms. The rotation seems 
to occur as though the nebula were a rigid body. 

The spectra of spiral riebuloe are continuous with superposed 
absorption lines. They are therefore not gaseous objects. 
The spectral type corresponds to the spectral classes F5 to K. 
It is impossible to decide at present whether the absorption 
lines are merely the integrated spectra of stars associated with 
the nebula, the nebulosity being attributed to matter in a 
state of meteoric dust, or whether the nebula is in reality 
merely an aggregation of stars. Against the latter supposition 
is the fact that the most powerful telescopes have failed to 
resolve even the large nebula in Andromeda into discrete stars. 

234. Distances of Spiral Nebulae. Nothing definite 
can be stated with regard to the distance of spiral nebulae. 


Their distances are certainly too great to be determined by 
trigonometrical methods, and as yet no indirect method has 
been found which will enable them to be determined with a 
sufficiently small percentage error to be considered at all 
reliable. It is possible, however, to obtain some qualitative 
ideas as to the distances of these objects. 

The average velocity of the spirals across the line of sight 
must, in view of the values found for their radial motions, 
be at least several hundred km. per second. At various 
distances a velocity of 700 km. per second would produce 
annual proper motions as follows : 

Distance in light-years 1,000 10,000 100,000 1,000,000 

Annual proper motion 0"-48 0"-048 0"-005 0"-0005 

A value of 0"-4S would have been detected long ago and may 
be excluded. A value of 0"-048 would not probably have 
escape I detection with photographs extending over twenty- 
five years. It would appear from this argument that the 
distances are not less than about 10,000 light-years, but they 
may be much greater. 

Comparing the linear velocities of rotation of spirals seen 
edgewise with the angular velocity of rotation for M 101, 
which is viewed broadside on, and making the assumption 
that the two rotations are comparable, the distance found for 
M 101 would be about 25,000 light-years. 

Another argument can be derived from the appearances of 
now stars in spiral nebulae. Comparison of photographs of 
several of the larger spirals, taken at intervals of some years, 
has revealed the presence of stars with the characteristics of 
new stars. Several such stars have appeared, for instance, in 
the Andromeda nebula, one of which reached the seventh mag- 
nitude. On the grounds of probability, it must be assumed 
that these stars are physically connected with the nebula and 
not merely seen in projection upon it, particularly as novae 
very rarely appear outside the galactic plane. If it is legiti- 
mate to assume that the new stars in the spirals have on the 
average the same absolute brightness at maximum as the 
galactic novae, it is possible to deduce a hypothetical parallax 
for a spiral in which novae have been detected. The dis- 
tances of only four galactic novae have been determined, and 

B B 


these correspond to a mean absolute magnitude for the nov^e 
in question of 3 m. at maximum and +7 m. at mimimum. 
Now 16 novae in the Andromeda nebula had a mean apparent 
magnitude at maximum of 17 m. If the distance of the 
nebulae is (i) 500,000 light-years, (ii) 20,000 light-years, the 
corresponding absolute magnitudes would be respectively 
4 m. and +3 m. This slender material would seem to 
imply a distance of the order of 500,000 rather than one of 
20,000 light-years, and the smaller spirals would presumably 
be at even greater distances ; but it is doubtful whether much 
reliance can be placed upon the underlying assumptions. If 
the measured angular rotations of spirals can bo accepted as 
of the right order of magnitude, distances of this order would 
correspond to impossibly large velocities. On the whole, the 
balance of evidence at present, although indicating that the 
distance of the spirals is large, say of the order of 25,000 light- 
years or more, seems to be opposed to the acceptance of such 
large distances as are deduced from the magnitudes of nova). 

235. The Nature of Spiral Nebulae. Much controversy 
has raged as to the nature of spiral nebuloe. The view held 
recently, when our galactic system was thought to be of much 
smaller dimensions than is indicated by more modern evidence, 
was that they were separate galactic systems. In support of 
this view was advanced the oblate shape of the systems when 
seen edgewise on and the appearance of the spiral arms, which 
were held to be analogous to the Milky Way. The spectra 
were taken as indicating that the spirals consisted of discrete 
stars. The large velocities much greater than are found for 
any other celestial objects seemed not unnatural on this 
hypothesis, and their distribution, which shows no relationship 
to the galaxy, was not opposed to it. 

On the other hand, the distances found for globular clusters 
indicate that the dimensions of our galactic system are of the 
order of at least 300,000 light-years. The measures of rotation 
in spirals do not seem to permit, in some cases at least, of 
distances of this order, and this being so, the spirals must be 
members of our galactic system. Further, Searcs has made a 
careful study of the surface brightness of the Milky Way, from 
which it must be concluded that none of the known spiral 


nebulae has a surface brightness as small as that of our galaxy. 
Recent investigations of the distribution of light and colour 
in various spirals also seem to render it improbable that they 
can be stellar systems. If they are members of one great 
stellar universe, then the significance of their shape and large 
velocities remains obscure. 

Although the balance of evidence at present seems opposed 
to the island- universe theory, the question cannot be regarded 
as yet definitely closed. 

236. The Course of Stellar Evolution. The Harvard 
system of classification of stellar spectra was based solely upon 
a study of the spectra without any consideration of a possible 
bearing upon the order of stellar evolution. The sequence 
finally adopted, types B, A, F, G, K, M, corresponds, as we 
have previously seen, to a gradual and progressive change it! 
the nature of the spectra. The fact that the spectra of type 
B are allied to those of the Wolf-Rayet stars and to the gaseous 
nebulae suggested that the sequence corresponded also to the 
successive stages in the evolution of a star. The gradual pro- 
gression in average velocity with type and other kindred 
phenomena appeared to give support to this theory. In course 
of time, however, other results were derived which did not fit 
in with so simple a theory and necessitated its modification. 

In the early parallax investigations, stars with large proper 
motion were alone selected ; these may be expected to be on 
the average the nearest stars, and therefore the stars whose 
distances can be most easily determined. We have also seen 
that it is possible to derive mean distances for groups of stars 
of known proper motion, by using the components of the 
proper motions towards the solar apex, assuming the solar 
velocity known and that the peculiar motions of the stars 
are distributed at random. The surprising result is obtained 
from these two lines of investigation that in the former case, 
with few exceptions, all the stars of late type (K and M) are 
of low intrinsic brightness, whereas in the latter case the 
mean intrinsic brightness is high. The increase in accuracy 
in parallax determinations has confirmed the wide range in 
luminosity in stars of late types, but it has shown further that 
stars of these types are either intrinsically bright or intrinsi- 


cally faint, and that the separation into two classes previously 
indicated is a physical reality and not a spurious result due to 
the selection of material. This is the basis of the hypothesis 
of " giant" and " dwarf " stars put forward originally by 
Hertzsprung and developed and confirmed by Russell. (See 
also 199.) 

According to this theory, each spectral type consists of two 
subdivisions, consisting respectively of stars of high and low 
luminosity, which are called for convenience "giants" and 
c< dwarfs " respectively. If the latter are more abundant than 
the former, the selection of the nearer stars will be mainly a 
selection of stars of low luminosity, whilst the stars in star 
catalogues, which are more distant and are selected by mag- 
nitude, will be mainly of high luminosity, and are the stars 
used for statistical investigations. The two classes are widely 
separated in the case of stars of type M, less widely separated 
for type K, for types F and G they begin to merge into one 
another, whilst for types B and A they cannot be distinguished. 
Thus, for instance, Adams and Joy find from spectroscopically 
determined parallaxes mean absolute magnitudes for the two 
classes as follows : 

Spectrum . . . Ma-Md K9-K4 K3-KO GO-GO F9 FO 
Giants .... 1-6 m. 1-4 m. 1-3 rn. 0-6 in. 1-1 m. 
Dwarfs .... 10-8 m. 7-8 m. 6-3 m. 5-3 m. 4-1 m. 

It will be seen that the absolute magnitudes of the " giants " 
is approximately the same for stars of all types, whereas the 
" dwarfs " increase in brightness with the sequence from M 
to B. 

These results may be correlated with the determinations of 
the densities of visual and spectroscopic binary systems. For 
types B and A the densities are mainly included within the 
range 0-50 to 0-05, that of water being taken as unity, but 
for types F, G, and K there is an increasing range in density, 
values exceeding 1-0 and less than 0-0001 occurring, the 
median values at the same time being absent. This suggests 
that the separation into giants and dwarfs may be a separation 
according to density. 

At this stage, the theoretical aspect of the question may be 
considered. According to theories advanced by Lane and 
Hitter, a star, assumed to be gaseous and in a highly diffused 


state, will gradually contract under the influence of its own 
gravitational attraction, and this contraction will be accom- 
panied by an increase in temperature due to the conversion 
of gravitational energy into heat. For a certain critical 
concentration, the loss of heat by radiation will be equal to 
the gain resulting from the contraction, and thereafter the 
temperature will decrease. We have seen that such a theory 
gives rise to difficulties with regard to the age of the Sun, but 
it is nevertheless probable that the course of evolution in a 
diffuse gaseous mass will be accompanied at first by a rise and 
subsequently by a fall in temperature. We have further seen 
that the spectrum of a star is conditioned mainly by ionization 
processes in its atmosphere and that these in turn depend 
mainly upon temperature. A classification of stars by spectra 
will therefore be mainly a classification according to tempera- 
ture (as we have indeed seen in 191), irrespective of physical 
state. Stars of the same type may therefore include diffuse 
masses of low density, or stars of relatively high density, and 
as we know that stars have all much the same mass, the stars 
of low density will have a much larger superficial area and a 
much higher luminosity than those of high density. 

This is the basis of the theory advanced by Russell and 
now generally accepted. Starting with a diffuse mass of gas 
of low density (of the order of one ten-thousandth that of the 
Sun) and temperature, the mass gradually contracts, the 
temperature increasing and the luminosity remaining approxi- 
mately constant. As the evolution proceeds the spectrum 
runs through the sequence M, K, G, F, A, B, the temperature 
increasing throughout. The stage is then reached at which 
the temperature commences to decrease, the density being of 
the order of one -tenth that of the Sun. The spectrum runs 
through the same sequence in the reverse order, the density 
increasing throughout. The maximum temperature attained 
will depend upon the mass of the star, so that a star of small 
mass may reach its maximum before arriving at the type-B 
stage. This seems to be confirmed by observation, for it is 
found that on the average the stars of type B are more massive 
than the bulk of the stars. Stars on the branch with in- 
creasing temperature are the " giant " stars ; they are rela- 
tively largo and of low density ; those on the decreasing- 



temperature branch are the " dwarf " stars, and are relatively 
small and of high density. 

It must not be assumed that the spectra of a giant and 
dwarf star of the same spectral type are identical. Certain 
differences in relative intensities of a few lines have been 
found, which have been made the basis of the spectroscopic 
determinations of absolute magnitudes. The lines in question 
are some which are very sensitive to pressure changes. More- 
over, there is a difference between the two classes as regards 
the general distribution of intensity throughout the spectrum, 
the giant stars being relatively the redder. 

The following table gives particulars of typical giant and 
dwarf stars, the diameters in terms of the Sun being deduced 
theoretically : 










= 1). 




ft Geminorum 



1 0-3 




Aldebaran .... 














r) Draconis .... 



- 0-7 




a Leporis .... 



- 1-0 











/5 Centauri .... 



- 1-3 

















+ 1-3 






+ 0-5 

+ 2-9 




A Serpentis .... 



+ 4-1 




11 Leo Minoris . 



+ 6-2 




61 Cygni 



+ 8-0 




Lacaillo 9,352 . . . 



+ 9-7 




Lalancle 21,185 . . . 



+ 10-7 





The table shows the approximate constancy of the absolute 
magnitudes of the giant stars and the progressive change with 
spectral type of those of the dwarfs. The large diameters of 
the giant red stars should also be noted. 

237. The Interior of a Star. Some physical support is 
given to this theory which was based by Russell upon purely 
observational evidence by a mathematical discussion of the 
processes which take place in the interior of a star. The 
latter has been made by Eddington, using as a basis Schwarz- 
schild's theory of radiative equilibrium. This theory supposes 
that the main factor responsible for the transfer of energy 
within a star is not conduction or convection but radiation. 
Consider a spherical layer within a star. Energy coming from 
the interior layers falls on this layer. Some of the energy is 
absorbed by the layer and the remainder is transmitted 
onwards. The layer, on the other hand, is itself sending out 
energy towards the outer layers. The energy received must 
balance that emitted in order that there may be a steady 
state. This provides one physical condition. Another is 
obtained by equating the difference of pressure on the two 
sides of the layer to the gravitational attraction towards the 
centre : but the pressure to be used is not the ordinary gas 
pressure ; to this has to be added the pressure due to the 
radiation. The latter, normally negligible, becomes impor- 
tant in the interior of a star where the temperature is high (at 
the centre it may amount to one million degrees), and the 
essential feature of Eddington's theory is the recognition of 
tliis fact. To determine the three quantities pressure, tem- 
perature, and density at any point a third relationship is 
necessary : in the case of a giant star it may be assumed that 
the ordinary gas-laws hold, and this provides the tliird con- 
dition. The equations so obtained hold everywhere within 
the star except near the surface, to a depth negligible compared 
with the radius, but deep from the point of view of the spec- 
troscopist. To complete the solution a few simplifying 
assumptions are made that the absorption coefficient is 
constant throughout the star and that the molecular weight 
may be taken as 2, on account of the completeness of the 
ionization at the high temperatures concerned. 


The theory then proves that under these assumptions the 
total radiation is independent of the stage of evolution, so that 
for giant stars the absolute magnitude remains constant what- 
ever the spectral type. We have seen that this is actually the 
case, and the agreement with theory may be taken as confir- 
mation of the underlying assumptions. It is also found that 
the effective temperature of the giant star will vary as the 
sixth root of its density. The theory seems in addition to 
point to the cause which determines the mass of a star. We 
know that the masses of the great majority of the stars fall 
within a comparatively narrow range from about the mass of 
the Sun to twenty times that amount. For the latter mass, 
the theory indicates that four-fifths of its gravitation is balanced 
by the pressure of radiation, and under the action of these two 
opposed and nearly balanced forces the star would probably 
be on the verge of instability. High radiation pressure, in 
fact, seems to produce a state in which a star may easily be 
broken up by rotation or other disturbing cause. The mass of 
the giant star determines its absolute magnitude : a giant star 
of mass one-half that of the Sun will have an absolute magni- 
tude of 2-6 m., and one of 4-5 times the Sun's mass will have 
an absohite magnitude of 2-4 m. 

If the density of the star is too great to enable the laws of 
a perfect gas to be applied, the theory becomes more uncertain 
because further assumptions must be made. Eddington 
assumed that van der Waal's equation holds in the interior, 
adjusting the constants to agree with observation at two points, 
and was able to calculate the effective temperatures of stars 
of various masses and densities. The results show that stars 
of small mass cannot rise to such high temperatures as more 
massive stars, which supports Russell's contention : in par- 
ticular, a star of mass less than one-seventh of the Sun could 
not attain an effective temperature of 3,000, and would there- 
fore never become visible. In this connection it is suggestive 
that no star is known with a mass less than one-tenth that of 
the Sun. The absence of type-M stars of a range of absolute 
magnitude between the giants and the dwarfs indicates that 
stars of less than half the Sun's mass are comparatively un- 
common. To reach the B-type stage, a mass at least 2-2 times 
the Sun's mass is necessary. Approximate determinations of 


the effective temperatures of stars of different types are deduced 
from the theory, which agree well with observation. The 
theory in fact correlates many phenomena, supplies a physical 
basis for Russell's theory, and gives a sufficiently accurate 
picture of the phenomena taking place in the interior of a 

238. Theories of Cosmogony. We have gained some 
insight into the nature of the interior of a star and of the 
processes of evolutionary development of an individual star. 
But wo have also seen that the Universe is not a mere collec- 
tion of isolated stars : there are more complex structures 
which are met with so frequently that involuntarily we ask 
ourselves the question how they have come to exist. Spiral 
nebulae, globular clusters, double and multiple stars so many 
examples of each type of system are known that in each case 
a common process of formation and development must have 
taken place. There is also our solar system itself : though 
unique in the stellar universe as far as we are aware, it is 
possible that there may be many similar systems which obser- 
vation can never reveal to us. How have these various 
structures been formed ? Many theories of cosmogony have 
been advanced to account for some of them. All are more or 
less speculative, though some are more plausible than others ; 
none are entirely free from objections. It would be outside 
the scope of this volume to give more than a brief summary of 
a few of these theories with an indication of the present posi- 
tion of the problem. Although most of the theories of cos- 
mogony were propounded originally with a view to explaining 
the structure of the solar system, the majority of them may 
more appropriately be utilized to explain some of the other 

239. The Theory of Laplace. Laplace supposed the 
solar system to have originated out of a flattened mass of 
gas or nebula, extending beyond the present orbit of Neptune, 
which was at the outset at a high temperature and in rotation. 
The mass gradually cooled by radiation at its surface, and at 
the same time contracted under the influence of its own 
gravitation. This resulted in a heating of the central portion 


and an increase in the angular velocity of rotation, since the 
angular momentum must necessarily have remained constant. 
With continual increase in the angular velocity, the centri- 
fugal force at the equator at length became greater than 
gravity ; Laplace supposed that as a result a ring of matter 
was left behind along the equator and that further contraction 
detached a series of such rings. Each ring was supposed to 
break up and condense into a gaseous planet which in turn 
went through a similar process of evolution on a smaller scale, 
resulting in the formation of satellites. The theory, it will 
be seen, is given a form which might explain the ring-system 
of Saturn, as well as the satellites of the other planets, but it 
does not attempt to explain why the supposed ring-systems 
should become unstable and break up, nor why the ring system 
of Saturn remained stable. 

The objections to this theory, as applied to the solar system, 
are numerous and strong. 

1. The angular momentum of the system must have 
remained constant during the evolution. It is found by 
mathematical investigation that this angular momentum 
would not have been sufficient to cause the detachment of 
matter when the system extended to any of the planets unless 
the original nebula was very strongly condensed towards the 

2. The matter left behind during the contraction would 
not form definite rings ; the separation of matter would be 
continuous and lead to another gaseous nebula, not in rotation. 

3. Even if the rings were produced as the theory requires, 
these rings could not condense into a planet. 

4. The theory is not able to account for satellites revolving 
in the opposite direction to their primaries, as do two of the 
satellites of Jupiter and one of Saturn. Nor can it account 
for satellites revolving in a shorter time than their primaries, 
as in the case of Phobos, one of the satellites of Mars. 

Laplace's theory presupposes that all the planets were 
formerly gaseous. Jeans has investigated whether by modifi- 
cation the theory might be made more plausible. He finds 
that before the ejected matter could form a ring it must have 
increased in density to the order of 200 times the density in the 
outer regions of the nebula. This is only possible by supposing 


that the ejected matter liquefied shortly after ejection. But 
this supposition in itself raises further difficulties : in order 
that this liquid mass should itself break up and produce 
satellites, an enormous shrinkage would be necessary, and even 
granting that this might have occurred, mathematical investiga- 
tion shows that the break-up would then be into masses of 
comparable size. Moreover, the central mass ought to have 
continued disintegrating until a double star was formed. It 
seems probable, therefore, that this theory must be abandoned 
as far as the origin of the solar system is concerned. 

240. The Planetesimal Theory. This theory has been 
developed by Chamberlin and Moult on as an alternative to 
the theory discussed in the previous section. It supposes an 
initial non-rotating gaseous mass which, under the influence 
of its own gravitation, would be spherical in shape. If a 
second body passes sufficiently near this mass, tidal forces 
are produced, causing tidal protuberances to be raised directly 
under and directly away from the second body. As the 
bodies approach one another more closely, their tides will 
rise in height until, according to the theory, tw<"> jets of matter 
will rush out from the two antipodal points where the tides 
are highest. As the tide-raising body passes on, it will be 
somewhat ahead of the diameter through the tidal protuber- 
ances and the resulting couple will set the primary body in 
rotation. The two jets of nebulous matter are therefore 
ejected from a slowly rotating body and the ejected matter 
will take the form of two spiral arms. It is supposed further 
that the ejection takes place by pulsations, corresponding to 
the nuclei observed in the arms of a spiral nebula. This 
portion of the theory appears to give a reasonable explanation 
of the typical spiral form which is of such common occurrence. 

The second part of the theory deals with the evolution of 
the ejected matter. It is supposed that the larger nuclei 
steadily grow by picking up the smaller ones, which are termed 
planetesimals, and so form planets. This process would result 
in a diminution of the eccentricity of the original highly- 
eccentric orbits. It is difficult to decide whether this would 
have happened or not, but it would seem that collisions between 
planetesimals must have been much more numerous than 


collisions between planets and planetesimals, and that as a 
result of such collisions, the planetesimals would have been 
turned to gas before the nuclei could have gained much by 

This theory, regarded as a theory of the origin of spiral 
nebulae, is open to strong objection, for it requires the close 
approach of two large bodies. Spiral nebulae, as we have 
seen, occur with very great frequency in the stellar system, 
whilst the close approach of two bodies required by the theory 
must be of very rare occurrence. On the grounds of prob- 
ability, it is impossible that spiral nebulae in general can have 
been so formed. If we regard the solar system as an abnormal 
system with few other parallels in the Universe, Jeans has 
shown that a tidal theory will give a plausible explanation 
of its origin : though not free from objections, it provides a 
more satisfactory explanation than any alternative theory. 

241. Jeans 's Theory. Jeans supposes that the relative 
velocity of the two bodies at the time of encounter was 
small. Matter was ejected slowly at first, but at a rate which 
increased gradually until the distance of closest approach was 
attained ; thereafter at a rate decreasing and finally diminish- 
ing to zero. The result would be an ejected filament of matter, 
with density small at the ends and greatest near the centre. 
Owing to radiation, its temperature would fall most rapidly 
at the ends and more slowly near the middle, so that in course 
of time liquefaction would commence near the ends and 
ultimately instability would set in and the mass break up 
into detached masses. Jeans shows that the smallest masses 
would form out of the densest matter, so that the theory 
accounts for the two largest planets, Jupiter and Saturn, being 
in the middle of the series and explains why the smaller planets 
must have been liquid or solid from birth (to which other 
considerations also point) whilst the larger planets were 
probably gaseous. The tidal forces exerted on the planets by 
the Sun the central mass resulted, in a similar way, in the 
creation of systems of satellites. On this theory, the direc- 
tion of revolution of the majority of the satellites is accounted 
for at once, and it also explains why their orbital planes are, 
with few exceptions, inclined at small angles to the orbital 


planes of the planets. The theory also accounts for the 
comparative crowding of the planets near to the Sun and for 
the corresponding phenomenon in the systems of Jupiter and 
Saturn. The planets at the two ends of the series, Mercury, 
Venus, Uranus, and Neptune, which condensed to a liquid 
state at a very early stage, would not be likely to be broken 
up by tidal friction, and we should therefore expect them 
not to have satellites. The satellites of Uranus, which have 
a high inclination, and that of Neptune, which has retrograde 
motion, may have been otherwise formed. The satellites of 
Mars and the Earth-Moon system present difficulties on this 
as on other theories. 

Other theories, such as the Capture Theory of See, have 
been put forward to account for the origin of the solar system, 
but they do not appear as plausible or to have so solid a 
foundation as the above and will therefore not be discussed here. 

242. The Origin of Spiral Nebulae . Jeans has developed 
a theory of the origin of spiral nebulae based upon a mathe- 
matical investigation of the behaviour of rotating, gravitating 
masses. He considers the permanent astronomical bodies as 
beginning existence as a gaseous mass in a state of extreme 
rarity. Such a mass, if out of the influence of other bodies, 
would assume a spherical form if not in rotation, or a spheroidal 
form if slowly rotating. Under the influence of the gravi- 
tational attractions of other masses, such a mass would be set 
in motion, and the tidal couples produced when any two masses 
pass in the neighbourhood of one another would gradually 
produce slow rotations. These would increase as the mass 
shrinks, the angular momentum remaining constant. The 
heavier elements will tend to collect near the centre, the 
lighter elements forming a surrounding atmosphere : there 
will thus be a central condensation of mass. Under such 
circumstances, with increasing rotation the mass will in time 
assume a lenticular figure with a sharp edge from which matter 
tends to be thrown off. Such a figure corresponds with the 
shape of certain nebula), such as N.G.C. 3,115 or N.G.C. 5,866. 
This figure will not rotate as a rigid body owing to viscosity, 
although the angular velocity will increase outwards from 
the centre. 


In the unstable state, when matter is about to be thrown 
off, the exact points at which the break-up will commence 
are conditioned by the slight external gravitational field due 
to other and distant bodies. The cross-section will become 
slightly elliptical and the ejection will occur at the two ends 
of the major axis of this ellipse. Should there be no external 
field, matter would be ejected as a ring which would be dis- 
integrated by its own rotation. In the case under considera- 
tion, the ejection will be in the equatorial plane, and theory 
shows that it will continue almost indefinitely from the same 
two antipodal points. The long streams of gas emitted must 
become longitudinally unstable and will tend to break up 
into condensations or nuclei under their own gravitational 
attraction, and in this way the nuclei observed on the arms of 
spiral nebulae might be accounted for. Jeans is able, from 
an examination of the conditions under which the ejected 
matter will condense into nuclei, to conclude that these 
nuclear condensations will have a mass comparable with that 
of the Sun, so that the masses of the spirals must be supposed 
to be enormously greater than that of the Sun. These results, 
which have a firm theoretical basis, are fully in accord with 

243. The Evolution of Star Clusters. It is thought by 
some that the final process of disintegration which is going 
on in spiral nebulae, and which we have seen is progressive, 
will be a globular star cluster. In this connection, it is 
significant that many star clusters are found to have galactic 
planes of symmetry. The conjecture is not free from objec- 
tions and cannot be regarded as in any sense established. 
But whether true or not, the question is suggested as to the 
final result of the evolution of star clusters. 

A star cluster may be compared with a mass of gas, the 
individual stars corresponding to molecules. In the mass of 
gas, the molecules are moving about in different directions, 
and continually colliding with one another, until rapidly a 
steady state is attained in which the molecular velocities are 
distributed at random and the mean square velocity is the 
same in any portion of the gas. Will such a steady state be 
attained in a star cluster in astronomical time ? 


In the star cluster the relative velocities of the stars arc 
comparatively small and the density of matter is also very 
small. With reasonable estimates of these quantities for our 
local stellar system, it is found that there would be, on the 
average, only one direct collision between stars in a period 
of about 4 x 10 12 years. As regards " encounters," when 
two stars approach sufficiently close for their mutual gravita- 
tion to influence their motion, it is found that there would 
on the average be only one encounter in 10 11 years which 
would cause a deviation in path of more than 1. These 
intervals of time are so long that the effect of only those 
feeble encounters which produce deviations of less than 1 
need be considered. The result of this investigation is to 
establish that the feeble encounters with neighbouring stars 
may be neglected, and the changes in stellar velocities may 
therefore l)e regarded as coming from the forces exerted by 
the Universe as a whole. The problem then becomes a problem 
of the kinetic theory of gases, simplified by the omission of 

Jeans finds that under these circumstances the only possible 
configurations when a steady state has been attained are 
those in which the stars form a spherically symmetrical figure 
or a figure of revolution symmetrical about an axis, but that 
the former could not have originated out of any system in 
which the angular momentum was not zero. In the latter 
case, the velocities at any point will not be distributed uni- 
formly for all directions in space, in other words, the phenome- 
non of star-streaming will occur, the direction of streaming 
being along the axis of the system. 

An investigation of the time required for a steady state to 
be attained for our Universe indicates that the approach to 
a final steady state, though fairly rapid in the early stages, 
must later be excessively slow after an interval of 10 million 
years, the courses of the stars in our Universe will be but 
little altered. It might be concluded, therefore, that though 
our system is approaching a steady state, it has not yet reached 
it. If the steady state had been attained, the kinetic energy 
of every star would be the same and there would be no star- 
streaming. These conditions are not obeyed. For the globular 
clusters, on the other hand, it is estimated a steady state will 


have been more nearly obtained than in our own system, 
but that they are hardly likely yet to have finally reached 
the steady state. 

244. The Origin of Binary Stars: The problem of the 
origin of binary and multiple systems is one beset with many 
difficulties, so that it is not easy to draw any definite con- 
clusions. Considering one of the nuclei of the original nebula, 
it follows that this nucleus, originally in rotation, will continue 
to contract, its rotation meanwhile becoming more rapid until 
at length ejection of matter commences. This matter will 
not be ejected at a sufficient rate to condense into nuclei ; 
it may either be dissipated into space or form an atmosphere 
about the star. Planetary nebula 1 ., nebulous stars, and ring 
nebula, may possibly be accounted for iri this way. The 
equatorial ejection of matter will continue until a further 
critical density is reached, when a pseudo-ellipsoidal form 
develops which, under certain circumstances, will change to 
a pear-shaped figure and finally will develop a neck in the 
middle and divide into two detached masses, rotating about 
one another in the atmosphere of ejected matter. This 
atmosphere will ultimately condense round the two stars, 
leaving a binary system. 

If the stars are regarded as masses of ordinary gas, fission 
cannot take place until the ideal gas laws are substantially 
departed from, at a density of something like J. It follows on 
this theory that no binary star which was formed by fission 
could have a density of less than about J, and also that no 
giant binary star could have been formed by fission. These 
conclusions cannot be reconciled with observation unless it is 
supposed that not all binaries have been formed by fission. 

The theory probably requires modification on account of 
the very high temperature prevailing in the interior of a star, 
which causes almost complete ionizatioii with a consequent 
reduction in the molecular weight. This might reduce the 
limit of the critical density from J to ^ {r 

Granting that fission has occurred, further shrinkage will 
result in an increase in the rate of rotation of each star. The 
rotations will become more rapid than the mutual period of 
revolution and tidal couples will be produced, the effect of 


which will be to tend to equalize the periods of rotation and 
of revolution. As a consequence of such tidal action, the 
evolution of the binary system should be accompanied 
by increasing separation, increasing period, and increasing 

Statistics of binary systems appear at first sight to accord 
admirably with these conclusions, as will be seen by 
reference to the table given in 219. A closer theoretical 
examination, however, indicates that there are limits to 
the possible increase with evolution of separation and 
period. The latus rectum of the orbit cannot increase 
by more than 90 per cent., and the greatest possible 
increase in period is one of 13*6 times with, however, in a very 
large majority of binaries, an increase not exceeding 4-4 times. 
The linear dimensions and period are not therefore subject 
to great changes. In the course of evolution, spectral type 
and ecccntrioity will vary, but the period remains of the same 
order. The theory therefore does not explain why short - 
period spectroscopic binaries have small eccentricities and are 
generally of early spectral type, whilst long -period binaries 
have high eccentricity and are generally of late spectral 

Such an explanation may possibly be found in the general 
cumulative effect of successive stellar encounters. Theory 
proves that these produce the same general effects as tidal 
friction increases in eccentricity, linear dimensions, and 
period but that the possible increases bear no relation to 
the original values. The dependence of eccentricities and 
periods upon spectral type may possibly find an explanation 
in the' hypothesis that the binary systems of late type were 
formed at a much earlier date than those of the B and A 
types, and that therefore close encounters have not only had 
a longer time in which to act but also in the early stages they 
would have been far more numerous. But this is barely more 
than conjecture. 

Whilst some binary stars have undoubtedly been formed 
by fission as just described, it is very probable that others 
are systems which have evolved from two adjacent nuclei in 
the original nebular arms and have remained permanently in 
mutual rotation under their gravitational attraction. 



245. Multiple Stars. The question of the evolution of 
multiple stains may be briefly discussed. We commence with 
a binary system ; each component will continue to shrink 
and the velocity of rotation to increase correspondingly. The 
effect of tidal friction will be to diminish somewhat the angular 
momentum of each component and therefore to delay a further 
fission. Neglecting such effect, theoretical investigation shows 
that a further fission cannot occur until the total increase in 
density since the first fission is 342 times, whilst the linear 
dimensions of the sub-system will be about one-seventh of 
those of the original system and its period will be about one- 
eighteenth that of the original system. The effect of tidal 
friction will be to increase these inequalities of dimensions, 
density, and period when fission occurs. For either component 
of the sub-system to divide again, the density must be at 
least (342) 2 or 11,700 times that at the original fission, and the 
period of the new sub-system will be less than 1/342 times 
that of the primary system. A typical multiple system is 
that of Polaris, which is composed of a spectroscopic triple 
system having periods of 4 days and 12 years in rotation 
about a fourth visible star with a period of the order of 
20,000 years. 


Abbot, 137 

Aberration, 64 ; * constant of, 66 ; 

chromatic, 163 ; diurnal, 68 ; 

spherical, 163 

Absorption of light in space, 360 
Acceleration, secular, of Moon, 88 
Adams, 208, 262 
Aerolites, 278 
Age of Moon, 79 
Albedo, 230 
Algol, 345 
Almucantar, 4 
Altazimuth telescope, 178 
Altitude, 4 
Annual equation, 85 
Annular eclipse, 150 
Anomalistic period, 84 ; year, 69 
Antarctic Circle, 41 
Aphelion, 39 
Apogee, 39, 81 
Apparent place of star, 202 
Arctic Circle, 41 
Argelander, 287 
Aries, First Point of, 7 
Aristotle, 266 
Asteroids, size and number of, 244 ; 

orbits of, 245 ; origin of, 245 
Azimuth, 4 ; determination of, 204 
Azimuth error of transit circle, 175 

Baily's Beads, 155 

Barnard, 239, 250, 363 

Bayer, 285 

Belopolsky, 235 

Bolot, 216 

Bessel, 203, 261, 311 

Binaries, spectroscopic, 327; orbits 

of, 328 
Binary Stars, 323; origin of, 349, 


Black body radiation, 137 
Bocle's Law, 216, 243 
Bohnenberger eyepiece, 174 
Bond, 255, 257 

Boss, 319 
Bo uguer, 19 
Bouvard, 261 
Boys, 21 
Bradley, 63, 311 
Bredichin, 275 
Brown, 88 

Calendar, Gregorian, 70 ; Julian, 70 
Calendar reform, 71 
Campbell, 241, 318, 328 
Canals on Mars, 239 
Cancer, Tropic of, 41 
Capricorn, Tropic of, 41 
Cassegrain telescope, 165 
Cassini, 238, 255, 257 
Cavendish experiment, 20 
Celestial Equator, 4 ; sphere, 1 
Cepheid variables, 340 : luminosity- 
period of, 342 ; pulsation theory 

of, 342 

Ceres, 243, 244 
Chamberlin, 379 
Chandler, 26 
Charlois, 244 
Chromosphere, 133 
Chronograph, 172, 179 
Clairaut, 87 
Cluster, k>9al, 359 
Clusters, ,globular, 356 ; distance of, 

357 ; system of, 358 ; open, 362 ; 

stellar, 355 ; evolution of, 382 
Collimation, 172 
Colour index, 289, 306 
Colour of sky, 28 
Coluro, equinoctial, 7 
Comets, 265 ; appearance of, 272 ; 

density of, 273 ; families of, 269 ; 

groups of, 270 ; number of, 265 ; 

old views on, 266 ; orbits of, 267 ; 

origin of, 268 ; periodic, 269 ; size 

and mass of, 272 ; spcctras , 274 ; 

tail of, 274 




Conjunction, 7G, 209 ; inferior, 209 ; 

superior, 209 

Constellations, stellar, 284 
Copernicus, 212, 284 
Corona, solar, 134 
Coronium, 135 
Cosmogony, theories of, 377 
Craters, lunar, 102 
Culmination, 7 
Curtis, 303 

Darwin, 100 

Date line, 55 

Dawes, 255 

Day numbers, 203 

Day, sidereal, 47 ; solar, 49 

Declination, 4 ; determination of, 201 

Deferent, 212 

Deimos, 242 

Denning, 248, 266, 332 

Do Witt, 113 

Diameters, angular, of Stars, 315 ; 

of Planets, 220 
Differential observations, 201 
Diffraction grating, 295 
Dip of horizon, 32 
Distance of globular clusters, 357 ; 

Milky Way, 354; Moon, 94; 

spiral nebulse, 368 ; Stars, 310 ; 

Sun, 108 

Doppler's principle, 120 
Double Stars 323 ; masses of, 326 ; 

measurement of, 324 ; orbits of, 

325 ; origin of, 349, 384 
Draconian period, 84 
Draper catalogue, 304 
Dwarf Stars, 321, 372 

Earth, age of, 139 ; atmosphere of, 
28 ; eccentricity of orbit of, 39 ; 
ellipticity of, 16; interior of, 24; 
mass of, 17 ; mean density of, 24 ; 
orbit of, 37 ; rotation of, 11, 
14, 15 ; shape of, 11 ; size of, 15 

Earth's way, 60 

Easter Day, date of, 73 

Eccentricity of orbits, 39 

Eclipse limits, 142 

Eclipses, cause of, 141 ; lunar, 148 ; 
number in one year, 145 ; recur- 
rence of, 147 ; solar, 150 ; of 
Jupiter's satellites, 197 

Eclipsing variables. 344 

Ecliptic, 36 ; obliquity of, 36 

Edclington, 375 

Effective temperature of Stars. 307 ; 

Sun, 137 

Elements of planet's orbit, 216 
Elements in Sun, 119 
Ellipticity, 16 
Elongation, 210 . 
Encke, 255 
Enhanced lines, 133 
Epicycle, planetary, 212 
Epoch, 216 
Equation, annual, 85 ; of centre, 81 ; 

personal, 192 ; of time, 50 
Equator, Celestial, 4 
Equatorial telescope, 182 
Equinoctial eolure, 7 
Equinox, 7, 36 ; autumnal and 

vernal, 7, 36 ; precession of, 58 
Eros, 113 

Error of clock, 182 
Evection, 83 
Evorshed, 131 
Eyepieces, 169; Bohnenberger, 174; 

Iluyghenitin, 169 ; Kamsden, 169 

Fabricius, 337 

Facula3, 122 

Families of comets, 269 

Filar micrometer, 185 

Fizeau, 315 

Flash spectrum, 133 

Flamsteed, 286 

Flocculi, 130 

Foueault's experiment, 13 

Fowle, 137 

Fraunhofer spectrum, 133 

Fundamental observations. 201 

Galactic condensation of Stars, 352, 


Galactic plane, 351 
Galaxy, 351 
Galileo, 102, 249, 254 
Gauss, 219, 243 
Giant Stars, 321, 372 
Gill, 112 
Globular clusters, 356 ; distance of, 

357 ; system of, 358 
Gnomon, 59, 195 
Golden number, 73 



Goodricke, 345, 347 

Graham, 181 

Gravitation, constant of, 17 ; deter- 
mination of, constant of, 22 ; 
law of 17 ; universality of, 208 

Gregorian calendar, 70 ; telescope, 

Groups of comets, 270 

Grubb, 184 

Guillaume, 181 

Hale, 123, 130, 168 

Halley, 267, 284 

Hansen, 88 

Harvest Moon, 98 

Heliacal risings, 59 

Heliometer, 188 

Helium, 121 

Helmholtz, 138 

Henderson, 311 

Herschol, J., 2$7 

Herschel, W., 243, 257, 259, 200, 300, 

323, 352, 362 
Hertzsprung, 372 
Hipparchus, 60, 287 
Horizon, 3 
Horrocks, 110 
Hour angle, 5 ; circle, 4 
Huggins, 274, 299 
Huyghens, 180, 238, 255, 257 
Huyghenian eyepiece, 160 

Imago, brightness of, 161 

Inferior conjunction, 209 

Interior of a Star, 375 

Invariable plane, 218 

loiiisatioii in stellar atmospheres, 308 

Irregular nebula?, 363 

Irregular variables, 339 

Janson, 333 

Joans, 380, 383 

Jones, 319 

Julian calendar, 70 ; date, 72 

Juno, 244 

Jupiter, eclipses of satellites of, 197 ; 
orbit and size of, 246 ; physical 
state of, 248 ; red spot on, 248 ; 
rotation period of, 247 ; satellites 
of, 249 ; telescopic appearance of, 

Kapteyn, 301 

Keeler, 256 
Kepler, 266, 333 
Kepler's Laws, 39, 205 
Kustner, 26 

Lagrange, 218 

Lane, 372 

Langley, 137 

Laplace, 218, 259, 377 

Latitude, celestial, 37 ; determina- 
tion of, 193 ; variation of, 26 

Law of equal areas, 206 

Law of gravitation, 17 ; universality 
of, 208 

Leonid meteor swarm, 281 

Level error of transit instrument, 172 

Leverrier, 208, 262, 263 

Libra, First Point of, 36 

Libratioii of Moon in latitude, 93 ; 
in longitude, 92 

Light ratio, 287 

Light, total, of Stars, 296 

Light-year, 310 

Local cluster, 359 

Local time, 54 

Lockyer, 121 

Longitude, celestial, 37 ; determina- 
tion of, 195 

Long period variables, 336 

Lowell, 235, 240 

Luminosities of Stars, 321 

Magnetic, field of Sun-spots, 123 ; 

storms, 126 
Magnitudes of Stars absolute, 313 ; 

determination of absolute, 314 ; 

photographic, 288 ; visual, 286 ; 

determination of apparent, 289 
Maintenance of Sun's heat, 137 
Mars, 236 ; atmosphere of, 240 ; 

canals on, 239 ; orbit of, 236 ; 

polar caps of, 238 ; rotation 

period of, 237 ; satellites of, 242 ; 

size and shape of, 236 ; telescopic 

appearance of, 237 ; temperature of, 


Maskelyne, 19 
Mass of double stars, 326 ; planets 


Maunder, 126 
Maxwell, 255 
Mayer, 138 

Mean place of Star, 203 
Mean Sun, 52 



Mercury, 228 ; atmosphere of, 230 ; 
markings on, 230 ; physical nature 
of, 230 ; rotation period of, 229 ; 
size of, 229 ; telescopic appear- 
ance of, 229 ; transits of, 157 

Meridian, 3 

Meridian Circle, 177 

Meridian photometer, 291 

Meteors, 276 ; connection with 
comets, 280 

Meteor radiants, 278 

Melon, 72 

Metonic cycle, 72 

Mieholl, 323 

Michelson, 221, 315 

Micrometer, filar, 185 

Milky Way, 351 ; distance of, 354 

Minor planets (see Asteroids) 

Month, lunar, 76 

Moon, ago of, 79 ; apparent motion 
of, 75 ; concavity of path to Sun, 
89 ; distance of, 94 ; harvest, 98 ; 
li brat ions of, 92 ; mass of, 96 ; orbit 
of, 81 ; origin of, 106 ; parallax of, 
95 ; phases of, 77 ; physical condi- 
tions, 104; rising and setting on, 97; 
rotation of, 91 ; secular acceleration 
on, 88 ; sidereal revolution on, 76 ; 
size of, 95 ; surface structure on, 
101 ; synodic revolution on, 76 ; 
tables of motion of, 87 ; tropical 
revolution on, 76 

Motion of planets, laws of, 205 

Motion in resisting medium, 224 

Moulton, 379 

Multiple Stars, 330 ; origin of, 386 

Nadir, 2 

Neap tides, 101 

Nebulas, 362 ; distance of, 368 ; 

irregular, 3(>3 ; nature of, 370 ; 

origin of, 381 ; planetary, 365 ; 

spiral, 367 
Nebulium, 364 
Neptune, 261 ; satellites of, 263 ; 

telescopic appearance of, 262 
Newton, 87, 206, 266 
Newtonian telescope, 165 
Nichols and Hull, 275 
Nodes of Moon's orbit, 81 
North polar distance, 4 
Novae, 331 ; in spiral nebulae, 369 ; 

spectral changes of, 334 ; theories 

of, 335 

Numbers of Stars of various magni- 
tudes, 296 
Nutation, 63 

Obliquity of ecliptic, 36 

Occultations, 156 

Opposition, 76 

Orbit, of binary star, 325 ; deter- 
mination of planet' s, 219; elements 
of planet's, 216 ; of Moon, 81 ; 
gpectroscopic binary, 328 

Origin of binary stars, 349 ; comets, 
268 ; Moon, 106 

Palisa, 244 

Pallas, 243 

Parabolic velocity, 226 

Parallax of Moon, 95; Stars, 310; 
Sun, 108 

Parsec, 310 

Perigee, 39, 81 

Perihelion, 39 

Periodic comets, 269 

Periodicity of Sun-spots, 124 

Perrotin, 239 

Personal equation, 192 

Phases of Moon, 77 ; Venus, 232 

Phobos, 242 

Photographic magnitudes, 288 ; 
determination of, 294 

Photometer, meridian, 291 ; Zollner, 

Piazzi, 213 

Pickering, 291, 328, 331, 338 

Planetary nebulae, 365 

Planotosimal theory, 379 

Planets, 205 ; apparent motion of, 
209 ; densities of, 223 ; diameters 
of, 220 ; determination of dia- 
meters of, 220 ; elements of orbits 
of, 216 ; determination of elements 
of, 219 ; laws of motion of, 205 ; 
mass of, 222 ; minor, 243 ; periods 
of, 221 ; statistics of, 227 ; surface 
gravities of, 223 

Pogson, 288 

Pole of celestial sphere, 2 

Poynting, 23, 241 

Procession, 58 ; physical cause of 
60 ; variation of, 62 

Pressure of radiation, 275 

Prime vertical, 3 

Prominences, solar, 130 

Proper motions, 297 



Ptolemaic system, 212, 233 

Ptolemy, 287 

Pulsation theory of cepheids, 342 
Purkinjo effect, 293 
Pyrheliometer, 136 

Quadrature, 76, 209 

Radiation, pressure of, 275 

Radial velocity, 298 

Ramsden eyepiece, 169 

Rate of clock, 182 

Red spot on Jupiter, 248 

Reflecting telescope, 165 

Refracting telescope, 160 

Refracting and reflecting telescopes, 
relative advantages of, 167 

Refraction, atmospheric, 29 ; ele- 
mentary theory of, 30 ; law of, 30 

Resisting medium, motion in, 224 

Resolving power, 161 

Retrograde motion of planets, 210 

Reversing layer, 134 

Riefler, 181 

Right Ascension, 7 ; determination 
of, 201 

Ring system of Saturn, 253 ; dimen- 
sions of, 255 ; structure of, 255 

Ritter, 372 

Roemer, 251 

Russell, 321, 372, 373 

Saha, 309 

St. John, 123, 235 

Saros, 143 

Satellites of Jupiter, 249 ; Mars, 

242 ; Neptune, 263 ; Saturn, 257 ; 

Uranus, 260 
Saturn, dimensions of rings, 255 ; ring 

system of, 253 ; rotation period of, 

254 ; satellites of, 257 ; size and 
orbit of, 253 ; structure of rings of, 

255 ; telescopic appearance of, 254 
Schiaparelli, 239 

Schwabe, 125 
Schwarzschild, 303 
Seasons, 42 ; length of, 43 
Secchi, 304 
Soeliger, 257 
Shadow bands, 155 
Shaploy, 354, 357 
Short period variables, 339 
Sidereal period, 214 ; time, 8, 47 ; 
year, 69 

Slipher, 235, 239 

Solar constant, 135 ; motion, 299 

Solstices, 36 

Spectra, 118 ; of comets, 274 

Spectral changes of novae, 334 

Spectral types of stars, 304, 306; 
phenomena associated with, 319 

Spectroheliograph, 129 

Spectroscope, 118, 186 

Spectroscopic binaries, 327 ; orbits 
of, 328 

Sphere, celestial, 1 

Spiral nebular?, 367 ; distance of, 368 ; 
nature of, 370 ; origin of, 381 

Spring tides, 101 

Stability of solar system, 217 

Standard time, 54 

Stars, absolute magnitudes of, 313 ; 
angular diameters of, 315 ; dist- 
ances of, 310 ; effective tempera- 
tures of, 307 ; evolution of, 371 ; 
galactic condensation of, 352 ; 
giant and dwarf, 321, 372 ; 
interior of, 375 ; lino of sight 
velocities of, 298 ; luminosities of, 
321 ; number of, 296 ; proper 
motions of, 297 ; spectral types of, 
304, 306 ; total light of, 296 ; 
variable, 331 ; velocities of, 321 

Star clusters, 355 ; evolution of, 

Star streams, 301 

Statistics of planets, 227 

Stellar magnitudes, 286 

Stefan's law, 137 

Stromgren, 267 

Struve, 311 

Sumner line, 199 

Sun, apparent motion of, 34 ; con- 
stitution of, 118; corona of, 134; 
declination of, 40 ; distance of, 108 ; 
mass of, 115; parallax of, 108; 
prominences on, 130 ; rotation of, 
117 ; size of, 115 ; spots on, 122 ; 
surface of, 121 ; temperature of, 
135 ; velocity of, 299 

Sunrise and sunset, times of, 53 

Sun spots, 122 ; magnetic field of, 
123 ; periodicity of, 124 ; spec- 
trum of, 122 ; vortices, 123 

Superior conjunction, 209 

Surface gravities of planets, 223 

Synodic period, 111, 214 



Talcott, 193 

Telescopes, 159 ; equatorial, 182 ; 
reflecting, 105 ; refracting, 160 ; 
resolving power of, 161 ; zenith, 

Temperature of Mars, 240 ,* Stars, 
307 ; Sun, 135 

Terminator, 79 

Thollon, 239 

Tides, 99 

Time, 46 ; apparent solar, 48 ; 
determination of, 191 ; determina- 
tion of, at sea, 192 ; equation of, 50 ; 
local, 54 ; moan solar, 49 ; sidereal, 
8, 47 ; standard, 54 

Titan, 257 

Torsion balance, 19 

Transit instrument, 171 ; adjust- 
ments of, 172 ; circle, 177 

Transits of Mercury, 157 ; Venus, 
109, 157 

Triple Stars, 330 

Tropics, 36 

Tropical year, 68 

Twilight, 29 ; duration of, 42 

Tycho Brahe, 266, 334 

Uranus, 258 ; rotation period of, 260 ; 
satellites of, 260 ; telescopic appear- 
ance of, 260 

Variability of Earth's rotation, 15 

Variable Stars, 331 ; eclipsing, 344 ; 
irregular, 339 ; long period, 336 ; 
short period, 339 

Variation of latitude, 26 

Variation, lunar, 86 

Velocity, from infinity, 225 ; line of 
sight, 298 ; parabolic, 225 ; in 
planetary orbit, 225 

Venus, 231 ; appearance of , 228, 233 ; 
brightness and phases of, 232 ; 
physical conditions of, 235 ; rota- 
tion period of, 233 ; size of, 232 j 
transits of, 157 

Vernal equinox, 7, 36 

Vertical, circles, 3 ; prime, 3 

Vesta, 244 

Visual magnitudes, 286 ; determina- 
tion of, 289 

Vulcan, 263 

Vortices, Sun-spot, 123 

Wolf, 244 

Wolf-Rayet Stars, 305, 354, 363, 366 

Year, anomalistic, 69 ; civil, 69 ; 
sidereal, 68 ; tropical, 68 

Zeeman effect, 123 

Zenith, 2 ; distance, 4 ; telescope, 


Zodiac, 36 

Zodiacal light, 281 ; spectrum of, 282 
Zollner photometer, 289 
Zone time, 55