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OSMANIA UNIVERSITY LIBRARY

Gall No. Accession No.

* i

Author

Title

This book should be returned on or before the date

last marked below.

GENERAL
ASTRONOMY

H. SPENCER JONES, M.A., B.Sc.

H.M. ASTRONOMER AT THE CAPE OF r,oon HOPS.
LATE FELLOW OF JESUS COLLEGE, CAMBRIDGE.

THIRD IMPRESSION

LONDON
EDWARD ARNOLD & CO.

1924

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

PREFACE

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
Astronomy.

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,
1922.

H. SPENCER JONES.

CHRISTMAS ISLAND.
1922.

CONTENTS

CHAP. PAGE

I THE CELESTIAL SPHERE ...... 1

II THE EARTH 11

III THE EARTH IN RELATION TO THE SUN ... 34

IV THE MOON 75

V THE SUN 108

VI ECLIPSES AND OCCULTATIONS . . . . .141

VII ASTRONOMICAL INSTRUMENTS . . . . .159

VIII ASTRONOMICAL OBSERVATIONS . . . . .191

IX THE PLANETARY MOTIONS ...... 205

X THE PLANETS 228

XI COMETS AND METEORS ...... 265

XII THE STARS 284

XIII DOUBLE AND VARIABLE STARS ..... 323

XIV THE STELLAR UNIVERSE . . . . . .351

INDEX e ........ 387

vii

LIST OF PLATES

FACING
NO. PAGE

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

viii

CHAPTER I
THE CELESTIAL SPHERE

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

2 GENERAL ASTRONOMY

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 CELESTIAL SPHERE

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
points.

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.

4 GENERAL ASTRONOMY

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
horizon.

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

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
Experiment.

14 GENERAL ASTRONOMY

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.

THE EARTH 15

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

16 GENERAL ASTRONOMY

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

1
298^3'

THE EARTH 17

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

18 GENERAL ASTRONOMY

" 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

THE EARTH 19

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
obtained.

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

GENERAL ASTRONOMY

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

THE EARTH

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

GENERAL ASTRONOMY

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
5448.

(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,

THE EARTH

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

24 GENERAL ASTRONOMY

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
5-493.

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,

THE EARTH 25

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

26 GENERAL ASTRONOMY

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-

THE EARTH

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

+0-30 -

+0-40 +0-30

-0-30

- -0-20

- -0-10

0-00 -

- +0-10

- +0-20

-+0-30

-030

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.

28 GENERAL ASTRONOMY

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

THE EARTH 29

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

GENERAL ASTRONOMY

P f p

FIG. 11. Elementary Theory of
Refraction.

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

THE EARTH 31

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
pronounced.

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

GENERAL ASTRONOMY

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
sea-level,

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-

27~>
Xfc

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

1 -

or A ~

THE EARTH 33

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

3438

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.

CHAPTER III
THE EARTH IN RELATION TO THE SUN

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

THE EARTH IN RELATION TO THE SUN 35

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
Earth.

36 GENERAL ASTRONOMY

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
Sun.

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

THE EARTH IN RELATION TO THE SUN 37

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*

GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 39

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

40 GENERAL ASTRONOMY

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.

THE EARTH IN RELATION TO THE SUN 41

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

42 GENERAL ASTRONOMY

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
twilight.

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

THE EARTH IN RELATION TO THE SUN 43

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

GENERAL ASTRONOMY

AuturnnaA
_1 Equinox

autumn and winter are of longer duration than spring and
summer.

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.

Summer

Vernal
equinox

FLO.

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

THE EARTH IN RELATION TO THE SUN 53

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,

54 GENERAL ASTRONOMY

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
phenomenon.

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 EARTH IN RELATION TO THE SUN 55

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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58 GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 59

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
determined.

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

GO GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 61

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

62 GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 63

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.

64 GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 65

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.

66 GENERAL ASTRONOMY

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.
c

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

THE EARTH IN RELATION TO THE SUN 67

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
observation.

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

68 GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 69

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
described.

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

70 GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 71

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

72 GENERAL ASTRONOMY

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

THE EARTH IN RELATION TO THE SUN 73

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

74 GENERAL ASTRONOMY

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.

CHAPTER IV
THE MOON

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

76 GENERAL ASTRONOMY

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

THE MOON

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

GENERAL ASTRONOMY

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
Crescent.

THE MOON 79

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)

80 GENERAL ASTRONOMY

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.

THE MOON 81

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

82 GENERAL ASTRONOMY

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

THE MOON 83

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
revolution.

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

84 GENERAL ASTRONOMY

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
days.

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

THE MOON 85

conjunction, opposition or quadrature (E = 0, 90, 180,
270).

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

GENERAL ASTRONOMY

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.

THE MOON 87

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

88 GENERAL ASTRONOMY

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

THE MOON 89

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

GENERAL ASTRONOMY

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.

THE MOON

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.

GENERAL ASTRONOMY

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
Longitude.

THE MOON 93

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

94 GENERAL ASTRONOMY

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

THE MOON

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

96 GENERAL ASTRONOMY

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

THE MOON 97

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

98 GENERAL ASTRONOMY

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
usual.

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 MOON 99

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
mankind.

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

100

GENERAL ASTRONOMY

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
tides.

Fid. 40. Tho Diurnal Inequality
in the Tides.

THE MOON 101

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

102 GENERAL ASTRONOMY

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

THE MOON 103

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 :

104 GENERAL ASTRONOMY

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

THE MOON 105

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

106 GENERAL ASTRONOMY

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

THE MOON 107

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
Earth.

CHAPTER V
THE SUN

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

108

THE SUN

109

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
48
50
52
54

Solar Parallax.

8"-808
799
790
782
773

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~

Date.

Int

1631,

Dec.

1639,

Dec.

1761,

June

1769,

June

1874,

Doc.

105J

1882,

Dec.

110 GENERAL ASTRONOMY

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
observation.

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

112

GENERAL ASTRONOMY

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

114 GENERAL ASTRONOMY

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

116 GENERAL ASTRONOMY

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,

118 GENERAL ASTRONOMY

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

THE SUN

119

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 :

Iron
Nickel

Neodymium
Yttrium

Aluminium
Cadmium

Bismuth (?)
Tellurium

Titanium

Lanthanum

Rhodium

Indium

Manganese
Chromium
Cobalt
Carbon
Vanadium

Niobium
Molybdenum
Palladium
Magnesium
Sodium

Erbium
Zinc
Copper
Silver
Germanium

Oxygen
Tungsten
Mercury (?)
Helium
Ytterbium

Zirconium
Cerium
Calcium

Silicon
Hydrogen
Strontium

Glucinum
Tin
Lead

Europium
Radium (?)

Scandium

Barium

Potassium

120 GENERAL ASTRONOMY

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
lines.

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.

122 GENERAL ASTRONOMY

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
disappear.

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

124

GENERAL ASTRONOMY

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

Year.

by Spots.

Year.

by Spots.

Year.

by Spots.

1889 .

. . 78

1900 .

. . 75

1911

. . . 64

1890 .

. . 97

1901 .

. . 29

1912

... 37

1891 .

. . 421

1902 .

. . 63

1913

. . . 7

1892 .

. . 1,214

1903 .

. . 339

1914

, . . 152

1893 .

. . 1,458

1904 .

. . 488

1915

. . . 697

1894 .

. . 1,282

1905 .

. . 1,191

1916

. . . 724

1895 .

. . 974

1906 .

. . 778

1917

. . . 1,537

1896 .

. . 543

1907 .

. . 1,082

1918

. . . 1,118

1897 .

. . 514

1908 .

. . 697

1919

. . . 1,052

1898 .

. . 376

1909 ,

. . 692

1920

. . . 618

1899 .

. . Ill

1910 .

. . 264

1921

. . . 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

THE SUN

125

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

Cycle.

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
atitudes.

126 GENERAL ASTRONOMY

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.

128

GENERAL ASTRONOMY

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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

130 GENERAL ASTRONOMY

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
noticeable.

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
studied.

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

132 GENERAL ASTRONOMY

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
observed.

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

134 GENERAL ASTRONOMY

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

136 GENERAL ASTRONOMY

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
investigated.

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

138 GENERAL ASTRONOMY

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
controversy.

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

140 GENERAL ASTRONOMY

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.

CHAPTER VI
ECLIPSES AND OCCULTATIONS

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,

141

142 GENERAL ASTRONOMY

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.

ECLIPSES AND OCCUPATIONS 143

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.

Year.

Moon.

Sun.

Moon,

Sun.

1914

Mar cli 1 1

Feb. 25

Sept. 4

Aug. 21

1915

Feb. 14

Aug. 10

1916

Jan. 20

(Feb. 3 \
(Dec. 241

July 15

July 30

1917

(Jan. 8 )
(Dec. 28 j

( Jan. 23)
( Dec. 14)

July 4

(Juno 19|
(July 19 /

1918

Dec. 3

June 24

June 8

1919

Nov. 8

Nov. 22

May 29

1920

Oct. 27

Nov. 10

May 3

May 18

1921

Oct. 16

Oct. 1

Apr. 22

Apr. 8

1922

Sept. 21

March 28

1923

Aug. 26

Sept. 10

March 3

March 17

1924

Aug. 14

(July 31)

(Aug. 30{

Feb. 20

March 5

1925

Aug. 4

July 20

Feb. 8

Jan. 24

1926

July 9

Jan. 14

1927

June 15

June 29

Dec. 8

(Jan. 3 )

JDcc. 24)

1928

Juno 3

(May 19 \
(June 17)

Nov. 27

Nov. 12

1929

May 9

Nov. 1

1930

April 13

April 28

Oct. 7

Oct. 21

1931

April 2

April 18

Sept. 26

(Sept. 12 [
(Oct. 11 J

1932

March 22

March 7

Sept. 14

Aug. 31

1933

Feb. 24

Aug. 21

1934

Jan. 30

Fob. 14

July 26

Aug. 10

1935

Jan. 19

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

July 16

/June 30 )
(July 30 }

1936

Jan. 8

Dec. 13

July 4

June 19

1937

Nov. 18

Dec. 2

June 6

1938

Nov. 7

Nov. 22

May 14

May 29

1939

Oct. 28

Oct. 12

May 3

April 19

1940

Oct. 1

April 7

1941

Sept. 5

Sept. 21

March 13

March 27

1942

Aug. 26

(Aug. 12 }
(Sept. 10)

March 3

March 16

1943

Aug. 15

Aug. 1

Feb. 20

Feb. 4

1944

July 20

Jan. 25

1945

Juno 25

July 9

Dec. 19

Jan. 14

1946

June 14

/May 30)
(Juno 29 j

Dec. 8

(Jan. 3 )

(Nov. 23 j"

1947

Juno 3

May 20

Nov. 12

1948

April 23

May 9

Nov. 1

1949

April 13

April 28

Oct. 7

Oct. 21

1950

April 2

March 18

Sept. 26

Sept. 12

144

ECLIPSES AND OCCULTATIONS 145

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

146 GENERAL ASTRONOMY

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
impossible.

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.

ECLIPSES AND OCCULTATIONS 147

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

148 GENERAL ASTRONOMY

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

ECLIPSES AND OCCULTATIONS 149

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
occur.

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

150

GENERAL ASTRONOMY

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
Eclipse.

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

ECLIPSES AND OCCULT ATIONS 151

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
End.

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.

152 GENERAL ASTRONOMY

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

ECLIPSES AND OCCULTATIONS

153

154 GENERAL ASTRONOMY

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-
tion.

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
minutes.

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

ECLIPSES AND OCCULTATIONS 155

escaped observation at all the eclipses which have been well
observed.

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
Moon.

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

156 GENERAL ASTRONOMY

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

ECLIPSES AND CGWITATIONS 157

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
secured.

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

158

GENERAL ASTRONOMY

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
transit.

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.

CHAPTER VII
ASTRONOMICAL INSTRUMENTS

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.

159

160 GENERAL ASTRONOMY

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

ASTRONOMICAL INSTRUMENTS 1G1

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

162 GENERAL ASTRONOMY

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

-e

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,

ASTRONOMICAL INSTRUMENTS

163

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

164.

GENERAL ASTRONOMY

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

ASTRONOMICAL INSTRUMENTS 165

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

166

GENERAL ASTRONOMY

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

(U)

(d)

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

ASTRONOMICAL INSTRUMENTS 167

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

168 GENERAL ASTRONOMY

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-
loidal.

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

ASTRONOMICAL INSTRUMENTS

160

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

170

GENERAL ASTRONOMY

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

ASTRONOMICAL INSTRUMENTS

171

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
declination.

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
Instrument.

172 GENERAL ASTRONOMY

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
eliminated.

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

ASTRONOMICAL INSTRUMENTS

173

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.

174

GENERAL ASTRONOMY

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.

ASTRONOMICAL INSTRUMENTS

175

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
Instrument.

176

GENERAL ASTRONOMY

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
Instrument.

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.

ASTRONOMICAL INSTRUMENTS 177

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

178 GENERAL ASTRONOMY

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

ASTRONOMICAL INSTRUMENTS

179

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-
strument.

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

180 GENERAL ASTRONOMY

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

ASTRONOMICAL INSTRUMENTS 181

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
pressure.

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
devised.

182 GENERAL ASTRONOMY

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
telescope.

ASTRONOMICAL INSTRUMENTS 183

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
body.

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

184 GENERAL ASTRONOMY

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-
tained.

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

ASTRONOMICAL INSTRUMENTS 185

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
floor.

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

186 GENERAL ASTRONOMY

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,

ASTRONOMICAL INSTRUMENTS

187

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

Colfirnator

Eyepiece

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-

188

GENERAL ASTRONOMY

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

ASTRONOMICAL INSTRUMENTS 189

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

190 GENERAL ASTRONOMY

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.

CHAPTER VIII
ASTRONOMICAL OBSERVATIONS

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

191

102 GENERAL ASTRONOMY

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

ASTRONOMICAL OBSERVATIONS 193

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

194 GENERAL ASTRONOMY

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.

ASTRONOMICAL OBSERVATIONS 195

(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
foundation.

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

196 GENERAL ASTRONOMY

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

ASTRONOMICAL OBSERVATIONS 197

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

198

GENERAL ASTRONOMY

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
semi-diameter.

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
Point.

ASTRONOMICAL OBSERVATIONS 199

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
position.

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

200 GENERAL ASTRONOMY

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

ASTRONOMICAL OBSERVATIONS 201

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
azimuth.

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

202 GENERAL ASTRONOMY

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

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

ASTRONOMICAL OBSERVATIONS 203

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 :

204: GENERAL ASTRONOMY

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
purpose.

CHAPTER IX
THE PLANETARY MOTIONS

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

205

206 GENERAL ASTRONOMY

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

THE PLANETARY MOTIONS

207

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
equal.

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

7/3

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

208 GENERAL ASTRONOMY

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

THE PLANETARY MOTIONS

209

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

'Quadrature

W. Quadrature

Opposition
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.

210

GENERAL ASTRONOMY

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
motion.

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
Motions.

THE PLANETARY MOTIONS 211

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
affected.

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
QQ'.

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

212

GENERAL ASTRONOMY

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 PLANETARY MOTIONS 213

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
Earth.

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

214 GENERAL ASTRONOMY

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

THE PLANETARY MOTIONS

215

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
P

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

Planet.

Mercury
Vonus
Earth .
Mars
Jupiter .
Saturn .
Uranus .
Neptune

Sidereal Period.

Synodic Period.

88 clays

116 days

225

584

365

687

780

12 years

3iK)

378

370

165

368

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

216 GENERAL ASTRONOMY

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
foundation.

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

THE PLANETARY MOTIONS

217

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
Orbit.

218 GENERAL ASTRONOMY

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 :

THE PLANETARY MOTIONS

219

Eccentricity.

Inclination relative
to invariable Plane.

Min.

Max.

Present
Value.

Miri.

Max.

9 11
3 16

Mercury ....
Venus

0-1215
0000

0-2317

0706

0-2056

0068

i /
4 44

Earth
Mars

-0000
0185
0255
0124
0118

0694
1397

0608
0843

0780

0167
0933
0483
0559
0463

14
47
54

3 6

5 56
29
1 1
1 7

Jupiter ....
Saturn

Uranus ....

Neptune ....

0056

0145

0090

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.

220 GENERAL ASTRONOMY

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

THE PLANETARY MOTIONS 221

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

222 GENERAL ASTRONOMY

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,

whence

M + m __ t

\'J

THE PLANETARY MOTIONS 223

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-

224 GENERAL ASTRONOMY

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.

THE PLANETARY MOTIONS 225

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
motion.

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-
a

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

226 GENERAL ASTRONOMY

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.

THE PLANETARY MOTIONS

227

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.

Semi-

Inclin-

Moan

Reciprocal

Planet.

Period.

Axis
Major.

Eccen-
tricity.

ation to
Ecliptic.

Daily
Motion.

of Mass
(Sun-1)

o /

Mercury.

87-97

0-387

2056

14732-4

9,700,000

Venus

224-70

0-723

0068

3 24

5767-7

408,000

Earth

365-26

1-000

0167

3548-2

333,432

Mars.

686-98

1-524

0933

1 51

1886-5

3,093,500

Jupiter

4332-59

5-203

0483

1 18

299-1

1047-36

Saturn .

10759-23

9-539

0559

2 30

120-5

3501-6

Uranus

30688-45

19-191

0471

42-2

22,869

Neptune

60181-3

30-071

0085

1 47

21-5

19,314

CHAPTER X
THE PLANETS

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

228

THE PLANETS 229

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

230 GENERAL ASTRONOMY

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

THE PLANETS 231

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

232

GENERAL ASTRONOMY

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.

THE PLANETS 233

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

234 GENERAL ASTRONOMY

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

THE PLANETS 235

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

236 GENERAL ASTRONOMY

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

THE PLANETS 237

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

238 GENERAL ASTRONOMY

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

THE PLANETS 239

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

240 GENERAL ASTRONOMY

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

THE PLANETS 241

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
atmosphere.

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

242 GENERAL ASTRONOMY

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

THE PLANETS 243

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

244 GENERAL ASTRONOMY

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.

THE PLANETS 245

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.

246 GENERAL ASTRONOMY

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
amounts.

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

THE PLANETS 247

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

248 GENERAL ASTRONOMY

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

THE PLANETS 249

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

250

GENERAL ASTRONOMY

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 :

Distance

Satel-
lite.

in
terms of
Jupiter's
Radius.

Distance in

Miles.

Period.

Diameter,
Miles.

Stellar
Magni-
tude.

Year of
Dis-
covery.

d. li. rn.

5-9

267,000

1 18 28

2,470

5-6

1610

9-4

421,000

3 13 14

2,000

5-7

1610

15-0

078,000

7 3 43

3,580

5-0

1610

26-4

1,191,000

16 16 32

3,300

6-3

1610

2-5

115,000

11 57

13-

1S92

!(>()

7,230,000

251

14-

1904

167-

7,540,000

265

1905

330-

14,910,000

739

16-

1908

345-

15,560,000

745

19-

1914

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

THE PLANETS

251

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
disc.

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.

252 GENERAL ASTRONOMY

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

THE PLANETS 253

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.

254 GENERAL ASTRONOMY

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 PLANETS 255

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
miles.

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

256

GENERAL ASTRONOMY

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 PLANETS

257

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
edges.

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 :

Name.

Distance in
terms of
Saturn's

Distance in
Miles.

Period.

Stellar
Magni-

Radius.

tude.

ro.

rn.

Mimas

3-1

113,000

Enceladus

3-9

145,000

Tethys. . . .

4-9

180,000

Dione ....

6-2

231,000

Bhea ....

8-7

322,000

Titan ....

20-2

746,000

Themis

24-2

892,000

Hyperion .

24-5

903,000

Japetus

58-9

2,172,000

Phoebe . . .

214-4

7,900,000

550

258 GENERAL ASTRONOMY

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
elements.

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
Saturn.

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
motion.

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

THE PLANETS 259

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-
tening.

Uranus possesses four satellites, so that its mass can be
determined with accuracy. It is 1/22869 of that of the Sun,

260 GENERAL ASTRONOMY

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
hours.

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 :

THE PLANETS

261

Name.

Distance in
terms of tho
Radius of

Distance
in Miles.

Period.

Uranus.

h. m.

Ariel

7-0

111,000

12 29

Umbriel ....

9-9

156,000

3 28

Titania

16-1

253,000

16 56

Oberon

21-5

339,000

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

262 GENERAL ASTRONOMY

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

THE PLANETS 263

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

264 GENERAL ASTRONOMY

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.

CHAPTER XI
COMETS AND METEORS

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

265

266 GENERAL ASTRONOMY

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
six.

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,

COMETS AND METEORS 267

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

268 GENERAL ASTRONOMY

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

COMETS AND METEORS 269

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

270 GENERAL ASTRONOMY

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

COMETS AND METEORS

271

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
1

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.

272 GENERAL ASTRONOMY

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 AND METEORS 273

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

274 GENERAL ASTRONOMY

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
sunlight.

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.

COMETS AND METEORS

275

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.

276 GENERAL ASTRONOMY

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

COMETS AND METEOKS 277

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

278 GENERAL ASTRONOMY

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

COMETS AND METEOPxS 279

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

280 GENERAL ASTRONOMY

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 :

COMETS AND METEORS

281

Kadi ant in
Constellation.

Meteor Shower.

Date of Shower.

Comet.

Period.

Lyra

Lyrids

April 20-21

1861 I

415 years

Persons

Porseids

August 10-11

1802 III

120

Leo

Leonids

November 14-15

1866 I

Andromeda

Bielids

November 23

Biola

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,

282 GENERAL ASTRONOMY

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

COMETS AND METEORS 2S3

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.

CHAPTER XII
THE STARS

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

284

THE STARS 285

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

286 GENERAL ASTRONOMY

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
designated.

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

THE STARS 287

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-

288 GENERAL ASTRONOMY

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-
minations.

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

THE STARS 289

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

290

GJKJNJSKAJL ASTKUJNUMY

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

THE STARS 201

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

292

GENERAL ASTRONOMY

FIG. 88. The Meridian Photometor.

THE STARS 293

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

294 GENERAL ASTRONOMY

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

THE STARS 295

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.

296

GENERAL ASTRONOMY

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.

(Visual,)

(Photographic.)

138

111

454

300

1,480

950

4,750

3,150

14,960

9,810

45,710

32,360

134,000

97,400

373,000

271,800

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

THE STARS 297

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

298

GENERAL ASTRONOMY

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.

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

8-5
4-5
5-3
9-2
0-2
7-3

4-49
4-08
3-75
3-72
3-68
3-44

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
hundred.

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
years.

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

THE STARS

299

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

300 GENERAL ASTRONOMY

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,

THE STARS 301

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

302

GENERAL ASTRONOMY

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

THE STARS 303

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

304 GENERAL ASTRONOMY

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

THE STARS 305

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

306

GENERAL ASTRONOMY

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

King.

Parkhurst.

-0-31

-0-48

0-00

0-06

+ 0-32

-r-0-40

-fO-71

+ 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

THE STARS 307

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

308 GENERAL ASTRONOMY

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
low.

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

THE STARS 309

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-

310 GENERAL ASTRONOMY

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-
years.

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.

THE STARS

311

Parallax.

Distance.

Morlorri Observations.

Parallax.

Distance.

a
61
a

Centauri (Henderson)
Cygni (Bossel) .
Lyrae (Struve) .

1-0

<K*i4

0-262

200,000
640,000
760,000

0-750
0-285
0-100

270,000
700,000
2,000,000

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

312 GENERAL ASTRONOMY

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
thousand.

Particulars of the stars with largest known parallaxes are
given in the following table ;

THE STARS

313

Name.

Annual
Proper
Motion.

Parallax.

Distance
in Light-
Years.

Lumin-
osity.
Sim = l.

Spectral
Type.

Proxima Centauri

//
3-85

0-79

4-1

0-0001

a Contauri ....

3-68

0-76

4-3

1-3

Munich 15,040 . . .

10-29

0-53

6-2

0-0005

Lalando 21,185 . . .

4-74

0-41

7-9

0-0054

Sirius

1-32

0-38

8-6

Cordoba, V h. 243 . .

8-75

0-32

10-2

0-0022

i Coti

1-92

0-32

10-2

0-35

e Eridani ....

0-97

0-31

10-5

0-31

Procyon

1-24

0-30

10-9

7-0

61 Cygni

5-24

0-30

10-9

0-064

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

314 GENERAL ASTRONOMY

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

THE STARS 315

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
1-33.

316

GENERAL ASTRONOMY

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
telescope.

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

THE STARS 317

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.

318

GENERAL ASTRONOMY

Type.
Vis. Mag.

0-0

0034

0054

0098

0219

0859

2-0

0014

0022

0039

0087

0342

4-0

0005

0009

0016

0035

0136

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 . . .

6-52

225

10-95

177

14-37

185

14-97

128

K
M

16-8
17*1

382
73

The velocities given in this table are not corrected for the star

THE STARS

319

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 :

Type.

Cross Linear Motion,
km. per sec.

No. of Stars.

6-3

490

10-2

1,647

F
G

16-2
18-6

656
444

15-1

1,227

17-1

222

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
velocity.

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

320

GENERAL ASTRONOMY

B A F G K M N

+ 2

H-4

.+6

4-10

+12

-4

. -;-

: : '"

r> \

\ i
\

) .

+ 9

\
\

N
\

*X x

4-A.

\ *:

.v .
\ ,

\ ,

V i

: : \

: .\

v :;

''\,

x \
\

: K

: . \

f-Q

1 Q

*
\

V *

:\.

+ c>

4-in

\
\
\

t \
t
,

+1U
4 a

114

FIG, 95. Giant and Dwarf Stars (Russell),

THE STARS 321

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
respectively.

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

322 GENERAL ASTRONOMY

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
type.

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.

CHAPTER XIII
DOUBLE AND VARIABLE STARS

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,

323

324 GENERAL ASTRONOMY

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

DOUBLE AND VARIABLE STARS 325

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,

326 GENERAL ASTRONOMY

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
give

(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.

DOUBLE AND VARIABLE STARS

327

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.

Angular

Separation.

OM.

0*-05.

0*-01.

(T-005.

0-25

2J years

8 years

90 years

250 years

0-50

250

700

1-00

700

2,000

2-00

180

2,000

5,600

5-00

250

700

7,900

22,000

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
are

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

328 GENERAL ASTRONOMY

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

DOUBLE AND VARIABLE STARS 329

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

330 GENERAL ASTRONOMY

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

DOUBLE AND VARIABLE STARS 331

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
variables.

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
variables.

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

332 GENERAL ASTRONOMY

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

DOUBLE AND VARIABLE STARS

333

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-

334 GENERAL ASTRONOMY

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

DOUBLE AND VARIABLE STARS 335

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.

336 GENERAL ASTRONOMY

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

DOUBLE AND VARIABLE STARS 337

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
period.

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,

33$

GENERAL ASTRONOMY

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.

DOUBLE AND VARIABLE STARS 339

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

340

GENERAL ASTRONOMY

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

m.
3-6

3-8

4-0

4-Z

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

DOUBLE AND VARIABLE STARS

341

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) ;

m.
3-5

3-7

4-1
4-3

4-5
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

342 GENERAL ASTRONOMY

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

DOUBLE AND VARIABLE STARS

343

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

344 GENERAL ASTRONOMY

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,

DOUBLE AND VARIABLE STARS 345

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

346

GENERAL ASTRONOMY

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

c-oo

jg-0'40
^

^-0-60
'-0.60

rl-ZO

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

DOUBLE AND VARIABLE STARS

347

m
3-4

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-

4-0

4-2

FIG. 102. The Light Curve of /3 Lyrae.

348 GENERAL ASTRONOMY

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-
tively.

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.

DOUBLE AND VARIABLE STARS

349

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
variables.

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 :

Period.

Eccentricity.

(

3 days

0-05

Spectroscopic Binaries . . J

8
23

0-15
0-32

1J years

0-35

(

0-42

Visual Binaries . . , . J

0-51

170

0-54

220. The Origin of Binary Systems. The principal
phenomena to be taken into account in formulating a theory

350 GENERAL ASTRONOMY

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.)

CHAPTER XIV
THE STELLAR UNIVERSE

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-

351

352

GENERAL ASTRONOMY

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
Magnitude.

6'0 m.

9-0 m.

14-0 m.

+ 90

to +70

0-113

2-78

107

+ 70

, +50

122

3-03

154

+ 50

, +30

-124

3-54

281

+ 30

, +10

145

5-32

560

+ 10

, -10

160

8-17

2,019

- 10

, -30

154

6-07

672

-30

, -50

129

3-71

261

-50

, -70

124

3-21

154

-70

, -90

125

3-14

111

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

THE STELLAR UNIVERSE

353

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.

Total.

189

176

569

184

174

562

26:3

212

754

206

323

266

1,067

161

382

116

239

1,027

lf>8

276

117

100

247

967

101

5'J

203

633

107

202

627

-1-02-3
-1-41-3
H-21-0
+ 9-2
7-0
-22-2
-38-2
-62-3

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

354 GENERAL ASTRONOMY

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
nebula).

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

THE STELLAR UNIVERSE 355

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-

356 GENERAL ASTRONOMY

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 STELLAR UNIVERSE 357

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-

358 GENERAL ASTRONOMY

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

THE STELLAR UNIVERSE 359

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

360 GENERAL ASTRONOMY

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

THE STELLAR UNIVERSE 361

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

362 GENERAL ASTRONOMY

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 STELLAR UNIVERSE 363

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
arms.

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

364 GENERAL ASTRONOMY

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-

THE STELLAR UNIVERSE 365

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
15,000.

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.

366 GENERAL ASTRONOMY

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-

THE STELLAR UNIVERSE 367

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
numerous.

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

368 GENERAL ASTRONOMY

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.

THE STELLAR UNIVERSE 369

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

370 GENERAL ASTRONOMY

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

THE STELLAR UNIVERSE 371

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-

372 GENERAL ASTRONOMY

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

THE STELLAR UNIVERSE 373

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-

374

GENERAL ASTRONOMY

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 :

GIANTS.

Linear

Star.

Parallax.

Visual
Mng.

Abs.
Mag.

Spec-
trum.

Angular
Dium.

Diain.
(Sun

= 1).

111.

ft Geminorum

026

3-2

1 0-3

016

Aldebaran ....

055

1-1

~0-2

024

Arcturus

095

0-2

[0-1

021

r) Draconis ....

019

2-9

- 0-7

004

a Leporis ....

018

2-7

- 1-0

002

Regulus

033

1-3

1-1

002

/5 Centauri ....

037

0-9

- 1-3

001

DWARFS.

Linear

Star.

Parallax.

Visual

Mag.

Abs.
Mag.

Spec-
trum.

Angular
JJmrn.

Diam.
(Sun

Sirius

376

m.
1-6

M.
+ 1-3

007

2-0

Procyon

304

+ 0-5

+ 2-9

006

2-0

A Serpentis ....

086

4-4

+ 4-1

001

1-5

11 Leo Minoris .

140

5-5

+ 6-2

002

1-5

61 Cygni

303

5-6

+ 8-0

003

Lacaillo 9,352 . . .

290

7-4

+ 9-7

002

0-9

Lalancle 21,185 . . .

414

7-6

+ 10-7

002

0-6

THE STELLAR UNIVERSE 375

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.

376 GENERAL ASTRONOMY

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 STELLAR UNIVERSE 377

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
star.

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
structures.

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

378 GENERAL ASTRONOMY

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
centre.

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

THE STELLAR UNIVERSE 379

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

380 GENERAL ASTRONOMY

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
accretion.

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

THE STELLAR UNIVERSE 381

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.

382 GENERAL ASTRONOMY

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
observation.

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 ?

THE STELLAR UNIVERSE 383

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
collisions.

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

384 GENERAL ASTRONOMY

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

THE STELLAR UNIVERSE 385

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
eccentricity.

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
type.

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.

386 GENERAL ASTRONOMY

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.

INDEX

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,

384

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

387

388

INDEX

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,

353

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

INDEX

389

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,
165

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,
247

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,

240

Maskelyne, 19
Mass of double stars, 326 ; planets

222

Maunder, 126
Maxwell, 255
Mayer, 138

Mean place of Star, 203
Mean Sun, 52

390

INDEX

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,
289

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

INDEX

391

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,
382

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

392

INDEX

Talcott, 193

Telescopes, 159 ; equatorial, 182 ;
reflecting, 105 ; refracting, 160 ;
resolving power of, 161 ; zenith,
189

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,

189

Zodiac, 36

Zodiacal light, 281 ; spectrum of, 282
Zollner photometer, 289
Zone time, 55

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