**Darwin and Modern Science** (1909)

Edited by A.C. Seward

**XXVIII. THE GENESIS OF DOUBLE STARS.**

By SIR GEORGE DARWIN, K.C.B., F.R.S.

Plumian Professor of Astronomy and Experimental Philosophy in the

University of Cambridge.

n ordinary speech a system of any sort is said to be stable when it cannot
be upset easily, but the meaning attached to the word is usually somewhat
vague. It is hardly surprising that this should be the case, when it is
only within the last thirty years, and principally through the
investigations of M. Poincare, that the conception of stability has, even
for physicists, assumed a definiteness and clearness in which it was
previously lacking. The laws which govern stability hold good in regions
of the greatest diversity; they apply to the motion of planets round the
sun, to the internal arrangement of those minute corpuscles of which each
chemical atom is constructed, and to the forms of celestial bodies. In the
present essay I shall attempt to consider the laws of stability as relating
to the last case, and shall discuss the succession of shapes which may be
assumed by celestial bodies in the course of their evolution. I believe
further that homologous conceptions are applicable in the consideration of
the transmutations of the various forms of animal and of vegetable life and
in other regions of thought. Even if some of my readers should think that
what I shall say on this head is fanciful, yet at least the exposition will
serve to illustrate the meaning to be attached to the laws of stability in
the physical universe.

I propose, therefore, to begin this essay by a sketch of the principles of
stability as they are now formulated by physicists.

I.

If a slight impulse be imparted to a system in equilibrium one of two
consequences must ensue; either small oscillations of the system will be
started, or the disturbance will increase without limit and the arrangement
of the system will be completely changed. Thus a stick may be in
equilibrium either when it hangs from a peg or when it is balanced on its
point. If in the first case the stick is touched it will swing to and fro,
but in the second case it will topple over. The first position is a stable
one, the second is unstable. But this case is too simple to illustrate all
that is implied by stability, and we must consider cases of stable and of
unstable motion. Imagine a satellite and its planet, and consider each of
them to be of indefinitely small size, in fact particles; then the
satellite revolves round its planet in an ellipse. A small disturbance
imparted to the satellite will only change the ellipse to a small amount,
and so the motion is said to be stable. If, on the other hand, the
disturbance were to make the satellite depart from its initial elliptic
orbit in ever widening circuits, the motion would be unstable. This case
affords an example of stable motion, but I have adduced it principally with
the object of illustrating another point not immediately connected with
stability, but important to a proper comprehension of the theory of
stability.

The motion of a satellite about its planet is one of revolution or
rotation. When the satellite moves in an ellipse of any given degree of
eccentricity, there is a certain amount of rotation in the system,
technically called rotational momentum, and it is always the same at every
part of the orbit. (Moment of momentum or rotational momentum is measured
by the momentum of the satellite multiplied by the perpendicular from the
planet on to the direction of the path of the satellite at any instant.)

Now if we consider all the possible elliptic orbits of a satellite about
its planet which have the same amount of "rotational momentum," we find
that the major axis of the ellipse described will be different according to
the amount of flattening (or the eccentricity) of the ellipse described. A
figure titled "A 'family' of elliptic orbits with constant rotational
momentum" (Fig. 1) illustrates for a given planet and satellite all such
orbits with constant rotational momentum, and with all the major axes in
the same direction. It will be observed that there is a continuous
transformation from one orbit to the next, and that the whole forms a
consecutive group, called by mathematicians "a family" of orbits. In this
case the rotational momentum is constant and the position of any orbit in
the family is determined by the length of the major axis of the ellipse;
the classification is according to the major axis, but it might have been
made according to anything else which would cause the orbit to be exactly
determinate.

I shall come later to the classification of all possible forms of ideal
liquid stars, which have the same amount of rotational momentum, and the
classification will then be made according to their densities, but the idea
of orderly arrangement in a "family" is just the same.

We thus arrive at the conception of a definite type of motion, with a
constant amount of rotational momentum, and a classification of all members
of the family, formed by all possible motions of that type, according to
the value of some measurable quantity (this will hereafter be density)
which determines the motion exactly. In the particular case of the
elliptic motion used for illustration the motion was stable, but other
cases of motion might be adduced in which the motion would be unstable, and
it would be found that classification in a family and specification by some
measurable quantity would be equally applicable.

A complex mechanical system may be capable of motion in several distinct
modes or types, and the motions corresponding to each such type may be
arranged as before in families. For the sake of simplicity I will suppose
that only two types are possible, so that there will only be two families;
and the rotational momentum is to be constant. The two types of motion
will have certain features in common which we denote in a sort of shorthand
by the letter A. Similarly the two types may be described as A + a and A +
b, so that a and b denote the specific differences which discriminate the
families from one another. Now following in imagination the family of the
type A + a, let us begin with the case where the specific difference a is
well marked. As we cast our eyes along the series forming the family, we
find the difference a becoming less conspicuous. It gradually dwindles
until it disappears; beyond this point it either becomes reversed, or else
the type has ceased to be a possible one. In our shorthand we have started
with A + a, and have watched the characteristic a dwindling to zero. When
it vanishes we have reached a type which may be specified as A; beyond this
point the type would be A - a or would be impossible.

Following the A + b type in the same way, b is at first well marked, it
dwindles to zero, and finally may become negative. Hence in shorthand this
second family may be described as A + b,...A,...A - b.

In each family there is one single member which is indistinguishable from a
member of the other family; it is called by Poincare a form of bifurcation.
It is this conception of a form of bifurcation which forms the important
consideration in problems dealing with the forms of liquid or gaseous
bodies in rotation.

But to return to the general question,--thus far the stability of these
families has not been considered, and it is the stability which renders
this way of looking at the matter so valuable. It may be proved that if
before the point of bifurcation the type A + a was stable, then A + b must
have been unstable. Further as a and b each diminish A + a becomes less
pronouncedly stable, and A + b less unstable. On reaching the point of
bifurcation A + a has just ceased to be stable, or what amounts to the same
thing is just becoming unstable, and the converse is true of the A + b
family. After passing the point of bifurcation A + a has become definitely
unstable and A + b has become stable. Hence the point of bifurcation is
also a point of "exchange of stabilities between the two types." (In order
not to complicate unnecessarily this explanation of a general principle I
have not stated fully all the cases that may occur. Thus: firstly, after
bifurcation A + a may be an impossible type and A + a will then stop at
this point; or secondly, A + b may have been an impossible type before
bifurcation, and will only begin to be a real one after it; or thirdly,
both A + a and A + b may be impossible after the point of bifurcation, in
which case they coalesce and disappear. This last case shows that types
arise and disappear in pairs, and that on appearance or before
disappearance one must be stable and the other unstable.)

In nature it is of course only the stable types of motion which can persist
for more than a short time. Thus the task of the physical evolutionist is
to determine the forms of bifurcation, at which he must, as it were, change
carriages in the evolutionary journey so as always to follow the stable
route. He must besides be able to indicate some natural process which
shall correspond in effect to the ideal arrangement of the several types of
motion in families with gradually changing specific differences. Although,
as we shall see hereafter, it may frequently or even generally be
impossible to specify with exactness the forms of bifurcation in the
process of evolution, yet the conception is one of fundamental importance.

The ideas involved in this sketch are no doubt somewhat recondite, but I
hope to render them clearer to the non-mathematical reader by homologous
considerations in other fields of thought (I considered this subject in my
Presidential address to the British Association in 1905, "Report of the
75th Meeting of the British Assoc." (S. Africa, 1905), London, 1906, page
3. Some reviewers treated my speculations as fanciful, but as I believe
that this was due generally to misapprehension, and as I hold that
homologous considerations as to stability and instability are really
applicable to evolution of all sorts, I have thought it well to return to
the subject in the present paper.), and I shall pass on thence to
illustrations which will teach us something of the evolution of stellar
systems.

States or governments are organised schemes of action amongst groups of
men, and they belong to various types to which generic names, such as
autocracy, aristocracy or democracy, are somewhat loosely applied. A
definite type of government corresponds to one of our types of motion, and
while retaining its type it undergoes a slow change as the civilisation and
character of the people change, and as the relationship of the nation to
other nations changes. In the language used before, the government belongs
to a family, and as time advances we proceed through the successive members
of the family. A government possesses a certain degree of stability--
hardly measurable in numbers however--to resist disintegrating influences
such as may arise from wars, famines, and internal dissensions. This
stability gradually rises to a maximum and gradually declines. The degree
of stability at any epoch will depend on the fitness of some leading
feature of the government to suit the slowly altering circumstances, and
that feature corresponds to the characteristic denoted by a in the physical
problem. A time at length arrives when the stability vanishes, and the
slightest shock will overturn the government. At this stage we have
reached the crisis of a point of bifurcation, and there will then be some
circumstance, apparently quite insignificant and almost unnoticed, which is
such as to prevent the occurrence of anarchy. This circumstance or
condition is what we typified as b. Insignificant although it may seem, it
has started the government on a new career of stability by imparting to it
a new type. It grows in importance, the form of government becomes
obviously different, and its stability increases. Then in its turn this
newly acquired stability declines, and we pass on to a new crisis or
revolution. There is thus a series of "points of bifurcation" in history
at which the continuity of political history is maintained by means of
changes in the type of government. These ideas seem, to me at least, to
give a true account of the history of states, and I contend that it is no
mere fanciful analogy but a true homology, when in both realms of thought--
the physical and the political--we perceive the existence of forms of
bifurcation and of exchanges of stability.

Further than this, I would ask whether the same train of ideas does not
also apply to the evolution of animals? A species is well adapted to its
environment when the individual can withstand the shocks of famine or the
attacks and competition of other animals; it then possesses a high degree
of stability. Most of the casual variations of individuals are
indifferent, for they do not tell much either for or against success in
life; they are small oscillations which leave the type unchanged. As
circumstances change, the stability of the species may gradually dwindle
through the insufficiency of some definite quality, on which in earlier
times no such insistent demands were made. The individual animals will
then tend to fail in the struggle for life, the numbers will dwindle and
extinction may ensue. But it may be that some new variation, at first of
insignificant importance, may just serve to turn the scale. A new type may
be formed in which the variation in question is preserved and augmented;
its stability may increase and in time a new species may be produced.

At the risk of condemnation as a wanderer beyond my province into the
region of biological evolution, I would say that this view accords with
what I understand to be the views of some naturalists, who recognise the
existence of critical periods in biological history at which extinction
occurs or which form the starting-point for the formation of new species.
Ought we not then to expect that long periods will elapse during which a
type of animal will remain almost constant, followed by other periods,
enormously long no doubt as measured in the life of man, of acute struggle
for existence when the type will change more rapidly? This at least is the
view suggested by the theory of stability in the physical universe. (I
make no claim to extensive reading on this subject, but refer the reader
for example to a paper by Professor A.A.W. Hubrecht on "De Vries's theory
of Mutations", "Popular Science Monthly", July 1904, especially to page
213.)

And now I propose to apply these ideas of stability to the theory of
stellar evolution, and finally to illustrate them by certain recent
observations of a very remarkable character.

Stars and planets are formed of materials which yield to the enormous
forces called into play by gravity and rotation. This is obviously true if
they are gaseous or fluid, and even solid matter becomes plastic under
sufficiently great stresses. Nothing approaching a complete study of the
equilibrium of a heterogeneous star has yet been found possible, and we are
driven to consider only bodies of simpler construction. I shall begin
therefore by explaining what is known about the shapes which may be assumed
by a mass of incompressible liquid of uniform density under the influences
of gravity and of rotation. Such a liquid mass may be regarded as an ideal
star, which resembles a real star in the fact that it is formed of
gravitating and rotating matter, and because its shape results from the
forces to which it is subject. It is unlike a star in that it possesses
the attributes of incompressibility and of uniform density. The difference
between the real and the ideal is doubtless great, yet the similarity is
great enough to allow us to extend many of the conclusions as to ideal
liquid stars to the conditions which must hold good in reality. Thus with
the object of obtaining some insight into actuality, it is justifiable to
discuss an avowedly ideal problem at some length.

The attraction of gravity alone tends to make a mass of liquid assume the
shape of a sphere, and the effects of rotation, summarised under the name
of centrifugal force, are such that the liquid seeks to spread itself
outwards from the axis of rotation. It is a singular fact that it is
unnecessary to take any account of the size of the mass of liquid under
consideration, because the shape assumed is exactly the same whether the
mass be small or large, and this renders the statement of results much
easier than would otherwise be the case.

A mass of liquid at rest will obviously assume the shape of a sphere, under
the influence of gravitation, and it is a stable form, because any
oscillation of the liquid which might be started would gradually die away
under the influence of friction, however small. If now we impart to the
whole mass of liquid a small speed of rotation about some axis, which may
be called the polar axis, in such a way that there are no internal currents
and so that it spins in the same way as if it were solid, the shape will
become slightly flattened like an orange. Although the earth and the other
planets are not homogeneous they behave in the same way, and are flattened
at the poles and protuberant at the equator. This shape may therefore
conveniently be described as planetary.

If the planetary body be slightly deformed the forces of restitution are
slightly less than they were for the sphere; the shape is stable but
somewhat less so than the sphere. We have then a planetary spheroid,
rotating slowly, slightly flattened at the poles, with a high degree of
stability, and possessing a certain amount of rotational momentum. Let us
suppose this ideal liquid star to be somewhere in stellar space far removed
from all other bodies; then it is subject to no external forces, and any
change which ensues must come from inside. Now the amount of rotational
momentum existing in a system in motion can neither be created nor
destroyed by any internal causes, and therefore, whatever happens, the
amount of rotational momentum possessed by the star must remain absolutely
constant.

A real star radiates heat, and as it cools it shrinks. Let us suppose then
that our ideal star also radiates and shrinks, but let the process proceed
so slowly that any internal currents generated in the liquid by the cooling
are annulled so quickly by fluid friction as to be insignificant; further
let the liquid always remain at any instant incompressible and homogeneous.
All that we are concerned with is that, as time passes, the liquid star
shrinks, rotates in one piece as if it were solid, and remains
incompressible and homogeneous. The condition is of course artificial, but
it represents the actual processes of nature as well as may be,
consistently with the postulated incompressibility and homogeneity.
(Mathematicians are accustomed to regard the density as constant and the
rotational momentum as increasing. But the way of looking at the matter,
which I have adopted, is easier of comprehension, and it comes to the same
in the end.)

The shrinkage of a constant mass of matter involves an increase of its
density, and we have therefore to trace the changes which supervene as the
star shrinks, and as the liquid of which it is composed increases in
density. The shrinkage will, in ordinary parlance, bring the weights
nearer to the axis of rotation. Hence in order to keep up the rotational
momentum, which as we have seen must remain constant, the mass must rotate
quicker. The greater speed of rotation augments the importance of
centrifugal force compared with that of gravity, and as the flattening of
the planetary spheroid was due to centrifugal force, that flattening is
increased; in other words the ellipticity of the planetary spheroid
increases.

As the shrinkage and corresponding increase of density proceed, the
planetary spheroid becomes more and more elliptic, and the succession of
forms constitutes a family classified according to the density of the
liquid. The specific mark of this family is the flattening or ellipticity.

Now consider the stability of the system, we have seen that the spheroid
with a slow rotation, which forms our starting-point, was slightly less
stable than the sphere, and as we proceed through the family of ever
flatter ellipsoids the stability continues to diminish. At length when it
has assumed the shape shown in a figure titled "Planetary spheroid just
becoming unstable" (Fig. 2.) where the equatorial and polar axes are
proportional to the numbers 1000 and 583, the stability has just
disappeared. According to the general principle explained above this is a
form of bifurcation, and corresponds to the form denoted A. The specific
difference a of this family must be regarded as the excess of the
ellipticity of this figure above that of all the earlier ones, beginning
with the slightly flattened planetary spheroid. Accordingly the specific
difference a of the family has gradually diminished from the beginning and
vanishes at this stage.

According to Poincare's principle the vanishing of the stability serves us
with notice that we have reached a figure of bifurcation, and it becomes
necessary to inquire what is the nature of the specific difference of the
new family of figures which must be coalescent with the old one at this
stage. This difference is found to reside in the fact that the equator,
which in the planetary family has hitherto been circular in section, tends
to become elliptic. Hitherto the rotational momentum has been kept up to
its constant value partly by greater speed of rotation and partly by a
symmetrical bulging of the equator. But now while the speed of rotation
still increases (The mathematician familiar with Jacobi's ellipsoid will
find that this is correct, although in the usual mode of exposition,
alluded to above in a footnote, the speed diminishes.), the equator tends
to bulge outwards at two diametrically opposite points and to be flattened
midway between these protuberances. The specific difference in the new
family, denoted in the general sketch by b, is this ellipticity of the
equator. If we had traced the planetary figures with circular equators
beyond this stage A, we should have found them to have become unstable, and
the stability has been shunted off along the A + b family of forms with
elliptic equators.

This new series of figures, generally named after the great mathematician
Jacobi, is at first only just stable, but as the density increases the
stability increases, reaches a maximum and then declines. As this goes on
the equator of these Jacobian figures becomes more and more elliptic, so
that the shape is considerably elongated in a direction at right angles to
the axis of rotation.

At length when the longest axis of the three has become about three times
as long as the shortest (The three axes of the ellipsoid are then
proportional to 1000, 432, 343.), the stability of this family of figures
vanishes, and we have reached a new form of bifurcation and must look for a
new type of figure along which the stable development will presumably
extend. Two sections of this critical Jacobian figure, which is a figure
of bifurcation, are shown by the dotted lines in a figure titled "The
'pear-shaped figure' and the Jocobian figure from which it is derived"
(Fig. 3.) comprising two figures, one above the other: the upper figure is
the equatorial section at right angles to the axis of rotation, the lower
figure is a section through the axis.

Now Poincare has proved that the new type of figure is to be derived from
the figure of bifurcation by causing one of the ends to be prolonged into a
snout and by bluntening the other end. The snout forms a sort of stalk,
and between the stalk and the axis of rotation the surface is somewhat
flattened. These are the characteristics of a pear, and the figure has
therefore been called the "pear-shaped figure of equilibrium." The firm
line shows this new type of figure, whilst, as already explained, the
dotted line shows the form of bifurcation from which it is derived. The
specific mark of this new family is the protrusion of the stalk together
with the other corresponding smaller differences. If we denote this
difference by c, while A + b denotes the Jacobian figure of bifurcation
from which it is derived, the new family may be called A + b + c, and c is
zero initially. According to my calculations this series of figures is
stable (M. Liapounoff contends that for constant density the new series of
figures, which M. Poincare discovered, has less rotational momentum than
that of the figure of bifurcation. If he is correct, the figure of
bifurcation is a limit of stable figures, and none can exist with stability
for greater rotational momentum. My own work seems to indicate that the
opposite is true, and, notwithstanding M. Liapounoff's deservedly great
authority, I venture to state the conclusions in accordance with my own
work.), but I do not know at what stage of its development it becomes
unstable.

Professor Jeans has solved a problem which is of interest as throwing light
on the future development of the pear-shaped figure, although it is of a
still more ideal character than the one which has been discussed. He
imagines an INFINITELY long circular cylinder of liquid to be in rotation
about its central axis. The existence is virtually postulated of a demon
who is always occupied in keeping the axis of the cylinder straight, so
that Jeans has only to concern himself with the stability of the form of
the section of the cylinder, which as I have said is a circle with the axis
of rotation at the centre. He then supposes the liquid forming the
cylinder to shrink in diameter, just as we have done, and finds that the
speed of rotation must increase so as to keep up the constancy of the
rotational momentum. The circularity of section is at first stable, but as
the shrinkage proceeds the stability diminishes and at length vanishes.
This stage in the process is a form of bifurcation, and the stability
passes over to a new series consisting of cylinders which are elliptic in
section. The circular cylinders are exactly analogous with our planetary
spheroids, and the elliptic ones with the Jacobian ellipsoids.

With further shrinkage the elliptic cylinders become unstable, a new form
of bifurcation is reached, and the stability passes over to a series of
cylinders whose section is pear-shaped. Thus far the analogy is complete
between our problem and Jeans's, and in consequence of the greater
simplicity of the conditions, he is able to carry his investigation
further. He finds that the stalk end of the pear-like section continues to
protrude more and more, and the flattening between it and the axis of
rotation becomes a constriction. Finally the neck breaks and a satellite
cylinder is born. Jeans's figure for an advanced stage of development is
shown in a figure titled "Section of a rotating cylinder of liquid" (Fig.
4.), but his calculations do not enable him actually to draw the state of
affairs after the rupture of the neck.

There are certain difficulties in admitting the exact parallelism between
this problem and ours, and thus the final development of our pear-shaped
figure and the end of its stability in a form of bifurcation remain hidden
from our view, but the successive changes as far as they have been
definitely traced are very suggestive in the study of stellar evolution.

Attempts have been made to attack this problem from the other end. If we
begin with a liquid satellite revolving about a liquid planet and proceed
backwards in time, we must make the two masses expand so that their density
will be diminished. Various figures have been drawn exhibiting the shapes
of two masses until their surfaces approach close to one another and even
until they just coalesce, but the discussion of their stability is not
easy. At present it would seem to be impossible to reach coalescence by
any series of stable transformations, and if this is so Professor Jeans's
investigation has ceased to be truly analogous to our problem at some
undetermined stage. However this may be this line of research throws an
instructive light on what we may expect to find in the evolution of real
stellar systems.

In the second part of this paper I shall point out the bearing which this
investigation of the evolution of an ideal liquid star may have on the
genesis of double stars.

II.

There are in the heavens many stars which shine with a variable brilliancy.
Amongst these there is a class which exhibits special peculiarities; the
members of this class are generally known as Algol Variables, because the
variability of the star Beta Persei or Algol was the first of such cases to
attract the attention of astronomers, and because it is perhaps still the
most remarkable of the whole class. But the circumstances which led to
this discovery were so extraordinary that it seems worth while to pause a
moment before entering on the subject.

John Goodricke, a deaf-mute, was born in 1764; he was grandson and heir of
Sir John Goodricke of Ribston Hall, Yorkshire. In November 1782, he noted
that the brilliancy of Algol waxed and waned (It is said that Georg
Palitzch, a farmer of Prohlis near Dresden, had about 1758 already noted
the variability of Algol with the naked eye. "Journ. Brit. Astron. Assoc."
Vol. XV. (1904-5), page 203.), and devoted himself to observing it on every
fine night from the 28th December 1782 to the 12th May 1783. He
communicated his observations to the Royal Society, and suggested that the
variation in brilliancy was due to periodic eclipses by a dark companion
star, a theory now universally accepted as correct. The Royal Society
recognised the importance of the discovery by awarding to Goodricke, then
only 19 years of age, their highest honour, the Copley medal. His later
observations of Beta Lyrae and of Delta Cephei were almost as remarkable as
those of Algol, but unfortunately a career of such extraordinary promise
was cut short by death, only a fortnight after his election to the Royal
Society. ("Dict. of National Biography"; article Goodricke (John). The
article is by Miss Agnes Clerke. It is strange that she did not then seem
to be aware that he was a deaf-mute, but she notes the fact in her
"Problems of Astrophysics", page 337, London, 1903.)

It was not until 1889 that Goodricke's theory was verified, when it was
proved by Vogel that the star was moving in an orbit, and in such a manner
that it was only possible to explain the rise and fall in the luminosity by
the partial eclipse of a bright star by a dark companion.

The whole mass of the system of Algol is found to be half as great again as
that of our sun, yet the two bodies complete their orbit in the short
period of 2d 20h 48m 55s. The light remains constant during each period,
except for 9h 20m when it exhibits a considerable fall in brightness
(Clerke, "Problems of Astrophysics" page 302 and chapter XVIII.); the curve
which represents the variation in the light is shown in a figure titled
"The light-curve and system of Beta Lyrae" (Fig. 7.).

The spectroscope has enabled astronomers to prove that many stars, although
apparently single, really consist of two stars circling around one another
(If a source of light is approaching with a great velocity the waves of
light are crowded together, and conversely they are spaced out when the
source is receding. Thus motion in the line of sight virtually produces an
infinitesimal change of colour. The position of certain dark lines in the
spectrum affords an exceedingly accurate measurement of colour. Thus
displacements of these spectral lines enables us to measure the velocity of
the source of light towards or away from the observer.); they are known as
spectroscopic binaries. Campbell of the Lick Observatory believes that
about one star in six is a binary ("Astrophysical Journ." Vol. XIII. page
89, 1901. See also A. Roberts, "Nature", Sept. 12, 1901, page 468.); thus
there must be many thousand such stars within the reach of our
spectroscopes.

The orientation of the planes of the orbits of binary stars appears to be
quite arbitrary, and in general the star does not vary in brightness.
Amongst all such orbits there must be some whose planes pass nearly through
the sun, and in these cases the eclipse of one of the stars by the other
becomes inevitable, and in each circuit there will occur two eclipses of
unequal intensities.

It is easy to see that in the majority of such cases the two components
must move very close to one another.

The coincidence between the spectroscopic and the photometric evidence
permits us to feel complete confidence in the theory of eclipses. When
then we find a star with a light-curve of perfect regularity and with a
characteristics of that of Algol, we are justified in extending the theory
of eclipses to it, although it may be too faint to permit of adequate
spectroscopic examination. This extension of the theory secures a
considerable multiplication of the examples available for observation, and
some 30 have already been discovered.

Dr Alexander Roberts, of Lovedale in Cape Colony, truly remarks that the
study of Algol variables "brings us to the very threshold of the question
of stellar evolution." ("Proc. Roy. Soc. Edinburgh", XXIV. Part II.
(1902), page 73.) It is on this account that I propose to explain in some
detail the conclusion to which he and some other observers have been led.

Although these variable stars are mere points of light, it has been proved
by means of the spectroscope that the law of gravitation holds good in the
remotest regions of stellar space, and further it seems now to have become
possible even to examine the shapes of stars by indirect methods, and thus
to begin the study of their evolution. The chain of reasoning which I
shall explain must of necessity be open to criticism, yet the explanation
of the facts by the theory is so perfect that it is not easy to resist the
conviction that we are travelling along the path of truth.

The brightness of a star is specified by what is called its "magnitude."
The average brightness of all the stars which can just be seen with the
naked eye defines the sixth magnitude. A star which only gives two-fifths
as much light is said to be of the seventh magnitude; while one which gives
2 1/2 times as much light is of the fifth magnitude, and successive
multiplications or divisions by 2 1/2 define the lower or higher
magnitudes. Negative magnitudes have clearly to be contemplated; thus
Sirius is of magnitude minus 1.4, and the sun is of magnitude minus 26.

The definition of magnitude is also extended to fractions; for example, the
lights given by two candles which are placed at 100 feet and 100 feet 6
inches from the observer differ in brightness by one-hundredth of a
magnitude.

A great deal of thought has been devoted to the measurement of the
brightness of stars, but I will only describe one of the methods used, that
of the great astronomer Argelander. In the neighbourhood of the star under
observation some half dozen standard stars are selected of known invariable
magnitudes, some being brighter and some fainter than the star to be
measured; so that these stars afford a visible scale of brightness.
Suppose we number them in order of increasing brightness from 1 to 6; then
the observer estimates that on a given night his star falls between stars 2
and 3, on the next night, say between 3 and 4, and then again perhaps it
may return to between 2 and 3, and so forth. With practice he learns to
evaluate the brightness down to small fractions of a magnitude, even a
hundredth part of a magnitude is not quite negligible.

For example, in observing the star RR Centauri five stars were in general
used for comparison by Dr Roberts, and in course of three months he secured
thereby 300 complete observations. When the period of the cycle had been
ascertained exactly, these 300 values were reduced to mean values which
appertained to certain mean places in the cycle, and a mean light-curve was
obtained in this way. Figures titled "Light curve of RR Centauri" (Fig. 5)
and "The light-curve and system of Beta Lyrae" (Fig. 7) show examples of
light curves.

I shall now follow out the results of the observation of RR Centauri not
only because it affords the easiest way of explaining these investigations,
but also because it is one of the stars which furnishes the most striking
results in connection with the object of this essay. (See "Monthly notices
R.A.S." Vol. 63, 1903, page 527.) This star has a mean magnitude of about
7 1/2, and it is therefore invisible to the naked eye. Its period of
variability is 14h 32m 10s.76, the last refinement of precision being of
course only attained in the final stages of reduction. Twenty-nine mean
values of the magnitude were determined, and they were nearly equally
spaced over the whole cycle of changes. The black dots in Fig. 5 exhibit
the mean values determined by Dr Roberts. The last three dots on the
extreme right are merely the same as the first three on the extreme left,
and are repeated to show how the next cycle would begin. The smooth dotted
curve will be explained hereafter, but, by reference to the scale of
magnitudes on the margins of the figure, it may be used to note that the
dots might be brought into a perfectly smooth curve by shifting some few of
the dots by about a hundredth of a magnitude.

This light-curve presents those characteristics which are due to successive
eclipses, but the exact form of the curve must depend on the nature of the
two mutually eclipsing stars. If we are to interpret the curve with all
possible completeness, it is necessary to make certain assumptions as to
the stars. It is assumed then that the stars are equally bright all over
their disks, and secondly that they are not surrounded by an extensive
absorptive atmosphere. This last appears to me to be the most dangerous
assumption involved in the whole theory.

Making these assumptions, however, it is found that if each of the
eclipsing stars were spherical it would not be possible to generate such a
curve with the closest accuracy. The two stars are certainly close
together, and it is obvious that in such a case the tidal forces exercised
by each on the other must be such as to elongate the figure of each towards
the other. Accordingly it is reasonable to adopt the hypothesis that the
system consists of a pair of elongated ellipsoids, with their longest axes
pointed towards one another. No supposition is adopted a priori as to the
ratio of the two masses, or as to their relative size or brightness, and
the orbit may have any degree of eccentricity. These last are all to be
determined from the nature of the light-curve.

In the case of RR Centauri, however, Dr Roberts finds the conditions are
best satisfied by supposing the orbit to be circular, and the sizes and
masses of the components to be equal, while their luminosities are to one
another in the ratio of 4 to 3. As to their shapes he finds them to be so
much elongated that they overlap, as exhibited in his figure titled "The
shape of the star RR Centauri" (Fig. 6.). The dotted curve shows a form of
equilibrium of rotating liquid as computed by me some years before, and it
was added for the sake of comparison.

On turning back to Fig. 5 the reader will see in the smooth dotted curve
the light variation which would be exhibited by such a binary system as
this. The curve is the result of computation and it is impossible not to
be struck by the closeness of the coincidence with the series of black dots
which denote the observations.

It is virtually certain that RR Centauri is a case of an eclipsing binary
system, and that the two stars are close together. It is not of course
proved that the figures of the stars are ellipsoids, but gravitation must
deform them into a pair of elongated bodies, and, on the assumptions that
they are not enveloped in an absorptive atmosphere and that they are
ellipsoidal, their shapes must be as shown in the figure.

This light-curve gives an excellent illustration of what we have reason to
believe to be a stage in the evolution of stars, when a single star is
proceeding to separate into a binary one.

As the star is faint, there is as yet no direct spectroscopic evidence of
orbital motion. Let us turn therefore to the case of another star, namely
V Puppis, in which such evidence does already exist. I give an account of
it, because it presents a peculiarly interesting confirmation of the
correctness of the theory.

In 1895 Pickering announced in the "Harvard Circular" No. 14 that the
spectroscopic observations at Arequipa proved V Puppis to be a double star
with a period of 3d 2h 46m. Now when Roberts discussed its light-curve he
found that the period was 1d 10h 54m 27s, and on account of this serious
discrepancy he effected the reduction only on the simple assumption that
the two stars were spherical, and thus obtained a fairly good
representation of the light-curve. It appeared that the orbit was circular
and that the two spheres were not quite in contact. Obviously if the stars
had been assumed to be ellipsoids they would have been found to overlap, as
was the case for RR Centauri. ("Astrophysical Journ." Vol. XIII. (1901),
page 177.) The matter rested thus for some months until the spectroscopic
evidence was re-examined by Miss Cannon on behalf of Professor Pickering,
and we find in the notes on page 177 of Vol. XXVIII. of the "Annals of the
Harvard Observatory" the following: "A.G.C. 10534. This star, which is
the Algol variable V Puppis, has been found to be a spectroscopic binary.
The period 1d.454 (i.e. 1d 10h 54m) satisfies the observations of the
changes in light, and of the varying separation of the lines of the
spectrum. The spectrum has been examined on 61 plates, on 23 of which the
lines are double." Thus we have valuable evidence in confirmation of the
correctness of the conclusions drawn from the light-curve. In the
circumstances, however, I have not thought it worth while to reproduce Dr
Roberts's provisional figure.

I now turn to the conclusions drawn a few years previously by another
observer, where we shall find the component stars not quite in contact.
This is the star Beta Lyrae which was observed by Goodricke, Argelander,
Belopolsky, Schur, Markwick and by many others. The spectroscopic method
has been successfully applied in this case, and the component stars are
proved to move in an orbit about one another. In 1897, Mr. G.W. Myers
applied the theory of eclipses to the light-curve, on the hypothesis that
the stars are elongated ellipsoids, and he obtained the interesting results
exhibited in Fig. 7. ("Astrophysical Journ." Vol. VII. (1898), page 1.)

The period of Beta Lyrae is relatively long, being 12d 21h 47m, the orbit
is sensibly eccentric, and the two spheroids are not so much elongated as
was the case with RR Centauri. The mass of the system is enormous, one of
the two stars being 10 times and the other 21 times as heavy as our sun.

Further illustrations of this subject might be given, but enough has been
said to explain the nature of the conclusions which have been drawn from
this class of observation.

In my account of these remarkable systems the consideration of one very
important conclusion has been purposely deferred. Since the light-curve is
explicable by eclipses, it follows that the sizes of the two stars are
determinable relatively to the distance between them. The period of their
orbital motion is known, being identical with the complete period of the
variability of their light, and an easy application of Kepler's law of
periodic times enables us to compute the sum of the masses of the two stars
divided by the cube of the distance between their centres. Now the sizes
of the bodies being known, the mean density of the whole system may be
calculated. In every case that density has been found to be much less than
the sun's, and indeed the average of a number of mean densities which have
been determined only amounts to one-eighth of that of the sun. In some
cases the density is extremely small, and in no case is it quite so great
as half the solar density.

It would be absurd to suppose that these stars can be uniform in density
throughout, and from all that is known of celestial bodies it is probable
that they are gaseous in their external parts with great condensation
towards their centres. This conclusion is confirmed by arguments drawn
from the theory of rotating masses of liquid. (See J.H. Jeans, "On the
density of Algol variables", "Astrophysical Journ." Vol. XXII. (1905), page
97.)

Although, as already explained, a good deal is known about the shapes and
the stability of figures consisting of homogeneous incompressible liquid in
rotation, yet comparatively little has hitherto been discovered about the
equilibrium of rotating gaseous stars. The figures calculated for
homogeneous liquid can obviously only be taken to afford a general
indication of the kind of figure which we might expect to find in the
stellar universe. Thus the dotted curve in Fig. 5, which exhibits one of
the figures which I calculated, has some interest when placed alongside the
figures of the stars in RR Centauri, as computed from the observations, but
it must not be accepted as the calculated form of such a system. I have
moreover proved more recently that such a figure of homogeneous liquid is
unstable. Notwithstanding this instability it does not necessarily follow
that the analogous figure for compressible fluid is also unstable, as will
be pointed out more fully hereafter.

Professor Jeans has discussed in a paper of great ability the difficult
problems offered by the conditions of equilibrium and of stability of a
spherical nebula. ("Phil. Trans. R.S." Vol. CXCIX. A (1902), page 1. See
also A. Roberts, "S. African Assoc. Adv. Sci." Vol. I. (1903), page 6.) In
a later paper ("Astrophysical Journ." Vol. XXII. (1905), page 97.), in
contrasting the conditions which must govern the fission of a star into two
parts when the star is gaseous and compressible with the corresponding
conditions in the case of incompressible liquid, he points out that for a
gaseous star (the agency which effects the separation will no longer be
rotation alone; gravitation also will tend towards separation...From
numerical results obtained in the various papers of my own,...I have been
led to the conclusion that a gravitational instability of the kind
described must be regarded as the primary agent at work in the actual
evolution of the universe, Laplace's rotation playing only the secondary
part of separating the primary and satellite after the birth of the
satellite."

It is desirable to add a word in explanation of the expression
"gravitational instability" in this passage. It means that when the
concentration of a gaseous nebula (without rotation) has proceeded to a
certain stage, the arrangement in spherical layers of equal density becomes
unstable, and a form of bifurcation has been reached. For further
concentration concentric spherical layers become unstable, and the new
stable form involves a concentration about two centres. The first sign of
this change is that the spherical layers cease to be quite concentric and
then the layers of equal density begin to assume a somewhat pear-shaped
form analogous to that which we found to occur under rotation for an
incompressible liquid. Accordingly it appears that while a sphere of
liquid is stable a sphere of gas may become unstable. Thus the conditions
of stability are different in these two simple cases, and it is likely that
while certain forms of rotating liquid are unstable the analogous forms for
gas may be stable. This furnishes a reason why it is worth while to
consider the unstable forms of rotating liquid.

There can I think be little doubt but that Jeans is right in looking to
gravitational instability as the primary cause of fission, but when we
consider that a binary system, with a mass larger than the sun's, is found
to rotate in a few hours, there seems reason to look to rotation as a
contributory cause scarcely less important than the primary one.

With the present extent of our knowledge it is only possible to reconstruct
the processes of the evolution of stars by means of inferences drawn from
several sources. We have first to rely on the general principles of
stability, according to which we are to look for a series of families of
forms, each terminating in an unstable form, which itself becomes the
starting-point of the next family of stable forms. Secondly we have as a
guide the analogy of the successive changes in the evolution of ideal
liquid stars; and thirdly we already possess some slender knowledge as to
the equilibrium of gaseous stars.

From these data it is possible to build up in outline the probable history
of binary stars. Originally the star must have been single, it must have
been widely diffused, and must have been endowed with a slow rotation. In
this condition the strata of equal density must have been of the planetary
form. As it cooled and contracted the symmetry round the axis of rotation
must have become unstable, through the effects of gravitation, assisted
perhaps by the increasing speed of rotation. (I learn from Professor Jeans
that he now (December 1908) believes that he can prove that some small
amount of rotation is necessary to induce instability in the symmetrical
arrangement.) The strata of equal density must then become somewhat pear-
shaped, and afterwards like an hour-glass, with the constriction more
pronounced in the internal than in the external strata. The constrictions
of the successive strata then begin to rupture from the inside
progressively outwards, and when at length all are ruptured we have the
twin stars portrayed by Roberts and by others.

As we have seen, the study of the forms of equilibrium of rotating liquid
is almost complete, and Jeans has made a good beginning in the
investigation of the equilibrium of gaseous stars, but much more remains to
be discovered. The field for the mathematician is a wide one, and in
proportion as the very arduous exploration of that field is attained so
will our knowledge of the processes of cosmical evolution increase.

From the point of view of observation, improved methods in the use of the
spectroscope and increase of accuracy in photometry will certainly lead to
a great increase in our knowledge within the next few years. Probably the
observational advance will be more rapid than that of theory, for we know
how extraordinary has been the success attained within the last few years,
and the theory is one of extreme difficulty; but the two ought to proceed
together hand in hand. Human life is too short to permit us to watch the
leisurely procedure of cosmical evolution, but the celestial museum
contains so many exhibits that it may become possible, by the aid of
theory, to piece together bit by bit the processes through which stars pass
in the course of their evolution.

In the sketch which I have endeavoured to give of this fascinating subject,
I have led my reader to the very confines of our present knowledge. It is
not much more than a quarter of a century since this class of observation
has claimed the close attention of astronomers; something considerable has
been discovered already and there seems scarcely a discernible limit to
what will be known in this field a century from now. Some of the results
which I have set forth may then be shown to be false, but it seems
profoundly improbable that we are being led astray by a Will-of-the-Wisp.

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