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.