THE PROCESS OF FISSION

The motion of our hypothetical mass of nebulous matter has now been traced out through its earlier stages in which it formed a rotating nebula, and through its later stages in which this nebula condensed into stars. In the last chapter we considered the general nature of the motion to be expected in the cluster of stars so formed; the present chapter will be devoted to the further history of individual stars.

We have supposed that an individual star comes into existence as a condensation in a nebular arm. In this earliest period of its existence its mean density is very low, being perhaps of the order of 10-17 grammes per cubic centimetre, and its surrounding atmosphere is contiguous with that of the neighbouring stars. At this stage it shares in the rotation of the nebula of which it forms part, the period of this rotation being perhaps of the order of 160,000 years.

As time proceeds the arms of the nebula expand while individual stars contract, so that the stars become continually more distinct from one another until finally they may be regarded as entirely separate bodies, each describing its independent orbit under the gravitational attractions of the other stars.

The best-known configurations of equilibrium of a rotating homogeneous mass, namely Maclaurin's spheroids and Jacobi's ellipsoids, are both of the ellipsoidal form, and this form will prove to be of primary importance in all the cosmogonical problems we shall attempt to solve. We accordingly devote a chapter to the subject of ellipsoidal configurations.

Looked at merely from the point of view of convenience in the development of the subject, the ellipsoidal form has the advantage that the potential of an ellipsoidal mass is known and is comparatively simple, and that the ellipsoidal configurations provide admirably clear examples of Poincaré's theory of linear series and stability. These reasons alone might justify our studying ellipsoidal configurations in some detail, but there are weightier reasons, as we shall soon see.

Throughout this chapter and the three succeeding chapters the matter under discussion will be supposed homogeneous and incompressible; the more complicated problems presented by non-homogeneous and compressible masses will be attacked in Chapter VII.

We shall deal in turn with three distinct problems–the first, that of a mass of liquid rotating freely under its own gravitational forces; the second, that of a mass devoid of rotation but acted on tidally by another mass; the third that of two masses rotating round one another and acting tidally on one another.

The present essay is primarily an attempt to follow up a line of research initiated by Laplace and Maclaurin, and extended in various directions by Roche, Lord Kelvin, Jacobi, Poincaré and Sir G. Darwin. Within two years of the close of his life, Darwin remarked that the way to further progress in cosmogony was blocked by our ignorance of the figures of equilibrium of rotating gaseous masses. He wrote as follows (Darwin and Modern Science, p. 563, and Tides, 3rd edition, p. 401):

“As we have seen, the study of the forms of equilibrium of rotating liquids is almost complete, and a good beginning has been made in the investigation of the equilibrium of gaseous stars, but much more remains to be discovered.”

“As a beginning we should like to know how a moderate degree of compressibility would alter the results for liquid, and…to understand more as to the manner in which rotation affects the equilibrium and stability of rotating gas. 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….

“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.”

The sequence of events to be expected in a mass of astronomical matter left solely to the influence of its own rotation has now been traced out with tolerable completeness.

Of the five uniformities of structure mentioned in our introductory chapter we have found that two fall naturally into their places in the scheme of evolution of a rotating mass, these two being the spiral nebulae and the binary and multiple stars. Two others, namely the planetary and ring nebulae and the globular and moving star-clusters, seem at least to be capable of explanation in terms of a rotational theory of evolution, although our interpretation of these formations was largely conjectural.

The fifth uniformity was that observed in the solar system, and for this no place has been found in the rotational scheme of evolution. It is true that we found (§ 257) that planets might possibly form out of the atmosphere thrown off equatorially from a rotating mass of gas, but several objections present themselves against any attempt to explain the origin of our solar system in this way–primarily the objection that the next stage in evolution ought to be for the central mass to break up into an ordinary binary star, whereas our sun and planets are not binary. Also the arrangement of the components of typical multiple stars such as can have been formed by rotation (cf. fig. 45, p. 265) does not in the least resemble that observed in the solar system.

From a purely theoretical discussion of the evolution of a mass of rotating gas we have been led to the hypothesis that the spiral nebulae are merely masses of rotating gas which have reached a stage of disintegration, the rotation having become so great through shrinkage that configurations of equilibrium are no longer possible. It would be of the utmost interest to follow out dynamically the different processes of this disintegration but unfortunately the mathematical difficulties have so far proved to be too great.

We have, however, found that the masses of these spirals must be supposed to be enormously greater than that of our sun, and the general nature of the disintegration has been seen to consist of the formation in the nebular arms of condensing nuclei each of mass just about comparable with that of our sun. Thus the hypothesis which has already been adopted seems to lead irresistibly to the conclusion that the final result of the process of disintegration which we see going on in the spiral nebulae must be the formation of star-clusters.

As to the features to be expected in these final star-clusters our dynamical analysis has so far told us almost nothing. It seems not unreasonable to expect that the star-clusters will be of the type we have described as “globular” – thus we may conjecture that the observed spiral nebulae are forming star-clusters similar to observed globular star-clusters and that the observed globular clusters have originated out of spiral nebulae.

In the last chapter we examined the sequence of changes which would occur in a mass of gas left to its own gravitation at rest in space. We found that matter once in existence would either disperse into space or contract continually. Masses which disperse into space would have but a transitory existence; the permanent bodies in the heavens must be supposed to be contracting.

We accordingly think of the permanent astronomical bodies as beginning existence in a state of extreme rarity. If one such mass existed alone in the universe, it would tend to assume a spherical form if devoid of rotation, or a spheroidal or pseudo-spheroidal form if endowed with a small amount of rotation. Observation, however, does not encourage the view that the whole universe originated out of a single mass of gas; we shall find it more profitable to think of a number of separate and detached nebular masses as forming the earliest stage in the process of cosmic evolution.

Whether these masses ought to be thought of as being originally endowed with motion, either of translation or of rotation, we do not know. In any case they must in time be set into motion by their mutual gravitational attractions.

The last chapter contained a discussion of the ellipsoidal configurations which can occur in the various problems we have had under consideration, and it was found possible to investigate their stability or instability subject to their remaining ellipsoidal. A configuration which is unstable when subject to an ellipsoidal constraint will of course remain unstable when this constraint is removed, but a configuration which is stable before the constraint is removed will not necessarily remain stable. We can only discuss whether such a configuration is stable or not when we have a complete knowledge of all configurations of equilibrium adjacent to the ellipsoidal configurations; we then know the positions of the various points of bifurcation on the ellipsoidal series, and the stability of this series is immediately determined.

A first condition for being able to discover configurations of equilibrium of any type is that we shall be able to write down the potential of the mass when in these configurations. Thus it appears that before being able to discuss in a general way the configurations of equilibrium adjacent to ellipsoidal configurations, we must be able to write down the potential of a distorted ellipsoid.

The method of ellipsoidal harmonics at once suggests itself. It has been used by Poincaré Darwin, and Schwarzschild to determine configurations of equilibrium adjacent to the equilibrium configurations. In this way the various points of bifurcation on the ellipsoidal series we have had under discussion are readily determined.

SURVEY OF THE PROBLEM

The Solar System

In 1543 Copernicus published his treatise “De Revolutionibus Orbium Coelestium” in which the apparent motion of the planets was explained by the simple hypothesis that they all described orbits about the Sun at rest. Two thirds of a century later, in the early days of 1610, Galileo first observed the satellites of Jupiter revolving around their primary, and so obtained what amounted almost to direct visual proof of the truth of the Copernican system of astronomy. But in verifying Copernicus' solution of one problem, Galileo had opened up another. For it now became clear that there were at least two systems of almost exactly similar formation in the universe, and a philosophic mind could not but conclude that they had probably originated from similar causes, and would be impelled to conjecture as to what those causes might be.

In this way the problem of scientific cosmogony had its origin. To the modern astronomer the problem is much richer, wider- and more definite, in proportion as the mass of observational material within his knowledge is greater than that with which Galileo was acquainted. In the solar system alone, we know that in addition to the eight great planets, there are upwards of 900 minor planets or asteroids, and all these 908- or more bodies shew the same regularity in their motion.

The result obtained in the last chapter for the rotational problem combined with those previously obtained in Chapter III for the tidal and double-star problems, has now established that

In all the three problems under consideration there are no figures of stable equilibrium except ellipsoids and spheroids.

In each of these problems the succession of states has been determined by the continuous variation of a parameter–the angular momentum in the rotational and double-star problems, and the distance R in the tidal problem. And in each case it is quite possible for this parameter to vary to beyond the limits within which stable configurations are possible. We must accordingly try to obtain what information we can as to the changes to be expected after this-limit is passed.

Poincaré, writing with special reference to the rotational problem, remarks that if the pear-shaped figure proved to be unstable, “la masse fluide devrait se dissoudre par un cataclysme subit.” The pear-shaped figure has now been proved to be unstable, and we must examine the nature of the cataclysm. The situation is similar in the two other problems; when the two masses concerned in either approach one another to within less than a certain distance no configurations of stable equilibrium are possible, and a cataclysm occurs.

The field of dynamical astronomy is a wide one and it is obvious that it will be impossible to consider even in the most elementary manner all branches of it; for it embraces all those effects in the heavens which may be attributed to the effects of gravitation. In the most extended sense of the term it may be held to include theories of gravitation itself. Whether or not gravitation is an ultimate fact beyond which we shall never penetrate is as yet unknown, but Newton, whose insight into physical causation was almost preternatural, regarded it as certain that some further explanation was ultimately attainable. At any rate from the time of Newton down to to-day men have always been striving towards such explanation—it must be admitted without much success. The earliest theory of the kind was that of Lesage, promulgated some 170 years ago. He conceived all space to be filled with what he called ultramundane corpuscles, moving with very great velocities in all directions. They were so minute and so sparsely distributed that their mutual collisions were of extreme rarity, whilst they bombarded the grosser molecules of ordinary matter. Each molecule formed a partial shield to its neighbours, and this shielding action was held to furnish an explanation of the mutual attraction according to the law of the inverse square of the distance, and the product of the areas of the sections of the two molecules.