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.