THE FISSION OF ROTATING GLOBES
Few people need to be told that a rotating fluid mass is shaped very much like an orange. It assumes the form of a compressed sphere. And the reason for its compression is obvious. It is that the power of gravity, being partially neutralized by the centrifugal tendency due to axial speed, decreases progressively from the poles, where that speed has a zero value, to the equator, where it attains a maximum. Here, then, the materials of the rotating body are virtually lighter than elsewhere, and consequently retreat furthest from the centre. The 'figure of equilibrium' thus constituted is a spheroid, a body with two unequal axes. In other words, its meridional contour—that passing through the poles—is an ellipse, while its equator is circular.
Now we know familiarly, not only that a spinning sphere becomes a spheroid, but that the spheroid grows more oblate the faster it spins. The flattened disc of Jupiter, for instance, compared with the round face of Mars, at once suggests a disparity in the rate of gyration. But there must be a limit to the advance of bulging, or the spheroid, accelerated ad infinitum, would at last cease to exist in three dimensions. Clearly this unthinkable outcome must be anticipated; at some given point the process of deformation must be interrupted. A breach of continuity intervenes; the train is shunted on to a branch line. Nor is it difficult to divine, in a general way, how this comes to pass. Equilibrium, beyond doubt, breaks down when rotation attains a certain critical velocity, varying according to circumstances, and the spheroid either alters fundamentally in shape or goes to pieces.
So much plain common-sense teaches, yet the precise determination of the course of events is one of the most arduous tasks ever grappled with by mathematicians. M. Poincaré essayed it in 1885;[34] it was independently undertaken a little later by Professor Darwin;[35] and the subject has now been prosecuted for eighteen years, chiefly by these two eminent men, with a highly interesting alternation of achievement, one picking up the thread dropped by the other, and each in turn penetrating somewhat further into the labyrinth. The results, nevertheless, are still to some extent inconclusive; they indicate, rather than indite, the genetic history of systems. A strong light is, indeed, thrown upon it; but in following its guidance, the limitations of the inquiry have to be borne in mind. The chief of these are, first, that the assumed spheroid is liquid; secondly, that it is homogeneous. Neither of these conditions, however, is really prevalent in nature, so that inferences based upon them can only be accepted under reserve. They were adopted, not by choice, but through the necessities of the case. There was no possibility of dealing mathematically with bodies in any other than the liquid state. The equilibrium of gaseous globes defies treatment, except under arbitrary restrictions.[36] Nor is it possible to cope with the intricacies of calculation introduced by variations of interior density. Cosmical masses, as they actually exist, are nevertheless strongly heterogeneous, so that at the utmost only an approximation to the genuine course of their evolution can be arrived at by the most skilful analysis. Yet even an approximate solution of such a problem is of profound interest. We can here only attempt briefly to specify its nature.
The course of change by which the equilibrium of a rotating liquid spheroid is finally overthrown has, at any rate, been satisfactorily tracked. When its spinning quickens to a disruptive pitch, it acquires three unequal axes instead of two. The equator becomes elliptical like the meridians. A 'Jacobian ellipsoid' is constituted. To this new form, it would seem, a long spell of stability must be attributed; only its major axis becomes more and more protracted as cooling progresses, and with cooling, contraction, and with contraction the increase of axial velocity. Then at last a crisis once more supervenes; there is a collapse of equilibrium, and its re-establishment involves the sacrifice of the last vestige of symmetry. An 'apioid,' or pear-shaped body, replaces the antecedent ellipsoid; and its apparent incipient duality suggested to M. Poincaré that the furrow unequally dividing it might deepen, with still accelerated gyration, into a cleft, splitting the primitively single mass into a planet and satellite. But this eventuality, he was careful to note, had no direct bearing on Laplace's hypothesis, which dealt with a nebula condensed towards the centre, while the fissured apioid was liquid and homogeneous.[37]
Professor Darwin followed out the conditions of this remarkable pear-shaped body to a closer degree of approximation than its original investigator had done, and succeeded in virtually demonstrating its conditional stability. But his analysis tended to smooth away the characteristic peculiarities of its shape, and, so far, to diminish the probability of its ultimate disruption. Mr. Jeans, on the other hand, from an elaborate study of a series of cigar-shaped figures which in theory follow a parallel course of development to that pursued by ellipsoids, derived, by strict mathematical reasoning, the actual separation of a satellite from one end of a parent-cylinder. The representative figures reminded Professor Darwin 'of some such phenomenon as the protrusion of a filament of protoplasm from a mass of living matter.' 'In this almost life-like process' he saw 'a counterpart to at least one form of the birth of double stars, planets, and satellites.'[38]
But the resemblance, when examined dispassionately, seems shadowy and evasive, especially when we confront it with the case of double stars. Here, indeed, an entirely different set of conditions comes into play from that postulated by Poincaré and Darwin, since stars are certainly not liquid bodies.[39] They are most likely gaseous to the core, though the indefinite diffusiveness incident to gaseity is restricted by their condensed photospheric surfaces. This circumstance intimates the possibility that the results arrived at for liquid globes by mathematical analysis may, with qualifications, be extended to stars; but the necessary qualifications, unfortunately, are vague and large; for too little is known regarding the physical condition of stellar spheres to warrant assumptions that might provide a secure basis for research.
The evolution of binary stars can then only be treated of inferentially, not rigorously; and we must, at the outset, discard the idea that it is illustrated by the phenomena of double nebulæ. Many such objects thought to supply clinching visual arguments for the actual effectiveness of slow cosmic fission proved, on the application to them of the late Professor Keeler's searching photographic methods, to be knots on spiral formations. Their mutual relations are then entirely different from what had been supposed by telescopic observers; they are, in fact, still structurally connected, and the mode of their origin, however inviting to conjecture, scarcely comes within the scope of definitely conducted inquiries. Their future destiny is no more accessible to it than their past history, and only by a daring flight of imagination can we see in spiral nebulæ the prototypes of double stars.
Questions as to the mode of genesis of these latter systems have, in recent years, acquired extraordinary interest. Conclusive answers cannot, indeed, at present be given to them, because the terms in which they are couched lack distinctness, owing to our lack of knowledge; but probable answers may legitimately supply their place, at least ad interim, above all when their probability is heightened almost to certainty by the accumulation of circumstantial evidence.