FOOTNOTES:
[28] Sämmtliche Werke, Bd. VI., pp. 5-12, 1839.
[29] G. H. Darwin, Encyclopædia Britannica, article on 'Tides.'
[30] Dynamik des Himmels, p. 49.
[31] Darwin, Philosophical Transactions, vol. clxxii., p. 528.
[32] Philosophical Transactions, vol. clxxi., p. 876.
[33] Harvard Annals, vol. liii., p. 58.
[CHAPTER VI]
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.
Observations and investigations of stellar eclipses have created a new department of astrophysics, and have vastly widened the domain of cosmogony. They have brought to notice a number of systems, not merely in a primitive, but seemingly in an inchoate stage of development. The periods of occulting stars are nearly all of them less than seven days, although one extending to thirty-one has lately been recognised; and the comparative length of the intervals of obscuration shows them to be produced by the circulation in narrow orbits of distended globes. These are characteristic symptoms of juvenility, for, as we have seen, orbits widen and periods lengthen with the efflux of time through the frictional power of bodily tides.
Now the class of stars which obviously and certainly undergo eclipses has some outlying members of a still dubious nature. And their marginal position serves greatly to enhance the present, the prospective, and the retrospective interest attaching to them. These remarkable objects vary in light continuously. Their phases are not, like those of Algol, mere interruptions to a regular course of steady shining. They progress without a moment's sensible pause; they are represented graphically by a smoothly-flowing, symmetrical curve. The eclipses by which they are occasioned—if they are so occasioned—must, accordingly, succeed each other in a strictly unbroken series. No sooner has one terminated than the next commences. One star passes first behind, then in front of its companion, and their combined brightness is seen undimmed only during the few moments of actual maximum. This means that they revolve in contact; they are separated by no sensible gap of space.
Goodricke's variable, β Lyræ, is held to be thus constituted. The possibility, at least, of employing the 'satellite-theory' to account for its changes was demonstrated some years ago by Mr. G. W. Myers, of Indiana.[40] He found the system to be composed of two barely separated ellipsoids, circulating in the visual plane, and producing, by their successive transits, two unequal eclipses in the course of each period of 12·91 days. The joint mass of the pair is just thirty times that of our sun, but their mean density has the almost incredibly small value of 1/1200 that of water. Their real existence is conditional upon the possibility that masses much more tenuous than atmospheric air should radiate with the intensity of true suns. Spectroscopic observations are not wholly unfavourable to Mr. Myers's hypothesis, but their interpretation is hampered by discrepancies so numerous and perplexing that no secure inference can be derived from them. Moreover, the star supposed to be alone presented to view at the principal minimum is that giving the bright-line spectrum; yet it is compulsorily assumed, in order to meet the exigencies of the situation, to be much more massive, while much less intrinsically bright, than its companion. This is disquieting, but nearly everything connected with β Lyræ is more or less disquieting.
A variable of the same type, but much fainter, was made the subject of a similar inquiry by Mr. Myers in 1898.[41] U Pegasi never attains ninth magnitude; hence, spectroscopic complications equally with spectroscopic verification remain at present out of sight. The star, nevertheless, excites keen interest, and claims sustained attention. Its light-curve has been laid down with exquisite accuracy at Harvard College, and shows two slightly unequal minima to be comprised within a period of nine hours, signifying, on the adopted theory, the occurrence of alternating eclipses at intervals of four and a half hours. The distance from centre to centre of the occulting stars, the smaller of which is of about eight-tenths the brightness of the larger, 'does not materially differ,' Mr. Myers tells us, 'from the sum of their radii, suggesting the probable existence of the "apioidal" form of Poincaré.' If they do not actually coalesce, the component bodies revolve in contact, and rotate synchronously. Thus, it is hard to say whether U Pegasi should be accounted as a single pear-shaped mass spinning in the time of light-change, or as a close couple circulating freely in that identical period. The mean density of the system appears to lie between one-third and one-fourth that of the sun.
Another specimen of the 'dumb-bell' system is possibly met with in R2 Centauri. The narrow range of its variation makes it a delicate object to observe; but Mr. A. W. Roberts, who first noticed its peculiarity in 1896, has since accumulated an extensive series of wonderfully accurate visual determinations of its fluctuating brightness, and has besides rendered them the basis of an able and exhaustive theoretical discussion.[42] The double period of R2 Centauri is restricted to fourteen hours thirty-two minutes. Within this brief span quadruple phases are included—that is to say, two evenly balanced maxima and two slightly disparate minima. These result, Mr. Roberts concludes, from the mutual eclipses of interpenetrating ellipsoids, one somewhat more luminous than the other, revolving—if they can properly be said to revolve—in an orbit inclined 32° to the visual plane. They are of just one-third the solar density, and the forms satisfying photometric requirements by the varying areas of luminous surface presented to sight in different sections of their path show a surprising agreement with the bi-prolate figure given by Professor Darwin's analysis as the shape of a body on the verge of disruption through accelerated rotatory movement. The inference is, then, almost irresistible that R2 Centauri really exemplifies the nascent stage of binary stars. To establish this completely, however, spectroscopic data are needed; and they are difficult to procure for a star below the seventh magnitude.
No such obstacle impedes the investigation of the analogous, but much brighter object, V Puppis. Detected as a spectroscopic binary by Professor Pickering in 1895, this star traverses so wide an orbit in the short period of thirty-five hours as to imply—if the published details are correct—that the pair possess no less than 348 times the gravitational power of the sun. They are, nevertheless, according to Mr. Roberts, fifty times more tenuous, and each globe should have a diameter of about 16½ million miles; yet nothing of all this is incredible. The light-curve of V Puppis, as traced by Mr. Roberts, is closely modelled upon that of U Pegasi. And he postulates similar conditions of eclipse. It rests with the spectroscope to determine whether those conditions are realized or not.
Probably all short-period variables are binaries, with coincident orbital- and light-cycles. But all are not occulting binaries. There are some—we are still ignorant of their proportionate numbers—which undergo a course of light-change, apparently compatible with an occulting hypothesis, yet certainly escape eclipse. Professor Campbell has made it unmistakably clear that ζ Geminorum is thus constituted.[43] Two stars are present, but their plane of motion is inclined at an unknown angle to the line of sight; it does not approximate to coincidence with it. Now the possibility is not excluded that V Puppis belongs to the same class. Mr. Roberts's assumptions are, indeed, in themselves plausible, and they may at any moment be proved, by a few well-timed spectrograms, to be undeniably true.
The one conclusive test of their truth is the cessation of radial movement at epochs of minimum. Evidently, if the diminution in lustre be due to an eclipse, the eclipsing and eclipsed bodies must be crossing the line of sight just when the obscuration is deepest. There is no evading this geometrical requirement, and it must be rigorously complied with in the circular orbits traversed by bodies revolving in contact. Before, then, Mr. Roberts's theory of V Puppis can be accepted with implicit confidence, it has to be ascertained whether a zero of radial speed is reached concurrently with the photometric minima. If so, these may be unhesitatingly set down as eclipse-phenomena; if, on the contrary, the decline in brightness prove to be unrelated to a slackening of speed, then the supposition that it accompanies and indicates a transit must be peremptorily discarded. Moreover, the spectroscopic verdict as regards V Puppis can safely be applied to stars with similar light-curves, especially to R2 Centauri and U Pegasi, and may serve to clear away some of the intricacies connected with the exceptional system of β Lyræ. The measurement of a single spectrographic plate might thus, by deciding the test-case of the binary in the poop of Argo, be made essentially to supply the lack of desirable, but at present unattainable determinations as regards a considerable number of analogous objects.
The existence of stellar systems of the 'dumb-bell' type would violate no mechanical law. 'Roche's limit' does not apply to globes comparable in size. The range of disparity within which it holds good has not, indeed, been theoretically established; but it may be said, in general terms, to concern the relations of planets and satellites (to use a purposely vague phrase), not those of double stars. What the law asserts is that a subordinate small body cannot revolve intact at a less distance than 2·44 radii of its primary from that primary's centre, if their mean density be the same. For satellites of slighter consistence the limit should be extended. Our own moon, for instance, could never have circulated, without being rent in pieces by tidal strains, in an orbit less than 22,000 miles in diameter.[44]
Bodies of co-ordinate mass are, however, exempted from the prohibitive rule against mutual approach. No analytical veto is imposed upon the origin by fission of double stars, or upon the subsistence of stellar Siamese twins. The inequalities of their mutual attractions avail to distort, not to disrupt, such embryo globes. Their individuality, therefore, once created, is in a manner indestructible. It tends, in fact, to become more pronounced as the orbital span gradually widens through the reactive effects of tidal friction. The 'dumb-bell' condition may then be regarded as in a manner transitory. Nor can we be assured of its actuality otherwise than by the peculiar nature of the eclipses attending upon it, taken in connection with correlated spectroscopic observations proving eclipses of the kind veritably to take place. The disclosure, by such means, of systems so strangely conditioned promises to afford a deeper insight than would else have been possible into the cosmical order, and fills a blank page in the marvellous history of sidereal birth and growth.