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 16-1/2 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.
FOOTNOTES:
[Footnote 34: _Acta Mathematica_, vol. vii., Stockholm, 1885.]
[Footnote 35: _Proceedings Royal Society_, vols. xlii., lxxi.;
_Philosophical Transactions_, vols. cxcviii., cxcix., Series A.]
[Footnote 36: J. H. Jeans, _Philosophical Transactions_, vol. cxcix.,
Series A, p. 1.]
[Footnote 37: _Figures d'Équilibre d'une Masse Fluide_, p. 172.]
[Footnote 38: _Proceedings Royal Society_, vol. lxxi., p. 183.]
[Footnote 39: J. H. Jeans, _Astrophysical Journ._, vol. xxii., p. 93.]
[Footnote 40: _Astrophysical Journal_, vol. vii., p. 1.]
[Footnote 41: _Astrophysical Journal_, vol. viii., p. 163.]
[Footnote 42: _Monthly Notices_, vol. lxiii., p. 627.]
[Footnote 43: _Astrophysical Journal_, vol. xiii., p. 90; _Science_,
July 27, 1900.]
[Footnote 44: G. H. Darwin, _The Tides_, p. 327.]