FOOTNOTES:
[16] Mémoires de l'Académie de Montpellier, tom. i.
[17] The Tides, p. 327.
[18] Mémoires de l'Académie de Montpellier, tom. viii.
[19] C. Wolf, Bulletin Astronomique, tom. i., p. 596.
[20] American Journal of Science, vol. xxxviii., p. 3.
[21] G. H. Darwin, Nature, vol. xxxi., p. 506.
[22] Proceedings of the American Philosophical Society, vol. xxii., p. 109.
[23] De Vries, Die Mutationstheorie, Bd. II., p. 714.
[24] Formation Mécanique du Système du Monde. See also Le Problème Solaire, by the Abbé Th. Moreux, p. 63 et seq.
[25] Revue des Questions Scientifiques, January, 1904.
[26] The Earth's Beginnings, p. 247.
[27] Astrophysical Journal, vol. xi., p. 130.
[CHAPTER V]
TIDAL FRICTION AS AN AGENT IN COSMOGONY
The effects of tidal friction are of almost infinite complexity. How it will act in each particular case cannot be predicted offhand; it is a matter for detailed inquiry. Mutually countervailing influences have to be taken into account, nor is the balance easy to strike. The manner of its inclination may, indeed, often depend upon qualities and relations of the bodies concerned which lie outside the range of what can be distinctly ascertained. All that may be hoped for, then, is to arrive at estimates neither misleading by their ostensible precision, nor yet so vague as to be wholly uninstructive, of the part played by tidal forces in moulding the history of connected globes.
The assumption that they attract one another as if the mass of each were collected at its centre, is one of those convenient fictions without which the advancing feet of science would be impeded by tangled thickets of illusory refinements and superfluous elaborations. The fiction would correspond with fact only if the globes were truly spherical, and they could be truly spherical only if they were ideally rigid. Cosmic bodies, however—suns and planets alike—are actually plastic spheroids; they can, to be sure, be treated without sensible error as attractive points when their distances are very great relatively to their diameters; but upon a closer approach inequality of action supervenes. The component parts of the gravitating masses respond, each individually, and in a measure independently, to the graduated pulls exercised upon them, and tidal strains begin variously to take effect.
Their historical significance was in part divined by Kant. His penetration of so recondite a secret is truly astonishing. A struggling young pedagogue in a remote Prussian province, profoundly learned, though no more than half skilled in technical acquirements, saw by intuition what escaped the acumen of all the great geometers of the eighteenth century—namely, that the moon turns one perpetual face towards the earth, because its primitive rotation was stopped by the friction of earth-raised tides. He perceived besides that a reciprocal action of the same kind must affect the earth, and will continue to affect it until the day coincides in length with the month. Nor did he fail to point out that, in a molten state of the globes, the process would advance with comparative rapidity. To one solitary thinker, then, it became apparent, already in 1754,[28] that oceanic tides are, in cosmogony, of negligible importance compared with bodily tides.
There is no substance in nature that will not change its shape through prolonged stress, and the more readily the nearer it approaches to the fluid condition. The heaping-up of the waters on the earth's surface at the bidding of the moon is thus a differential effect. Continents heave and subside as well as oceans, though not nearly to the same extent. The measurable rise of water serves to gauge the relative mobilities of the solid globe and of its liquid envelope. If the former did not yield at all to the pull so readily obeyed by the latter, the tides would, in fact, be greater than they actually are in the proportion of about three to two, the ratio indicating for the earth an effective rigidity at least equal to that of steel.[29] Were there no discrepancy in rigidity between the various parts of our terraqueous world, tides would fail to be perceptible. The ocean and the bed of the ocean would rise and fall together, and to the same extent. In the far past there was no discrepancy. The viscous earth took, as a whole, the form momentarily impressed upon it by the unequal attractions of the sun and moon on its variously distant sections, with the upshot of bringing the year, month, and day into relations so familiar as to appear inevitable.
Tidal friction does not merely act as a check upon rotational speed. One element of motion in a system cannot be altered without some counter-change in the others. They are coupled up together like a train of geared wheels. From the principle of the conservation of moment of momentum, we know with certainty that a loss in one direction must be compensated by a gain in some other. Tidal friction had, then, reactive consequences. They were first adverted to by Julius Robert Mayer in 1848,[30] and were brought prominently into view in the series of investigations begun by Professor Darwin in 1879. The rotational momentum removed from the earth by the drag of a circulating wave of deformation must assuredly have reappeared in some other part of the system. It was restored, all but the percentage wasted as heat, by the widening of the lunar orbit.[31] Concomitantly with the slackening of the earth's axial rate, the moon retreated from its surface, pulled forward by the tidal crest continually in advance of its position. This redressed the balance by augmenting orbital momentum, while at the same time diminishing the moon's linear velocity. The importance of this secondary frictional effect in the history of the earth-moon system was the virtual discovery of Professor Darwin.
That system occupies a critical situation in the solar cortège. The planets interior to it have no satellites; the planets exterior to it (Neptune making probably only an apparent exception to the rule) have two or more. The earth alone is truly binary; and the moon is not only its solitary companion, but it is by far the largest companion-body, relatively to the mass of its primary, to be found within the precincts of the solar domain. These circumstances are certainly not disconnected one from the other, and they obviously depend upon a single cause. Solar tidal friction was here the determining factor. The apportionment of satellites to the various planets was, beyond doubt, in great measure prescribed by the degrees of retarding power exerted on their axial movement through the agency of sun-raised tides in their still plastic bodies. Hence, the disruptive rate of spinning needed for the separation of satellites was never attained by either Mercury or Venus; they remained moonless for all time, and exposed, through the cutting down of their rotational velocity, to uncompensated extremes of temperature. How the earth was to fare in both respects long hung in the balance. Rightly to forecast its destiny would, indeed, have demanded no common perspicuity in an intelligent onlooker from some other sphere. Although the solar brake acted upon terrestrial rotation with no more than one-eleventh the power brought to bear upon that of Venus, it nevertheless sufficed during uncounted ages to hinder acceleration from reaching the pitch involving instability.
Our embryonic planet had long ceased to be nebulous, and had, in fact, shrunk by cooling nearly to its present dimensions before the die was cast. Then, at last, the hurrying effects of contraction prevailed over the slowing down due to tidal resistance, axial speed overbore equilibrium, and the spheroid divided. Now globes thus far advanced in condensation are apt to split less unequally than globes in a more primitive stage; and the moon, because late-born, was of large size. Its mass is 1/81 that of the earth; the masses of Titan and Saturn are as 1 to 4,600; while Jupiter's third and greatest satellite contains only 1/11300 part of the matter englobed in the parent-body. Moreover, Professor Darwin has made it clear that the satellites of Jupiter and Saturn revolve now in orbits not widely remote from those at first pursued by them; while the moon, on the contrary, started on its career almost, if not quite, from grazing contact with its primary. Owing to these two exceptional circumstances—its considerable relative mass and its close initial vicinity—the moon wielded over the earth tidal influence incomparably more powerful than that exerted by any of its compeers in the sun's realm.
The lunar-terrestrial system offers, accordingly, an example unique among those in solar subordination of a pair of globes, the mechanical relations of which have been settled on their present basis by the predominating agency of bodily tides. It holds forth, too, the one case in which origin by fission was possible. Professor Darwin's communication to the Royal Society in 1879 occasioned on this point a remarkable diversion of ideas. Saturn's rings were at last, through the reasonings contained in it, perceived to be illustrative of only one among many feasible modes of cosmic growth. It became clear that a single cut-and-dried method would not answer all the infinitely varied purposes of creative design. Annulation might have served its turn, but there were alternatives. A fresh standpoint was virtually attained, and the wide prospect commanded by it begins already to spread out invitingly before the gaze of investigators.
But whether the moon emerged from the earth as a protuberance, or was abandoned by it as an irregular equatorial ring, it was revolving, when our theoretical acquaintance with it begins, in a period of not less than two and not more than four hours, quite close to the earth's surface; while the nearly isochronous rotation of the earth was conducted with all but disruptive rapidity. The situation is so suggestive that it needs only a short and tolerably safe leap in the dark to reach the conclusion that the two masses had very recently been one. With their division, at an epoch estimated to have been about sixty million years ago, the process began by which the moon was pushed back along a widening spiral course to its present position, the vanished rotational momentum of the earth cropping up again in the augmented orbital momentum of the moon. And the transformation is, at least in theory, still going on.
Tidal friction has further capabilities. The transference of momentum from one part of a system to another is only the most obvious among the crowd of its results. Scarcely an element of movement escapes its influence. It increases, as a rule, orbital eccentricity. The smallest initial deviation from circularity develops, through the inequality of accelerative action thence ensuing, into pronounced ovalness. That of the moon's path can in this way be accounted for. Moreover, its plane was, in all probability, shifted simultaneously and under compulsion of the same power, from its original coincidence with the earth's equatorial plane to the level it now occupies. The obliquity of the ecliptic, too, is partially explicable on the same principle. 'The present motion of the two bodies' (to quote Professor Darwin's words), are 'completely co-ordinated by the theory that tidal friction was the ruling power in their evolution.' Holding this clue, we are enabled to trace them back to the start of their dual existence, and to follow the insensible modifications by which their state was moulded to its actual form.
In no other satellite-system is this possible. No moon besides our own possesses a stock of orbital momentum large enough to intimate for it an analogous history. Planetary attendants elsewhere travel nearly in their original tracks; the fluid ripples raised by them on the surfaces of their primaries lacked power to displace them sensibly. Their own rotation, indeed, seems to have been completely destroyed. Destroyed, that is, relatively to the destroying body. There is a certainty that some, there is the strongest likelihood that all, of the Jovian and Saturnian satellites turn unchangingly the same face towards their primaries. They rotate in the period of their several revolutions, just as our moon does, and as a consequence of the same cause. Tidal friction, however, appears to have been otherwise of subordinate importance in shaping their dynamical relations.
The agency will not, then, serve in all cases for a deus ex machinâ. It is not indiscriminately efficacious. The modes of its action have, in each of the systems considered, to be delicately distinguished. The stage of development arrived at by the bodies affected, their degree of viscosity, their comparative mass and bulk, their modes of motion, all avail profoundly, and it may be incalculably, to modify the outcome. The facility of error in estimates of the kind is illustrated by Professor Darwin's remark that the magnitude of the tide-raising force is only one factor of the product.[32] The other is relative movement. Now, in the case of the moon the former continually augmented retrospectively, while the latter fell off. Tidal generative power varies inversely as the cube of the distance; in antique times, then, when the earth and moon revolved contiguously, the bodily distortions they mutually produced were beyond question on an extremely large scale. Yet, because of the near coincidence of the periods of the globes, they must have been almost inoperative for frictional purposes. The travelling of the piled-up matter over their surfaces was too slow to lend it much power as a friction-brake. The insignificant waves raised by the sun were, we are led to believe, because of their swift relative motion, more influential at that early epoch in checking terrestrial rotation than the colossal, but nearly stationary waves due to the moon.
Numerical calculations, where they are practicable, afford the only safe guide to this intricate field of inquiry. It does not suffice to show that tidal action would have been of the kind required—would have taken the right direction—for bringing about some apparently anomalous result. Proof must, besides, be forthcoming that the action would have been of adequate power. Plausible guesses on the subject may be entirely fallacious. The machine, even if properly constructed for the end in view, may work too feebly for its attainment. We are, for instance, assured that no difficulty connected with the sense of planetary rotation need impede acceptance of the theory of planetary origin from separated rings, since even if the embryo globes gyrated the wrong way at the outset, solar tidal friction would promptly have reduced them to conformity with the general current of movement. This is true in principle, but will it bear quantitative investigation? Many promising hypotheses have broken down under the weight of figures; whether this particular one is strong enough to survive their application remains to be seen. We are, indeed, sure of its validity as regards Mercury, but the efficacy of tidal friction decreases as the sixth power of increasing distance, and the actual rotation of Venus furnishes an enigma sufficiently perplexing to discourage scrutiny of its dimly discerned antecedent conditions. As regards the earth and the exterior planets, the question could only be answered with the help of information which is not forthcoming.
The unexpected circumstance that the newly-discovered ninth Saturnian moon circulates from east to west can thus be no more than tentatively explained by invoking this agency of change. Admitting (as we seem bound to do) that satellites are the offspring of the planets they attend, there is no evading the conclusion that the small body under discussion was thrown off from a primary endowed with a rotation opposite to that now possessed by it. And the reversal must have been completely brought to pass before the eighth satellite, Japetus, came into existence. The crux is most arduous; there is no other resource for meeting it but to consider the effects on planetary rotation of solar tides, and this Professor W. H. Pickering, the discoverer of Phœbe, has done.[33] But a cause may be true without being sufficient; and close calculation will be needed to determine, in this instance, how the matter stands.
Professor Darwin's researches were fruitful just because they were definite. They demonstrated, once for all, the diverse faculties of tidal friction as a cosmogonic agency, and indicated clearly the departments of cosmogonic change in which its competence lay. They availed, moreover, to determine for the earth-moon system the amount of work actually done by tidal friction in these several departments, and to prove its large excess over the corresponding output in any other sub-system falling within the sphere of observation. This memorable result suggests that our terrestrial home may be singular, not only in its evolutionary history, but in the innumerable adjustments fitting it to be the abode of life.
The relations of the earth and moon adumbrate, and scarcely more than adumbrate, the physical influences mutually exerted upon each other by numerous twin-globes in stellar space. Tidal friction is of maximum power in systems formed of equal masses; and those of double stars are seldom widely disparate. Most, if not all of them, were, besides, primitively very near neighbours, so that their symmetry must have been marred by conspicuous tidal deformations. The results upon their development have been expounded in detail by Dr. See. One of the most remarkable is the high average eccentricity of their orbits. Visual binaries, with few exceptions, travel in considerably elongated ellipses, while spectroscopic binaries as a rule pursue approximately circular paths. Dr. See's argument that the eccentricity of the more spacious systems was acquired under the influence of tidal friction, during the long course of progressive separation, is well-nigh irresistible.
True, this line of explanation is not wholly clear of obstacles and incongruities. Yet they may probably be described as of a complicating, rather than of a contradictory kind. The theory of tidal friction is not a universal solvent of the difficulties encountered in the study of double stars. That the mode of action it deals with had a contributory share towards regulating their mechanical arrangements may, nevertheless, be regarded as certain, while the potency and perhaps even the manner of its operation varied extensively from system to system. What precisely it effected in each lies beyond our range of determination. For the data available regarding the viscosity, density, and axial movements of embryo star-pairs must always be too scanty and insecure to provide a basis for rigorous computations. The mystery of the fore-time can never be entirely dissipated. Enough if we can look at it through a glass which darkens, without distorting, the objects presented in its field of view.