Time and Tide by Sir Robert Stawell Ball (read with me txt) 📖

by Jupiter will tend to augment the sun's period of rotation. Both Jupiter and the sun will therefore lose some moment of momentum. We cannot, however, repeat too often the dynamical truth that the total moment of momentum must remain constant, therefore what is lost by the rotation must be made up in the revolution; the orbit of Jupiter around the sun must accordingly be swelling. So far the reasoning appears similar to that which led to such startling consequences in regard to the moon.

But now for the fundamental difference between the two cases. The moon, it will be remembered, always revolves with the same face towards the earth. The tides have ceased to operate there, and consequently the moon is not able to contribute any moment of momentum, to be applied to the enlargement of its distance from the earth; all the moment of momentum necessary for this purpose is of course drawn from the single supply in the rotation of the earth on its axis. But in the case of the system consisting of the sun and Jupiter the circumstances are quite different--Jupiter does not always bend the same face to the sun; so far, indeed, is this from being true, that Jupiter is eminently remarkable for the rapidity of his rotation, and for the incessantly varying aspect in which he would be seen from the sun. Jupiter has therefore a store of available moment of momentum, as has also of course the sun. Thus in the sun and planet system we have in the rotations two available stores of moment of momentum on which the tides can make draughts for application to the enlargement of the revolution. The proportions in which these two available sources can be drawn upon for contributions is not left arbitrary. The laws of dynamics provide the shares in which each of the bodies is to contribute for the joint purpose of driving them further apart.

Let us see if we cannot form an estimate by elementary considerations as to the division of the labour. The tides raised on Jupiter by the sun will be practically proportional to the sun's mass and to the radius of Jupiter. Owing to the enormous size of the sun, the efficiency of these tides and the moment of the friction-brake they produce will be far greater on the planet than will the converse operation of the planet be on the sun. Hence it follows that the efficiency of the tides in depriving Jupiter of moment of momentum will be greatly superior to the efficiency of the tides in depriving the sun of moment of momentum. Without following the matter into any close numerical calculation, we may assert that for every one part the sun contributes to the common object, Jupiter will contribute at least a thousand parts; and this inequality appears all the more striking, not to say unjust, when it is remembered that the sun is more abundantly provided with moment of momentum than is Jupiter--the sun has, in fact, about twenty thousand times as much.

The case may be illustrated by supposing that a rich man and a poor man combine together to achieve some common purpose to which both are to contribute. The ethical notion that Dives shall contribute largely, according to his large means, and Lazarus according to his slender means, is quite antagonistic to the scale which dynamics has imposed. Dynamics declares that the rich man need only give a penny to every pound that has to be extorted from the poor man. Now this is precisely the case with regard to the sun and Jupiter, and it involves a somewhat curious consequence. As long as Jupiter possesses available moment of momentum, we may be certain that no large contribution of moment of momentum has been obtained from the sun. For, returning to our illustration, if we find that Lazarus still has something left in his pocket, we are of course assured that Dives cannot have expended much, because, as Lazarus had but little to begin with, and as Dives only puts in a penny for every pound that Lazarus spends, it is obvious that no large amount can have been devoted to the common object. Hence it follows that whatever transfer of moment of momentum has taken place in the sun-Jupiter system has been almost entirely obtained at the expense of Jupiter. Now in the solar system at present, the orbital moment of momentum of Jupiter is nearly fifty thousand times as great as his present store of rotational moment of momentum. If, therefore, the departure of Jupiter from the sun had been the consequence of tidal evolution, it would follow that Jupiter must once have contained many thousands of times the moment of momentum that he has at present. This seems utterly incredible, for even were Jupiter dilated into an enormously large mass of vaporous matter, spinning round with the utmost conceivable speed, it is impossible that he should ever have possessed enough moment of momentum. We are therefore forced to the conclusion that the tides alone do not provide sufficient explanation for the retreat of Jupiter from the sun.

There is rather a subtle point in the considerations now brought forward, on which it will be necessary for us to ponder. In the illustration of Dives and Lazarus, the contributions of Lazarus of course ceased when his pockets were exhausted, but those of Dives will continue, and in the lapse of time may attain any amount within the utmost limits of Dives' resources. The essential point to notice is, that so long as Lazarus retains anything in _his_ pocket, we know for certain that Dives has not given much; if Lazarus, however, has his pocket absolutely empty, and if we do not know how long they may have been in that condition, we have no means of knowing how large a portion of wealth Dives may not have actually expended. The turning-point of the theory thus involves the fact that Jupiter still retains available moment of momentum in his rotation; and this was our sole method of proving that the sun, which in this case was Dives, had never given much. But our argument must have taken an entirely different line had it so happened that Jupiter constantly turned the same face to the sun, and that therefore his pockets were entirely empty in so far as available moment of momentum is concerned. It would be apparently impossible for us to say to what extent the resources of the sun may not have been drawn upon; we can, however, calculate whether in any case the sun could possibly have supplied enough moment of momentum to account for the recession of Jupiter. Speaking in round numbers, the revolutional moment of momentum of Jupiter is about thirty times as great as the rotational moment of momentum at present possessed by the sun. I do not know that there is anything impossible in the supposition that the sun might, by an augmented volume and an augmented velocity of rotation, contain many times the moment of momentum that it has at this moment. It therefore follows that if it had happened that Jupiter constantly bent the same face to the sun, there would apparently be nothing impossible in the fact that Jupiter had been born of the sun, just as the moon was born of the earth. These same considerations should also lead us to observe with still more special attention the development of the earth-moon system. Let us restate the matter of the earth and moon in the light which the argument with respect to Jupiter has given us. At present the rotational moment of momentum of the earth is about a fifth part of the revolutional moment of momentum of the moon. Owing to the fact that the moon keeps the same face to us, she has now no available moment of momentum, and all the moment of momentum required to account for her retreat has of late come from the rotation of the earth; but suppose that the moon still had some liquid on its surface which could be agitated by tides, suppose further that it did not always bend the same face towards us, that it therefore had some available moment of momentum due to its rotation on which the tides could operate, then see how the argument would have been altered. The gradual increase of the moon's distance could be provided for by a transfer of moment of momentum from two sources, due of course to the rotational velocities of the two bodies. Here again the moon and the earth will contribute according to that dynamical but very iniquitous principle which regulated the appropriations from the purses of Dives and Lazarus. The moon must give not according to her abundance, but in the inverse ratio thereof--because she has little she must give largely. Nor shall we make an erroneous estimate if we say that nine-tenths of the whole moment of momentum necessary for the enlargement of the orbit would have been exacted from the moon; that means that the moon must once have had about five or six times as much moment of momentum as the earth possesses at this moment. Considering the small size of the moon, this could only have arisen by terrific velocity of rotation, which it is inconceivable that its materials could ever have possessed.

This presents the demonstration of tidal evolution in a fresh light. If the moon now departed to any considerable extent from showing the constant face to the earth, it would seem that its retreat could not have been caused by tides. Some other agent for producing the present configuration would be necessary, just as we found that some other agent than the tides has been necessary in the case of Jupiter.

But I must say a few words as to the attitude of this question with regard to the entire solar system. This system consists of the sun presiding at the centre, and of the planets and their satellites in revolution around their respective primaries, and each also animated by a rotation on its axis. I shall in so far depart from the actual configuration of the system as to transform it into an ideal system, whereof the masses, the dimensions, and the velocities shall all be preserved; but that the several planes of revolution shall be all flattened into one plane, instead of being inclined at small angles as they are at present; nor will it be unreasonable for us at the same time to bring into parallelism all the axes of rotation, and to arrange that their common directions shall be perpendicular to the plane of their common orbits. For the purpose of our present research this ideal system may pass for the real system.

In its original state, whatever that state may have been, a magnificent endowment was conferred upon the system. Perhaps I may, without derogation from the dignity of my subject, speak of the endowment as partly personal and partly entailed. The system had of course different powers with regard to the disposal of the two portions; the personal estate could be squandered. It consisted entirely of what we call energy; and considering how frequently we use the expression conservation of energy, it may seem strange to say now that this portion of the endowment has been found capable of alienation, nay, further, that our system has been squandering it persistently from the first moment until now. Although the doctrine of the conservation of energy is, we have every reason to believe, a fundamental law affecting the whole universe, yet it would be wholly inaccurate to say that any particular system such as our solar system shall invariably preserve precisely the same quantity of energy without alteration. The circumstance that heat is a form of energy indeed negatives this supposition. For our system possesses energy of