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

all the different kinds: there is energy due to the motions both of rotation and of revolution; there is energy due to the fact that the mutually attracting bodies of our system are separated by distances of enormous magnitude; and there is also energy in the form of heat; and the laws of heat permit that this form of energy shall be radiated off into space, and thus disappear entirely, in so far as our system is concerned. On the other hand, there may no doubt be some small amount of energy accruing to our system from the other systems in space, which like ours are radiating forth energy. Any gain from this source, however, is necessarily so very small in comparison to the loss to which we have referred, that it is quite impossible that the one should balance the other. Though it is undoubtedly true that the total quantity of energy in the universe is constant, yet the share of that energy belonging to any particular system such as ours declines steadily from age to age.

I may indeed remark, that the question as to what becomes of all the radiant energy which the millions of suns in the universe are daily discharging offers a problem apparently not easy to solve; but we need not discuss the matter at present, we are only going to trace out the vicissitudes of our own system; and whatever other changes that system may exhibit, the fact is certain that the total quantity of energy it contains is declining.

Of the two endowments of energy and of moment of momentum originally conferred on our system the moment of momentum is the entailed estate. No matter how the bodies may move, no matter how their actions may interfere with one another, no matter how this body is pulled one way and the other body that way, the conservation of moment of momentum is not imperilled, nor, no matter what losses of heat may be experienced by radiation, could the store of moment of momentum be affected. The only conceivable way in which the quantity of moment of momentum in the solar system could be tampered with is by the interference of some external attracting body. We know, however, that the stars are all situated at such enormous distances, that the influences they can exert in the perturbation of the solar system are absolutely insensible; they are beyond the reach of the most delicate astronomical measurements. Hence we see how the endowment of the system with moment of momentum has conferred upon that system a something which is absolutely inalienable, even to the smallest portion.

Before going any further it would be necessary for me to explain more fully than I have hitherto done the true nature of the method of estimating moment of momentum. The moment of momentum consists of two parts: there is first that due to the revolution of the bodies around the sun; there is secondly the rotation of these bodies on their axes. Let us first think simply of a single planet revolving in a circular orbit around the sun. The momentum of that planet at any moment may be regarded as the product of its mass and its velocity; then the moment of momentum of the planet in the case mentioned is found by multiplying the momentum by the radius of the path pursued. In a more general case, where the planet does not revolve in a circle, but pursued an elliptic path, the moment of momentum is to be found by multiplying the planet's velocity and its mass into the perpendicular from the sun on the direction in which the planet is moving.

These rules provide the methods for estimating all the moments of momentum, so far as the revolutions in our system are concerned. For the rotations somewhat more elaborate processes are required. Let us think of a sphere rotating round a fixed axis. Every particle of that sphere will of course describe a circle around the axis, and all these circles will lie in parallel planes. We may for our present purpose regard each atom of the body as a little planet revolving in a circular orbit, and therefore the moment of momentum of the entire sphere will be found by simply adding together the moments of momentum of all the different atoms of which the sphere is composed. To perform this addition the use of an elaborate mathematical method is required. I do not propose to enter into the matter any further, except to say that the total moment of momentum is the product of two factors--one the angular velocity with which the sphere is turning round, while the other involves the sphere's mass and dimensions.

To illustrate the principles of the computation we shall take one or two examples. Suppose that two circles be drawn, one of which is double the diameter of the other. Let two planets be taken of equal mass, and one of these be put to revolve in one circle, and the other to revolve in the other circle, in such a way that the periods of both revolutions shall be equal. It is required to find the moments of momentum in the two cases. In the larger of the two circles it is plain that the planet must be moving twice as rapidly as in the smaller, therefore its momentum is twice as great; and as the radius is also double, it follows that the moment of momentum in the large orbit will be four times that in the small orbit. We thus see that the moment of momentum increases in the proportion of the squares of the radii. If, however, the two planets were revolving about the same sun, one of these orbits being double the other, the periodic times could not be equal, for Kepler's law tells us that the square of the periodic time is proportional to the cube of the mean distance. Suppose, then, that the distance of the first planet is 1, and that of the second planet is 2, the cubes of those numbers are 1 and 8, and therefore the periodic times of the two bodies will be as 1 to the square root of 8. We can thus see that the velocity of the outer body must be less than that of the inner one, for while the length of the path is only double as large, the time taken to describe that path is the square root of eight times as great; in fact, the velocity of the outer body will be only the square root of twice that of the inner one. As, however, its distance from the sun is twice as great, it follows that the moment of momentum of the outer body will be the square root of twice that of the inner body. We may state this result a little more generally as follows--

In comparing the moments of momentum of the several planets which revolve around the sun, that of each planet is proportional to the product of its mass with the square root of its distance from the sun.

Let us now compare two spheres together, the diameter of one sphere being double that of the other, while the times of rotation of the two are identical. And let us now compare together the moments of momentum in these two cases. It can be shown by reasoning, into which I need not now enter, that the moment of momentum of the large sphere will be thirty-two times that of the small one. In general we may state that if a sphere of homogeneous material be rotating about an axis, its moment of momentum is to be expressed by the product of its angular velocity by the fifth power of its radius.

We can now take stock, as it were, of the constituents of moments of momentum in our system. We may omit the satellites for the present, while such unsubstantial bodies as comets and such small bodies as meteors need not concern us. The present investment of the moment of momentum of our system is to be found by multiplying the mass of each planet by the square root of its distance from the sun; these products for all the several planets form the total revolutional moment of momentum. The remainder of the investment is in rotational moment of momentum, the collective amount of which is to be estimated by multiplying the angular velocity of each planet into its density, and the fifth power of its radius if the planet be regarded as homogeneous, or into such other power as may be necessary when the planet is not homogeneous. Indeed, as the denser parts of the planet necessarily lie in its interior, and have therefore neither the velocity nor the radius of the more superficial portions, it seems necessary to admit that the moments of momentum of the planets will be proportional to some lower power of the radius than the fifth. The total moment of momentum of the planets by rotation, when multiplied by a constant factor, and added to the revolutional moment of momentum, will remain absolutely constant.

It may be interesting to note the present disposition of this vast inheritance among the different bodies of our system. The biggest item of all is the moment of momentum of Jupiter, due to its revolution around the sun; in fact, in this single investment nearly sixty per cent. of the total moment of momentum of the solar system is found. The next heaviest item is the moment of momentum of Saturn's revolution, which is twenty-four per cent. Then come the similar contributions of Uranus and Neptune, which are six and eight per cent. respectively. Only one more item is worth mentioning, as far as magnitude is concerned, and that is the nearly two per cent. that the sun contains in virtue of its _rotation_. In fact, all the other moments of momentum are comparatively insignificant in this method of viewing the subject. Jupiter from his rotation has not the fifty thousandth part of his revolutional moment of momentum, while the earth's rotational share is not one ten thousandth part of that of Jupiter, and therefore is without importance in the general aspect of the system. The revolution of the earth contributes about one eight hundredth part of that of Jupiter.

These facts as here stated will suffice for us to make a forecast of the utmost the tides can effect in the future transformation of our system. We have already explained that the general tendency of tidal friction is to augment revolutional moment of momentum at the expense of rotational. The total, however, of the rotational moment of momentum of the system barely reaches two per cent. of the whole amount; this is of course almost entirely contributed by the sun, for all the planets together have not a thousandth part of the sun's rotational moment. The utmost therefore that tidal evolution can effect in the system is to distribute the two per cent. in augmenting the revolutionary moment of momentum. It does not seem that this can produce much appreciable derangement in the configuration of the system. No doubt if it were all applied to one of the smaller planets it would produce very considerable effect. Our earth, for instance, would have to be driven out to a distance many hundreds of times further than it is at present were the sun's disposable moment of momentum ultimately to be transferred to the earth alone. On the other hand, Jupiter could absorb the whole of the sun's share by quite an insignificant enlargement of its present path. It does not seem likely that the distribution that must ultimately take place can much affect the present configuration of the system.

We thus see that the tides do not appear to have exercised