The Story of the Heavens, Sir Robert Stawell Ball [snow like ashes series TXT] 📗

revolution has been accomplished, the pole will be at a distance of twice the radius of the circle from the Pole Star. It will then happen that the pole will be near the bright star Vega or a Lyrae, so that our successors some 12,000 years hence may make use of Vega for many of the purposes for which the Pole Star is at present employed! Looking back into past ages, we see that some 2,000 or 3,000 years B.C. the star a Draconis was suitably placed to serve as the Pole Star, when b and d of the Great Bear served as pointers. It need hardly be added, that since the birth of accurate astronomy the course of the pole has only been observed over a very small part of the mighty circle. We are not, however, entitled to doubt that the motion of the pole will continue to pursue the same path. This will be made abundantly clear when we proceed to render an explanation of this very interesting phenomenon.

The north pole of the heavens is the point of the celestial sphere towards which the northern end of the axis about which the earth rotates is directed. It therefore follows that this axis must be constantly changing its position. The character of the movement of the earth, so far as its rotation is concerned, may be illustrated by a very common toy with which every boy is familiar. When a peg-top is set spinning, it has, of course, a very rapid rotation around its axis; but besides this rotation there is usually another motion, whereby the axis of the peg-top does not remain in a constant direction, but moves in a conical path around the vertical line. The adjoining figure (Fig. 101) gives a view of the peg-top. It is, of course, rotating with great rapidity around its axis, while the axis itself revolves around the vertical line with a very deliberate motion. If we could imagine a vast peg-top which rotated on its axis once a day, and if that axis were inclined at an angle of twenty-three and a half degrees to the vertical, and if the slow conical motion of the axis were such that the revolution of the axis were completed in about 26,000 years, then the movements would resemble those actually made by the earth. The illustration of the peg-top comes, indeed, very close to the actual phenomenon of precession. In each case the rotation about the axis is far more rapid than that of the revolution of the axis itself; in each case also the slow movement is due to an external interference. Looking at the figure of the peg-top (Fig. 101) we may ask the question, Why does it not fall down? The obvious effect of gravity would seem to say that it is impossible for the peg-top to be in the position shown in the figure. Yet everybody knows that this is possible so long as the top is spinning. If the top were not spinning, it would, of course, fall. It therefore follows that the effect of the rapid rotation of the top so modifies the effect of gravitation that the latter, instead of producing its apparently obvious consequence, causes the slow conical motion of the axis of rotation. This is, no doubt, a dynamical question of some difficulty, but it is easy to verify experimentally that it is the case. If a top be constructed so that the point about which it is spinning shall coincide with the centre of gravity, then there is no effect of gravitation on the top, and there is no conical motion perceived.

If the earth were subject to no external interference, then the direction of the axis about which it rotates must remain for ever constant; but as the direction of the axis does not remain constant, it is necessary to seek for a disturbing force adequate to the production of the phenomena which are observed. We have invariably found that the dynamical phenomena of astronomy can be accounted for by the law of universal gravitation. It is therefore natural to enquire how far gravitation will render an account of the phenomenon of precession; and to put the matter in its simplest form, let us consider the effect which a distant attracting body can have upon the rotation of the earth.

To answer this question, it becomes necessary to define precisely what we mean by the earth; and as for most purposes of astronomy we regard the earth as a spherical globe, we shall commence with this assumption. It seems also certain that the interior of the earth is, on the whole, heavier than the outer portions. It is therefore reasonable to assume that the density increases as we descend; nor is there any sufficient ground for thinking that the earth is much heavier in one part than at any other part equally remote from the centre. It is therefore usual in such calculations to assume that the earth is formed of concentric spherical shells, each one of which is of uniform density; while the density decreases from each shell to the one exterior thereto.

A globe of this constitution being submitted to the attraction of some external body, let us examine the effects which that external body can produce. Suppose, for instance, the sun attracts a globe of this character, what movements will be the result? The first and most obvious result is that which we have already so frequently discussed, and which is expressed by Kepler's laws: the attraction will compel the earth to revolve around the sun in an elliptic path, of which the sun is in the focus. With this movement we are, however, not at this moment concerned. We must enquire how far the sun's attraction can modify the earth's rotation around its axis. It can be demonstrated that the attraction of the sun would be powerless to derange the rotation of the earth so constituted. This is a result which can be formally proved by mathematical calculation. It is, however, sufficiently obvious that the force of attraction of any distant point on a symmetrical globe must pass through the centre of that globe: and as the sun is only an enormous aggregate of attracting points, it can only produce a corresponding multitude of attractive forces; each of these forces passes through the centre of the earth, and consequently the resultant force which expresses the joint result of all the individual forces must also be directed through the centre of the earth. A force of this character, whatever other potent influence it may have, will be powerless to affect the rotation of the earth. If the earth be rotating on an axis, the direction of that axis would be invariably preserved; so that as the earth revolves around the sun, it would still continue to rotate around an axis which always remained parallel to itself. Nor would the attraction of the earth by any other body prove more efficacious than that of the sun. If the earth really were the symmetrical globe we have supposed, then the attraction of the sun and moon, and even the influence of all the planets as well, would never be competent to make the earth's axis of rotation swerve for a single second from its original direction.

We have thus narrowed very closely the search for the cause of the "precession." If the earth were a perfect sphere, precession would be inexplicable. We are therefore forced to seek for an explanation of precession in the fact that the earth is not a perfect sphere. This we have already demonstrated to be the case. We have shown that the equatorial axis of the earth is longer than the polar axis, so that there is a protuberant zone girdling the equator. The attraction of external bodies is able to grasp this protuberance, and thereby force the earth's axis of rotation to change its direction.

There are only two bodies in the universe which sensibly contribute to the precessional movement of the earth's axis: these bodies are the sun and the moon. The shares in which the labour is borne by the sun and the moon are not what might have been expected from a hasty view of the subject. This is a point on which it will be desirable to dwell, as it illustrates a point in the theory of gravitation which is of very considerable importance.

The law of gravitation asserts that the intensity of the attraction which a body can exercise is directly proportional to the mass of that body, and inversely proportional to the square of its distance from the attracted point. We can thus compare the attraction exerted upon the earth by the sun and by the moon. The mass of the sun exceeds the mass of the moon in the proportion of about 26,000,000 to 1. On the other hand, the moon is at a distance which, on an average, is about one-386th part of that of the sun. It is thus an easy calculation to show that the efficiency of the sun's attraction on the earth is about 175 times as great as the attraction of the moon. Hence it is, of course, that the earth obeys the supremely important attraction of the sun, and pursues an elliptic path around the sun, bearing the moon as an appendage.

But when we come to that particular effect of attraction which is competent to produce precession, we find that the law by which the efficiency of the attracting body is computed assumes a different form. The measure of efficiency is, in this case, to be found by taking the mass of the body and dividing it by the _cube_ of the distance. The complete demonstration of this statement must be sought in the formulae of mathematics, and cannot be introduced into these pages; we may, however, adduce one consideration which will enable the reader in some degree to understand the principle, though without pretending to be a demonstration of its accuracy. It will be obvious that the nearer the disturbing body approaches to the earth the greater is the _leverage_ (if we may use the expression) which is afforded by the protuberance at the equator. The efficiency of a given force will, therefore, on this account alone, increase in the inverse proportion of the distance. The actual intensity of the force itself augments in the inverse square of the distance, and hence the capacity of the attracting body for producing precession will, for a double reason, increase when the distance decreases. Suppose, for example, that the disturbing body is brought to half its original distance from the disturbed body, the leverage is by this means doubled, while the actual intensity of the force is at the same time quadrupled according to the law of gravitation. It will follow that the effect produced in the latter case must be eight times as great as in the former case. And this is merely equivalent to the statement that the precession-producing capacity of a body varies inversely as the cube of the distance.

It is this consideration which gives to the moon an importance as a precession-producing agent to which its mere attractive capacity would not have entitled it. Even though the mass of the sun be 26,000,000 times as great as the mass of the moon, yet when this number is divided by the cube of the relative value of the distances of the bodies (386), it is seen that the efficiency of the moon is more than twice as great as that of the sun. In other words, we may say that one-third of the movement of precession is due to the sun, and two-thirds to the moon.

For the study of the joint precessional effect due to the sun and the moon acting simultaneously, it will be advantageous