THE PRECESSION AND NUTATION OF THE EARTH'S AXIS

The position of the pole of the heavens is most conveniently indicated by the bright star known as the Pole Star, which lies in its immediate vicinity. Around this pole the whole heavens appear to rotate once in a sidereal day; and we have hitherto always referred to the pole as though it were a fixed point in the heavens. This language is sufficiently correct when we embrace only a moderate period of time in our review. It is no doubt true that the pole lies near the Pole Star at the present time. It did so during the lives of the last generation, and it will do so during the lives of the next generation. All this time, however, the pole is steadily moving in the heavens, so that the time will at length come when the pole will have departed a long way from the present Pole Star. This movement is incessant. It can be easily detected and measured by the instruments in our observatories, and astronomers are familiar with the fact that in all their calculations it is necessary to hold special account of this movement of the pole. It produces an apparent change in the position of a star, which is known by the term "precession."

Fig. 100.

The movement of the pole is very clearly shown in the accompanying figure (Fig. 100), for which I am indebted to the kindness of the late Professor C. Piazzi Smyth. The circle shows the track along which the pole moves among the stars.

The centre of the circle in the constellation of Draco is the pole of the ecliptic. A complete journey of the pole occupies the considerable period of about 25,867 years. The drawing shows the[Pg 494] position of the pole at the several dates from 4000 b.c. to 2000 a.d. A glance at this map brings prominently before us how casual is the proximity of the pole to the Pole Star. At present, indeed, the distance of the two is actually lessening, but afterwards the distance will increase until, when half of the 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 α Lyræ, 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 α Draconis was suitably placed to serve as the Pole Star, when β and δ 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[Pg 495] 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.

Fig. 101.—Illustration of the Motion of Precession.

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[Pg 496] 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[Pg 497] 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[Pg 498] 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 formulæ 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[Pg 499] 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 to consider the effect produced by the two bodies separately; and as the case of the sun is the simpler of the two, we shall take it first. As the earth travels in its annual path around the sun, the axis of the earth is directed to a point in the heavens which is 23-1⁄2° from the pole of the ecliptic. The precessional effect of the sun is to cause this point—the pole of the earth—to revolve, always preserving the same angular distance from the pole of the ecliptic; and thus we have a motion of the type represented in the diagram. As the ecliptic occupies a position which for our present purpose we may regard as fixed in space, it follows that the pole of the ecliptic is a fixed point on the surface of the heavens; so that the path of the pole of the earth must be a small circle in the heavens, fixed in its position relatively to the surrounding stars. In this we find a motion strictly analogous to that of the peg-top. It is the gravitation of the earth acting upon the peg-top which forces it into the conical motion. The immediate effect of the gravitation is so modified by the rapid rotation of the top, that, in obedience to a profound dynamical principle, the axis of the top revolves in a cone rather than fall down, as it would do were the top not spinning. In a similar manner the immediate effect of the sun's attraction on the protuberance at the equator would be to bring the pole of the earth's axis towards the pole of the ecliptic, but the rapid rotation of the earth modifies this into the conical movement of precession.

The circumstances with regard to the moon are much more[Pg 500] complicated. The moon describes a certain orbit around the earth; that orbit lies in a certain plane, and that plane has, of course, a certain pole on the celestial sphere. The precessional effect of the moon would accordingly tend to make the pole of the earth's axis describe a circle around that point in the heavens which is the pole of the moon's orbit. This point is about 5° from the pole of the ecliptic. The pole of the earth is therefore solicited by two different movements—one a revolution around the pole of the ecliptic, the other a revolution about another point 5° distant, which is the pole of the moon's orbit. It would thus seem that the earth's pole should make a certain composite movement due to the two separate movements. This is really the case, but there is a point to be very carefully attended to, which at first seems almost paradoxical. We have shown how the potency of the moon as a precessional agent exceeds that of the sun, and therefore it might be thought that the composite movement of the earth's pole would conform more nearly to a rotation around the pole of the plane of the moon's orbit than to a rotation around the pole of the ecliptic; but this is not the case. The precessional movement is represented by a revolution around the pole of the ecliptic, as is shown in the figure. Here lies the germ of one of those exquisite astronomical discoveries which delight us by illustrating some of the most subtle phenomena of nature.

The plane in which the moon revolves does not occupy a constant position. We are not here specially concerned with the causes of this change in the plane of the moon's orbit, but the character of the movement must be enunciated. The inclination of this plane to the ecliptic is about 5°, and this inclination does not vary (except within very narrow limits); but the line of intersection of the two planes does vary, and, in fact, varies so quickly that it completes a revolution in about 18-2⁄3 years. This movement of the plane of the moon's orbit necessitates a corresponding change in the position of its pole. We thus see that the pole of the moon's orbit must be actually revolving around the pole of the ecliptic, always remaining at the same distance of 5°, and completing its revolution in 18-2⁄3[Pg 501] years. It will, therefore, be obvious that there is a profound difference between the precessional effect of the sun and of the moon in their action on the earth. The sun invites the earth's pole to describe a circle around a fixed centre; the moon invites the earth's pole to describe a circle around a centre which is itself in constant motion. It fortunately happens that the circumstances of the case are such as to reduce considerably the complexity of the problem. The movement of the moon's plane, only occupying about 18-2⁄3 years, is a very rapid motion compared with the whole precessional movement, which occupies about 26,000 years. It follows that by the time the earth's axis has completed one circuit of its majestic cone, the pole of the moon's plane will have gone round about 1,400 times. Now, as this pole really only describes a comparatively small cone of 5° in radius, we may for a first approximation take the average position which it occupies; but this average position is, of course, the centre of the circle which it describes—that is, the pole of the ecliptic.

We thus see that the average precessional effect of the moon simply conspires with that of the sun to produce a revolution around the pole of the ecliptic. The grosser phenomena of the movements of the earth's axis are to be explained by the uniform revolution of the pole in a circular path; but if we make a minute examination of the track of the earth's axis, we shall find that though it, on the whole, conforms with the circle, yet that it really traces out a sinuous line, sometimes on the inside and sometimes on the outside of the circle. This delicate movement arises from the continuous change in the place of the pole of the moon's orbit. The period of these undulations is 18-2⁄3 years, agreeing exactly with the period of the revolution of the moon's nodes. The amount by which the pole departs from the circle on either side is only about 9·2 seconds—a quantity rather less than the twenty-thousandth part of the radius of the sphere. This phenomenon, known as "nutation," was discovered by the beautiful telescopic researches of Bradley, in 1747. Whether we look at the theoretical interest of the subject or at the refinement of the observations involved, this achievement of the "Vir incomparabilis," as Bradley has[Pg 502] been called by Bessel, is one of the masterpieces of astronomical genius.

The phenomena of precession and nutation depend on movements of the earth itself, and not on movements of the axis of rotation within the earth. Therefore the distance of any particular spot on the earth from the north or south pole is not disturbed by either of these phenomena. The latitude of a place is the distance of the place from the earth's equator, and this quantity remains unaltered in the course of the long precession cycle of 26,000 years. But it has been discovered within the last few years that latitudes are subject to a small periodic change of a few tenths of a second of arc. This was first pointed out about 1880 by Dr. Küstner, of Berlin, and by a masterly analysis of all available observations, made in the course of many years past at various observatories, Dr. Chandler, of Boston, has shown that the latitude of every point on the earth is subject to a double oscillation, the period of one being 427 days and the other about a year, the mean amplitude of each being O´´·14. In other words, the spot in the arctic regions, directly in the prolongation of the earth's axis of rotation, is not absolutely fixed; the end of the imaginary axis moves about in a complicated manner, but always keeping within a few yards of its average position. This remarkable discovery is not only of value as introducing a new refinement in many astronomical researches depending on an accurate knowledge of the latitude, but theoretical investigations show that the periods of this variation are incompatible with the assumption that the earth is an absolutely rigid body. Though this assumption has in other ways been found to be untenable, the confirmation of this view by the discovery of Dr. Chandler is of great importance.