Part 30 (1/2)
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 ill.u.s.trates a point in the theory of gravitation which is of very considerable importance.
The law of gravitation a.s.serts that the intensity of the attraction which a body can exercise is directly proportional to the ma.s.s 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 ma.s.s of the sun exceeds the ma.s.s 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 a.s.sumes a different form.
The measure of efficiency is, in this case, to be found by taking the ma.s.s 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 ent.i.tled it. Even though the ma.s.s of the sun be 26,000,000 times as great as the ma.s.s 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 s.p.a.ce, 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 a.n.a.logous 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 circ.u.mstances with regard to the moon are much more 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...o...b..t. 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...o...b..t. 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...o...b..t 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 ill.u.s.trating 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...o...b..t, 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...o...b..t necessitates a corresponding change in the position of its pole. We thus see that the pole of the moon's...o...b..t 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 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 circ.u.mstances 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...o...b..t. 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 92 seconds--a quant.i.ty 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 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 lat.i.tude of a place is the distance of the place from the earth's equator, and this quant.i.ty 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 lat.i.tudes 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. Kustner, of Berlin, and by a masterly a.n.a.lysis of all available observations, made in the course of many years past at various observatories, Dr. Chandler, of Boston, has shown that the lat.i.tude 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 lat.i.tude, but theoretical investigations show that the periods of this variation are incompatible with the a.s.sumption that the earth is an absolutely rigid body. Though this a.s.sumption has in other ways been found to be untenable, the confirmation of this view by the discovery of Dr. Chandler is of great importance.
CHAPTER XXV.
THE ABERRATION OF LIGHT.
The Real and Apparent Movements of the Stars--How they can be Discriminated--Aberration produces Effects dependent on the Position of the Stars--The Pole of the Ecliptic--Aberration makes Stars seem to Move in a Circle, an Ellipse, or a Straight Line according to Position--All the Ellipses have Equal Major Axes--How is this Movement to be Explained?--How to be Distinguished from Annual Parallax--The Apex of the Earth's Way--How this is to be Explained by the Velocity of Light--How the Scale of the Solar System can be Measured by the Aberration of Light.
We have in this chapter to narrate a discovery of a recondite character, which ill.u.s.trates in a forcible manner some of the fundamental truths of Astronomy. Our discussion of it will naturally be divided into two parts. In the first part we must describe the nature of the phenomenon, and then we must give the extremely elegant explanation afforded by the properties of light. The telescopic discovery of aberration, as well as its explanation, are both due to the ill.u.s.trious Bradley.
The expression _fixed_ star, so often used in astronomy, is to be received in a very qualified sense. The stars are, no doubt, well fixed in their places, so far as coa.r.s.e observation is concerned. The lineaments of the constellations remain unchanged for centuries, and, in contrast with the ceaseless movements of the planets, the stars are not inappropriately called fixed. We have, however, had more than one occasion to show throughout the course of this work that the expression ”fixed star” is not an accurate one when minute quant.i.ties are held in estimation. With the exact measures of modern instruments, many of these quant.i.ties are so perceptible that they have to be always reckoned with in astronomical enquiry. We can divide the movements of the stars into two great cla.s.ses: the real movements and the apparent movements. The proper motion of the stars and the movements of revolution of the binary stars const.i.tute the real movements of these bodies. These movements are special to each star, so that two stars, although close together in the heavens, may differ in the widest degree as to the real movements which they possess. It may, indeed, sometimes happen that stars in a certain region are animated with a common movement. In this phenomenon we have traces of a real movement shared by a number of stars in a certain group. With this exception, however, the real movements of the stars seem to be governed by no systematic law, and the rapidly moving stars are scattered here and there indiscriminately over the heavens.
The apparent movements of the stars have a different character, inasmuch as we find the movement of each star determined by the place which it occupies in the heavens. It is by this means that we discriminate the real movements of the star from its apparent movements, and examine the character of both.
In the present chapter we are concerned with the apparent movements only, and of these there are three, due respectively to precession, to nutation, and to aberration. Each of these apparent movements obeys laws peculiar to itself, and thus it becomes possible to a.n.a.lyse the total apparent motion, and to discriminate the proportions in which the precession, the nutation, and the aberration have severally contributed.
We are thus enabled to isolate the effect of aberration as completely as if it were the sole agent of apparent displacement, so that, by an alliance between mathematical calculation and astronomical observation, we can study the effects of aberration as clearly as if the stars were affected by no other motions.
Concentrating our attention solely on the phenomena of aberration we shall describe its particular effect upon stars in different regions of the sky, and thus ascertain the laws according to which the effects of aberration are exhibited. When this step has been taken, we shall be in a position to give the beautiful explanation of those laws dependent upon the velocity of light.
At one particular region of the heavens the effect of aberration has a degree of simplicity which is not manifested anywhere else. This region lies in the constellation Draco, at the pole of the ecliptic. At this pole, or in its immediate neighbourhood, each star, in virtue of aberration, describes a circle in the heavens. This circle is very minute; it would take something like 2,000 of these circles together to form an area equal to the area of the moon. Expressed in the usual astronomical language, we should say that the diameter of this small circle is about 409 seconds of arc. This is a quant.i.ty which, though small to the unaided eye, is really of great relative magnitude in the present state of telescopic research. It is not only large enough to be perceived, but it can be measured, with an accuracy which actually does not admit of a doubt, to the hundredth part of the whole. It is also observed that each star describes its little circle in precisely the same period of time; and that period is one year, or, in other words, the time of the revolution of the earth around the sun. It is found that for all stars in this region, be they large stars or small, single or double, white or coloured, the circles appropriate to each have all the same size, and are all described in the same time. Even from this alone it would be manifest that the cause of the phenomenon cannot lie in the star itself. This unanimity in stars of every magnitude and distance requires some simpler explanation.
Further examination of stars in different regions sheds new light on the subject. As we proceed from the pole of the ecliptic, we still find that each star exhibits an annual movement of the same character as the stars just considered. In one respect, however, there is a difference. The apparent path of the star is no longer a circle; it has become an ellipse. It is, however, soon perceived that the shape and the position of this ellipse are governed by the simple law that the further the star is from the pole of the ecliptic the greater is the eccentricity of the ellipse. The apparent path of the stars at the same distance from the pole have equal eccentricity, and of the axes of the ellipse the shorter is always directed to the pole, the longer being, of course, perpendicular to it. It is, however, found that no matter how great the eccentricity may become, the major axis always retains its original length. It is always equal to about 409 seconds--that is, to the diameter of the circle of aberration at the pole itself. As we proceed further and further from the pole of the ecliptic, we find that each star describes a path more and more eccentric, until at length, when we examine a star on the ecliptic, the ellipse has become so attenuated that it has flattened into a line. Each star which happens to lie on the ecliptic oscillates to and fro along the ecliptic through an amplitude of 409 seconds. Half a year accomplishes the journey one way, and the other half of the year restores the star to its original position. When we pa.s.s to stars on the southern side of the ecliptic, we see the same series of changes proceed in an inverse order. The ellipse, from being actually linear, gradually grows in width, though still preserving the same length of major axis, until at length the stars near the southern pole of the ecliptic are each found to describe a circle equal to the paths pursued by the stars at the north pole of the ecliptic.
The circ.u.mstance that the major axes of all those ellipses are of equal length suggests a still further simplification. Let us suppose that every star, either at the pole of the ecliptic or elsewhere, pursues an absolutely circular path, and that all these circles agree not only in magnitude, but also in being all parallel to the plane of the ecliptic: it is easy to see that this simple supposition will account for the observed facts. The stars at the pole of the ecliptic will, of course, show their circles turned fairly towards us, and we shall see that they pursue circular paths. The circular paths of the stars remote from the pole of the ecliptic will, however, be only seen somewhat edgewise, and thus the apparent paths will be elliptical, as we actually find them. We can even calculate the degree of ellipticity which this surmise would require, and we find that it coincides with the observed ellipticity.