THE PRECESSION AND NUTATION OF THE EARTH'S AXIS.
The Pole is not a Fixed Point--Its Effect on the Apparent Places of
the Stars--The Illustration of the Peg-Top--The Disturbing Force
which acts on the Earth--Attraction of the Sun on a Globe--The
Protuberance at the Equator--The Attraction of the Protuberance by
the Sun and by the Moon produces Precession--The Efficiency of the
Precessional Agent varies inversely as the Cube of the
Distance--The Relative Efficiency of the Sun and the Moon--How the
Pole of the Earth's Axis revolves round the Pole of the
Ecliptic--Variation of Latitude.
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."
[Illustration: 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 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 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.
[Illustration: 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 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 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 deg. 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 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 deg. 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 deg.
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 deg., 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 deg.,
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
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 deg. 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
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. Kuestner, 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.