Precession and Nutation—Their Effects on the Apparent Places of the
Fixed Stars.
IT has been shown that the axis of rotation is invariable on the surface
of the earth; and observation as well as theory prove that, were it not
for the action of the sun and moon on the matter at the equator, it
would remain exactly parallel to itself in every point of its orbit.
The attraction of an external body not only draws a spheroid towards it,
but, as the force varies inversely as the square of the distance, it
gives it a motion about its centre of gravity, unless when the
attracting body is situated in the prolongation of one of the axes of
the spheroid. The plane of the equator is inclined to the plane of the
ecliptic at an angle of 23° 27ʹ 28ʺ·29; and the inclination of the lunar
orbit to the same is 5° 8ʹ 47ʺ·9. Consequently, from the oblate figure
of the earth, the sun and moon, acting obliquely and unequally on the
different parts of the terrestrial spheroid, urge the plane of the
equator from its direction, and force it to move from east to west, so
that the equinoctial points have a slow retrograde motion on the plane
of the ecliptic of 50ʺ·41 annually. The direct tendency of this action
is to make the planes of the equator and ecliptic coincide, but it is
balanced by the tendency of the earth to return to stable rotation about
the polar diameter, which is one of its principal axes of rotation.
Therefore the inclination of the two planes remains constant, as a top
spinning preserves the same inclination to the plane of the horizon.
Were the earth spherical, this effect would not be produced, and the
equinoxes would always correspond with the same points of the ecliptic,
at least as far as this kind of motion is concerned. But another and
totally different cause which operates on this motion has already been
mentioned. The action of the planets on one another and on the sun
occasions a very slow variation in the position of the plane of the
ecliptic, which affects its inclination to the plane of the equator, and
gives the equinoctial points a slow but direct motion on the ecliptic of
0ʺ·31 annually, which is entirely independent of the figure of the
earth, and would be the same if it were a sphere. Thus the sun and moon
by moving the plane of the equator cause the equinoctial points to
retrograde on the ecliptic: and the planets by moving the plane of the
ecliptic give them a direct motion, though much less than the former.
Consequently the difference of the two is the mean precession, which is
proved both by theory and observation to be about 50ʺ·1 annually
(N. 146).
As the longitudes of all the fixed stars are increased by this quantity,
the effects of precession are soon detected. It was accordingly
discovered by Hipparchus in the year 128 before Christ, from a
comparison of his own observations with those of Timocharis 155 years
before. In the time of Hipparchus the entrance of the sun into the
constellation Aries was the beginning of spring; but since that time the
equinoctial points have receded 30°, so that the constellations called
the signs of the zodiac are now at a considerable distance from those
divisions of the ecliptic which bear their names. Moving at the rate of
50ʺ·1 annually, the equinoctial points will accomplish a revolution in
25,868 years. But, as the precession varies in different centuries, the
extent of this period will be slightly modified. Since the motion of the
sun is direct, and that of the equinoctial points retrograde, he takes a
shorter time to return to the equator than to arrive at the same stars;
so that the tropical year of 365^d 5^h 48^m 49^s·7 must be increased
by the time he takes to move through an arc of 50ʺ·1, in order to have
the length of the sidereal year. The time required is 20^m 19^s·6, so
that the sidereal year contains 365^d 6^h 9^m 9^s·6 mean solar days.
The mean annual precession is subject to a secular variation; for,
although the change in the plane of the ecliptic in which the orbit of
the sun lies be independent of the form of the earth, yet, by bringing
the sun, moon, and earth into different relative positions from age to
age, it alters the direct action of the two first on the prominent
matter at the equator: on this account the motion of the equinox is
greater by 0ʺ·455 now than it was in the time of Hipparchus.
Consequently the actual length of the tropical year is about 4^s·21
shorter than it was at that time. The utmost change that it can
experience from this cause amounts to 43 seconds.
Such is the secular motion of the equinoxes. But it is sometimes
increased and sometimes diminished by periodic variations, whose periods
depend upon the relative positions of the sun and moon with regard to
the earth, and which are occasioned by the direct action of these bodies
on the equator. Dr. Bradley discovered that by this action the moon
causes the pole of the equator to describe a small ellipse in the
heavens, the axes of which are 18ʺ·5 and 13ʺ·674, the longer being
directed towards the pole of the ecliptic. The period of this inequality
is about 19 years, the time employed by the nodes of the lunar orbit to
accomplish a revolution. The sun causes a small variation in the
description of this ellipse; it runs through its period in half a year.
Since the whole earth obeys these motions, they affect the position of
its axis of rotation with regard to the starry heavens, though not with
regard to the surface of the earth; for in consequence of precession
alone the pole of the equator moves in a circle round the pole of the
ecliptic in 25,868 years, and by nutation alone it describes a small
ellipse in the heavens every 19 years, on each side of which it deviates
every half-year from the action of the sun. The real curve traced in the
starry heavens by the imaginary prolongation of the earth’s axis is
compounded of these three motions (N. 147). This nutation in the earth’s
axis affects both the precession and obliquity with small periodic
variations. But in consequence of the secular variation in the position
of the terrestrial orbit, which is chiefly owing to the disturbing
energy of Jupiter on the earth, the obliquity of the ecliptic is
annually diminished, according to M. Bessel, by 0ʺ·457. This variation
in the course of ages may amount to 10 or 11 degrees; but the obliquity
of the ecliptic to the equator can never vary more than 2° 42ʹ or 3°,
since the equator will follow in some measure the motion of the
ecliptic.
It is evident that the places of all the celestial bodies are affected
by precession and nutation. Their longitudes estimated from the equinox
are augmented by precession; but, as it affects all the bodies equally,
it makes no change in their relative positions. Both the celestial
latitudes and longitudes are altered to a small degree by nutation;
hence all observations must be corrected for these inequalities. In
consequence of this real motion in the earth’s axis, the pole-star,
forming part of the constellation of the Little Bear, which was formerly
12° from the celestial pole, is now within 1° 24ʹ of it, and will
continue to approach it till it is within 1/2°, after which it will
retreat from the pole for ages; and 12,934 years hence the star α Lyræ
will come within 5° of the celestial pole, and become the polar star of
the northern hemisphere.