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° 27' 28·29; and the inclination of the lunar orbit to the same is 5° 8' 47·9. 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 50·41 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 0·31 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 50·1 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 50·1 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 50·1, 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 0·455 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 18·5 and 13·674, 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 0·457. 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° 42' 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° 24' 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 a LyrÆ will come within 5° of the celestial pole, and become the polar star of the northern hemisphere.