Axial Precession - Equations

Equations

The tidal force on Earth due a perturbing body (Sun, Moon or planet) is the result of the inverse-square law of gravity, whereby the gravitational force of the perturbing body on the side of Earth nearest it is greater than the gravitational force on the far side. If the gravitational force of the perturbing body at the center of Earth (equal to the centrifugal force) is subtracted from the gravitational force of the perturbing body everywhere on the surface of Earth, only the tidal force remains. For precession, this tidal force takes the form of two forces which only act on the equatorial bulge outside of a pole-to-pole sphere. This couple can be decomposed into two pairs of components, one pair parallel to Earth's equatorial plane toward and away from the perturbing body which cancel each other, and another pair parallel to Earth's rotational axis, both toward the ecliptic plane. The latter pair of forces creates the following torque vector on Earth's equatorial bulge:

where

Gm = standard gravitational parameter of the perturbing body

r = geocentric distance to the perturbing body

C = moment of inertia around Earth's axis of rotation

A = moment of inertia around any equatorial diameter of Earth

C − A = moment of inertia of Earth's equatorial bulge (C > A)

δ = declination of the perturbing body (north or south of equator)

α = right ascension of the perturbing body (east from vernal equinox).

The three unit vectors of the torque at the center of the Earth (top to bottom) are x on a line within the ecliptic plane (the intersection of Earth's equatorial plane with the ecliptic plane) directed toward the vernal equinox, y on a line in the ecliptic plane directed toward the summer solstice (90° east of x), and z on a line directed toward the north pole of the ecliptic.

The value of the three sinusoidal terms in the direction of x (sinδ cosδ sinα) for the Sun is a sine squared waveform varying from zero at the equinoxes (0°, 180°) to 0.36495 at the solstices (90°, 270°). The value in the direction of y (sinδ cosδ (−cosα)) for the Sun is a sine wave varying from zero at the four equinoxes and solstices to ±0.19364 (slightly more than half of the sine squared peak) halfway between each equinox and solstice with peaks slightly skewed toward the equinoxes (43.37°(−), 136.63°(+), 223.37°(−), 316.63°(+)). Both solar waveforms have about the same peak-to-peak amplitude and the same period, half of a revolution or half of a year. The value in the direction of z is zero.

The average torque of the sine wave in the direction of y is zero for the Sun or Moon, so this component of the torque does not affect precession. The average torque of the sine squared waveform in the direction of x for the Sun or Moon is:

where

= semimajor axis of Earth's (Sun's) orbit or Moon's orbit

e = eccentricity of Earth's (Sun's) orbit or Moon's orbit

and 1/2 accounts for the average of the sine squared waveform, accounts for the average distance cubed of the Sun or Moon from Earth over the entire elliptical orbit, and (the angle between the equatorial plane and the ecliptic plane) is the maximum value of δ for the Sun and the average maximum value for the Moon over an entire 18.6 year cycle.

Precession is:

where ω is Earth's angular velocity and Cω is Earth's angular momentum. Thus the first order component of precession due to the Sun is:

whereas that due to the Moon is:

where i is the angle between the plane of the Moon's orbit and the ecliptic plane. In these two equations, the Sun's parameters are within square brackets labeled S, the Moon's parameters are within square brackets labeled L, and the Earth's parameters are within square brackets labeled E. The term accounts for the inclination of the Moon's orbit relative to the ecliptic. The term (C−A)/C is Earth's dynamical ellipticity or flattening, which is adjusted to the observed precession because Earth's internal structure is not known with sufficient detail. If Earth were homogeneous the term would equal its third eccentricity squared,

where a is the equatorial radius (6378137 m) and c is the polar radius (6356752 m), so e''2 = 0.003358481.

Applicable parameters for J2000.0 rounded to seven significant digits (excluding leading 1) are:

which yield

dψ_S/dt = 2.450183×10−12 /s

dψ_L/dt = 5.334529×10−12 /s

both of which must be converted to "/a (arcseconds/annum) by the number of arcseconds in 2π radians (1.296×106"/2π) and the number of seconds in one annum (a Julian year) (3.15576×107s/a):

dψ_S/dt = 15.948788"/a vs 15.948870"/a from Williams

dψ_L/dt = 34.723638"/a vs 34.457698"/a from Williams.

The solar equation is a good representation of precession due the Sun because Earth's orbit is close to an ellipse, being only slightly perturbed by the other planets. The lunar equation is not as good a representation of precession due to the Moon because its orbit is greatly distorted by the Sun.