## Precession and Nutation

Consider the motion of a spherical Moon orbiting the nonspherical Earth. Based on the discussion in Section 2.3, the gradient of the gravitational potential given by Eq. (2.3.12), multiplied by the mass of the Moon, yields the gravitational force experienced by the Moon. By Newton's Third Law, the Earth will experience an equal and opposite force; however, this force will not, in general, be directed through the center of mass of the Earth. As a consequence, the Earth will experience a gravitational torque, T, given by

where U' is the Earth's gravitational potential, VP is the gradient operator with respect to the coordinates of the Moon (rp), and MP is the mass of the Moon. The torque acting on the Earth produces changes in the inertial orientation of the Earth.

In the case of the Earth, both the Moon and the Sun are significant contributors to this gravitational torque. A dominant characteristic of the changes in inertial orientation is a conical motion of the Earth's angular velocity vector in inertial space, referred to as precession. Superimposed on this conical motion are small oscillations, referred to as nutation.

Consider the two angles in Fig. 2.4.1, W and e. The angle W is measured in the ecliptic from a fix ed direction in space, X*, to the instantaneous intersection of the Earth's equator and the ecliptic. As discussed in Section 2.2.3, the intersection between the equator and the ecliptic where the Sun crosses the equator from the southern hemisphere into the northern hemisphere is the vernal equinox. For this discussion, the ecliptic is assumed to maintain a fix ed orientation in space, although this is not precisely the case because of gravitational perturbations from the planets. The angle e represents the angle between the angular velocity vector and an axis Zec, which is perpendicular to the ecliptic. The gravitational interaction between the Earth's mass distribution, predominantly represented by the oblateness term, J2, and the Sun or Moon produces a change in W, which can be shown to be

2 \ CJ ap where the subscript p represents the Sun or the Moon, the parameters in the parentheses refer to Earth, C is the polar moment of inertia of Earth, and e is the mean obliquity of the ecliptic. This motion, known as the precession of the equinoxes, has been known since antiquity. The combined effect of the Sun and the Moon produce W = —50 arcsec/yr; that is, a westward motion of the vernal equinox amounting to 1.4° in a century. Thus, the conical motion of about Zec requires about 26,000 years for one revolution.

The obliquity of the ecliptic, e, is e = e + Ae, where e represents the mean value, approximately 23.5°. The nutations are represented by Ae, which exhibits periodicities of about 14 days (half the lunar period, or semimonthly) and 183 days (half the solar period, or semiannual), plus other long period variations. The 5° inclination of the Moon's orbit with respect to the ecliptic, iM, produces a long period nutation given by:

where the subscript M refers to the Moon, and the amplitude of the coefficient of cos(lM — 1!f) in Eq. (2.4.3) is about 9 arcsec with a period of 18.6 years. The 9 arcsec amplitude of this term is known as the constant of nutation. The amplitudes of the semimonthly and semiannual terms are smaller than the constant of nutation, but all are important for high-precision applications.

The precession/nutation series for a rigid body model of the Earth, including the ecliptic variations, were derived by Woolard (1953). Extensions that include the effects of nonrigidity have been developed. Nevertheless, small discrepancies between observation and theory exist, believed to be associated with geophysical effects in the Earth's interior (Herring et al., 1991; Wahr, 1981). Such observational evidence will lead, in time, to the adoption of new series representing the precession and nutation. The current 1980 International Astronomical Union

(IAU) series are available with the lunar, solar, and planetary ephemerides in an efficient numerical form based on results generated by the Jet Propulsion Laboratory (Standish etal., 1997).

As described in Section 2.2.3, the J2000.0 mean equator and equinox specifies a specific point on the celestial sphere on January 1, 2000, 12 hrs. The true of date (TOD) designation identifies the true equator and equinox at a specified date. The J2000.0 equinox is fix ed in space, but the location of the TOD equinox changes with time.

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