## 2. Averaged equations for the precession of the Earth with planetary perturbationsThe equations of precession of the Earth are derived from a
Hamiltonian function We suppose here that the Earth is an homogeneous rigid body with moments of inertia and we assume that its spin axis is also the principal axis of inertia. It is convenient here to use canonical Andoyer's action variables and their conjugate angles (Andoyer, 1923, Kinoshita, 1977) (Fig. 2). is the modulus of rotational angular momentum of the Earth with rotation angular velocity ; , is the projection of the angular momentum on the normal to the ecliptic, at obliquity ; is the hour angle between the equinox of the date and a fixed point of the equator; the opposite of the general precession angle (see Fig. 2).
Let be a reference frame fixed with respect to the Earth, a reference frame linked to the orbital plane of the perturbing body (Sun or Moon) around the Earth (see Fig. 2), and the rotation such that According to Tisserand (1891) or Smart (1953) the potential energy
of the torque exerted by a perturbing body of
mass at distance where The motion of around the Earth is determined by its elliptical orbital elements defined with respect to the fixed ecliptic , with reference direction toward the fixed equinox . Let us denote its argument of perigee, its true anomaly, the true longitude of date, where is the longitude of ascending node of the apparent orbit of on if is the Sun, and on if it is the Moon. We first build the precession equations due to the perturbation of the Sun only. When is the Sun (subscript ), we have: and the transformation from the equatorial frame to the ecliptic one is: where the rotations and are defined as Hence We retain only the contribution of terms with no spherical symmetry, which gives with Andoyer's variables Let be the mean anomaly of the Sun. The fast angles and are removed by taking the average unless a spin-orbit resonance occurs, i.e. when the angle is librating (Peale, 1969). This leads to the following expression for the averaged potential due to the Sun: where is called the dynamical ellipticity.
The contribution of the Moon (subscript where Assuming a constant rate for the precession of the orbit of the Moon (node and perihelion), one can also average the subsequent on , and , which gives: where is the inclination of the lunar orbit on the ecliptic. The full averaged Hamiltonian function of the described motion is then obtained by adding the rotational kinetic energy , which gives where is the "precession constant": For a fast rotating planet like the Earth, can be considered as proportional to ; this correspond to the hydrostatic equilibrium (see for example Lambeck, 1980). In this approximation, is proportional to . Now, when considering the perturbations of the other planets, the
ecliptic is not an inertial plane any more and
the kinetic energy Let be Andoyer's variables relative to the
fixed ecliptic , and the
variables relative to the ecliptic . Then (see
for example Kovalevsky, 1963), if is canonical if, and only if, there exists a total differential
form The expression is the searched energy
In the previous section, where where is the obliquity relative to . As (Danjon, 1959) and , we finally obtain which leads to and or with and where and The
canonical equations and
then give the precession equations on the form (Laskar, 1986, Laskar
As was already done by Laskar which moves the singularity to and depend on fundamental frequencies of the solar system and they are implicitly given by the integration of the planetary motions. In this context, is obviously not a constant: is also given by the integration of the solar system; , , and have to be determined as functions of time because of the dissipation. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |