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Astron. Astrophys. 318, 975-989 (1997)

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2. Averaged equations for the precession of the Earth with planetary perturbations

The equations of precession of the Earth are derived from a Hamiltonian function H which is the sum of the kinetic energy and of the potential energy [FORMULA] of the torque exerted by the Sun and the Moon on the equatorial bulge of the Earth.

We suppose here that the Earth is an homogeneous rigid body with moments of inertia [FORMULA] 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 [FORMULA] and their conjugate angles [FORMULA] (Andoyer, 1923, Kinoshita, 1977) (Fig. 2). [FORMULA] is the modulus of rotational angular momentum of the Earth with rotation angular velocity [FORMULA] ; [FORMULA], is the projection of the angular momentum on the normal to the ecliptic, at obliquity [FORMULA] ; [FORMULA] is the hour angle between the equinox of the date [FORMULA] and a fixed point of the equator; [FORMULA] the opposite of the general precession angle (see Fig. 2).

[FIGURE] Fig. 2. Reference frames for the definition of precession. [FORMULA] and [FORMULA] are the mean equator and ecliptic of the date with equinox [FORMULA]. [FORMULA] is the fixed J2000 ecliptic, with equinox [FORMULA]. i is the inclination of the ecliptic [FORMULA] on [FORMULA]. The general precession in longitude, [FORMULA] is defined by [FORMULA], where N is the ascending node of the ecliptic of date on the fixed ecliptic. [FORMULA] is the obliquity, and [FORMULA] the hour angle of the equinox of date [FORMULA].

Let [FORMULA] be a reference frame fixed with respect to the Earth, [FORMULA] a reference frame linked to the orbital plane of the perturbing body (Sun or Moon) around the Earth (see Fig. 2), and [FORMULA] the rotation such that [FORMULA]

According to Tisserand (1891) or Smart (1953) the potential energy of the torque exerted by a perturbing body [FORMULA] of mass [FORMULA] at distance r from the Earth [FORMULA], limited to its largest component is

[EQUATION]

where R is the Earth's radius. [FORMULA] denotes the Earth's moment of inertia around the radius vector [FORMULA], and is given by

[EQUATION]

The motion of [FORMULA] around the Earth is determined by its elliptical orbital elements defined with respect to the fixed ecliptic [FORMULA], with reference direction toward the fixed equinox [FORMULA]. Let us denote [FORMULA] its argument of perigee, [FORMULA] its true anomaly, [FORMULA] the true longitude of date, where [FORMULA] is the longitude of ascending node of the apparent orbit of [FORMULA] on [FORMULA] if [FORMULA] is the Sun, and on [FORMULA] if it is the Moon.

We first build the precession equations due to the perturbation of the Sun only. When [FORMULA] is the Sun (subscript [FORMULA]), we have:

[EQUATION]

and the transformation from the equatorial frame to the ecliptic one [FORMULA] is:

[EQUATION]

where the rotations [FORMULA] and [FORMULA] are defined as

[EQUATION]

[EQUATION]

Hence

[EQUATION]

We retain only the contribution of terms with no spherical symmetry, which gives with Andoyer's variables

[EQUATION]

Let [FORMULA] be the mean anomaly of the Sun. The fast angles [FORMULA] and [FORMULA] are removed by taking the average

[EQUATION]

unless a spin-orbit resonance occurs, i.e. when the angle [FORMULA] is librating (Peale, 1969). This leads to the following expression for the averaged potential due to the Sun:

[EQUATION]

where [FORMULA] is called the dynamical ellipticity. The contribution of the Moon (subscript M) to the Hamiltonian follows the same procedure with

[EQUATION]

where [FORMULA] Assuming a constant rate for the precession of the orbit of the Moon (node and perihelion), one can also average the subsequent [FORMULA] on [FORMULA], [FORMULA] and [FORMULA], which gives:

[EQUATION]

where [FORMULA] 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 [FORMULA], which gives

[EQUATION]

where [FORMULA] is the "precession constant":

[EQUATION]

For a fast rotating planet like the Earth, [FORMULA] can be considered as proportional to [FORMULA] ; this correspond to the hydrostatic equilibrium (see for example Lambeck, 1980). In this approximation, [FORMULA] is proportional to [FORMULA].

Now, when considering the perturbations of the other planets, the ecliptic [FORMULA] is not an inertial plane any more and the kinetic energy E of its driving has to be added. We refer here to Kinoshita (1977).

Let [FORMULA] be Andoyer's variables relative to the fixed ecliptic [FORMULA], and [FORMULA] the variables relative to the ecliptic [FORMULA]. Then (see for example Kovalevsky, 1963), if K is the Hamiltonian of the system, function of the variables [FORMULA] relative to the moving [FORMULA], and F the Hamiltonian written with the variables [FORMULA] relative to [FORMULA], the transformation

[EQUATION]

is canonical if, and only if, there exists a total differential form dW such that

[EQUATION]

The expression [FORMULA] is the searched energy E.

In the previous section, H was the Hamiltonian F written with the new variables [FORMULA]. Then, the new Hamiltonian [FORMULA] can be obtain by identifying E and dW in the Eq. (2). Thanks to Danjon (1959), one can establish the following relation:

[EQUATION]

where i is the inclination of [FORMULA] on the fixed plane [FORMULA]. Then, if the obliquity [FORMULA] is oriented from the rotation axis [FORMULA] to the orbit normal [FORMULA], we have

[EQUATION]

where [FORMULA] is the obliquity relative to [FORMULA]. As [FORMULA] (Danjon, 1959) and [FORMULA], we finally obtain

[EQUATION]

which leads to

[EQUATION]

and

[EQUATION]

or

[EQUATION]

with

[EQUATION]

and where [FORMULA] and [FORMULA] The canonical equations [FORMULA] and [FORMULA] then give the precession equations on the form (Laskar, 1986, Laskar et al., 1993a-b):

[EQUATION]

As was already done by Laskar et al. (1993b) and Laskar and Robutel (1993), since the contribution of the planetary perturbations to [FORMULA] is singular for [FORMULA], we use for numerical integrations, instead of [FORMULA], the complex variable

[EQUATION]

which moves the singularity to [FORMULA]

[FORMULA] and [FORMULA] depend on fundamental frequencies of the solar system and they are implicitly given by the integration of the planetary motions. In this context, [FORMULA] is obviously not a constant: [FORMULA] is also given by the integration of the solar system; [FORMULA], [FORMULA], [FORMULA] and [FORMULA] have to be determined as functions of time because of the dissipation.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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