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Astron. Astrophys. 329, 329-338 (1998)

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3. Equations of motion

3.1. Basic equations

We built our analytical solution resolving the following differential equations of the second order (Bretagnon et al., 1997)






3.2. Equations for numerical integration

We used numerical integration to test our analytical development. Eqs. (6) are transformed in the form 1


Eqs. (9) are strictly equivalent to the basic equations.

3.3. Model used

The torques on the oblate rigid Earth due to the gravitational attraction of the Moon, the Sun and the planets from Mercury to Neptune are considered. The effects of the zonal harmonics [FORMULA] with [FORMULA] and of the tesseral harmonics [FORMULA], [FORMULA], [FORMULA], [FORMULA] with [FORMULA], [FORMULA], [FORMULA] are computed.

3.4. Construction of the analytical solution

For the motion of the Sun and the planets we use the solution VSOP87A (Bretagnon, Francou, 1988). For the motion of the Moon we use the solution ELP 2000-82B which involves the theory ELP 2000-82 (Chapront-Touzé, Chapront, 1983) and the arguments of the theory ELP 2000-85 (Chapront-Touzé, Chapront, 1988). We use also the derivatives with respect to the different constants for obtaining the same physical constants and the same tidal model as in DE403/LE403.

3.5. Numerical integration

From Eqs. (9) we run two numerical integrations. The first one uses numerical solutions of the motion of the Moon, the Sun and the planets computed from VSOP87A et ELP 2000; the second one uses DE403/LE403. We put DE403/LE403 in the inertial ecliptic and dynamical equinox by the two following rotations:

a) a rotation [FORMULA] in the equator plane,

b) a rotation [FORMULA]

The Euler angles [FORMULA] are reckoned positively in positive rotation. The initial conditions of the numerical integrations, computed for [FORMULA] (J2000) are


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© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997