6. The obliquity rate
The obliquity with respect to a fixed ecliptic (that we call here with respect to the ecliptic of J2000.0) itself has a quasi linear term, which is a part of long-periodic terms. According to Williams (1995) the time derivative of the long periodic terms in is called the obliquity rate. The quasi-linear term in the obliquity has two main origins: an indirect planetary origin and a direct planetary origin. Furthermore the indirect planetary origin is divided into two parts. One is the motion of the mean ecliptic, which is caused by the mutual perturbations among planets and whose motion is expressed of combinations of long-periodic terms (the period is ranged from 50 thousand years to 2 million years). This slow ecliptic motion disturbs the motion of the Moon and then causes a very long periodic perturbation in obliquity. Other indirect origin is due to the fact that the Sun is not located in the mean ecliptic and moves up and down with respect to the mean ecliptic with short period. This motion also causes a small long periodic change in the obliquity. These two perturbations were neglected in Kinoshita & Souchay (1990). The direct planetary part originates from the fact that the planetary orbital planes are precessing due to the mutual planetary perturbations. This small effect was neglected in Kinoshita & Souchay (1990). Williams (1995) calls these perturbations tilt effects and also obtained the obliquity rate from another approach.
We recalculate the two effects above, starting from Hamiltonian equations and the recent ephemerides ELP2000 for the Moon (Chapront-Touzé & Chapront 1988) and VSOP87 (Bretagnon & Francou 1988). They are used to calculate the obliquity rate coming respectively from the first and second origins above.
Concerning the planetary tilt-effect, our results give: , and after replacing by its polynomial developement in function of time (Lieske et al. 1977), this leads to the following rate: , which is exactly in accordance with the value determined by Williams (1994), that is to say: .
The obliquity rate related to the direct planetary torque is characterized for each planet by expressions in the form and , which can be assimilated to a polynomial expression in function of time, after a developement starting from J2000.0 (with a value of set to at this date).
Table 7. Influence of the direct torque exerted by the planets on the precession in obliquity
At last another contribution to the variation of has been identified in the present paper, following the output values of nutation (Souchay & Kinoshita 1996) when computing the effect of the solar potential. More precisely a term, with the following expression, in milliarsecond: , can be converted in a secular variation of as above, that is to say: .
The combination of the three components of leads to the following total amount of: . Williams (1994), by combinating the two first components, got: , and he added to this value the contribution of tidal torques amounting to: . Then his final estimation is: .
Notice that the long-periodic variation of the obliquity has been recently confirmed by VLBI analysis extending for 15 years of data (Souchay et al. 1995; Charlot et al. 1995). In the first work, the observational determination is , whereas for the second one it ranges between and These values are quite in agreement with the analytical ones above.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998