1 Introduction

Considering the increasing accuracy of the determination of the coefficients of nutation by modern techniques such as VLBI, and the large correction to the conventional IAU1976 value of the general precession in longitude $p_{\rm a}$ as given by Lieske et al. (1977), Souchay & Kinoshita (1996) made important corrections for the largest coefficients due to the leading luni-solar effect coming from the J₂ Earth's geopotential, in particular for the leading terms of period 18.6 y, 9.3 y, 1 y, 182 d, 13.66 d, with respect to their corresponding values in Kinoshita & Souchay (1990). They also confirmed the presence and the value of an out-of-phase component for the 18.6y and 9.3y terms both in longitude and obliquity, already pointed out by Williams (1994). At last they made some corrections to the tables listed in Kinoshita & Souchay (1990) according to some remarks made by Williams (private communication).

In a second paper, Souchay & Kinoshita (1997) calculated again the coefficients of nutation due to the second-order J₃, J₄, C_2,2, and S_2,2 coefficients of the Earth's geopotential, and also the direct action of the planets on nutation, with a truncation limit of 0.1 $\mu$as for the coefficients of $\Delta$ $\psi$ cos $\varepsilon$and $\Delta$$\varepsilon$, that is to say 50 times smaller than the truncation limit of the series in Kinoshita & Souchay (1990).

The results were listed by Souchay & Kinoshita (1997) and compared with Hartmann & Soffel (1995) and Williams (1995) respectively for each of the two kinds of effects mentioned above. In the two comparisons the agreement is remarkable, for the absolute difference in the amplitude of the coefficients does not exceed 1 $\mu$as except for a few ones, although the total number of these coefficients is much larger than in the Kinoshita & Souchay series. Notice that the three ways of determination of the coefficients are quite different: Hartmann & Soffel (1995) compute them from tidal waves, Williams (1995) is using the torque approach and Souchay & Kinoshita (1997) use Hamiltonian equations.

The fact that the results are very close together is a very probing confirmation of the validity of the terms found. The present paper is the third and final one in the scope of a global check of the coefficients of nutation for a rigid Earth model (Souchay & Kinoshita 1996, 1997). It is devoted to the last and more delicate part concerning the computation at the $0.1 ~\rm\mu as$ level of second-order coupling effects which can be divided into two categories: the first one which can be quoted as the spin-orbit coupling effect is the interaction between the orbital motion of the Moon and the J₂ component of the Earth geopotential characterizing the ellipticity of the Earth. The second one, which can be called the crossed-nutation effect, is the influence of the nutation itself on the torque exerted by the Moon and the Sun: in a few words, when calculating this torque, we must take into account the small contribution due to the displacement of the figure axis coming from the nutation itself.

The first coupling effect above has been pointed out for the first time by Kubo (1982) who made a rough calculation of the perturbations on the longitude and latitude coordinates $\lambda$ and $\beta$ of the Moon, caused by the Earth's flattening, then simultaneously of the perturbations on the nutation. In the frame of a global reconstruction of the theory of the nutation for a rigid Earth model, Kinoshita & Souchay (1990) found a few terms down to their level of truncation of their series, that is to say 0.005 $\mu$as (millliarcsecond).

The second coupling effect was already partially computed by Kinoshita (1977) when elaborating a new theory of nutation starting from Hamiltonian formalism. It was more accurately re-calculated by Kinoshita & Souchay (1990) down to 0.005 $\mu$as.

In the following we will compute the two kinds of second-order coupling effects which have just been explained, with a double objective: one is to catch all the coupling terms down to 0.1$~\rm\mu as$ instead of 5$~\rm\mu as$ (Kinoshita & Souchay 1990). The other one is to carry out all calculations with a computer instead of manually as it was the case in this last paper. The advantage, in addition of avoiding miscalculations, is to push farer the development of the luni-solar potential instead of keeping only its leading terms, and thus to take into account some possible coupling interactions previously neglected.

Moreover one of the best way to check the validity of the series of nutation determined analytically is to carry out a numerical integration of the nutation, and to study the residuals between the results given by these two methods. This was already done by Souchay & Kinoshita (1991) who showed that the residuals were about 20 times smaller, both in longitude and in obliquity, than those found by Kubo & Fukushima (1988), as well as by Schastock et al. (1989) before the reconstruction of the analytical theory by Kinoshita & Souchay (1990). This proved that the relatively important second-order analytical corrections due to the coupling effects described above and calculated in this last paper were justified, and was a probing confirmation of the theory. Moreover a new comparison between our new series and numerical integration using the numerical ephemeris DE403 of the JPL, is on the way (Souchay 1998).

In the following we describe the methods from which we computed the coefficients of the nutation related to the crossed-nutation and to the spin-orbit coupling effect. Then, we present our final tables of nutation REN-2000 for a rigid Earth model, including all the improvements done previously (Souchay & Kinoshita 1996, 1997) and in the present paper. Notice that in order to be complete at the level of $0.1 ~\rm\mu as$ our tables REN-2000 include also the diurnal and sub-diurnal components of the nutation related to the C_3,i and S_3,i of the geopotential, as calculated by Folgueira et al. (1998a) and those related to the C_4,i and S_4,i coefficients, as calculated by Folgueira et al. (1998b). These new contributions not included in previous tables (Kinoshita & Souchay 1990) will be presented in the end of the present paper.