9 Orientation of the celestial axes

We gather in Table 9 our new determinations of the position angles $\phi$, $\epsilon$ and $\psi$ (see Sect. 4). The angle $\psi$ is obtained through the 2 different versions of the mean longitude of the Moon W₁, which is evaluated in the ICRS (with Sol. 2), and in the reference system R, R being the MCEP with Sol. 1, or the reference system of any JPL numerical integration DEn with the solution fit to the lunar ephemeris of DEn. Such a solution is deduced from an analysis of the same nature as those using the LLR observations themselves, by considering the JPL lunar ephemeris as an "observational model''. Hence, we compute the difference:

$$\begin{eqnarray*}\psi(R)&=& W_{1}(ICRS)- W{_1}(R) \\ \hphantom{\psi(R)}&=& W_{1... ...\hphantom{\psi(R)}&&+[W_{1}^{(2)}(ICRS) - W_{1}^{(2)}(R)]t^{2}, \end{eqnarray*}$$

where t is the time at a mean epoch, reckoned in centuries from J2000.0.

The mean epoch for MCEP arises directly from the least-squares fit; it is the weighted time related to the distribution of weights of the sub-groups. For the JPL ephemerides, the mean epochs are mentioned in the literature and correspond to JPL's fits. In $\psi (R)$ the linear term corresponds to the difference of sidereal mean motions in the 2 systems: $\nu(ICRS)-\nu(R)$; the quadratic term corresponds to half the difference between the tidal parts of the acceleration of the mean longitude in the 2 systems: W₁^(2,T)(ICRS)-W₁^(2,T)(R), the non tidal parts being the same.

The $\psi$ function that we obtain for DE405 is:
$\psi(DE405) = 0.00832 + 0.01793 t - 0.01555 t^{2}$ (arcsecond).
From (Standish 2000), the reference system tied to the ephemeris is based on VLBI observations (Magellan spacecraft to Venus and Phobos approach to Mars) made between 1989 and 1994. Hence, we have arbitrarily chosen 1990 Jan. 1 for the mean epoch of DE405.

Using the quantities $\phi$ and $\psi$ of Table 9, we make the projection on the ICRS "equator'' of the origin of right ascension o(DE405) which is distant from o(ICRS) by less than one mas. We find:

o(ICRS)o(DE405) = 0.7 mas (Epoch, 1990 Jan. 1).

For DE403 we obtain:

o(ICRS)o(DE403) = 1.9 mas (Epoch, 1985 Jan. 1).

These results concerning DE405 agree with the fact that the numerical integration is oriented onto the ICRS (Standish 1998).