3 The fitted parameters

We list below the parameters that are fit in the solutions S1998 and S2000. All the angles and mean motions are referred to J2000.0.
- The geocentric lunar orbital parameters W₁⁽⁰⁾, W₂⁽⁰⁾, W₃⁽⁰⁾ (constants of the mean longitude and mean longitudes of perigee and node), $\nu=W_{1}^{(1)}$, $\Gamma$, E (sidereal mean motion, constants for inclination and eccentricity).
- The heliocentric orbital parameters of the Earth-Moon barycenter T⁽⁰⁾, $\varpi ^{(0)}$ (constants of the mean longitude and mean longitude of perihelion), n', e' (sidereal mean motion and eccentricity).
- The bias parameters $\Delta W_{1}^{(2)}$, $\Delta W_{2}^{(1)}$, $\Delta W_{3}^{(1)}$ (observed corrections to the computed coefficient of the quadratic term of the lunar mean longitude, and the computed mean motions of perigee and node). $\Delta W_{1}^{(2)}$ yields an observed value of W₁^(2,T), the tidal part of the coefficient of the quadratic term of the mean longitude (half tidal secular acceleration).
- The 6 free libration parameters (parameters tied to the coefficients of the main free libration terms and values of the free libration arguments).
- The $3\times4$ reflector coordinates. The reflector coordinates are referred to lunar principal axes of inertia.
- The position angles $\phi$, $\epsilon$ and $\psi$ with respect to different systems of axes. Figure 2 illustrates the relative positions of various systems presented in Sect. 4.
- A correction to precession $\Delta p$: optional parameter.

In the solution S2001 we keep the same list as above and we add $5\times3\times 2$ optional parameters giving the positions and velocities of the 5 stations: McDONALD 2.70 m, MLRS1 and MLRS2, CERGA and Haleakala. The parameters are the equatorial rectangular coordinates X, Y, Z in the ITRF (position) and their derivatives $\dot{X}$, $\dot{Y}$, $\dot{Z}$ (velocity). Note that simultaneously fitting all the parameters has not been possible. The fits have been performed in several steps, but tests have been made in order to check the stability of the results. Indeed, strong correlations exist among some parameters that may weaken the accuracy of our determinations; in particular, it is the case of the variables related to the reference frame ($\phi$ and $\epsilon$) and the positions of the stations (X, Y, Z). $\Delta p$ (precession) and $\dot{\epsilon}$ (obliquity rate) are correlated with the velocities of the stations ($\dot{X}$, $\dot{Y}$, $\dot{Z}$); $\Delta p$ and the principal nutation term are also difficult to separate. We have adopted the following strategy. First, we determine the whole set of parameters mentioned above except the positions and velocities of the stations. Then fixing the value of $\phi$, we add the positions of the stations to the whole set and make a new improvement. Next we determine the velocities of the stations separately. Finally, fixing all the parameters, we perform a last analysis including $\Delta p$ and the principal term of $\Delta\psi$ (nutation in longitude) (see Sect. 10). At each step of the process we verify the coherence of the determinations; for example we verify that the introduction of the fitted values of X, Y, Z does not change significantly the value of $\phi$ if the first step is reiterated.

Figure 2: Relative positions of the mean inertial ecliptic of J2000.0 with respect to ICRS, MCEP and JPL.