5 Orbital motion

 

 

Table 2: Solution S2001: corrections to the nominal values of the orbital parameters formerly fit to DE200 (in arcsecond, except for $\nu$ and n', in arcsecond/cy).

Variable	Sol. 1 (MCEP)	Sol. 2 (ICRS)
W1(0)	-0.1218 $\pm$ 0.0002	-0.0775 $\pm$ 0.0002
W2(0)	-0.0673 $\pm$ 0.0002	-0.0229 $\pm$ 0.0002
W3(0)	-0.1155 $\pm$ 0.0005	-0.0717 $\pm$ 0.0005
$\nu$	-0.3978 $\pm$ 0.0008	-0.4033 $\pm$ 0.0008
$\Gamma$	0.0008 $\pm$ 0.0000	0.0009 $\pm$ 0.0000
E	0.0002 $\pm$ 0.0000	0.0002 $\pm$ 0.0000
T(0)	-0.0770 $\pm$ 0.0002	-0.0326 $\pm$ 0.0002
$\varpi'^{(0)}$	-0.0587 $\pm$ 0.0003	-0.0140 $\pm$ 0.0003
n'	0.0304 $\pm$ 0.0008	0.0258 $\pm$ 0.0008
e'	0.0000 $\pm$ 0.0000	0.0000 $\pm$ 0.0000

 

 

Table 3: Solution S2001: fitted value of tidal part of the quadratic term of the mean longitude (in arcsecond/cy²) and observed corrections to the mean motion of perigee and node (in arcsecond/cy).

Variable	Sol. 1 (MCEP)	Sol. 2 (ICRS)
W1(2,T)	-12.9257 $\pm$ 0.0016	-12.9290 $\pm$ 0.0016
$\Delta W_{2}^{(1)}$	0.0334 $\pm$ 0.0008	0.0317 $\pm$ 0.0008
$\Delta W_{3}^{(1)}$	-0.3806 $\pm$ 0.0097	-0.3611 $\pm$ 0.0097

The main results are summarized in Tables 2 and 3. They can be compared to analogous tables in Chapront et al. (1999b, 2000). The uncertainties reported in these tables are formal errors.

Table 2 shows the corrections to the nominal values of the orbital parameters of the Moon which were formerly fit to JPL ephemerides DE200. These basic parameters are regarded as nominal values in our lunar ephemeris ELP2000. They have been used in our work since 1982 (Chapront-Touzé & Chapront 1983, 1988) and maintained to facilitate further comparison. The origin of angles is $\gamma_{2000}^{I}({\rm MCEP})$ in Sol. 1 (MCEP) and $\gamma _{2000}^{I}({\rm ICRS})$ in Sol. 2 (ICRS). The differences between the angles W₁⁽⁰⁾, W₂⁽⁰⁾, W₃⁽⁰⁾ in Sol. 1 (MCEP) and in Sol. 2 (ICRS) are close to the value of :

$$\begin{eqnarray*}\psi ({\rm MCEP}) = \gamma_{2000}^{I}({\rm ICRS}) \gamma_{2000}^{I}({\rm MCEP}) = 0.0445'' \end{eqnarray*}$$

given below (see Table 9). For all the quantities, the differences, corrected by $\psi$ for angles, are less than 3$\sigma$ except for $\nu$ and n'. We note that the differences $\nu _{{\rm MCEP}}-\nu _{{\rm ICRS}}$ and $n'_{{\rm MCEP}}-n'_{{\rm ICRS}}$ are almost equal, around 0.0050''/cy, and do not vary significantly with the time interval of observations in the intermediate solutions (1996-2001) (see Sect. 1). This suggests a slight drift between the two frames that is not yet completely clarified.

 

 

Table 4: Angular mean elements of the Moon and the Earth-Moon barycenter and Delaunay arguments deduced from S2001. The origin of the angles is $\gamma _{2000}^{I}({\rm ICRS})$ (see Fig. 2); t is the time in Julian centuries reckoned from J2000.0.

W1	=	$218^{\circ}18'59\farcs878~2 ~+1~732~559~343\farcs332~8~t ~-\hphantom{1}6\farcs870~0~t^2 ~+0\farcs006~604~t^3 ~-0\farcs000~031~69~t^4$
W2	=	$\hphantom{1}83^{\circ}21'11\farcs651~8 ~+\hphantom{1}\hphantom{1}~14~643~420\... ...s330~4~t ~-38\farcs263~9~t^2 ~-0\farcs045~047~t^3 ~+0\farcs000~213~01~t^4$
W3	=	$125^{\circ}02'40\farcs326~5 ~-\hphantom{1}\hphantom{1}\hphantom{1}~6~967~919\... ...~+\hphantom{1}6\farcs359~3~t^2 ~+0\farcs007~625~t^3 ~-0\farcs000~035~86~t^4$
T	=	$100^{\circ}27'59\farcs188~0 ~+\hphantom{1}~129~597~742\farcs301~6~t ~-\hphantom{1}0\farcs020~2~t^2 ~+0\farcs000~009~t^3 ~+0\farcs000~000~15~t^4$
$\varpi'$	=	$102^{\circ}56'14\farcs413~6 ~+\hphantom{1}\hphantom{1}\hphantom{1}\hphantom{1... ...tom{1}0\farcs532~7~t^2 ~-0\farcs000~138~t^3 \hphantom{~+0.000~000~00''~t^0}$
l	=	$134^{\circ}57'48\farcs226~4 ~+1~717~915~923\farcs002~4~t ~+31\farcs393~9~t^2 ~+0\farcs051~651~t^3 ~-0\farcs000~244~70~t^4$
l'	=	$357^{\circ}31'44\farcs774~4 ~+\hphantom{1}~129~596~581\farcs073~3~t ~-\hphantom{1}0\farcs552~9~t^2 ~+0\farcs000~147~t^3 ~+0\farcs000~000~15~t^4$
F	=	$\hphantom{1}93^{\circ}16'19\farcs551~7 ~+1~739~527~263\farcs217~9~t ~-13\farcs229~3~t^2 ~-0\farcs001~021~t^3 ~+0\farcs000~004~17~t^4$
D	=	$297^{\circ}51'00\farcs690~2 ~+1~602~961~601\farcs031~2~t ~-\hphantom{1}6\farcs849~8~t^2 ~+0\farcs006~595~t^3 ~-0\farcs000~031~84~t^4$

Table 3 shows the observed value W₁^(2,T), and the bias in the mean motions W₂⁽¹⁾, W₃⁽¹⁾. It is worthwhile to note the agreement of the three quantities determined independently in the two solutions; it was not the case in S1998 for W₁^(2,T) and W₃⁽¹⁾.

Table 4 gives the expressions of the angular mean elements of the Moon and of the Earth-Moon barycenter, and the corresponding Delaunay arguments deduced from the S2001 Sol. 2 (ICRF):
- the mean longitude and the mean longitudes of perigee and node of the Moon (W₁, W₂, W₃);
- the heliocentric orbital parameters of the Earth-Moon barycenter (T, $\varpi'$);
- the Delaunay arguments: l=W₁-W₂, $l'=T-\varpi '$, F= W₁- W₃ and $D=W_1-T+180^{\circ}$.
The mean motions are referred to J2000.0 and the origin of the angles is $\gamma _{2000}^{I}({\rm ICRS})$ (see Fig. 2).

 

 

Table 5: Fit of the positions and velocities of the two observing LLR stations CERGA (Yag) and McDONALD (MLRS2). Units: positions X, Y, Z in meters; velocities $\dot{X},~\dot{Y},~\dot{Z}$ in meter/year.

CERGA (Yag)	X	Y	Z	$\dot{X}$	$\dot{Y}$	$\dot{Z}$
References
ITRF1994 Epoch 1993.0	4581692.254	556195.961	4389355.019	-0.0122	0.0194	0.0084
ITRF1996 Epoch 1997.0	4581692.217	556196.039	4389355.075	-0.0120	0.0189	0.0106
ITRF2000 Epoch 1997.0	4581692.181	556196.024	4389355.072	-0.0131	0.0189	0.0101
Corrections
S2001 Sol. 2 (ICRS)	-0.012	-0.006	-0.001	-0.0008	0.0000	0.0036
Comparisons
ITRF96 - ITRF2000	0.036	0.015	0.003
S2001*  - ITRF96	-0.027	-0.008	-0.017
S2001*  - ITRF2000	0.009	0.007	-0.014
S2001** - ITRF2000	0.005	0.007	0.001
McDONALD (MLRS2)	X	Y	Z	$\dot{X}$	$\dot{Y}$	$\dot{Z}$
References
ITRF1994 Epoch 1993.0	-1330021.390	-5328403.336	3236481.726	-0.0116	-0.0036	-0.0073
ITRF1996 Epoch 1997.0	-1330021.431	-5328403.340	3236481.672	-0.0117	-0.0034	-0.0064
ITRF2000 Epoch 1997.0	-1330021.440	-5328403.330	3236481.675	-0.0125	-0.0001	-0.0065
Corrections
S2001 Sol. 2 (ICRS)	-0.006	0.005	-0.031	-0.0004	-0.0007	-0.0002
Comparisons
ITRF96 - ITRF2000	0.009	-0.010	-0.003
S2001*  - ITRF96	-0.015	0.009	-0.003
S2001*  - ITRF2000	-0.006	-0.001	-0.006
S2001** - ITRF2000	-0.008	-0.004	-0.007