Position of Earth and Moon

Tab.5 gives also ecliptic positions of Earth and Moon. DE200 and VSOP87 give only the heliocentric position $\mathbf{r_{EMB}}$ of the EMB. DE200 gives in addition the geocentric position $\mathbf{{r_g}_{M}}$ of the Moon. If both values are given the position of the Earth $\mathbf{r_E}$ can be calculated exactly (within the IAU1976 system) using the mass ratio Moon/Earth of $\mu_M = 0.01230002$[A. K6] (or its respective value used for the ephemeris) by:

$$\mathbf{r_E} = \mathbf{r_{EMB}} - \mathbf{r_{gM}}\frac{\mu_M}{1+\mu_M}$$ (31)

The velocity vector has the same transformation. For the VSOP87 system we apply following formula by J.L.Simon [personal communication] describing the rotation of the Earth around the EMB:

$$\lambda_E = \lambda_{EMB} + 6''.468 \sin D \hspace{2cm} r_E = r_{EMB} + 4613 \cos D \mbox{[km]},$$ (32)

where $D$ is the Delauney argument from eqn.3.5 in Simon et al. (1994):

$$D = 297^\circ.8502 + P \cdot T_0,$$ (33)

where the rotation period is $P= 445267^\circ.11/\mbox{century}$. The Earth velocity vector $\mathbf{v_E}$ can be calculated by

$$\mathbf{v_E} = \mathbf{v_{EMB}} + \Omega_E \times \mathbf{r_D},$$ (34)

where $\mathbf{r_D} = \mathbf{r_E}-\mathbf{r_{EMB}}$ and the ecliptic angular velocity vector is given by $\Omega_E = 2\pi/P \cdot (0,0,1)$.

If only $\mathbf{r_{EMB}}$ and $\mathbf{r_E}$ are given the ecliptic position and velocity of the Moon can then be calculated by

$$\mathbf{r_{M}} = ((1+\mu_M)\mathbf{r_{EMB}} - \mathbf{r_E})/\mu_M.$$ (35)

But the precision of the resulting lunar velocity is rather low (190 m/s). Neglecting the difference between $\mathbf{r_{EMB}}$ and $\mathbf{r_E}$ increases the total error in the Earth position to $14''$ (Earth-EMB in Tab.5).

For slightly lower precision without solving the Kepler equation (26) the geometric ecliptic longitude of the Earth can be calculated by the approximation given for the Solar longitude in [A. C24]:

$$\lambda_{geo} = \lambda_{mean} + 1^\circ.915\sin g + 0^\circ... ...{geo} = 1.00014 - 0.01671\cos g - 0.00014 \cos 2g \mbox{[AU]},$$ (36)

where $\lambda_{mean}$ and the mean anomaly $g = \lambda_{mean} - \varpi$ for the EMB can be taken from Tab.4. This approximation has also been used by Hapgood (1992). The respective precision is $34''$ (Earth$_{approx}$ in Tab.5).