PETREL19: a new numerical solution of planetary and lunar ephemeris

Abstract

The aim of this work is to develop a new numerical ephemeris of the Sun, the eight planets, the Pluto and the Moon. We first construct a dynamical model, which consists of translational equations of motion for the major bodies and 343 asteroids and of rotational equations of motion for a two-layered Moon. By aligning initial state parameters of the considered bodies and physical parameters in the dynamical model to the JPL ephemeris DE430, we evaluated the adopted dynamical model through a detailed comparison with DE430. After the test, a weighted least square method is applied to fit ephemeris parameters to planetary and lunar observations from 1925 to 2021 simultaneously, and an initial version of our planetary and lunar ephemeris PETREL19 is built. Mass parameters of the 343 asteroids are determined along with other ephemeris parameters by an iteration procedure.

Appendices

Appendix A: The difference between DE430 and $\hbox {PETREL}_\textrm{TEST}$

See Figs. 30 and 31.

Fig. 30

Comparisons in solar system barycentric positions of the eight planets and the Pluto between $\hbox {PETREL}_\textrm{TEST}$ and DE430 from 1869 to 2069

Fig. 31

Comparisons in three lunar Euler angles and in solar system barycentric positions of the Moon, Earth and Sun between $\hbox {PETREL}_\textrm{TEST}$ and DE430 from 1869 to 2069

Appendix B: On transformation between SCRS and BCRS

The results of Brumberg–Kopeikin and Darmour–Soffel–Xu formalisms (see, for example, Brumberg and Kopejkin 1989; Damour et al. 1991) allow us to define a local reference frame for the Moon, and Selenocentric Reference System (SCRS) in a similar way as local kinematically non-rotating GCRS is defined in IAU Resolution B1.3. In analogy to transformation from BCRS to GCRS (e.g. Appendix A Soffel et al. 2003), the transformation from BCRS (t = TCB, ${\varvec{x}}$) to SCRS (T = TCS, ${\varvec{X}}= {\varvec{r}}_\textrm{TCS}$) is written as (only linear terms about ${\varvec{r}}_\textrm{M}= {\varvec{x}}- {\varvec{x}}_\textrm{M}$ remain),

$$\begin{aligned} T= & {} t - {1 \over c^2} \biggl [ A(t) + {\varvec{v}}_\textrm{M}\cdot {\varvec{r}}_\textrm{M}\biggr ] + {1 \over c^4} \biggl [B(t) + \textbf{B}(t) \cdot {\varvec{r}}_\textrm{M}\biggr ] + O(c^{-5})\,, \end{aligned}$$

(B.1)

$$\begin{aligned} {\varvec{X}}= & {} {\varvec{r}}_\textrm{M}+ {1 \over c^2} \biggl [ {1 \over 2} ({\varvec{v}}_\textrm{M}\cdot {\varvec{r}}_\textrm{M}) {\varvec{v}}_\textrm{M}+ w_\textrm{ext}({\varvec{x}}_\textrm{M}) {\varvec{r}}_\textrm{M}\biggr ] + O(c^{-4}) \end{aligned}$$

(B.2)

where Eq. B.1 on time transformation is in the same form as Eq. 1 for TCG, but the functions A(t), B(t), $\textbf{B}(t)$ and $w_\textrm{ext}$ are expressed in terms of corresponding quantities of the Moon.

The transformation from the position in SCRS (TCS-compatible) to BCRS (TDB-compatible), Eq. 21, can be derived directly by combining the linear relation in Eq. 7 and the inversion of the relation given in Eq. B.2. Several previous works (e.g. Manche 2011; Biskupek 2015) have adopted Eq. B.2; meanwhile, in some works (e.g. Pavlov et al. 2016; Williams and Boggs 2022), the transformation

$$\begin{aligned} {\varvec{r}}_\textrm{TDB} = {\varvec{r}}_\textrm{TCS} \left( 1 - \frac{w_\textrm{ext}({\varvec{x}}_\textrm{M}) }{c^2} - L_M \right) - {1 \over 2} \left( \frac{{\varvec{v}}_{M} \cdot {\varvec{r}}_\textrm{TCS}}{c^2} \right) {\varvec{v}}_{M} , \end{aligned}$$

(B.3)

was used with a scale factor $L_M$, which is different to $L_B = 1.550519768 \times 10^{-8}$ in Eq. 21. Two specific factors, $L_M = 0$ and $L_M = 1.4825\times 10^{-8}$ are adopted in Eq.23 of Pavlov et al. (2016) and Eq.17 of Williams and Boggs (2022), respectively. The maximum difference ($=1.550519768 \times 10^{-8}$) between three scale factors may give rise to about 3 cm before fit in position of the LLR retro-reflectors on the lunar surface. The scale factor directly relates to the scale of the lunar reference frame realized by five LLR retro-reflectors and should be fixed in the definition of a conventional lunar reference frame with cm-level accuracy.