## 2 Theoretical Analysis of Moon-Based Earth Observational Angles
### Definition and Calculation of Observational Angles
#### 2.2.1 Calculation Method for Lunar and Solar Nadirs  
Because the orbital elements of the Sun and Moon are defined in the Earth-centered ecliptic coordinate system, we first calculated the ecliptic latitude and longitude and distances of the Moon and Sun from Earth. We mainly adopted a previously described analytical method for the ecliptic coordinates of the Moon and Sun, which are accurate to about 0.1-1% [38]. The ecliptic longitude \(\lambda_{m}\) of the Moon is  
\[\lambda_{m}=L_{m}+dl_{m}, \tag{1}\]  
where  
\[dl_{m}=2e_{m}sin(l_{m})+\frac{5}{4}e_{m}{}^{2}sin2l_{m}-4586^{\prime}sin(l_{m} -2D)+2370^{\prime\prime}sin(2D). \tag{2}\]  
The ecliptic latitude \(\beta_{m}\) is  
\[\beta_{m}=i_{m}sin(F+dl_{m}+\Delta F)+N, \tag{3}\]  
where  
\[\Delta F=412^{\prime\prime}sin(2F)+541^{\prime\prime}sin(l_{s}), \tag{4}\]  
\[N=-526^{\prime\prime}sin(F-2D)+44^{\prime\prime}sin(l+F-2D)-31^{\prime\prime} sin(-l+F-2D)+\cdots. \tag{5}\]  
The Earth-Moon distance \(r_{m}\) is  
\[r_{m}=a_{m}(1+\frac{1}{2}e_{m}{}^{2}-e_{m}cos(l_{m})-\frac{1}{2}e_{m}{}^{2} cos(2l_{m})-3699 cos(2D-l)+\cdots, \tag{6}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Star/Planet** & **Orbital Parameters** & **Symbol** \\ \hline \multirow{7}{*}{Sun} & Semimajor axis & \(a_{s}\) \\  & Eccentricity & \(e_{s}\) \\  & Obliquity of the ecliptic & \(\epsilon_{s}\) \\  & Mean ecliptic longitude of the perigee & \(\Gamma_{s}\) \\  & Mean ecliptic longitude of the Sun & \(L_{s}\) \\  & Mean anomaly of the Sun & \(l_{s}=L_{s}-\Gamma_{s}\) \\  & Semimajor axis & \(a_{m}\) \\  & Eccentricity & \(e_{m}\) \\ Earth & Inclination & \(i_{m}\) \\  & Mean ecliptic longitude of the ascending node & \(\Omega_{m}\) \\  & Mean ecliptic longitude of the perigee & \(\Gamma_{m}\) \\  & Mean ecliptic longitude of the Moon & \(L_{m}\) \\  & Mean anomaly of the Moon & \(l_{m}=L_{m}-\Gamma_{m}\) \\  & Mean distance of the Moon from the ascending node & \(F_{m}=L_{m}-\Omega_{m}\) \\  & Mean elongation of the Moon & \(D=L_{m}-\Gamma_{s}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Orbital parameters of the Sun and Moon.  
Similarly, the solar ecliptic longitude is  
\[\lambda_{s}=L_{s}+dl_{s}, \tag{7}\]  
where  
\[dl_{s}=2e_{s}sin(l_{s})+\frac{5}{4}e_{s}^{2}sin2l_{s}. \tag{8}\]  
The Sun-Earth distance is  
\[r_{s}=a_{s}(1+\frac{1}{2}e_{s}^{2}-e_{s}cos(l_{s})-\frac{1}{2}e_{s}^{2}cos(2l_{ s})). \tag{9}\]  
The solar ecliptic latitude is zero in the Earth-centered ecliptic coordinate system.  
However, for Moon-based Earth observations, the ecliptic coordinates of the Moon and Sun must be converted to equatorial coordinates because the position of the observed point is generally defined in the equatorial coordinate system. The equatorial longitude \(\alpha_{m}\) and equatorial latitude \(\kappa_{m}\) of the Moon can be deduced from the ecliptic longitude \(\lambda_{m}\) and ecliptic latitude \(\beta_{m}\).  
\[sin(\kappa_{m})=cos(\epsilon_{s})sin(\beta_{m})+sin(\epsilon_{s})cos(\beta_{m} )sin(\lambda_{m}) \tag{10}\]  
\[cos(\alpha_{m})=\frac{cos(\beta_{m})cos(\lambda_{m})}{cos(\delta_{m})} \tag{11}\]  
\[sin(\alpha_{m})=\frac{-sin(\beta_{m})sin(\epsilon_{s})+cos(\beta_{m})cos( \epsilon_{s})sin(\lambda_{m})}{cos(\delta_{m})} \tag{12}\]  
Similarly, by substituting the ecliptic longitude \(\lambda_{m}\) and ecliptic latitude \(\beta_{m}\) for the ecliptic longitude \(\lambda_{s}\) and ecliptic latitude \(\beta_{s}\) in Equations (14)-(16), the solar equatorial coordinates can be calculated.  
\[sin(\kappa_{s})=cos(\epsilon_{s})sin(\beta_{s})+sin(\epsilon_{s})cos(\beta_{s })sin(\lambda_{s}) \tag{13}\]  
\[cos(\alpha_{s})=\frac{cos(\beta_{s})cos(\lambda_{s})}{cos(\delta_{s})} \tag{14}\]  
\[sin(\alpha_{s})=\frac{-sin(\beta_{s})sin(\epsilon_{s})+cos(\beta_{s})cos( \epsilon_{s})sin(\lambda_{s})}{cos(\delta_{s})} \tag{15}\]