## 2 Theoretical Analysis of Moon-Based Earth Observational Angles
### Definition and Calculation of Observational Angles
#### 2.2.2 Definition and Calculation of Observational Angles  
The three observational angles, including the viewing zenith angle, solar zenith angle, and relative azimuth angle, are introduced in the topocentric reference system, as shown in Figure 1. The topocentric reference system refers to the Cartesian reference system, whose origin is the observation point. Its \(z\)-axis is perpendicular to the ground, and the fundamental plane is the horizontal plane of the observation point. Based on the topocentric reference system, the viewing zenith angle refers to the angle from the sensor's nadir to the line of sight. Similarly, the solar zenith angle refers to the angle from solar nadir to the line of sight of the Sun. The relative azimuth angle is the angle between the line of sight of the Sun and that of a Moon-based platform in the horizontal plane. To describe the relative positions of the observed point and solar and lunar nadirs, an Earth-centered Earth-fixed spherical coordinate system is introduced based on the topocentric reference system. The Earth-centered Earth-fixed spherical coordinate system refers to a coordinate system in which the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. In the Earth-centered Earth-fixed spherical coordinate system, the position of the Moon can be described as follows:  
\[M_{\text{SCS}}=\left[ \begin{array}{c}r_{m}sin(\theta_{m})cos(\varphi_{m})\\ r_{m}sin(\theta_{m})sin(\varphi_{m})\\ r_{m}cos(\theta_{m})\end{array} \right]. \tag{16}\]  
The two parameters of the spherical coordinates can be derived based on the equatorial longitude and equatorial latitude. The \(\theta_{m}\) can be expressed as follows:  
\[\theta_{m}=\kappa_{m}+\frac{\pi}{2}=\frac{\pi}{2}+arcsin(cos(\epsilon)sin( \beta_{m})+sin(\epsilon)sin(L_{m})cos(\beta_{m})) \tag{17}\]  
To calculate \(\varphi_{m}\), we must introduce the Greenwich sidereal time \(S\). \[\varphi_{m}=\left\{ \begin{array}{c}\alpha_{m}-S\\ 2\pi+\alpha_{m}-S\end{array} \right.=\left\{ \begin{array}{c}arctan(\frac{- sin(\beta_{m})sin(\epsilon)+cos(\beta_{m})cos(\epsilon)sin(L_{m})}{cos( \beta_{m})cos(L_{m})})-S\\ 2\pi+arctan(\frac{-sin(\beta_{m})sin(\epsilon)+cos(\beta_{m})cos(\epsilon) sin(L_{m})}{cos(\beta_{m})cos(L_{m})})-S\end{array} \right. \tag{18}\]  
Similarly, we can calculate the spherical coordinates (\(\theta_{s},\ \varphi_{s}\)) of the Sun by using Equations (16)-(18) and replacing the equatorial longitude and equatorial latitude. The position of the Sun can then be expressed as follows:  
\[S_{\text{SCS}}=\left[ \begin{array}{c}r_{s}sin(\theta_{s})cos(\varphi_{s})\\ r_{s}sin(\theta_{s})sin(\varphi_{s})\\ r_{s}cos(\theta_{s})\end{array} \right]. \tag{19}\]  
Figure 1: Schematic of the observational angles. Given the position of the observed point, its spherical coordinates can also be determined. \[P_{SCS}=\left[ \begin{array}{c}Rsin\big{(}\theta_{p}\big{)}cos\big{(}\varphi_{p} \big{)}\\ Rsin\big{(}\theta_{p}\big{)}sin\big{(}\varphi_{p}\big{)}\\ Rcos\big{(}\theta_{p}\big{)}\end{array} \right], \tag{20}\]  
where \(R\) is the radius of Earth. The spherical coordinates of the lunar and solar nadirs and observed point are \((\theta_{m},\ \varphi_{m},r_{m})\), \((\theta_{s},\ \varphi_{s},r_{s})\), and \((\theta_{p},\ \varphi_{p},R)\), respectively. Finally, the three observational angles for the Moon-based Earth observation can be derived from the spherical coordinates of the Moon, Sun, and observed point.