## 2 Methodology
### SRP Model  
#### 2.1.1 ABW Model  
The ABW model decomposes the SRP acceleration of the satellite into two components: \(a_{\text{wing}}\) arising from the solar panels and \(a_{\text{box}}\) resulting from the satellite body. The SRP acceleration related to the satellite body and solar panel at 1 astronomical unit (AU, 1 AU = 149,597,871 km) can be written as [14]:  
\[\begin{array}{lcl}a_{wing}&=&\frac{-AS_{0}}{mc}\cos\theta\big{[}(1+\rho+ \frac{2}{3}\delta)\cdot e_{\mathbb{O}}\big{]}\\ a_{box}&=&\frac{-AS_{0}}{mc}cos\theta\big{[}(\alpha+\delta)\cdot\big{(}e_{ \mathbb{O}}+\frac{2}{3}e_{N}\big{)}+2\rho cos\theta\cdot e_{N}\big{]}\end{array} \tag{1}\]where \(A\) and \(m\) are the surface area and the mass of the satellite, respectively. \(S_{0}\) is solar irradiance at 1 AU (\(\sim\)1361 W/m\({}^{2}\)). \(c\) is the speed of light. \(\alpha\), \(\rho\), and \(\delta\) represent the optical reflective properties of a satellite surface, named the fraction of the absorbed, diffusely scattered, and reflected photons. They are satisfied with an implicit condition of \(\alpha+\rho+\delta=1\). \(e_{N}\) is the normal of the satellite surface. \(e_{\bigodot}\) is a unit vector pointing from the satellite to the Sun. \(\theta\) is the angle between \(e_{N}\) and \(e_{\bigodot}\). For the purpose of parameter estimation, the optical parameters of the solar sail panel are further denoted by parameter \(SP\), and the optical parameters of the satellite body are denoted by parameters \(AD\) and \(R\).  
\[SP =1+\rho+\frac{2}{3}\delta \tag{2}\] \[AD =\alpha+\delta\] \[R =\rho\]  
Thus, the partial derivative of the SRP acceleration resulting from the solar panel is  
\[\frac{\partial a_{wing}}{\partial SP}=\frac{-AS_{0}}{mc}\cos\theta e_{\bigodot}. \tag{3}\]  
The partial derivatives of the satellite body acceleration to parameters \(AD\) and \(R\) are obtained as follows:  
\[\frac{\partial a_{wing}}{\partial AD}=\frac{-AS_{0}}{mc}cos\theta\big{(}e_{ \bigodot}+\frac{2}{3}e_{N}\big{)} \tag{4}\]  
\[\frac{\partial a_{box}}{\partial R}=\frac{-AS_{0}}{mc}2 cos^{2}\theta e_{N}.\]  
Although there are six surfaces for a satellite body, only the surfaces of \(\pm\)Z and +X are exposed to the Sun during yaw-steering attitude mode. Thus six SRP parameters (+\(XAD\), +\(XR\), \(\pm ZAD\), and \(\pm ZR\)) need to be estimated. In addition to the SRP acceleration relative to the satellite's surface, the residual acceleration in the \(Y\)-axis direction and the impact of solar panel rotation lag also need to be considered [14]. Generally, a constant parameter \(Y_{0}\) is employed to correct for the \(Y\)-axis bias. The solar panel rotation lag (\(SB\)) can be estimated using the following partial derivative:  
\[\frac{\partial a_{wing}}{\partial SB}=\frac{-AS_{0}}{mc}2\bigg{(}\rho+\frac{ \delta}{3}\bigg{)}\text{sign}\big{(}\dot{\varepsilon}\big{)}e_{B} \tag{5}\]  
where \(\dot{\varepsilon}\) means time derivative of the Sun-satellite-Earth angle, \(e_{B}\) is a unit vector in a Sun-oriented DYB frame.  
Therefore, the ABW model contains a total of nine SRP parameters, of which six parameters are related to the reflective properties of the satellite buses, two parameters are related to the solar panel, and one constant parameter is caused by the \(Y\)-axis bias. Since the AWB model fully considers the geometric structure and reflective properties of the satellite and the observations, it can be directly applied to the POD of the navigation satellite without an empirical model. In this paper, the ABW model is used to calibrate the surface optical parameters of the cuboid body of the BDS-3 MEO satellite.