# Advancing the Solar Radiation Pressure Model for BeiDou-3 IGSO Satellites
## 2 Materials and Methods
### Box-Wing Model  
The ECOM1 model can also be applied if a proper box-wing model is involved to consider the systematic impact caused by the stretched satellite shape. The structure of a GNSS satellite can be simplified to a cuboid body with six faces (box) plus solar panels (wings). Total SRP acceleration can be theoretically obtained by summing the SRP acceleration for each illuminated satellite surface and solar panel. This modeling method, commonly referred to as the box-wing model, was originally established for Topex/Poseidon POD [25]. According to [PERSON] et al. [26], the acceleration produced by the physical interaction between the solar radiation and a flat surface of the satellites can be formulated by:  
\[a=-\frac{A}{M}\frac{S_{0}}{c}\cos\theta\bigg{[}(\alpha+\delta)\overset{\to}{ \epsilon}_{D}+2(\frac{\delta}{3}+\rho\cos\theta)\overset{\to}{\epsilon}_{N} \bigg{]} \tag{3}\]  
where \(S_{0}\) is the total solar irradiance at the 1 AU, and \(M\) is the satellite's mass. The parameter \(c\) is the velocity of light and \(A\) is the illuminated surface area. The terms \(\overset{\to}{\epsilon}_{D}\) and \(\overset{\to}{\epsilon}_{N}\) indicate the satellite-Sun unit vector, and the normal vector of the illuminated surface, respectively. The term \(\theta\) is the angle between \(\overset{\to}{\epsilon}_{D}\) and \(\overset{\to}{\epsilon}_{N}\), and the parameters \(\alpha\), \(\delta\) and \(\rho\) (with \(\alpha+\delta+\rho=1\), [PERSON] et al. [26]) are the absorption, diffuse and specular reflection coefficients of illuminated surface, respectively. Under the assumption of mostly balanced thermal re-mission from the front and back-side of the solar panels, the SRP acceleration \(a_{sp}\) on the satellite solar panels can be described by using Equation (3) [16]. For the nominal attitude [27], the satellite solar panels are perpendicular to the Sun direction with \(\cos\theta_{sp}=1\) and \(\overset{\to}{\epsilon}_{N,sp}=\overset{\to}{\epsilon}_{D}\). Equation (3) can be reformulated as [16]:  
\[a_{sp}=-\frac{A}{M}\frac{S_{0}}{c}\bigg{[}(1+\rho+\frac{2}{3}\delta)\overset{ \to}{\epsilon}_{D}\bigg{]} \tag{4}\]  
The satellite bus is covered by multilayer insulation for thermal protection. According to [PERSON]'s law, the accelerations \(a_{i,th}\) aroused by the immediate thermal re-radiation from the satellite bus surfaces can be obtained by [7]:  
\[a_{i,th}=-\frac{A}{M}\frac{S_{0}}{c}\cos\theta\frac{2}{3}\overset{\to}{ \epsilon}_{N} \tag{5}\]  
where subscript \(i\) represents the illuminated satellite body surface. According to the IGS convention [27], for satellite bodies in nominal attitude, only their \(+X\), \(+Z\), \(-Z\) surfaces are illuminated by the Sun. By adding the instantaneous thermal re-radiation accelerations \(a_{i,th}\) to Equation (3), the accelerations \(a_{b}\) acting on satellite bus are expressed as [16]:  
\[a_{b}=-\frac{A}{M}\frac{S_{0}}{c}\cos\theta\bigg{[}(\alpha+\delta)(\overset{ \to}{\epsilon}_{D}+\frac{2}{3}\overset{\to}{\epsilon}_{N})+2\rho\cos\theta \overset{\to}{\epsilon}_{N}\bigg{]} \tag{6}\]  
Currently, only the absorption coefficient \(\alpha\) for BeiDou-3 IGSO surfaces is disclosed by CSNO [17], which hinders the formation of a proper BW model. Therefore, in the study, the adjustable box-wing model developed by [16] is used to solve the optical coefficients for BeiDou-3 IGSOs with real tracking measurements. For the satellites in nominal attitude, there are nine estimated parameters in the ABW model, i.e., the absorption plus diffuse reflection \((\alpha+\delta)\) as well as the specular reflection \((\rho)\) for each illuminated satellite surface \((+X,+Z,-Z)\), the scale parameter for solar panels \(1+\rho+\frac{2}{3}\delta\), and two non-optical parameters \(Y\) bias \((Y_{0})\) and solar panel rotation lag angle \((SB)\).