# Solar Radiation Pressure Models for BeiDou-3 I2-S Satellite: Comparison and Augmentation
## 2 Characteristics and SRP Models for BeiDou-3 I2-S Satellite
### Box-Wing Model  
Based on the analytical physical theory, the SRP acceleration generated by a surface of area A depends on the relative alignment of the Sun direction and the surface normal \(\mathbf{e}_{N}\) as well as the fraction of absorbed photons (\(\alpha\)), of diffusely reflected photons (\(\delta\)), and of specularly reflected photons (\(\rho\)) [15; 16]. For \(\cos\theta=\mathbf{e}_{\odot}^{T}\cdot\mathbf{e}_{N}>0\), that is, for an illuminated surface, the acceleration \(\mathbf{a}\) at a distance of 1 AU can be accounted as  
\[a=-\frac{A}{M}\frac{S_{0}}{c}\cos\theta\bigg{[}(1-\rho)\mathbf{e}_{\odot}+2( \frac{\delta}{3}+\rho\cos\theta)\mathbf{e}_{N}\bigg{]}, \tag{5}\]  
where \(S_{0}=1367\) W/m\({}^{2}\) is the solar flux at 1 AU, \(c\) is the vacuum velocity of light, and \(\mathbf{M}\) is the total mass of the satellite. This equation can be used for SRP calculation on SPs. For a satellite bus panel, which is covered by multilayer insulation for thermal protection, the thermal re-radiation must be accounted for. Hence, the modified formulation for the acceleration [17] on the satellite bus surfaces is  
\[a=-\frac{A}{M}\frac{S_{0}}{c}\cos\theta\bigg{[}(\alpha+\delta)(\mathbf{e}_{ \odot}+\frac{2}{3}\mathbf{e}_{N})+2\rho\cos\theta\cdot\mathbf{e}_{N}\bigg{]} \tag{6}\]  
By simply treating the satellite as a combination of box as two wings, the SRP acceleration acting on the satellite can be easily calculated by summing the SRP accelerations of each illustrated satellite bus panels and SPs. As the changes in the a priori geometrical or optical properties of the satellite, or attitude deviations from the nominal, result in deficiencies of the above analytical model in modeling actual SRP perturbation, the adjustable box-wing model (ABW) has been proposed [17]. By fitting the optical coefficients and other parameters with real-tracking measurements, this model provides precise orbit solution with a clear physical understanding of SRP. For the satellites with nominal YS attitude, up to nine parameters are adjusted, including the absorption plus diffusion (\(\alpha+\delta\)) as well as reflection (\(\rho\)) for the illustrated surfaces (+X, +Z, and \(-\)Z), scale parameter of SPs (\(1+\rho+\frac{2}{3}\delta\)), SP rotation lag (denoted as SB hereafter), and Y bias. From Equation (6), the partial derivatives of the acceleration w.r.t the optical properties of the satellite surfaces can be obtained as  
\[ \begin{array}{l}\frac{\partial a}{\partial(\alpha+\delta)}=-\frac{A}{M} \frac{S_{0}}{c}\cos\theta(\mathbf{e}_{\odot}+\frac{2}{3}\mathbf{e}_{N}),\\ \frac{\partial a}{\partial\rho}=-\frac{A}{M}\frac{S_{0}}{c}2\cos^{2}\theta \cdot\mathbf{e}_{N}.\end{array}  \tag{7}\]  
For the nominal YS attitude, the SPs are perpendicular to the Sun direction with \(\cos\theta=1\) and \(\mathbf{e}_{N}=\mathbf{e}_{\odot}\), the partial derivatives of the acceleration w.r.t scale parameter of SPs and SB are expressed as  
\[\frac{\partial a}{\partial(1+\rho+\frac{2}{3}\delta)}=-\frac{A}{M}\frac{S_{0} }{c}\mathbf{e}_{\odot}, \tag{8}\]  
\[\frac{\partial a}{\partial SB}=-\frac{A}{M}\frac{S_{0}}{c}2\bigg{(}\frac{ \delta}{3}+\rho\bigg{)}sign(\dot{\epsilon})\mathbf{e}_{B}. \tag{9}\]  
The correlation between parameters of ABW model is strong, so reasonable results are achieved only with a priori constraints being put on most of those parameters. To avoid the disadvantages of the ABW model, the box-wing model was reformulated [6]. The deduced model employs a specific parameterization that isolates distinct contributions of the satellite bus, in which the SRP accelerations depends only on the \(\epsilon\) angle. Without considering the SRP acting on SPs, the model can be expressed in \(D\) and \(B\) directions in the DYB frame as  
\[