# Solar Radiation Pressure Models for BeiDou-3 I2-S Satellite: Comparison and Augmentation
## 2 Characteristics and SRP Models for BeiDou-3 I2-S Satellite
### ECOM Model  
As an empirical SRP model, the ECOM model was originally proposed in the 1990s [4], and further develops the reduced and extended version [5; 10]. The ECOM model decomposes the SRP acceleration into three orthogonal directions: the satellite-to-Sun direction (\(\mathbf{e}_{D}\)), the nominal SP axis in YS mode (\(\mathbf{e}_{Y}\)), and the orthogonal axis (\(\mathbf{e}_{B}\))  
\[\begin{array}{l}\mathbf{e}_{D}=\mathbf{e}_{\odot}\\ \mathbf{e}_{Y}=\frac{\mathbf{e}_{\odot}\times\mathbf{r}}{\|\mathbf{e}_{\odot }\times\mathbf{r}\|}\\ \mathbf{e}_{B}=\mathbf{e}_{D}\times\mathbf{e}_{Y}.\end{array} \tag{2}\]  
\[\begin{array}{l}a_{D}=D_{0}+D_{c}\cos\mu+D_{s}\sin\mu\\ a_{Y}=Y_{0}+Y_{c}\cos\mu+Y_{s}\sin\mu\\ a_{B}=B_{0}+B_{c}\cos\mu+B_{s}\sin\mu,\end{array} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Panel** & **Area [m\({}^{2}\)]** & \(\boldsymbol{\alpha}\) & \(\rho\) & \(\delta\) \\ \hline +X & 7.20 & 0.350 & 0.650 & 0.0 \\ \(-\)X & 7.20 & 0.350 & 0.650 & 0.0 \\ +Y & 9.00 & 0.114 & 0.856 & 0.0 \\ \(-\)Y & 9.00 & 0.114 & 0.856 & 0.0 \\ +Z & 5.00 & 0.350 & 0.650 & 0.0 \\ \(-\)Z & 5.00 & 0.350 & 0.650 & 0.0 \\ SPs & 40.56 & 0.720 & 0.280 & 0.0 \\ \hline \end{tabular}
\end{table}
Table 1: The coarse values of optical (absorption \(\alpha\), reflectivity \(\rho\), and diffusion \(\delta\)) and geometrical properties of satellite bus and SPs for BeiDou-3 I2-S satellite.  
Within ECOM model, the empirical SRP accelerations in the three directions at 1 astronomical unit (AU) are expressed as Equation (3) in a right-handed DYB frame for YS attitude in term of three constant accelerations (\(D_{0}\), \(Y_{0}\), and \(B_{0}\)) and six one-cycle-per-revolution (1-cpr) accelerations (\(Dc\), \(Ds\), \(Yc\), \(Ys\), \(Bc\), and \(Bs\)), where \(\mu\) is the orbit angle; i.e., the argument of satellite with respect to the midnight point, i.e., the position where the satellite is far away from the Sun. Although this model can incorporate with an a priori model, good performance can also be obtained without such a priori model. Hence, the empirical reduced 5-parameter ECOM (called ECOM1 hereafter) is widely used to generate GNSS orbits. In this case, only three constant accelerations and two 1-cpr accelerations in \(B\) direction are estimated.  
In recent years, however, it became obvious that the ECOM1 could introduce large draconitic error in GNSS geodetic products [13; 14]. In order to reduce the error in the time series of IGS (International GNSS Service) products, ECOM2 model has been proposed by additional introduction of high-order harmonics in Equation (3) [10], and expressed as  
\[\begin{split} a_{D}&=D_{0}+\sum\limits_{i=1}^{n_{D}} \{D_{2i,c}\cos(2i\Delta u)+D_{2i,s}\sin(2i\Delta u)\}\\ a_{Y}&=Y_{0}\\ a_{B}&=B_{0}+\sum\limits_{i=1}^{n_{B}}\{B_{2i-1,c} \cos((2i-1)\Delta u)+B_{2i-1,s}\sin((2i-1)\Delta u)\}\end{split} \tag{4}\]  
where the upper limit values, \(n_{D}\), \(n_{Y}\), and \(n_{B}\) are defined by users, and \(D_{2i,c}\), \(D_{2i,s}\), \(B_{2i-1,c}\), and \(B_{2i-1,s}\) are the SRP coefficients to be estimated, and \(\Delta u\) is the difference of the satellite's argument of the latitude and the Sun's argument of the latitude. As recommendation by [10], the 2-cpr and 4-cpr terms in \(D\) and the 1-cpr terms in \(B\) direction is better to account for.