# The Impact of Eclipsing GNSS Satellites on the Precise Point Positioning
## Materials and Methods  
### The Attitude Model for GNSS Satellites  
The nominal yaw angle \(\varphi_{n}\) can be computed by the sun angle \(\beta\) (the elevation angle of the sun above the orbital plane [23], \(-90^{\circ}\)-\(90^{\circ}\)) and the orbit angle \(\mu\) (the geocentric angle between the sun and orbit midnight, growing with the satellite motion, 0-360\({}^{\circ}\)):  
\[\varphi_{n}=\text{atan2}(-\tan\beta,\sin\mu), \tag{1}\]  
where the function \(\text{atan2}(y,x)\) in C/C++ returns the principal value of the arc tangent of y/x, expressed in radians (in the interval \([-\pi,\pi]\)). According to the current IGS convention, the unit vector of sun and the \(X\)-axis direction of the yaw-steering frame are always pointing towards the same hemisphere, including GPS IIR and GALILEO satellites [3]. Therefore, Equation (1) is used for all GNSS satellites to calculate the nominal yaw angles and the sign of \(\phi_{n}\) is opposite to that of \(\beta\). Ignoring the small time variation of the sun angle \(\beta\), the yaw rate \(\dot{\varphi}_{n}\) can be derived as follows:  
\[\dot{\varphi}_{n}=\frac{\dot{\mu}\tan\beta\cos\mu}{\sin^{2}\mu+\tan^{2}\beta}, \tag{2}\]  
where \(\dot{\mu}\) denotes the orbital angular velocity. According to the orbital period, the average orbital angular velocities for GNSS are summarized in Table 1. Substituting \(\mu=0\) and the maximum hardware yaw rate \(R\) into Equation (2), the approximate maximum sun angle limit \(\beta_{0}\) can be expressed as follows:  
\[\beta_{0}=\text{atan}\big{(}\dot{\mu}/R\big{)}, \tag{3}\]  
where the function \(\text{atan}(x)\) in C/C++ returns the arc tangent of x, expressed in radians (in the interval \([-\pi/2,\pi/2]\)). For GPS II/IIA satellites, the averaged JPL reprocessing yaw-rate solutions during 1996-2008 [24] are provided, with the average value of \(0.1176^{\circ}/\)s (\(0.0815^{\circ}/\)s-\(0.1401^{\circ}/\)s). The average hardware yaw rate for GPS IIR, GPS IIF and GLONASS-M satellites are \(0.20^{\circ}/\)s, \(0.11^{\circ}/\)s and \(0.25^{\circ}/\)s, respectively. Different from GPS and GLONASS satellites, the GALILEO (IOV and FOC) and BDS (IGSO and MEO) satellites adopt the fixed threshold values. Then, the sun angle limits of noon maneuver are showed in the second column of Table 2. Only when the satellite moves to near the noon and \(|\beta|\leq\beta_{0}\), it will experience noon maneuver or switch to orbit normal mode (BDS MEO and IGSO satellites). Different from the noon maneuver, as soon as the satellite enters into the shadow regime, the solar sensors of some satellites are unable to perceive the position of the sun. According to the mean radius of earth and the altitude of the satellite orbit, the average sun angle limits for shadow crossing regime are shown in the third column of Table 2. It is noteworthy that only the midnight maneuver is considered during the shadow crossing regime for GPS IIR and GALILEO (IOV and FOC) satellites. At this time, a smaller value of \(\beta_{0}\) causes a shorter period of shadow crossing. Except for GPS II/IIA satellites, there is no post-shadow recovery for other satellite types, including GPS IIF satellites. It is recommended to delete the 30 min of data for GPS II/IIA satellites after the exit of shadow crossing. Different from GPS/GLONASS/GALILEO satellites, BDS satellites adopt the yaw steering mode and orbit normal mode. It is important to know the attitude switching time because of the big differences of yaw angle between these two attitude modes (the yaw angle is always 0 for orbit normal mode). Considering that the yaw angle cannot complete an abrupt change in a short time, the preliminary attitude switch model, established by [PERSON] et al. (2012), is adopted in this contribution:  
* The condition for YS- > ON: (1) \(|\beta|\leq 4^{\circ}\) and \(|\varphi_{n}|\leq 5^{\circ}\); or (2) \(|\beta|\leq 4^{\circ}\) and \(5^{\circ}<|\varphi_{n}|<20^{\circ}\) when \(\varphi_{n}\dot{\varphi}_{n}>0\).