## 3 Analysis of Reaching Solar Heat Flux of Lunar Relay Satellite
### Analysis of External Heat Flux for Relay Satellites in Elliptic Orbit  
Parameters of the elliptic orbit can be obtained from [25]. The elliptic orbit is shown in Figure 5. In this diagram, the dash line denotes the schematic representation of the direction from the satellite to the lunar south pole. Besides the attitude, the included angle between the solar vector and orbital plane, namely beta angle \(\beta\), also has a great influence on the reaching solar heat flux [10]. For the inclined orbit mentioned above, \(\beta\) varies from \(-56^{\circ}\) to +56\({}^{\circ}\). To analyze extreme conditions, we considered \(\beta\) as \(0^{\circ}\), \(-56^{\circ}\), and \(+56^{\circ}\). In this chapter, the satellite model is the same as the model established in Section 3.1. Figure 4: Results of the solar heat flux on \(\pm\)Z plane. Figure 5: Schematic diagram of elliptic frozen orbit. (1) \(\beta=0^{\circ}\)  
Calculation results of the solar heat flux on different surfaces when \(\beta=0^{\circ}\) are shown in Figure 6. In this case, the transient solar heat flux of two orbital periods was calculated to show the variety. For the proposed pointing attitude, the +Y axis is perpendicular to the solar vector. Hence, the solar heat flux on the \(\pm\)Y plane is 0. According to the geometric relationship, the included angle between the +X axis and solar vector is smaller than \(90^{\circ}\), which means that the +X plane cannot be illuminated. The variety of the solar heat flux on the \(\pm\)Z plane and \(-\)X plane can be seen in Figure 6. When \(\beta=0^{\circ}\), there will be a moment when the \(-\)X plane is perpendicular to the solar vector, and the \(\pm\)Z plane is parallel to the solar vector at the same time. Therefore, the maximum solar heat flux on the \(-\)X plane is equal to the solar constant. The \(-\)Z plane and +Z plane cannot be illuminated simultaneously, but the variety of the solar heat flux is symmetrical about the sub-solar point, and the maximum is close to the solar constant. (2) \(\beta=-56^{\circ}\)  
Calculation results when \(\beta=-56^{\circ}\) are shown in Figure 7. Because the attitude is the same as that when \(\beta=0^{\circ}\), the \(\pm\)Y plane and +X plane are not illuminated, either. Figure 6: Results of solar heat flux on different surfaces when \(\beta=0^{\circ}\). Figure 7: Results of solar heat flux on different surfaces when \(\beta=-56^{\circ}\). (3) \(\beta=56^{\circ}\)  
From Figures 6-8, it can be seen that the reaching solar heat flux on different surfaces is significantly influenced by the angle of \(\beta\). When \(\beta=0^{\circ}\), the maximum solar heat flux on \(\pm\)Z plane is greater, but the mean value of a period on the \(+\)Z plane is greater when \(\beta=56^{\circ}\) and on the \(-\)Z plane it is greater when \(\beta=-56^{\circ}\). Solar heat flux varies more violently when \(\beta=0^{\circ}\), and is greater when \(\beta=\pm 56^{\circ}\). Additionally, the variety of solar heat flux on the \(-\)X plane is the same when \(\beta=-56^{\circ}\) and \(\beta=56^{\circ}\). A comparison of external heat flux results between Halo and highly elliptical orbits reveals that, within one orbital period, both orbit types exhibit identical variation trends in heat flux on both \(+\)Z and \(-\)Z surfaces. Under pointing attitude conditions, the \(\pm\)Y surfaces remain unilluminated with zero absorbed heat flux. These analytical results provide valuable references for spacecraft radiator surface selection. Surfaces with higher absorbed heat flux should be thermally insulated to prevent heat ingress into the spacecraft interior, while surfaces with lower absorbed heat flux are more suitable as radiators. Consequently, the \(\pm\)Y surfaces may be considered as primary radiator surfaces, with \(\pm\)Z surfaces serving as auxiliary ones. Notably, orbital characteristics affect surface illumination differently: in Halo orbits the \(+\)X surface experiences periodic illumination, whereas in highly elliptical orbits it remains unilluminated.