## 2 Analysis Method for Direct Solar Heat Flux of Pointing Attitude  
Before introducing the theoretical methodology, the assumptions underlying this study and the constructed model are first elaborated. Most satellite bodies are typically a hexahedron. In thermal analysis, it is necessary to calculate the external heat flux incident on each surface. Therefore, in the external heat flux analysis of this paper, we create a hexahedron as a simplified model of the satellite. The model has the following characteristics:  
1. The satellite body is assumed to be a hexahedron;
2. All edge lengths of the satellite are set to 1 m;
3. All surfaces of the satellite are flat;
4. To intuitively evaluate the external heat flux incident on each surface, various protrusions on the satellite surface are neglected, and the satellite surface is assumed to be unobstructed with each face being a convex surface. The analytical method proposed in this paper will be specifically introduced below. The key to obtain the reaching direct solar heat flux on a certain surface is solving the angle between the solar vector and normal vector. Then, the solar heat flux reaching on different surfaces can be calculated by analytic method according to the geometry relationship. To determine the relationship, a coordinate system based on pointing attitude was first established. Considering the communication load is usually located on the +Z plane of the satellite, the pointing orientation was set as the positive direction of the Z coordinate axis. The three-axis stability nadir attitude is the basic attitude for satellites, and on the basis of the nadir attitude coordinate system, the pointing attitude coordinate system was constructed. First, the nadir attitude coordinate system was rotated around the Z axis. Then, the coordinate system was rotated around the Y axis. By these two rotations, the orientation of the +Z axis was changed from nadir to pointing direction, and the X and Y axis was also determined along with the Z axis. The pointing attitude coordinate system is shown in Figure 1. In this figure, A and B denote Satellite A and Satellite B, respectively, with the attitude vector directed from A to B; X, Y, and Z represent the axes of the nadir attitude coordinate system, respectively; and X', Y', and Z' represent the axes of the pointing attitude coordinate system. \(\Phi\) is the yaw angle and \(\theta\) is the pitch angle; the dashed arrowed line denotes the projection of the pointing vector onto the XOY plane. It can be seen that the key to solving the reaching solar heat flux is to obtain the yaw angle \(\varphi\) and the pitch angle \(\theta\). Once these two angles are certain, the coordinate system is fixed, and the relationship between solar vector and satellite can also be determined. Therefore, the core of this analysis method is coordinate system transformation, namely calculating the rotation angles. Firstly, the nadir attitude coordinate system is rotated around the Z axis, to make the +X axis reach the projection of pointing vector on the XOY plane. Assuming the angle between pointing vector and +X axis of nadir attitude coordinate system is \(\alpha\), the angle between the pointing vector and +Y axis of the nadir attitude coordinate system is \(\eta\), and the angle between the pointing vector and +Z axis is \(\gamma\), the yaw angle \(\varphi\) can be described as shown in Figure 2. In this Figure, a denotes the unit vector of the pointing vector. Considering that the projection of the pointing vector on the XOY plane can be in different quadrants, the calculation situation is divided into four parts:  
Figure 1: Schematic diagram of pointing attitude coordinate system. Figure 2: Schematic diagram of yaw angle.