## 2 Analysis Method for Direct Solar Heat Flux of Pointing Attitude
The projection is in the first quadrant, namely \(\eta<90^{\circ}\) and \(\alpha<90^{\circ}\): According to the spatial geometric relation, the yaw angle \(\varphi\) can be described as follows: \[\tan\varphi=\frac{\cos\eta}{\cos\alpha}\Rightarrow\phi=\arctan\frac{\cos\eta}{ \cos\alpha}\] (1) (2) The projection is in the second quadrant, namely \(\eta<90^{\circ}\) and \(\alpha>90^{\circ}\): \[\tan(\pi-\phi)=\frac{\cos\eta}{\cos(\pi-\alpha)}=-\frac{\cos\eta}{\cos\alpha}\] (2) Namely, \[\phi=\pi+\arctan\frac{\cos\eta}{\cos\alpha}\] (3) (3) The projection is in the third quadrant, namely \(\eta>90^{\circ}\) and \(\alpha>90^{\circ}\): \[\tan(\phi-\pi)=\frac{\cos(\pi-\eta)}{\cos(\pi-\alpha)}=\frac{\cos\eta}{\cos\alpha}\] (4) Namely, \[\phi=\pi+\arctan\frac{\cos\eta}{\cos\alpha}\] (5) (4) The projection is in the fourth quadrant, namely \(\eta>90^{\circ}\) and \(\alpha<90^{\circ}\): \[\tan(2\pi-\phi)=-\frac{\cos(\pi-\eta)}{\cos\alpha}=-\frac{\cos\eta}{\cos\alpha }\Rightarrow\tan(\phi-2\pi)=\frac{\cos\eta}{\cos\alpha}\] (6) Figure 2.0 (7) Thus, \[\phi=\left\{ \begin{array}{ll}\arctan\frac{\cos\eta}{\cos\alpha},&\eta<90^{ \circ},\alpha<90^{\circ}\\ \pi+\arctan\frac{\cos\eta}{\cos\alpha},&\alpha>90^{\circ}\\ 2\pi+\arctan\frac{\cos\eta}{\cos\alpha},&\eta>90^{\circ},\ \alpha<90^{\circ} \end{array} \right.\] (8)  
Through the above analysis, it can be seen that while \(\eta>90^{\circ}\), the projection of the pointing vector is in the third or fourth quadrant, and the yaw angle \(\varphi\) is greater than \(180^{\circ}\); conversely, \(\varphi\) is smaller than \(180^{\circ}\). Additionally, there are some special cases:  
1. While \(\alpha=90^{\circ}\), the pointing vector is in the YOZ plane. According to the geometric relation, the yaw angle \(\varphi=90^{\circ}\) (while \(\eta<90^{\circ}\)) or \(270^{\circ}\) (while \(\eta>90^{\circ}\)). 2. While \(\eta=90^{\circ}\), the pointing vector is in the XOZ plane. According to the geometric relation, the yaw angle \(\varphi=0^{\circ}\) (while \(\alpha\leq 90^{\circ}\)) or \(180^{\circ}\) (while \(\alpha>90^{\circ}\)). In this case, the pointing vector is in the orbital plane. According to the position of the satellite relative to the object, the pointing attitude can be divided into forward pointing attitude and backward pointing attitude. For the forward pointing attitude, the object is in front of the satellite, namely in the +X direction of the satellite. In this case, there is no need for yaw to realize the pointing attitude, which matches the situation of \(\alpha\leq 90^{\circ}\). As for the backward pointing attitude, the object is at the back, which means \(\alpha>90^{\circ}\). To point to the object, the satellite should turn from the flight direction to the opposite flight direction, hence the yaw angle is \(180^{\circ}\). Once the yaw is completed, the direction of coordinate system is certain. Then, the coordinate system should be rotated around the Y axis, and the pitch angle is determined as follows:  
\[\theta=\gamma \tag{9}\]By two rotations, the +Z axis is transferred into the pointing vector, and the pointing attitude coordinate system is obtained. With the established coordinate system, the reaching solar heat flux can be computed by analytic methods or commercial software. The formula for calculating direct solar heat flux is [18]:  
\[q_{\text{SUN}}=S\bullet\text{max}\left(M^{-1}\left[ \begin{array}{c}\cos\Phi\\ \sin\Phi\cos I\\ \sin\Phi\sin I\end{array} \right]\bullet\left[ \begin{array}{c}x\\ y\\ z\end{array} \right],0\right) \tag{10}\]  
In the equation, \(S\) is the solar constant, \(I\) is the obliquity of the ecliptic, \(\Phi\) is the solar ecliptic longitude, [_x,y,z_]T is the unit normal vector of a given surface point in the orbital coordinate system. For a three-axis stabilized Earth-pointing attitude, the transformation matrix from the spacecraft body frame to the orbital frame is a third-order identity matrix. The pointing attitude frame proposed in this study is derived from the Earth-pointing frame by applying pitch and yaw rotations.