## 4 Signal Attenuation Model for Inter-Satellite Terahertz Communication
### Signal Attenuation Model of Inter-Satellite Terahertz Communication under Pointing Error  
The satellite-borne antenna is fixedly connected with the satellite constellation coordinate system, and the attitude change of the satellite will inevitably cause the antenna pointing to change. In the inter-satellite terahertz link communication process, the transmitter controls the terahertz beam to point to the receiver. Due to the influence of orbital perturbation force and other factors, both the transmitter and the receiver have pointing errors. Although the influence of the perturbation force is small, the terahertz wave will have a certain offset when it reaches the receiving end through long-distance transmission, resulting in the attenuation of the received signal power and even the communication failure. The pointing error model of satellite communication system is based on several related coordinate systems such as inertial coordinate system OXYZ, satellite body coordinate system O\({}_{1}\)X\({}_{1}\)Y\({}_{1}\)Z\({}_{1}\) and antenna coordinate system O\({}_{2}\)X\({}_{2}\)Y\({}_{2}\)Z\({}_{2}\) and their mutual transformations. According to the six Keplerian numbers solved by TLE, the position of the satellite in the inertial coordinate system can be obtained by Equation (10). \[ \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} =\frac{a\big{(}1-e^{2}\big{)}}{1+e\cos f} \begin{bmatrix}\cos \Omega\cos u-\sin\Omega\sin u\cos f\\ \sin\Omega\cos u+\cos\Omega\sin u\cos i\\ \sin u\cos i\end{bmatrix}  \tag{10}\]  
Since this paper mainly considers the pointing error caused by satellite perturbation, in order to simplify the calculation, this paper assumes that there is no assembly error, and the antenna coordinate system coincides with the satellite body coordinate system. Define the angle between the inertial coordinate system and the satellite body coordinate system as \(\alpha\), \(\beta\), \(\gamma\), then the transformation matrix can be represented by Equation (11). \[A= \begin{bmatrix}\cos\alpha\cos\beta&\sin\alpha\cos\beta&-\sin\beta\\ -\sin\alpha\cos\gamma+\cos\alpha\sin\beta\sin\gamma&\cos\alpha\cos\gamma+\sin \alpha\sin\beta\sin\gamma&\sin\alpha\cos\beta\\ \sin\alpha\sin\gamma+\cos\alpha\sin\beta\cos\gamma&\sin\alpha\sin\beta\cos\gamma- \cos\alpha\sin\gamma&\cos\gamma\cos\beta\end{bmatrix}  \tag{11}\]In the antenna coordinate system, the azimuth angle \(\theta_{xy}\) and the elevation angle \(\theta_{xz}\) of the transmitter are shown in Figure 6a, and the rotation direction along the arrow is the positive direction. Assuming that the pointing vector of the transmitter in the antenna coordinate system is \(\overset{-}{r}=(x,y,z)\), the azimuth angle \(\theta_{xy}\) and the elevation angle \(\theta_{xz}\) of the transmitter can be described as Equation (12). \[\theta_{xy}=\arctan\frac{y}{x},\,\theta_{xz}=\arctan\frac{z}{x} \tag{12}\]  
Figure 6b shows the unit vector \(r_{0}\) of the terahertz wave emission direction and the unit vector \(r_{\mathrm{S}}\) of the terahertz wave emission direction under the influence of pointing error. The angle \(\theta\) between the pointing vector \(r_{0}\) without pointing error and the pointing vector \(r_{\mathrm{S}}\) with pointing error is the pointing error angle. Assuming that the perturbed position of the satellite receiving end is \(r_{1}(x_{1},y_{1},z_{1})\) and the offset displacement is \(\Delta r_{1}(\Delta x_{1},\Delta y_{1},\Delta z_{1})\), the azimuth pointing error angle \(\Delta\theta_{xy}\), and the pitch pointing error angle \(\Delta\theta_{xz}\) can be expressed as Equation (13). \[\Delta\theta_{xy}=\left[\frac{\cos^{2}\theta_{xy}-\sin^{2}\theta_{xy}}{y_{0} }\right]\left[ \begin{array}{c}\Delta y_{0}\\ \Delta x_{0}\end{array} \right]\]  
\[\Delta\theta_{xz}=\left[ \begin{array}{c}-\tan\theta_{xy}\cdot\zeta_{0}- \tan\theta_{xz}\cdot\zeta_{0}\\ \frac{\cos\theta_{yz}}{\zeta_{0}^{2}+\zeta_{0}^{2}}+\zeta_{0}^{2}\end{array}  \right.\frac{\cos\theta_{yz}}{\zeta_{0}^{2}+\zeta_{0}^{2}}\left[ \begin{array}[] {c}\Delta x_{0}\\ \Delta y_{0}\\ \Delta z_{0}\end{array}