# Joint Task Offloading and Power Allocation for Satellite Edge Computing Networks
## 3 System Model and Problem Formulation
### Doppler Frequency Shift Model  
We assume that all orbital elements (Semi-major Axis, Eccentricity, Inclination, Longitude of the ascending node, Argument of periapsis, and Mean Anomaly) of the LEO satellite are known to the users, which are represented as \([a,e,i,\omega,\Omega,\varphi]\). The orbital coordinate system can be transformed into a central celestial inertial system after cubic direction cosine matrix transformation, and the position vector of the LEO satellite can be formulated as  
\[\mathbf{r} =R_{3}(-\Omega)R_{1}(-i)R_{3}(-\omega)\begin{bmatrix}r\cos\varphi\\ r\sin\varphi\\ 0\end{bmatrix}\] \[=\frac{p}{1+e\cos\varphi}\begin{bmatrix}\cos\Omega\cos(\omega+ \varphi)-\sin\Omega\sin(\omega+\varphi)\cos i\\ \sin\Omega\cos(\omega+\varphi)+\cos\Omega\sin(\omega+\varphi)\cos i\\ \sin(\omega+\varphi)\sin i\end{bmatrix}, \tag{1}\]  
where \(R\) denotes the rotation matrix, and \(p\) and \(r\) represent the semi-latus rectum of the orbit, which satisfies \(p=a(1-e^{2})\) and the size of the position vector \(r=\frac{p}{1+e\cos\varphi}\), respectively.  
The velocity vector of the LEO satellite can be obtained by differentiating the position vector against variable \(\varphi\). The expression of the velocity vector is shown as  
\[\mathbf{v} =\sqrt{\frac{\mu}{p}}\begin{bmatrix}-\cos\Omega(\sin(\omega+\varphi )+e\sin\omega)-\sin\Omega(\cos(\omega+\varphi)+e\cos\omega)\cos i\\ -\sin\Omega(\sin(\omega+\varphi)+e\sin\omega)+\cos\Omega(\cos(\omega+\varphi)+ e\cos\omega)\cos i\\ (\cos(\omega+\varphi)+e\cos\omega)\sin i\end{bmatrix}. \tag{2}\]  
The motion of the LEO satellite generates the Doppler frequency shift. When the MS offloads to SSs, they initiate requests to each other and exchange their location information and velocity vectors. Subsequently, the SSs calculate the Doppler shift based on their relative velocity and position, and apply the necessary compensation to mitigate the challenges of demodulation at the satellite.  
The frequency received in the SSs can be calculated by  
\[f_{SS}=\frac{c+\mathbf{v_{SS}}\cdot\mathbf{r}}{c+\mathbf{v_{MS}}\cdot\mathbf{r}}f_{\text{MS}}, \tag{3}\]  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline c & The light speed \\ \(r_{0}\) & The Chip’s coefficient \\ \(s\) & Cycles needed to process 1 bit of data \\ \(\eta\) & The upper limit of the violation function \\ \(d_{k,i}\) & The distance between the \(k\)-th slave satellite \\  & and the master satellite \\ \(\alpha_{k,i}\) & The ratios of the offloading data to slave satellite \\  & \(k\) at slot \(i\) \\ \(t_{k,i}\) & The transmission delay to slave satellite \\  & \(k\) at slot \(i\) \\ \(t_{0,i}\) & The local computing time \\ \(\mu_{0}\) & The channel power gain at d = 1 m \\ \(\lambda,\rho\) & The Lagrange multiplier \\ \(P_{k,i}\) & The transmission power to slave satellite \\  & \(k\) at slot \(i\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._The Doppler frequency shift of the SSs can be denoted by  
\[f_{d} =f_{SS}-f_{MS} \tag{4}\] \[=\frac{\mathbf{v_{SS}}\cdot\mathbf{r}-\mathbf{v_{MS}}\cdot\mathbf{r}}{c+\mathbf{v_{MS}} \cdot\mathbf{r}}f_{MS}, \tag{5}\]  
where \(\mathbf{r}\) denote the unit vector in the direction of signal propagation (from the MS to the SSs), \(\mathbf{v_{SS}}\) and \(\mathbf{v_{MS}}\) denote the velocity vector of the SSs and MS, respectively, and \(f_{MS}\) represents the carrier frequency of the MS.