## 2 Stationary Orbits
The Jovian main ring is about 6440 km wide and probably less than 30 km thick. We denote this width by \(w\). Moreover, the distance, \(d\), of the ring from the center of mass of Jupiter is about 122,500 km, and the mass, \(m_{r}\), is about 1.0\(\times 10^{13}\) kg [32]. For satellites on stationary orbits with position \(\mathbf{r}_{s}\), the gravitational acceleration due to the Jovian main ring can be calculated as  
\[\mathbf{F}_{r}=G\int\int\int_{ring}\frac{(\mathbf{r}(\mathbf{P})-\mathbf{r}_{s })\rho(\mathbf{P})dV}{|\mathbf{r}(\mathbf{P})-\mathbf{r}_{s}|^{3}}, \tag{11}\]  
where \(\mathbf{r}(\mathbf{P}),\rho(\mathbf{P})\) denote the position vector and the density, respectively, which depend on the position of a point, \(\mathbf{P}\), belonging to the ring. We can see that  
\[|\mathbf{r}(\mathbf{P})-\mathbf{r}_{s}|\geq r_{0}-d-w\approx 160,240-122,500-64 40\approx 0.4378R,\,|\mathbf{F}_{r}|\leq\frac{Gm_{r}}{(0.4378R)^{2}}.\]  
Note that \(\frac{Gm_{r}}{(0.4378R)^{2}}/\left(\frac{GM_{J}}{r_{0}^{2}}\right)=\frac{m_{r} }{M_{J}}\cdot\left(\frac{r_{0}}{(0.4378R)}\right)^{2}\approx\frac{1.0\times 10^{ 3}}{1.9\times 10^{2}}\cdot\left(\frac{2.2414}{0.4378}\right)^{2}\approx 1.3 795\times 10^{-13}\), which is far smaller than \(J_{2}\) and \(J_{4}\) terms. Therefore, the effects of the Jovian main ring on stationary orbits can be neglected. Another perturbation that may affect stationary orbits is the magnetic field of Jupiter when the satellite is charged. Here, we briefly analyze the effects under some assumptions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Third Body** & \(m/M_{J}(\times 10^{-5})\) & \(\rho/r_{0}\) & \((a_{d}/a_{m})_{\max}(\times 10^{-6})\) \\ \hline Sun ([32]) & \(1.0473\times 10^{8}\) & 4856.8 & 0.01823 \\ Io ([30,32]) & 4.7047 & 2.6311 & 5.20000 \\ Europa ([30,32]) & 2.5283 & 4.1868 & 0.68904 \\ Ganymede ([30,32]) & 7.8066 & 6.6775 & 0.52379 \\ Callisto ([30,32]) & 5.6673 & 11.7511 & 0.06989 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The disturbing acceleration due to the third body. The magnetic field of Jupiter near stationary orbits when exterior terms are neglected can be written as  
\[\mathbf{B}(r,\phi,\lambda)=-\
abla V(r,\phi,\lambda), \tag{12}\]  
where  
\[V=R\sum_{l=1}^{\infty}\sum_{m=1}^{l}\left(\frac{R}{r}\right)^{l+1}[g_{l,m}\cos( m\lambda)+h_{l,m}\sin(m\lambda)]P_{l}^{m}(\cos\phi). \tag{13}\]  
Here, \(g_{l,m}\) and \(h_{l,m}\) are the geomagnetic Gauss coefficients (their values can be found in [33; 34]), and \(P_{l}^{m}(\cos\phi)\) is the normalized Legendre function. The acceleration due to the Lorentz force is  
\[F_{L}=\frac{q}{m}\left(\dot{\mathbf{r}}-\boldsymbol{\omega}\times\mathbf{r} \right)\times\mathbf{B}(r,\phi,\lambda), \tag{14}\]  
where \(q=4\pi\varepsilon_{0}\Phi s\varepsilon_{0}=8.854187817\times 10^{-12}\) F \(\cdot\) m\({}^{-1}\) is the vacuum permittivity, \(\Phi\) is the surface potential of satellites, \(s\) is the radius of the satellite, \(\boldsymbol{\omega}\) is the angular velocity vector of Jupiter. For a charged satellite with \(\Phi=500\) V (an in-depth study of the effect of spacecraft charging at Jupiter can be found in [35]), \(s=10\) m, and \(m=100\) kg, we have \(q=5.5633\times 10^{-7}\)C and \(\left|F_{L}\right|\leq 1.14\times 10^{-8}\)N. The Lorentz force compared with the lowest order of gravity provides the ratio  
\[\frac{\left|\mathrm{F}_{L}\right|}{\left|G_{0}\right|}=\frac{\left(n-n_{l} \right)Bqr^{3}}{GMm}\approx 2.3\times 10^{-11}\]  
This ratio is also far smaller than \(J_{2}\) and \(J_{4}\). Therefore, it can be concluded that the effects of Lorentz forces on a satellite can be neglected.