## 2 Stationary Orbits  
Satellites on stationary orbits are well-known for their stationary ground track. Therefore, stationary orbits are preferred for designing communications and navigation satellites. There exist numerous studies on stationary orbits of the Earth and Mars. However, the gravity field of Jupiter is significantly different. In this subsection, we will calculate the stationary orbit of Jupiter and investigate their stability in a spherical coordinate system. In the spherical coordinates of an inertial frame, \(O-r,\lambda,\phi\), where \(O\) is the center of mass of Jupiter, \(\lambda\) is the jovicentric longitude, \(r\) and \(\phi\) are the same as those in Equation (11), and the kinetic energy of the spacecraft can be written as  
\[T=\frac{1}{2}\bigg{(}\dot{r}^{2}+r^{2}\cos^{2}\phi\dot{\lambda}^{2}+r^{2}\dot {\phi}^{2}\bigg{)}, \tag{2}\]  
where \(\dot{r},\ \dot{\lambda},\ \dot{\phi}\) are the derivatives with respect to time. From the expressions of \(T\) and \(U\), one can see that \(\lambda\) is a cyclic variable. Let us introduce \(q=[r,\lambda,\phi]^{T}\), the Lagrangian can be written as \(L=T-U\). By Lagrange equations, we have  
\[\frac{d}{dt}\bigg{(}\frac{\partial T}{\partial\dot{q}}\bigg{)}-\frac{\partial T }{\partial q}=\frac{\partial U}{\partial q}. \tag{3}\]  
More precisely, the equations of motion can be presented as follows:  
\[\left\{ \begin{array}{l}\ddot{r}-r\cos^{2}\phi\dot{\lambda}^{2}-r\dot{\phi} ^{2}=-\frac{\mu}{r^{2}}+\frac{3\mu J_{2}R^{2}}{2r^{4}}\big{(}3\sin^{2}\phi-1 \big{)}+\frac{5\mu J_{2}R^{4}}{8r^{6}}\Big{(}35\sin^{4}\phi-30\sin^{2}\phi+3 \Big{)},\\ \frac{d}{dt}\Big{(}r^{2}\cos^{2}\phi\dot{\lambda}\Big{)}=0,\\ \frac{d}{dt}\Big{(}r^{2}\dot{\phi}\Big{)}+\frac{1}{2}r^{2}\sin(2\phi)\dot{ \lambda}^{2}=-\Big{[}\frac{3\mu J_{2}R^{2}}{2r^{3}}+\frac{\mu J_{4}R^{4}}{8r^{ 5}}\big{(}70\sin^{2}\phi-30\big{)}\Big{]}\sin 2\phi.\end{array} \right. \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Planet** & \(J_{2}(\times 10^{-3})\) & \(J_{3}(\times 10^{-6})\) & \(J_{4}(\times 10^{-5})\) & \(J_{22}(\times 10^{-6})\) & \(J_{3}/J_{2}(\times 10^{-3})\) & \(J_{4}/J_{2}(\times 10^{-2})\) \\ \hline Earth ([27]) & \(1.08263\) & \(-2.53266\) & \(-1.61962\) & \(1.81534\) & \(-2.3394\) & \(-1.4960\) \\ Mars ([28]) & \(1.95545\) & \(31.4498\) & \(-1.53774\) & \(63.0692\) & \(16.083\) & \(-0.7864\) \\ Jupiter ([26]) & \(14.6965\) & \(0.042\) & \(-58.661\) & \(0\) & \(0.0021378\) & \(-3.9923\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some harmonic coefficients and their ratios of the Earth, Mars, and Jupiter. From the second equation of (4), we see that the quantity \(r^{2}\cos^{2}\phi\dot{\lambda}\) is invariant, which can also be obtained by conservation of the angular momentum along the z-axis. Moreover, the zonal terms of the gravity field only lead to radial and North-South perturbations. The presence of these terms increases the radius of the stationary orbit with respect to the case of a spherical planet with the same mass of Jupiter. When the orbital plane coincides with the equatorial plane, namely \(\phi=0,\dot{\phi}=0\), the vertical perturbation vanishes. In order to find stationary orbits for Jupiter, let \(\dot{r}=\ddot{r}=0\), \(\dot{\lambda}=n_{J}\), \(\ddot{\lambda}=0\), \(\dot{\phi}=\ddot{\phi}=0\) in these equations; we get  
\[\left\{ \begin{array}{l}n_{J}^{2}=\frac{\mu}{r^{3}}+\frac{3\mu_{J}R^{2}}{2^{ \beta}}-\frac{15\mu_{J}R^{4}}{8r^{\prime}},\\ \sin 2\phi=0,\end{array} \right. \tag{5}\]  
or  
\[\left\{ \begin{array}{l}-\frac{\mu}{r^{2}}+\frac{3\mu_{J}R^{2}}{2r^{4}}\big{(} 3\sin^{2}\phi-1\big{)}+\frac{5\mu_{J}R^{4}}{8r^{6}}\Big{(}35\sin^{4}\phi-30 \sin^{2}\phi+3\Big{)}+r\cos^{2}\phi\eta_{J}^{2}=0,\\ \frac{1}{2}r^{2}\eta^{2}+\frac{3\mu_{J}R^{2}}{2r^{3}}+\frac{\mu_{J}R^{4}}{8r^ {\prime}}\big{(}70\sin^{2}\phi-30\big{)}=0,\end{array} \right. \tag{6}\]  
where \(\eta_{J}\) is the rotational angular velocity of Jupiter. One can verify that the left-hand side of the second equation in (6) is always positive when \(r\) is larger than \(R\).