## 2 Stationary Orbits
We computed the evolution of \(r\), \(\phi\), and \(\lambda\) when we selected initial values of \(r\) and \(\phi\) close to \(r_{0}\) and \(0\), respectively. Figure 2 shows a stationary orbit and four orbits obtained from \(r=0.99r_{0}\), \(\phi=\pm 0.1\) deg and \(r=1.01r_{0}\), \(\phi=\pm 0.1\) deg. We can see that these four disturbed orbits are no longer periodic. This can be explained by using Equation (9). Due to the presence of the terms that contain \(J_{2}\) and \(J_{4}\), the three frequencies are usually not equal or even commensurable. Figure 3 illustrates that the satellite oscillates in the radial and North-South directions. On the other hand, the satellite on stationary orbits does not drift in these directions. It only moves with a constant \(\dot{\lambda}\) along the orbit. Therefore, the drift of longitude would not occur for satellites on stationary orbits since only zonal harmonics are taken into account in the gravity model. Furthermore, there are no significant differences between points on the stationary orbit. As a result of the symmetry of the gravity field, it can be concluded that the points on stationary orbits are degenerate equilibrium points. Here, an equilibrium point is degenerate if, and only if, the matrix of the linearized equation for (4) is degenerate. This is the major difference with respect to stationary orbits of other planets, such as the Earth and Mars. For the Earth and Mars, there exist four equilibrium points on stationary orbits, among which two are stable and the other two are unstable. Figure 3: **(a,b)** Variations of orbital radius (\(r\)), (**c,d**) drift of longitude (\(\Delta\lambda\)), (**e,f**) evolution of latitude (\(\phi\)): green—stationary orbits; red, \(r=0.99r_{0}\) and \(\phi=-0.1\) deg; blue—\(r=0.99r_{0}\) and \(\phi=0.1\) deg; cyan—\(r=1.01r_{0}\) and \(\phi=-0.1\) deg; black—\(r=1.01r_{0}\), \(\phi=0.1\) deg, and \(T\) is a Jovian day. In the following, we show that the effects of third bodies, the Jovian ring and the magnetic field of Jupiter, are negligible compared to those of the \(J_{2}\) and \(J_{4}\) terms. Based on the data of the Sun and moons around Jupiter (see for example, [30]), one can calculate the maximal ratio of the disturbed acceleration (\(a_{d}\)) and central gravity acceleration (\(a_{m}\)) for satellites on stationary orbits, which is achieved when the satellite, Jupiter, and the third body are in a straight line. Therefore, the maximal ratio can be calculated as [31]  
\[\left(\frac{a_{d}}{a_{m}}\right)_{\max}=2\frac{m}{M_{J}}\left(\frac{r_{0}}{ \rho}\right)^{3}, \tag{10}\]  
where \(m\) is the mass of the third body, and \(\rho\) is the distance between Jupiter and the third body. Values of the maximum ratio (109) for different bodies are reported in Table 2. Since the \(J_{4}\) term of Jupiter is 100 times bigger than the ratio of acceleration for Io (which gives the highest value among the Galilean satellites), it is reasonable to ignore these effects when investigating the qualitive character of stationary orbits on short time scales. To verify the validity of these solutions, we use the numerical integration method to see the effect of the Galilean moon Io in 800 Jovian days. The orbital elements \((a,e,i,\Omega,\omega,M)\) of Io that we adopt here are taken from the 10 th China Trajectory Optimization Competition, i.e., \(\left( \begin{array}{c}422,029.687\text{ km}\\ \end{array} , \begin{array}{c}0.004308,0.04\text{ deg}\\ \end{array} , \begin{array}{c}-79.64\text{ deg},37.991\text{ deg}\\ \end{array} , \begin{array}{c}4.818\text{ deg}\\ \end{array} \right)\). Calculation results show that the drift of longitude (\(\Delta\lambda\)) for the satellite on stationary orbits is less than 0.1 deg in 800 Jovian days. The inclination changes no more than \(1.0\times 10^{-4}\) deg. The oscillation of the semi-major axis is less than 0.13% of the obital radius. However, it should be noted that the obtained solutions may not be correct on long time scales.