## 4 Orbits with Critical Inclination  
The variations of the eccentricity and the argument of perjove are mainly caused by the equatorial bulge of Jupiter. These usually produce negative effects on space missions to Jupiter. However, we can choose orbits with critical inclinations to avoid these disadvantages. According to the mean element theory, the mean variation rate of \(\omega\) caused by perturbations of the first and second order (36; 37) can be formulated as follows:  
\[ \begin{array}{l}\dot{\omega}=-\frac{3\eta j_{2}R^{2}}{2\dot{\sigma}^{2}(1-e^ {2})}\big{(}\frac{5}{2}\sin^{2}i-2\big{)}+\frac{9\eta j_{2}^{2}R^{4}}{p^{4}} \Big{\{}4+\frac{7}{12}^{2}+2\sqrt{1-e^{2}}-\sin^{2}i\Big{(}\frac{103}{12}+\frac {3e^{2}}{8}+\frac{11}{2}\sqrt{1-e^{2}}\Big{)}\\ +\sin^{4}i\Big{(}\frac{215}{48}-\frac{15}{32}e^{2}+\frac{15}{4}\sqrt{1-e^{2}} \Big{)}-\frac{35I_{4}}{18j_{2}^{2}}\Big{[}\frac{12}{7}+\frac{27}{14}^{2}-\sin ^{2}i\Big{(}\frac{93}{14}+\frac{27}{4}^{2}\Big{)}+\sin^{4}i\Big{(}\frac{21}{4}+ \frac{81E^{2}}{16}\Big{)}\Big{]}\bigg{\}}.\end{array}  \tag{22}\]  
To keep the invariance of the mean argument of perjove, we set \(\dot{\omega}=0\). Therefore, we obtain the following equation  
\[\Gamma_{4}\sin^{4}i+\Gamma_{5}\sin^{2}i+\Gamma_{6}=0, \tag{23}\]  
where  
\[\Gamma_{4}=\frac{9\eta j_{2}^{2}R^{4}}{4p^{4}}\Bigg{[}\frac{215}{48}-\frac{15 e^{2}}{32}+\frac{15}{4}\sqrt{1-e^{2}}-\frac{35I_{4}}{18j_{2}^{2}}\bigg{(} \frac{21}{4}+\frac{81e^{2}}{16}\bigg{)}\Bigg{]}, \tag{24}\]  
Figure 5: Evolution of \(\Omega\) and difference between \(\Delta\Omega=\Omega-\Omega_{0}\) and \(n_{s}t\) over \(25T\), where \(T\) is a Jovian day, \(\Omega_{0}=60\) deg is the initial condition (the blue curve was indeed obtained by numerical integration of osculating elements). (**a**) Evolution of \(\Omega\) for \(a=102755.451\) km, \(i=90.183\) deg. (**b**) Difference between \(\Delta\Omega\) and \(n_{s}t\) for \(a=102,755.451\) km, \(e=0.4\), \(i=90.183\) deg. (**c**) Evolution of for \(a=109,439.953\) km, \(e=0.1\), \(i=90.321\) deg. (**d**) Difference between \(\Delta\Omega\) and \(n_{s}t\) for \(a=109,439.953\) km, \(e=0.1\), \(i=90.321\) deg. \[\Gamma_{5}=\frac{3 nI_{2}R^{2}}{4p^{2}}\Bigg{[}-5-\frac{3I_{2}R^{2}}{p^{2}}\bigg{(} \frac{103}{12}+\frac{3e^{2}}{8}+\frac{11}{2}\sqrt{1-e^{2}}\bigg{)}+\frac{35I_{4}} {6I_{2}^{2}}\bigg{(}\frac{31}{14}+\frac{9e^{2}}{4}\bigg{)}\Bigg{]}, \tag{25}\]  
\[\Gamma_{6}=\frac{3 nI_{2}R^{2}}{4p^{2}}\Bigg{\{}4+\frac{3I_{2}R^{2}}{p^{2}} \Bigg{[}4+\frac{7e^{2}}{12}+2\sqrt{1-e^{2}}-\frac{5I_{4}}{6I_{2}^{2}}\bigg{(}4 +\frac{9e^{2}}{2}\bigg{)}\Bigg{]}\Bigg{\}}. \tag{26}\]  
Note that the left hand side of Equation (23) is a second-degree polynomial with respect to \(\sin^{2}i\). This polynomial in \(\sin^{2}i\) may have two real roots in [0,1], and so we may have up to four values of the inclination in \([0,\pi]\) that solve Equation (23). Let us set \(x=\sin^{2}i\) and write Equation (23) as  
\[h(x)=\Gamma_{4}x^{2}+\Gamma_{5}x+\Gamma_{6}=0. \tag{27}\]  
Noting that \(\Gamma_{4}>0\), \(\Gamma_{5}<0\), \(\Gamma_{6}>0\), we find that \(h(x)\) can have up to two real roots. Moreover, if \(h(1)>0\), they will fall in the interval [0,1]. If there exist four roots in \([0,\pi]\) of the critical inclination, then Equation (27) must have two different real roots lying in [0,1]. Observing that \(\Gamma_{4}>0\), \(\Gamma_{5}<0\), \(\Gamma_{6}>0\), we can easily see a necessary condition for Equation (27) to have two different roots lying in [0, 1] is  
\[h(0)=\Gamma_{6}>0,\ h(1)=\Gamma_{4}+\Gamma_{5}+\Gamma_{6}>0 \tag{28}\]  
Moreover, the condition (21) should also be satisfied to avoid the impact with Jupiter. The variation of \(h(1)\) for \(a\in[1.1R,9R]\) and \(e\in[0,1-R/a]\) are presented in Figure 6a. We note that \(h(1)<0\) in this domain of \((a,e)\), therefore, there exist two critical inclinations, and one of them corresponds to a retrograde orbit. The variation of critical inclinations of direct orbits as \(a\) increases from \(R\) to \(2R\) are presented in Figure 6b for different values of the eccentricity.