Analytical investigation about long-lifetime science orbits around Galilean moons

Abstract

It is desirable to design low-altitude and near-polar science orbits for missions to Galilean moons. However, the long-term perturbation from a distant perturber may lead such a kind of orbits to quick impacts, indicating that initial conditions of working orbits need to be well-designed. To this end, long-lifetime working orbits around oblate satellites are investigated in this work. Initially, numerical maps of lifetime around oblate satellites are produced under the long-term dynamical model and they show that initial conditions of long-lifetime orbits are distributed in the form of strips in the space spanned by initial longitude of ascending node and argument of pericenter. This phenomenon is known to be caused by the effect of nodal phasing due to the existence of the mother planet’s obliquity. To understand the mechanism of nodal phasing, we adopt Lie-series transformation to formulate an integrable Hamiltonian model, where the dynamical structures in phase space can be uncovered by phase portraits. Furthermore, we provide three constraints for solving the initial conditions of long-lifetime orbits: the first one states that the Hamiltonian of long-lifetime orbits should be equal to that of the stable manifold, the second and third ones are associated to given initial eccentricity and inclination. By solving these constraint equations and performing direct transformation, analytical strips are produced. It is shown that the analytical and numerical strips are in good agreement. At last, the analytical approach is applied to missions to Galilean moons.

Appendix A Detailed expressions of \({{\mathcal {K}}}\)

$$\begin{aligned} \begin{aligned} {{{\mathcal {K}}}}_0=&-\frac{0.000248472H^{*2}}{G^{*5}}+\frac{0.000082824}{G^{*3}}-\frac{0.000672026H^{*2}}{G^{*2}}-0.000112004G^{*2}\\&+0.000336013H^{*2}+0.000224009,\\ {{{\mathcal {K}}}}_1=&-\frac{0.000672026(1-0.833333G^{*2})(G^{*2}-H^{*2})\cos {(2g^*)}}{G^{*2}},\\ {{{\mathcal {K}}}}_2=&\frac{1.64381\times 10^{-8}G^*(G^{*2}-2)}{(G^{*5}-2G^{*3}-0.739472)^3}(-0.676158G^{*14}+G^{*12}(2.02848H^{*2}\\&+4.05695)+G^{*10}(-12.1709H^{*2}-8.1139)+G^{*9}+G^{*8}(24.3417H^{*2}\\&+5.40927)+G^{*7}(-3H^{*2}-4)-16.2278G^{*6}H^{*2}+G^{*5}(12H^{*2}+4)\\&-0.369736G^{*4}-12G^{*3}H^{*2}+G^{*2}(1.10921H^{*2}+0.739472)-2.21841H^{*2}). \end{aligned} \nonumber \\ \end{aligned}$$

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It is noted that ${{{\mathcal {K}}}}_3$ and higher-order Kamiltonian are too long, so they are not explicitly presented here.