The dynamics of a probe orbiting a moon can be significantly influenced by the non-coincidence between the moon's equatorial and orbital planes. Thus, we performed a general analysis about the effects of the angle (obliquity) between the above-mentioned planes and of the angle (nodal phasing) between the nodal lines of the mother planet's apparent orbit and the probe orbit on the lifetime of the probe. The lifetime, strictly correlated to the variations in eccentricity of the probe orbit, was evaluated starting from low values of the semi-major axis, moderate eccentricity, and high inclination to offer high ground spatial resolution and extend latitudinal coverage of the natural satellite. This investigation, carried out through numerical simulations, may be useful for identifying the optimal initial conditions of the probe's orbit elements, leading to an important increase in the probe lifetime in missions devoted to the exploration of natural satellites.

In this study, a dynamical model is developed to describe the secular evolution of navigation satellites under the geocentric reference frame with the Laplace orbit as the fundamental plane. The disturbing function, involving the effects of Earth's oblateness and lunisolar gravitational attraction, is averaged over the orbital periods of both the satellite and the perturbers. In the regions of medium-Earth orbits and geosynchronous orbits, there are varieties of lunisolar resonances for governing the secular dynamics of navigation satellites. Among these resonances, we are interested in the ones occurring at the critical inclinations as well as the lunar node resonances. For each resonance of interest, the resonant center and width are identified analytically. Finally, dynamical maps are compared with the analytical results.

In this work, two dynamical models are formulated to describe the secular dynamics of navigation satellites moving in the medium Earth orbit (MEO) and geosynchronous orbit (GSO) regions. In the dynamical models, the leading terms of the Earth’s oblateness and the luni-solar gravitational perturbations are considered. For convenience, the orbits of the Sun and the Moon are described in the geocentric ecliptic reference frame, where the regression of nodal line and precession of apsidal line of the lunar orbit can be approximated as linear functions of time. The disturbing function acting on navigation satellites is analytically averaged over the mean motions of both the satellite and the third body (the Sun or the Moon). Explicit expressions of the averaged disturbing function are provided in the geocentric ecliptic and equatorial reference frames, corresponding to averaged model 1 and averaged model 2, respectively. It is found that there are seven resonant arguments in averaged model 1, while there are thirty-two resonant arguments in averaged model 2. The associated resonance curves corresponding to these resonant arguments in each model form the dynamical backbone in the phase space, organizing secular behavior of navigation satellites. At last, the averaged models are numerically compared to the associated non-averaged model, and simulation results indicate that (a) the averaged models formulated in the geocentric ecliptic and equatorial reference frames are identical, and (b) both of these two averaged models are applicable in predicting secular behavior of navigation satellites.