Secular dynamics and the lifetimes of lunar artificial satellites under natural force-driven orbital evolution

Abstract

In this paper, we study the long-term (time scale of several years) orbital evolution of lunar satellites under the sole action of natural forces. In particular, we focus on secular resonances, caused either by the influence of the multipole moments of the lunar potential and/or by the Earth’s and Sun’s third-body effect on the satellite’s long-term orbital evolution. Our study is based on a simplified secular model obtained in ‘closed form’, i.e., without expansions in the satellite’s orbital eccentricity. Contrary to the case of artificial Earth satellites, in which many secular resonances compete in dynamical impact, we give numerical evidence that for lunar satellites only the $2g-$resonance ($\dot{\omega }=0$) affects significantly the orbits at secular timescales. We interpret this as a consequence of the strong effect of lunar mascons. We show that the lifetime of lunar satellites is, in particular, nearly exclusively dictated by the $2g$ resonance. By deriving a simple analytic model, we propose a theoretical framework which allows for both qualitative and quantitative interpretation of the structures seen in numerically obtained lifetime maps. This involves explaining the main mechanisms driving eccentricity growth in the orbits of lunar satellites. In fact, we argue that the re-entry process for lunar satellites is not necessarily a chaotic process (as is the case for Earth satellites), but rather due to a sequence of bifurcations leading to a dramatic variation in the structure of the separatrices in the $2g$ resonance’s phase portrait, as we move from the lowest to the highest limit in inclination (at each altitude) where the $2g$ resonance is manifested.