Dynamics of retrograde $1/n$ mean motion resonances: the $1/{-2}$, $1/{-3}$ cases

Abstract

In this paper, we investigate the dynamics of the exterior retrograde $1/n$ resonances within the framework of the planar circular restricted three-body problem (PCRTBP), taking the $1/{-2}$ and $1/{-3}$ resonances as examples. We prove that there is no asymmetric libration in retrograde $1/n$ resonances using analytical, numerical integrations, and semi-analytical methods, by mutual authenticating. For retrograde $1/n$ resonances, we calculate the magnitudes of the first- and second-harmonics of the expansion. The analytical results showed that the second-order harmonics in the expansion of the disturbing function can be neglected, which are the leading cause of the existence of the asymmetric libration in prograde $1/n$ resonances. Our results also conform to the latest qualitative criterion for the appearance of asymmetric libations proposed by Namouni and Morais (2016). And our analytical results are verified well by numerical integrations. By semi-analytical theory, we generate a series of phase-space portraits in the $e$ – $\phi $ polar plane for the full range of eccentricity to confirm the absence of the asymmetric librations in retrograde $1/n$ resonances. It is the first time the absence of the asymmetric librations in retrograde $1/n$ resonances is studied. We also analyze the dynamics of the pericentric and apocentric librations of the retrograde $1/n$ resonances. The stable resonant libration zones of the $1/{-2}$ , $1/{-3}$ , $1/{-4}$ and $1/{-5}$ resonances are illustrated in the $a$ – $e$ plane. Our research here reveals the differences between the dynamics of retrograde and prograde $1/n$ resonances.