On the stability of dust orbits in mean-motion resonances perturbed by from an interstellar wind

Abstract

Circumstellar dust particles can be captured in a mean motion resonance with a planet and simultaneously be affected by non-gravitational effects. It is possible to describe the secular variations of a particle orbit in the mean motion resonance analytically using averaged resonant equations. We derive the averaged resonant equations from the equations of motion in near-canonical form. The secular variations of the particle orbit depending on the orientation of the orbit in space are taken into account. The averaged resonant equations can be derived/confirmed also from Lagrange's planetary equations. We apply the derived theory to the case when the non-gravitational effects are the Poynting-Robertson effect, the radial stellar wind, and an interstellar wind. The analytical and numerical results obtained are in excellent agreement. We found that the types of orbits correspond to libration centers of the conservative problem. The averaged resonant equations can lead to a system of equations which hold for stationary points in a subset of resonant variables. Using this system we show analytically that for the considered non-gravitational effects, all stationary points should correspond to orbits which are stationary in interplanetary space after an averaging over a synodic period. In an exact resonance, the stationary orbits are stable. The stability is achieved by a periodic repetition of the evolution during the synodic period. Numerical solutions of this system show that there are no stationary orbits for either the exact or non-exact resonances.

Locations of stationary/periodic solutions in mean motion resonances according to the properties of dust grains

“…In the exact resonance the averaged mean motion of the particle and the mean motion of the planet have a ratio that is exactly equal to the ratio of two natural numbers. The periodicity of evolutions corresponding to the stationary solutions in the exact resonances with fixed longitude of pericenter was already showed in Pástor (2014a). In this work we take into account the possibility that the longitude of pericenter can be variable.…”

“…We use angle σ b defined so that the mean anomaly M can be computed from M = nt + σ b (Bate, Mueller & White 1971). In this case the following relations hold between the averaged partial derivatives of the disturbing function (Pástor 2014a)…”

“…In Pástor (2014a) only the exact resonances were considered. The condition nP (p + q)/p = (1/TS) T S 0 n dt valid for the exact resonances may not be valid for the non-exact resonances.…”

“…In the general case secular variations of a particle orbit, caused by the non-gravitational effects, can depend on spatial orientation of the orbit. In this case resonant equations averaged over a synodic period have form (Pástor 2014a):…”

Charged dust close to outer mean-motion resonances in the heliosphere

“…However, the role of it on motion close to mean motion-resonances is still unclear. Previous studies [Sicardy et al, 1993, Beauge and Ferraz-Mello, 1994, Beauge, 1994, Liou and Zook, 1997, Jancart et al, 2003, Klačka and Kocifaj, 2008, Pástor et al, 2009, Kocifaj and Kundracik, 2012, Pástor, 2014, Lhotka and Celletti, 2015, Pástor, 2016 investigated the dynamical problem under the assumption that dust is uncharged. This assumption is only valid in the case where the dust grain is heavy enough and where the magnitude of the Lorentz force is much smaller compared to the magnitudes of the other ones.…”