Linear stability analysis of resonant periodic motions in the restricted three-body problem

Abstract

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$, $0\leq \mu \leq 1/2$, that circle each other with period equal to $2\pi$. When $\mu=0$, the problem admits orbits for the massless particle that are ellipses of eccentricity $e$ with the primary of mass 1 located at one of the focii. If the period is a rational multiple of $2\pi$, denoted $2\pi p/q$, some of these orbits perturb to periodic motions for $\mu > 0$. For typical values of $e$ and $p/q$, two resonant periodic motions are obtained for $\mu > 0$. We show that the characteristic multipliers of both these motions are given by expressions of the form $1\pm\sqrt{C(e,p,q)\mu}+O(\mu)$ in the limit $\mu\to 0$. The coefficient $C(e,p,q)$ is analytic in $e$ at $e=0$ and $C(e,p,q)=O(e^{\abs{p-q}})$. The coefficients in front of $e^{\abs{p-q}}$, obtained when $C(e,p,q)$ is expanded in powers of $e$ for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass $1-\mu$.