Long term stability of planetary orbits in the eccentric binary star system HD 59686

Abstract

Up to April 2017, over 3400 exoplanets have been confirmed. Despite the large number of exoplanets detected, the number of planet hosting binary star systems is still less than 100. HD 59686 is one of the rare examples. The HD 59686 binary star system is discovered recently by radial velocity method (Ortiz et al. 2016). A giant planet is detected at a ∼ 1.1 AU despite the presence of an eccentric and close binary companion. Previous stability tests show the planet favors a coplanar retrograde configuration. In the first part of this thesis, the stability of circumstellar planetary orbits in the HD 59686 system, includingboth coplanar and inclined configurations, is studied. In the second part of this thesis, the stability of coplanar retrograde planets in general binary star systems was studied. Using Wisdom-Holman symplectic integrator and assuming the planet to be a test particle, we extensively studied the stability of planetary orbits in the HD 59686 system, including both coplanar and inclined orbits. I have found a bulk stable region in the (a, e) parameter space which is also observed in previous studies. Apart from the stable bulk region, I found that there are two more types stable region, which are the horizontal stable islands due to secular apsidal alignment and the vertical stable islands due to mean-motion resonance. These extra stable islands can present at distance beyond the stability boundary by 17% for prograde orbits and 39% for retrograde orbits. In a coplanar prograde configuration, I found that the apsidal alignment is crucial to planetary stability. A prograde planet can only be stable if it is in secular apsidal alignment where it lives in one of the horzontal stable islands due to secular apsidal alignment. These horizontal stable islands are also stable up to the age of the system and some are located near the real planet’s position within the error of measurements. Moreover, the planet is not in the vertical stable islands due to mean motion resonance if it is assumed to be stable. If assumed to be in a retrograde orbit, the planet locates well inside the stable region no matter whether it is in secular apsidal alignment. In an inclined configuration, the horizontal stable island due to secular apsi- dal alignment present up to i ∼ 20° and there are high inclination stable islands due to Kozai resonance. However the high inclination Kozai stable islands also have high planetary eccentricity values, which cannot account for the observed planet’s small eccentricity. The planet from the fitted data, which has a highmutual inclination, cannot be stable. In the second part of this study, I studied the stability of a retrograde test particle in the elliptic restricted three-body problem and derived an empirical formula similar to Holman & Wiegert (1999). I found that the retrograde/pro- grade stability boundary has a ratio of ∼ 1.6 over large part of binary parameter space and approaches 2 in the Hill’s regime. I further generalized (Innanen 1980; Hamilton & Burns 1991)’s work of generalized Hill’s sphere and successfully explained this ratio of 2.