Stability of a Planet in a Binary System: MACHO 97-BLG-41

Abstract

An extrasolar planet was detected in a binary system by gravitational microlensing. This is considered to be the first one detected orbiting both components of a binary star system. The event is now known as MACHO 97-BLG-41. Whether such a planetary system can last on a large timescale is a major concern to celestial mechanics. We investigate the stability of the planetary orbits with the observed mass ratios of the three bodies by taking the binary and planetary eccentricities as parameters. The eccentricities could hardly be determined by gravitational microlensing but may be estimated by long-term numerical integrations of the binary and planetary orbital motions. We performed such long-term numerical integrations of the coplanar elliptic restricted three-body problem with various initial conditions in order to see what initial conditions produce stable planetary orbits during the integration for 106 binary periods (2.8 × 106 yr). The results of our numerical integrations permit us to estimate the upper limit of binary eccentricity, which ensures stable planetary orbital motion to be about 0.5 in the cases of circular initial orbits of the planet. In the cases of elliptic initial orbits of the planet, the planetary orbital motion is found to be less stable; hence, the upper limit of the binary eccentricity is estimated to be smaller than that in the cases of circular initial orbits of the planet. The upper limit of the initial planetary eccentricity is estimated to be about 0.4 for stable planetary orbital motion. The results of similar integrations for retrograde orbits indicate that the planetary retrograde orbits are more stable than the prograde ones.