Secular Evolution of Hierarchical Planetary Systems

Abstract

Among the extrasolar planetary systems discovered to date are hierarchical systems such as HD 168443 where the orbital eccentricities of the two known substellar companions are large and the ratio of the orbital semimajor axes a1/a_2 is small. The octupole-level secular perturbation equations, which are based on an expansion to order (a1/a_2)3, should be applicable to coplanar hierarchical planetary systems with a wide range of companion masses and orbital eccentricities. We find that the octupole approximation describes well the secular evolution of stable coplanar hierarchical systems such as HD 168443 by comparison with direct numerical orbit integrations. The octupole approximation reduces the secular evolution of coplanar hierarchical systems to a problem of one degree of freedom, and the secular evolution of such systems can be understood by examining how the trajectories in diagrams of the inner-orbit eccentricity, e1, versus the difference in the longitudes of periapse, ϖ1 - ϖ2, depend on various parameters. There are usually two libration islands, one about a point at ϖ1 - ϖ2 = 0o and another about a point at ϖ1 - ϖ2 = 180o. The libration islands are large and large oscillations of both eccentricities are possible if L1 ≈ L2 (where Li is the maximum possible angular momentum of the ith orbit if the orbit were circular). The octupole-level equations do not provide any information on the stability of hierarchical systems. For hierarchical systems such as HD 168443 where the companion masses are relatively large compared to the primary mass, we find from direct numerical orbital integrations that such systems are unstable if the periapse distance of the outer orbit is less than about 3 times the semimajor axes of the inner orbit. The smallest periapse distance of the outer orbit for stability decreases slowly with decreasing companion masses.