Secular Evolution of Hierarchical Planetary Systems

We investigate the dynamical evolution of coplanar, hierarchical, two-planet systems where the ratio of the orbital semimajor axes α = a 1 / a 2 is small. Hierarchical two-planet systems are likely to be ubiquitous among extrasolar planetary systems. We show that the orbital parameters obtained from a multiple-Kepler fit to the radial velocity variations of a host star are best interpreted as Jacobi coordinates and that Jacobi coordinates should be used in any analyses of hierarchical planetary systems. An approximate theory that can be applied to coplanar, hierarchical, two-planet systems with a wide range of masses and orbital eccentricities is the octopole-level secular perturbation theory, which is based on an expansion to order α 3 and orbit averaging. It reduces the coplanar problem to 1 degree of freedom, with e 1 (or e 2 ) and ϖ 1 - ϖ 2 as the relevant phase-space variables (where e 1,2 are the orbital eccentricities of the inner and outer orbits, respectively, and ϖ 1,2 are the longitudes of periapse). The octopole equations show that if the ratio of the maximum orbital angular momenta, λ = L 1 / L 2 ≈ ( m 1 / m 2 )α 1/2 , for given semimajor axes is approximately equal to a critical value λ crit , then libration of ϖ 1 - ϖ 2 about either 0° or 180° is almost certain, with possibly large amplitude variations of both eccentricities. From a study of the HD 168443 and HD 12661 systems and their variants using both the octopole theory and direct numerical orbit integrations, we establish that the octopole theory is highly accurate for systems with α ≲ 0.1 and reasonably accurate even for systems with α as large as , provided that α is not too close to a significant mean-motion commensurability or above the stability boundary. The HD 168443 system is not in a secular resonance, and its ϖ 1 - ϖ 2 circulates. The HD 12661 system is the first extrasolar planetary system found to have ϖ 1 - ϖ 2 librating about 180°. The secular resonance means that the lines of apsides of the two orbits are on average antialigned, although the amplitude of libration of ϖ 1 - ϖ 2 is large. The libration of ϖ 1 - ϖ 2 and the large amplitude variations of both eccentricities in the HD 12661 system are consistent with the analytic results on systems with λ ≈ λ crit . The evolution of the HD 12661 system with the best-fit orbital parameters and sin i = 1 ( i is the inclination of the orbital plane from the plane of the sky) is affected by the close proximity to the 11 : 2 mean-motion commensurability, but small changes in the orbital period of the outer planet within the uncertainty can result in configurations that are not affected by mean-motion commensurabilities. The stability of the HD 12661 system requires sin i > 0.3.