Secular Evolution of Hierarchical Planetary Systems

(Abridged) We investigate the dynamical evolution of coplanar hierarchicaltwo-planet systems where the ratio of the orbital semimajor axes alpha=a_1/a_2is small. The orbital parameters obtained from a multiple Kepler fit to theradial velocity variations of a star are best interpreted as Jacobi coordinatesand Jacobi coordinates should be used in any analyses of hierarchical planetarysystems. An approximate theory that can be applied to coplanar hierarchicaltwo-planet systems with a wide range of masses m_j and orbital eccentricitiese_j is the octupole-level secular perturbation theory (OSPT). The OSPT showsthat if the ratio of the maximum orbital angular momenta, lambda \approx(m_1/m_2) alpha^{1/2}, for given a_j is approximately equal to a critical valuelambda_{crit}, then libration of the difference in the longitudes of periapse,w_1-w_2, about either 0 or 180 deg. is almost certain, with possibly largeamplitude variations of both e_j. We establish that the OSPT is highly accuratefor systems with alpha<0.1 and reasonably accurate even for systems with alphaas large as 1/3, provided that alpha is not too close to a significantmean-motion commensurability or above the stability boundary. The HD 168443system is not in a secular resonance and its w_1-w_2 circulates. The HD 12661system is the first extrasolar planetary system found to have w_1-w_2 libratingabout 180 deg. The libration of w_1-w_2 and the large-amplitude variations ofboth e_j in the HD 12661 system are consistent with the analytic results onsystems with lambda \approx lambda_{crit}. The HD 12661 system with the best-fit orbital parameters and sin i = 1 is affected by the close proximity to the11:2 commensurability, but small changes in the outer orbital period can resultin configurations that are not affected by mean-motion commensurabilities.Comment: 32 pages, including 8 figures; uses AASTeX v5.0; accepted for publication in Ap