Extrasolar planetary dynamics with a generalized planar laplace-lagrange secular theory

The dynamical evolution of nearly half of the known extrasolar planets in multiple-planet systems may be dominated by secular perturbations. The commonly high eccentricities of the planetary orbits calls into question the utility of the traditional Laplace-Lagrange (LL) secular theory in analyses of the motion. We analytically generalize this theory to fourth order in the eccentricities, compare the result with the second-order theory and octupole-level theory, and apply these theories to the likely secularly dominated HD 12661, HD 168443, HD 38529, and v And multiplanet systems. The fourth-order scheme yields a multiply branched criterion for maintaining apsidal libration and implies that the apsidal rate of a small body is a function of its initial eccentricity, dependencies which are absent from the traditional theory. Numerical results indicate that the primary difference the second- and fourth-order theories reveal is an alteration in secular periodicities and to a smaller extent amplitudes of the planetary eccentricity variation. Comparison with numerical integrations indicates that the improvement afforded by the fourth-order theory over the second-order theory sometimes is dwarfed by the improvement needed to reproduce the actual dynamical evolution. We conclude that LL secular theory, to any order, generally represents a poor barometer for predicting secular dynamics in extrasolar planetary systems, but does embody a useful tool for extracting an accurate long-term dynamical description of systems with small bodies and/or near-circular orbits.