Higher-order effects in the dynamics of hierarchical triple systems: Quadrupole-squared terms
Clifford M. Will
Abstract
We analyze the secular evolution of hierarchical triple systems to second order in the quadrupolar perturbation induced on the inner binary by the distant third body. The Newtonian three-body equations of motion, expanded in powers of the ratio of semimajor axes $a/A$, become a pair of effective one-body Keplerian equations of motion, perturbed by a sequence of multipolar perturbations, denoted quadrupole, $O[(a/A{)}^{3}]$, octupole, $O[(a/A{)}^{4}]$, and so on. In the Lagrange planetary equations for the evolution of the instantaneous orbital elements, second-order effects arise from obtaining the first-order solution for each element, consisting of a constant (or slowly varying) piece and an oscillatory perturbative piece, and reinserting it back into the equations to obtain a second-order solution. After an average over the two orbital timescales to obtain long-term evolutions, these second-order quadrupole (${Q}^{2}$) terms would be expected to produce effects of order $(a/A{)}^{6}$. However we find that the orbital average actually enhances the second-order terms by a factor of the ratio of the outer to the inner orbital periods, $\ensuremath{\sim}(A/a{)}^{3/2}$. For systems with a low-mass third body, the ${Q}^{2}$ effects are small, but for systems with a comparable-mass or very massive third body, such as a Sun-Jupiter system orbiting a solar-mass star, or a $100\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ binary system orbiting a ${10}^{6}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ massive black hole, the ${Q}^{2}$ effects can completely suppress flips of the inner orbit from prograde to retrograde and back that occur in the first-order solutions. These results are in complete agreement with those of Luo, Katz and Dong, derived using a ``corrected double averaging'' method.