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 , become a pair of effective one-body Keplerian equations of motion, perturbed by a sequence of multipolar perturbations, denoted quadrupole, , octupole, , 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 () terms would be expected to produce effects of order . 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, . For systems with a low-mass third body, the 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 binary system orbiting a massive black hole, the 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.
- Received 6 December 2020
- Accepted 11 February 2021
- Corrected 9 March 2021
DOI:https://doi.org/10.1103/PhysRevD.103.063003
© 2021 American Physical Society
Physics Subject Headings (PhySH)
Corrections
9 March 2021
Correction: A phrase appearing after Eq. (2.11) that defines a subscript symbol was misset during initial processing and then fixed incorrectly during proof stages; this phrase has been set properly now.