The dynamics of hierarchical triple systems: Two-timescale methods and dotriacontapole-order effects

We analyze the secular evolution of hierarchical three-body systems in Newtonian gravity to dotriacontapole order. We expand the Newtonian equations of motion in powers of the semimajor axes ε = a/A of the inner and outer orbits. The equations of motion take the effective form of two one-body Keplerian orbits that are perturbed by a series of multipolar perturbations that are labeled as quadrupole ε³, octupole ε⁴, hexadecapole ε⁵, dotriacontapole ε⁶ and so on. We use the Lagrange Planetary equations for the evolution of orbital elements and average over both orbital periods to obtain their secular evolutions. Effects of relevant order arise from the dotriacontapole term in the equations of motion, higher order corrections to the averaging of the quadrupole and octupole terms and second order terms derived from the two time-scale analysis. We also use a two-timescale method, expanded through second order in the quadrupolar perturbation timescale [O(ε⁶)]. Effects of relevant order arise from the dotriacontapole terms in the equations of motion, second order quadrupolar terms derived from the two time-scale analysis, contributions from corrections to the relationship between time and orbital phase, and second-order quadrupole-octupole “cross terms”, leading to an array of contributions at orders ε^9/2, ε⁵, ε^11/2 and ε⁶.