Perturbation of Binary Orbits in Hierarchical Triple Systems to Dotriocontopole Order*

We analyze the secular evolution of hierarchical triple body systems in Newtonian gravity to dotriocontopole order. We expand the Newtonian equations of motion in powers of the semimajor axes 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 (a/A)³ quadrupole, (a/A)⁴ octupole, (a/A)⁵ hexadecapole, (a/A)⁶ dotriocontopole and so on. We use the Lagrange Planetary equations and a two timescale method to obtain secular evolutions of the orbital elements.  Effects of dotriocontopole order arise from two places; the (a/A)⁶ terms in the expansion of the equations of motion, and higher order, "quadrupole-squared" effects arising from quadrupole perturbations feeding back into quadrupole terms, and from corrections to the procedure for averaging over both orbital periods.