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.