Higher-order effects in the dynamics of hierarchical triple systems.
  Quadrupole-squared terms
  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, $ \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 $100 \,
M_\odot$ binary system orbiting a $10^6 \, M_\odot$ 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.
