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 quadrupolar perturbation induced on inner binary by distant third body. The Newtonian three-body equations motion, expanded powers ratio semimajor axes $a/A$, become a pair effective one-body Keplerian perturbed sequence multipolar perturbations, denoted quadrupole, $O[(a/A)^3]$, octupole, $O[(a/A)^4]$, and so on. In Lagrange planetary for instantaneous orbital elements, effects arise from obtaining first-order solution each element, consisting constant (or slowly varying) piece an oscillatory perturbative piece, reinserting it back into obtain solution. After average over two timescales long-term evolutions, these quadrupole ($Q^2$) terms would be expected produce order $(a/A)^6$. However we find that actually enhances factor outer periods, $ \sim (A/a)^{3/2}$. For with low-mass body, $Q^2$ are small, but comparable-mass or very massive such as Sun-Jupiter system orbiting solar-mass star, $100 \, M_\odot$ $10^6 black hole, can completely suppress flips orbit prograde retrograde occur solutions. These results complete agreement those Luo, Katz Dong, derived using "Corrected Double-Averaging" method.