Higher-order effects in the dynamics of hierarchical triple systems. II. Second-order and dotriacontapole-order effects

We analyze the long-term evolution of hierarchical triple systems in Newtonian gravity to second order in the fundamental quadrupolar perturbation parameter, and to sixth order in $\ensuremath{\epsilon}=a/A$, the ratio of the semimajor axes of the inner and outer orbits. We apply the ``two-timescale'' method from applied mathematics to the Lagrange planetary equations for the inner and outer orbits, in which each osculating orbit element is split into an orbit averaged part that evolves on the long perturbative timescale, and an ``average-free'' part that is oscillatory in the orbital timescales. Averages over the two orbital timescales are performed using the well-known ``secular approximation,'' carefully adapted to account for the time integrals that produce the oscillatory solutions. We also incorporate perturbative corrections to the relation between time and the orbital phases, or ``anomalies.'' We place no restrictions on the masses of the three bodies, on the relative orbit inclinations or on the eccentricities, beyond the requirement that the quadrupolar perturbation parameter and $\ensuremath{\epsilon}$ both be ``small.'' The result is a complete set of long-timescale evolution equations for the averaged elements of the inner and outer orbits. At first order in perturbation theory, we obtain the dotriacontapole contributions explicitly at order ${\ensuremath{\epsilon}}^{6}$, augmenting earlier well-known results at quadrupole, octupole, and hexadecapole orders. At second order in perturbation theory, i.e. quadratic in the quadrupole perturbation amplitude, we find contributions that scale as ${\ensuremath{\epsilon}}^{9/2}$ (found in earlier work), ${\ensuremath{\epsilon}}^{5}$, ${\ensuremath{\epsilon}}^{11/2}$, and ${\ensuremath{\epsilon}}^{6}$. At first perturbative order and dotriacontapole order, the two averaged semimajor axes are constant in time (and we prove that this holds to arbitrary multipole orders); but at second perturbative order, beginning at $O({\ensuremath{\epsilon}}^{5})$, they are no longer constant. Nevertheless we verify that the total averaged energy of the system is conserved, and we argue that this behavior is not incompatible with classical theorems on secular evolution of the semimajor axes.