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We study the long-term evolution of selected hierarchical triple systems in Newtonian gravity. We employ analytic equations derived in Paper II for the evolution of orbit-averaged orbital elements for both inner and outer orbits, which include two classes of contributions. One class consists of linear-order contributions, including quadrupole, octupole, hexadecapole and dotriacontapole orders, the latter scaling as $\epsilon^6$, where $\epsilon = a/A$, the ratio of the semimajor axes of the inner and outer orbits. The second class consists of contributions at {\em second} order in the fundamental perturbation parameter; they contribute at orders $\epsilon^{9/2}$, $\epsilon^{5}$, $\epsilon^{11/2}$ and $\epsilon^{6}$. For well studied triples such as star-planet systems perturbed by a low-mass third body (``hot Jupiters''), second-order and dotriacontapole (SOD) effects induce only small corrections. For stellar-mass binaries orbiting supermassive black holes, SOD corrections can suppress orbital flips that are generated by purely first-order effects. Planets orbiting binary star systems are susceptible to significant variations in the planetary semimajor axis, an effect that does not occur at first perturbative order. SOD effects in triple black hole systems can induce migrations of the eccentricity to significantly larger values than predicted by first-order perturbations, with implications for the gravitational-wave induced inspiral of the inner binary. We also show that in most cases, evolutions using our SOD equations are in better agreement with those from direct integration of the N-body equations of motion than those from first-order perturbations through hexadecapole order., Comment: 12 pages, 11 figures, to be submitted to the Physical Review D15

We analyze the long-term evolution of hierarchical triple systems in Newtonian gravity to second order in the quadrupolar perturbation parameter, and to sixth order in $\epsilon = a/A$, 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. We also incorporate perturbative corrections to the relation between time and the orbital phases. We place no restrictions on the masses, on the relative orbit inclinations or on the eccentricities, beyond the requirement that the quadrupolar parameter and $\epsilon$ 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 $\epsilon^6$. At second order in perturbation theory, i.e. quadratic in the quadrupole perturbation amplitude, we find contributions that scale as $\epsilon^{9/2}$ (found in earlier work), $\epsilon^{5}$, $\epsilon^{11/2}$, and $\epsilon^{6}$. 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 $O( \epsilon^{5})$, 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., Comment: 21 pages, 2 figures, minor changes to align with published version