Modified double-averaged Hamiltonian in hierarchical triple systems

In this work, we introduce a modified double-averaging approach by considering the shortterm effects and formulate a more accurate double-averaged Hamiltonian (in comparison to the classical octupole-level Hamiltonian) for hierarchial triple systems. The Hamiltonian is expressed as a power series in the ratio of the semi-major axes of the inner and outer binaries. Both the Delaunay's elements and the classical orbit elements are adopted to describe the motion. To derive the secular Hamiltonian, the short-term oscillations in the Hamiltonian are averaged out by means of a double-averaging approach. In particular, during the average over the orbital period of the outer binary, the periodic corrections to the secular motion are taken into account. Based on the double-averaged Hamiltonian, we provide two versions of equations of secular motion, given in the form of canonic relations and Lagrange planetary equations. The resulting secular evolution equations can be utilized to reproduce the long-term behaviours for those physical systems where the perturbations coming from the disturbing bodies are relatively strong. To test the approach, we use the averaged Hamiltonian to predict the longterm motions of a planet in a stellar binary system and natural satellites in Sun-planet systems. Simulation results show that the modified Hamiltonian can reproduce secular behaviours with high accuracy. Additionally, the comparison of dynamical models truncated at different orders indicates that the secular Hamiltonian with inclusion of higher order terms has better accuracy in predicting long-term evolution.