Lagrange's planetary equations with time-dependent secular perturbations

TL;DR

This paper derives the evolution of the semi-major axis in Hamiltonian systems with explicit time dependence, analyzing specific astrophysical perturbations to understand long-term orbital dynamics.

Contribution

It introduces a method to analyze the semi-major axis evolution in time-dependent Hamiltonian systems, with applications to astrophysical perturbations.

Findings

Derived equations for semi-major axis evolution with time-dependent Hamiltonians

Analyzed harmonic and quadrupole perturbations in astrophysical contexts

Provided insights into energy conservation and orbital evolution

Abstract

The long-term evolution of astrophysical systems is driven by a Hamiltonian that is independent of the fast angle. As this Hamiltonian may contain explicitly time-dependent parameters, the conservation of mechanical energy is not guaranteed in such systems. We derive how the semi-major axis evolves in these cases. We analyze two astrophysically interesting examples, those of the harmonic and quadrupole perturbations.