Periastron advance for phase-dependent nonlocal-in-time perturbations

Develop a Poincaré–Lindstedt treatment for perturbed Binet equations whose perturbing functions depend explicitly on the orbital phase or mean anomaly, and evaluate the associated nonlocal-in-time 4PN tail contribution to the periastron advance, including the resulting infinite series of Hansen coefficients.

Background

The method developed in the paper assumes that the perturbing functions entering the Binet equation are independent of the phase angle. The genuinely nonlocal-in-time part of the conservative 4PN tail Hamiltonian violates this assumption because the perturbing function depends on both the radial variable and the mean anomaly.

For this case, the perturbing function is no longer necessarily even, so sine Fourier coefficients generally contribute. Computing the relevant first Fourier coefficient requires inverting the Kepler equation and handling functions evaluated at retarded and advanced times. The authors explain that this procedure leads to an infinite series of Hansen coefficients whose closed-form resummation is not currently available, leaving the calculation unresolved.