Revisiting the 2PN pericentre precession in view of possible future measurements of it

Abstract

At the second post-Newtonian (2PN) order, the secular pericentre precession $\dot\omega^\mathrm{2PN}$ of either a full two-body system made of well detached non-rotating monopole masses of comparable size and a restricted two-body system composed of a point particle orbiting a fixed central mass have been analytically computed so far with a variety of approaches. We offer our contribution by analytically computing $\dot\omega^\mathrm{2PN}$ in a perturbative way with the method of variation of elliptical elements by explicitly calculating both the direct contribution due to the 2PN acceleration ${\boldsymbol A}^\mathrm{2PN}$, and also an indirect part arising from the self-interaction of the 1PN acceleration ${\boldsymbol A}^\mathrm{1PN}$ in the orbital average accounting for the instantaneous shifts induced by ${\boldsymbol A}^\mathrm{1PN}$ itself. Explicit formulas are straightforwardly obtained for both the point particle and full two-body cases without recurring to simplifying assumptions on the eccentricity $e$. Two different numerical integrations of the equations of motion confirm our analytical results for both the direct and indirect precessions. The values of the resulting effects for Mercury and some binary pulsars are confronted with the present-day level of experimental accuracies in measuring/constraining their pericentre precessions. The supermassive binary black hole in the BL Lac object OJ 287 is considered as well. A comparison with some of the results appeared in the literature is made.