Post-Keplerian corrections to the orbital periods of a two-body system and their measurability

Abstract

The orbital motion of a binary system is characterized by various characteristic temporal intervals which, by definition, are different from each other: the draconitic, anomalistic and sidereal periods. They all coincide in the Keplerian case. Such a degeneracy is removed, in general, when a post-Keplerian acceleration is present. We analytically work out the corrections to such otherwise Keplerian periods which are induced by general relativity (Schwarzschild and Lense-Thirring) and, at the Newtonian level, by the quadrupole of the primary. In many astronomical and astrophysical systems, like exoplanets, one of the most accurately determined quantities is just the time span characterizing the orbital revolution, which is often measured independently with different techniques like the transit photometry and the radial velocities. Thus, our results could be useful, in principle, to either constrain the physical properties of the central body and/or perform new tests of general relativity, especially when no other standard observables like, e.g., the orbital precessions are accessible to observations. The difference of two independently measured periods would cancel out the common Keplerian term leaving just a post-Keplerian correction. Furthermore, by comparing the theoretically predicted post-Keplerian expressions $T^{(\textrm{pK})}$ with the experimental accuracy $σ_{T_\textrm{exp}}$ in measuring the orbital period(s) it is possible to identify those systems whose observations should be re-processed with genuine post-Keplerian models if $T^{(\textrm{pK})}>σ_{T_\textrm{exp}}$. It seems just the case for WASP-33 b since $σ_{T_\textrm{exp}}=0.04$ s, while $3$ s$\leq T_\textrm{dra}^{(J_2)}\leq 9.5$ s, $T_\textrm{dra}^{(\textrm{GE)}}=0.36$ s.