Classical and relativistic long-term time variations of some observables for transiting exoplanets

Abstract

We analytically work out the long-term, i.e. averaged over one orbital revolution, time variations of some direct observable quantities Y induced by classical and general relativistic dynamical perturbations of the two-body pointlike Newtonian acceleration in the case of transiting exoplanets moving along elliptic orbits. More specifically, the observables $Y$ with which we deal are the transit duration, the radial velocity and the time interval between primary and secondary eclipses. The dynamical effects considered are the centrifugal oblateness of both the star and the planet, their tidal bulges mutually raised on each other, a distant third body X, and general relativity (both Schwarzschild and Lense-Thirring). We take into account the effects due to the perturbations of all the Keplerian orbital elements involved in a consistent and uniform way. First, we explicitly compute their instantaneous time variations due to the dynamical effects considered and plug them in the general expression for the instantaneous change of Y; then, we take the overall average over one orbital revolution of the so-obtained instantaneous rate $\dot Y(t)$ specialized to the perturbations considered. Instead, somewhat hybrid expressions can be often found in literature: in them, the secular precession of, typically, the periastron only is straightforwardly inserted into instantaneous formulas. Numerical evaluations of the obtained results are given for a typical star-planet scenario and compared with the expected observational accuracies over a time span 10 yr long. Our results are, in principle, valid also for other astronomical scenarios. They may allow, e.g., for designing various tests of gravitational theories with natural and artificial bodies in our solar system. (Abridged)