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


  • L. Iorio

    Corresponding author
    1. Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR), Fellow of the Royal Astronomical Society (FRAS), Italy
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Present address: Viale Unità di Italia 68, 70125 Bari (BA), Italy.


We analytically work out the long-term, i.e. averaged over one orbital revolution, time variations inline image of some direct observable quantities Y induced by classical and general relativistic dynamical perturbations of the two-body point-like 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 Δtd, the radial velocity Vρ, the time interval Δtecl between primary and secondary eclipses, and the time interval Ptr between successive primary transits. 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 substitute 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 inline image specialized to the perturbations considered. In contrast, previous published works have often employed somewhat ‘hybrid’ expressions, in which the secular precession of, typically, the periastron only is straightforwardly inserted into instantaneous formulas. The transit duration is affected neither by the general relativistic Schwarzschild-type nor by the classical tidal effects, while the bodies’ centrifugal oblatenesses, a distant third-body X and the general relativistic Lense–Thirring-type perturbations induce non-vanishing long-term, harmonic effects on Δtd also for circular orbits. For exact edge-on configurations they vanish. Both Vρ and Δtecl experience non-vanishing long-term, harmonic variations, caused by all the perturbations considered, only for non-circular orbits. Also Ptr is affected by all of them with long-term signatures, but they do not vanish for circular orbits. 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 τ= 10 yr long. Also graphical investigations of the dependence of the effects considered on the semimajor axis and the eccentricity are presented. 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.