Post-Newtonian effects on some characteristic time-scales of transiting exoplanets

Abstract

ABSTRACT Some measurable characteristic time-scales {ttrn} of transiting exoplanets are investigated in order to check preliminarily if their cumulative shifts over the years induced by the post-Newtonian (pN) gravitoelectric (Schwarzschild) and gravitomagnetic (Lense–Thirring) components of the stellar gravitational field are, at least in principle, measurable. Both the primary (planet in front of the star) and the secondary (planet behind the star) transits are considered along with their associated characteristic time intervals: the total transit duration tD, the ingress/egress transit duration $\tau$, the full-width at half maximum primary transit duration tH, and also the time of conjunction tcj. For each of them, the net changes per orbit $\left\langle \Delta t_D\right\rangle ,\, \left\langle \Delta \tau\right\rangle ,\, \left\langle \Delta t_H\right\rangle ,\, \left\langle \Delta t_\mathrm{cj}\right\rangle$ induced by the aforementioned pN accelerations are analytically obtained; also the Newtonian effect of the star’s quadrupole mass moment $J_2^\star$ is worked out. They are calculated for a fictitious Sun-Jupiter system in an edge-on elliptical orbit, and the results are compared with the present-day experimental accuracies for the HD 286123 b exoplanet. Its pN gravitoelectric shift $\left\langle \Delta t_\mathrm{cj}^\mathrm{1pN}\right\rangle$ may become measurable, at least in principle, at a ≃8 × 10−5 level of (formal) relative accuracy after about 30 yr of continuous monitoring corresponding to about 1000 transits. Systematics like, e.g. confusing time standards, neglecting star-spots, neglecting clouds, additional planets in the system, etc. would likely deteriorate the actual accuracy. The method presented is general enough to be applied also to modified models of gravity.