Are we far from testing general relativity with the transiting extrasolar planet HD 209458b.pdf

a r X i v : g r - q c / 0 5 0 5 1 0 7 v 5 2 0 D e c 2 0 0 5 Are We Far from Testing General Relativ- ity with the Transiting Extrasolar Planet HD 209458b ‘Osiris’? L. Iorio Viale Unita? di Italia 68, 70125 Bari, Italy tel./fax 0039 080 5443144 e-mail: lorenzo.iorio@libero.it Abstract In this paper we investigate the possibility of measuring the general relativistic gravitoelectric correction P (GE) to the orbital period P of the transiting exoplanet HD 209458b ’Osiris’. It turns out that the predicted magnitude of such an effect is ～ 0.1 s, while the most recent determinations of the orbital period of HD 209458b with the photo- metric transit method are accurate to ～ 0.01 s. It turns out that the major limiting factor is the error in the measurement of the projected semiamplitude of the star’s radial velocity KM . Indeed, it affects the determination of the mass m of the planet which, in turn, induces a systematic error in the Keplerian period P (0) of ～ 8 s. However, the present situation may change if future one-two orders of magnitude improvements in the Doppler spectroscopy technique will occur. 1 Introduction In this paper we investigate the possibility of performing a test of general relativity, in its weak-field and slow-motion approximation, for the first time in a planetary system other than our Solar System1. More precisely, it has been shown (Soffel 1989; Mashhoon et al. 2001) that the general relativistic gravitoelectric (GE) part of the gravitational field of a static, spherically symmetric mass2 also induces an additional small correction P (GE) to the Keplerian orbital period P (0) of a test particle. For 1See e.g. http://www.obspm.fr/encycl/ for regularly updated information about the extrasolar planets. 2It is the Schwarzschild component of the space-time metric tensor which gives rise to various tested phenomena (Will 1993) like the well known secular Einstein pericenter advance, the deflection of the electromagnetic waves, the Shapiro time de