On the possibility of measuring the post-Newtonian gravitoelectric correction to the orbital period of a test body in a Solar System scenario

The possibility of measuring the post-Newtonian gravitoelectric correction to the orbital period of a test particle freely orbiting a spherically symmetric mass in the Solar System is analyzed. It should be possible, in principle, to detect it for Mercury at a precision level of 10. This level is mainly set by the unavoidable systematic errors due to the mismodelling in the Keplerian period which could not be reduced by accumulating a large number of orbital revolutions. Future missions like Messenger and BepiColombo should allow to improve it by increasing our knowledge of the Mercury’s orbital parameters. The observational accuracy is estimated to be 10 from the knowledge of the International Celestial Reference Frame (ICRF) axes. It could be improved by observing as many planetary transits as possible. It is not possible to measure such an effect in the gravitational field of the Earth by analyzing the motion of artificial satellites or the Moon because of the unavoidable systematic errors related to the uncertainties in the Keplerian periods. In the case of some recently discovered exoplanets the problems come from the observational errors which are larger than the relativistic effect.