Equations for the analysis of the Rossiter-McLaughlin effect in extrasolar planetary transits

Analytical equations are presented for the computation of the Rossiter-McLaughlin effect in the radial velocity curves of stars hosting extrasolar planets and showing periodic transits. This is a well-known effect in the radial velocity curves of eclipsing binaries that results from the fractional visibility of the stars and the subsequent loss of symmetry for the integrated rotational contribution. In this paper, general and accurate equations are presented, allowing for the computation of the effect for spherical components. The nonorbital component of the radial velocity curve, during the transit of the planet, can thus be used to determine the rotational properties of the star, including any deviation of its rotational axis from the perpendicular to the orbital plane. For this purpose, the observational effects of stellar limb darkening, as well as the geometrical and orbital elements of the system, are taken into account. Results can be used immediately for the analysis of the spectroscopic observations during the transit of extrasolar planets and allow for the simultaneous solution of the light and radial velocity curves, therefore improving the precision of the derived elements by using two completely different sources of information in an integrated form. The effects of changes in various of the involved parameters are also discussed using the analytical properties of the equations.