The Rossiter-McLaughlin effect and analytic radial velocity curves for transiting extrasolar planetary systems

A transiting extrasolar planet sequentially blocks off the light coming from the different parts of the disk of the host star in a time-dependent manner. Because of the spin of the star, this produces an asymmetric distortion in the line profiles of the stellar spectrum, leading to an apparent anomaly in the the radial velocity curves, known as the Rossiter-McLaughlin effect. Here, we derive approximate but accurate analytic formulae for the anomaly in the radial velocity curves, taking into account the stellar limb darkening. The formulae are particularly useful in extracting information on the projected angle between the planetary orbit axis and the stellar spin axis, lambda, and the projected stellar spin velocity, V sin I-s. We create mock samples for the radial curves for the transiting extrasolar system HD 209458 and demonstrate that constraints on the spin parameters (V sin I-s, lambda) can be significantly improved by combining our analytic template formulae and the precision velocity curves from high-resolution spectroscopic observations with 8-10 m class telescopes. Thus, future observational exploration of transiting systems using the Rossiter-McLaughlin effect will be one of the most important probes for a better understanding of the origin of extrasolar planetary systems, especially the origin of their angular momentum.