Optimizing Doppler estimates for extrasolar planet detection I. A specific algorithm for shifted spectra

We present an optimized algorithm to estimate from shifted stellar spectra the variations of radial velocity caused by a rotating planet. This algorithm is based on a rigorous approach in the spectral Fourier space that uses a weighted analysis of the cross spectrum phase between the high resolution spectra of the object and an appropriate reference. We give a simple method to calculate the best value of the stellar radial velocity in the presence of photon noise from a non linear least square fit, and we derive a formal expression for the velocity error. We show experimentally that the Fourier approach gives much better results than the classical cross-correlation technique.