How can they estimate exoplanet radial velocities using Doppler considering spectrograph resolving power? I read that spectrograph resolving powers, the ratio of wavelength uncertainty to wavelength are like 1000 or 10000. Plugging this into the non relativistic Doppler formula gives a velocity uncertainty like 30000 meters per second. So how can they claim one meter per second accuracy? And what about thermal broadening of spectral lines?
 A: Resolution tells you how well the spectrometer can separate lines with wavelengths close together, but not how precisely it can measure the wavelength of a single line. Measurement precision can be much better than the resolution. Then, techniques like template correlation can effectively average measurements of many lines, improving precision even more.
A: Typical resolving powers for exoplanet-finding spectrographs are 50000-100000, but nevertheless, this still means a resolution element has a FWHM of 3-6 km/s. This is to be compared with the radial velocity amplitudes caused by the planets of anywhere from 100 m/s for close-in hot Jupiters, to less than 1 m/s for Earth-like planets.
Measuring the precision of the shift of a spectral line against a wavelength scale boils down to how well you can estimate the centroid of the line. When you estimate the centroid of a Gaussian, the precision of your answer is not limited to the width of the Gaussian. It can, in principle, be of much higher quality. Roughly speaking, the uncertainty in the Gaussian centroid (or mean) is (to a small numerical factor) something like the FWHM divided by the signal-to-noise ratio of the flux in the Gaussian.
In addition, when measuring the spectral lines of stars there is a further $\sqrt{N}$ gain in precision from measuring $N$ spectral lines, since all the lines are shifted by the same velocity. $N$ can be of order 1000 for a sun-like star in a 100 nm spectral range.
The process is entirely analogous to estimating the mean of some distribution by taking repeated measurements. Whilst the estimated width (standard deviation or FWHM) remains approximately constant as the number of measurements increases, the standard error in the mean shrinks by a factor equal to the square root of the number of measurements.
