Precision of spectroscopy for astronomy?

Question

How precise can the measurements be when looking at spectral lines in astrophysics?For example, suppose I have a telescope in orbit, and I am looking at $H_\\alpha$ lines coming from a star at 613 pc. Is it realistic to observer the lines move from 626.282 nm to 656.282002854 nm? this would correspond to a radial velocity of about 1.3 m/s (I am not an astronomer, and I just want to get a feel for whether this is realistic to measure, or out of the question)

Answer 1

It had been know that the accuracy of the Doppler shift of a spectrum line is a few meters/sec.

As far as I know a few years ago, 2~3m/sec was the best limit using a special I2 gas-chamber in the optical path of the telescopes.