objective functions and Nyquist sampling

Adrian Price-Whelan and I discussed today some oddities that Matt Daunt (NYU) is finding while trying to measure radial velocities in extremely noisy, fast APOGEE sub-exposures. He finds that the objective function we are using is not obviously smooth on 10-ish km/s velocity scales. Why not? We don't know. But what we do know is that a spectrograph with resolution 22,500 cannot put sharp structures into a likelihood function on scales smaller than about 13 km/s.

There's a nice paradox here, in fact: The spectrograph can't see features on scales smaller than 13 km/s, and yet we can reliably measure radial velocities much better than this! How? The informal answer is that the radial-velocity precision is 13 km/s divided by a certain, particular signal-to-noise. The formal answer involves information theory—the Fisher information, to be precise.

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