Uncertainty of line-of-sight velocity measurement of faint stars from low and medium resolution optical spectra

Abstract

Massively multiplexed spectrographs will soon gather large statistical samples of stellar spectra. The accurate estimation of uncertainties on derived parameters, such as line-of-sight velocity $v_\mathrm{los}$, especially for spectra with low signal-to-noise ratios, is paramount. We generated an ensemble of simulated optical spectra of stars as if they were observed with low- and medium-resolution fiber-fed instruments on an 8-meter class telescope, similar to the Subaru Prime Focus Spectrograph, and determined $v_\mathrm{los}$ by fitting stellar templates to the simulations. We compared the empirical errors of the derived parameters -- calculated from an ensemble of simulations -- to the asymptotic error determined from the Fisher matrix, as well as from Monte Carlo sampling of the posterior probability. We confirm that the uncertainty of $v_\mathrm{los}$ scales with the inverse square root of $S/N$, but also show how this scaling breaks down at low $S/N$ and analyze the error and bias caused by template mismatch. We outline a computationally optimized algorithm to fit multi-exposure data and provide the mathematical model of stellar spectrum fitting that maximizes the so called significance, which allows for calculating the error from the Fisher matrix analytically. We also introduce the effective line count, and provide a scaling relation to estimate the error of $v_\mathrm{los}$ measurement based on the stellar type. Our analysis covers a range of stellar types with parameters that are typical of the Galactic outer disk and halo, together with analogs of stars in M31 and in satellite dwarf spheroidal galaxies around the Milky Way.