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Estimating Statistical Uncertainties of Internal Kinematics of Galaxies and Star Clusters Derived Using Full Spectrum Fitting

Pixel-space full spectrum fitting exploiting nonlinear χ2 minimization became a de facto standard way of deriving internal kinematics from absorption line spectra of galaxies and star clusters. However, reliable estimation of uncertainties for kinematic parameters remains a challenge and is usually addressed by running computationally expensive Monte-Carlo simulations. Here we derive simple formulae for the radial velocity and velocity dispersion uncertainties based solely on the shape of a template spectrum used in the fitting procedure and signal-to-noise information. Comparison with Monte-Carlo simulations provides agreement within a few per cent for different templates, signal-to-noise ratios and input velocity dispersions between 0.5 and 10 times the instrumental spectral resolution. For moderate template mismatch our technique returns uncertainties consistent within 20%–30% with those reported by Monte-Carlo simulations. We provide IDL and python implementations of our approach. The main applications are: (i) exposure time calculators; (ii) design of observational programs and estimates on expected uncertainties for spectral surveys of galaxies and star clusters; (iii) a cheap and accurate substitute for Monte-Carlo simulations when running them for large samples of thousands of spectra is unfeasible.