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

TL;DR

This paper presents simple, reliable formulas for estimating uncertainties in galaxy and star cluster kinematic measurements from full spectrum fitting, eliminating the need for computationally intensive simulations.

Contribution

It introduces analytical formulas for uncertainty estimation based on spectrum shape and S/N, validated against Monte-Carlo simulations, applicable to large spectral surveys.

Findings

Formulas agree with Monte-Carlo results across various conditions.

Applicable for quick uncertainty estimates in large spectral datasets.

Provides open-source implementations in IDL and Python.

Abstract

Pixel-space full spectrum fitting exploiting non-linear $\chi^2$ minimization became a \emph{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 perfect agreement for different templates, signal-to-noise ratios and velocity dispersion between 0.5 and 10 times of the instrumental spectral resolution. We provide {\sc IDL} and {\sc 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 or when uncertainties reported by a non-linear minimization algorithms are not considered reliable.