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