A numerical twist on the observational spin parameter, $\lambda_R$

A primary goal of integral field spectroscopic (IFS) surveys is to provide a statistical census of galaxies classified by their internal kinematics. As a result, the observational spin parameter, $\lambda_R$, has become one of the most popular methods of quantifying the relative importance of velocity dispersion and rotation in supporting a galaxy's inner structure. The goal of this paper is to examine the relationship between the observationally deduced $\lambda_R$ and one of the most commonly used theoretical spin parameters in the literature, the Bullock et al. (2001) $\lambda'$. Using a set of $N$-body realisations of galaxies from which we construct mock IFS observations, we measure $\lambda_R$ as an observer would, incorporating the effects of beam smearing and seeing conditions. Assuming parameters typical of current IFS surveys, we confirm that there are strong positive correlations between $\lambda_R$ and measurement radius, and strong negative correlations between $\lambda_R$ and size of the PSF, for late-type galaxies; these biases can be reduced using a recently proposed empirical correction. Once observational biases are corrected for, we find that $\lambda_R$ provides a good approximation to $\sim \sqrt{3}/2 \; \lambda'(\rm R_{\rm eff})$, where $\lambda'$ is evaluated for the galactic stellar component within 1 R$_{\rm eff}$.