$R_{\rm e}$, or not $R_{\rm e}$: Developing $R_5\equiv R_{-2}$ as a scale radius for galaxy sizes, masses, and mass-to-light ratios

Abstract

The effective half-light radius $R_{\rm e}$ marks an arbitrary 50-per-cent light boundary, and scaling relations involving such {\it effective} radii and their associated surface brightnesses, $\mu_{\rm e}$, systematically (and undesirably) vary as the percentage changes. Here, the projected radius $R_5\equiv R_{-2}$, where the logarithmic slope of the surface-brightness and intensity profile equals $5.00\,\text{mag\,dex}^{-1}$ and $-2$, respectively, and where the luminosity contributed per logarithmic radial interval is maximal, is developed as an alternative. It can be measured non-parametrically or with a parametrized fit. For the S\'ersic $R^{1/n}$ family, the exact relation $R_5=(2n/b_n)^n\,R_{\rm e}$ is derived, with $R_5/R_{\rm e}\rightarrow{\rm e}^{1/6}\approx1.181$ as $n\rightarrow\infty$. Reparameterizing the (now $b_n$-free) $R^{1/n}$ model in terms of the observable pair $(R_5,\mu_5)$ removes the non-linear $R_{\rm e}$--$n$ coupling, and because the local slope is $5\,\text{mag\,dex}^{-1}$ at $R_5$, correlated measurement errors in $R_5$ and $\mu_5$ largely cancel when deriving the inferred total magnitude. Additionally, an exact single-integral identity is provided to relate any projected light fraction to the fraction within a sphere of the same radius. The directly observable $R_5$ is shown to be connected, through a weakly $n$-dependent factor to the anisotropy-insensitive intrinsic radius $r_{-3}$, yielding a refined $n$-dependent Wolf-type mass estimator $M_{-3}$ and spatial mass-to-light ratio $(M_{\rm dyn}/L)_{-3}$. Specifically, $M_{-3}\approx4\,G^{-1}\langle\sigma_{\rm los}^2\rangle\,R_{-2}\approx4.72\,G^{-1}\langle\sigma_{\rm los}^2\rangle\,R_{\rm e}$. Past half-light substitutions in dynamical mass estimators introduce systematic S\'ersic-dependent offsets of 12--18 per~cent in enclosed mass and offsets spanning $>20$ per~cent in the mass-to-light ratio.