$R_{\mathrm{e}}$, or not $R_{\mathrm{e}}$: 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 $R_e$ marks an arbitrary 50-per-cent light boundary, and scaling relations involving such `effective' radii and their associated surface brightnesses, ${\mu }_e$, 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 mag/dex and -2, respectively, and where the luminosity per logarithmic radial interval is maximal, is developed as an alternative. It can be measured non-parametrically or with a parametrized fit. For the Sérsic $R^{1/n}$ family, $R_5=(2n/b_n{)}^n{R}_e$, with $R_5/R_e$->e${}^{1/6}\approx$1.181 as n->$\infty$. Reparameterizing the (now $b_n$-free) $R^{1/n}$ model in terms of $R_5$ and ${\mu }_5$ removes the non-linear $R_e$-n coupling, and because the local slope is 5 mag/dex at $R_5$, correlated measurement errors in $R_5$ and ${\mu }_5$ better cancel when deriving the inferred total magnitude ($\propto {\mu }_5-5\log R_5$). Additionally, an exact single-integral identity is provided to relate any projected light percentage to the percentage within a sphere of the same radius. The scale radius $R_5$ is shown to be connected, through a weakly n-dependent factor to the anisotropy-insensitive internal radius $r_{-3}$, yielding a refined n-dependent Wolf-type mass estimator $M_{-3}$ and spatial mass-to-light ratio $(M_{dyn}/L{)}_{-3}$. Specifically, $M_{-3}\approx 4G^{-1}<{\sigma }_{los}^2>R_{-2}\approx 4.72G^{-1}<{\sigma }_{los}^2>R_e$. Past half-light substitutions in dynamical mass estimators introduce systematic Sérsic-dependent offsets of 12-18 per~cent in enclosed mass and offsets spanning >20 per~cent in the mass-to-light ratio within $r_{-3}$. Dynamical mass estimators appropriate for (dark matter)-free stellar systems are also provided, and all relations are implemented in the companion Python package R5Tools.

arXiv comment22 pages, 3 figures, 5 tables. To appear in MNRAS. Companion Python package R5Tools available at https://github.com/A-Graham/R5Tools