R e, or not R e: Developing R 5 ≡ R −2 as a scale radius for galaxy sizes, masses, and mass-to-light ratios

Abstract The effective half-light radius $\text {$R_{\text {e}}$}$ marks an arbitrary 50 per cent light boundary, and scaling relations involving such effective radii and their associated surface brightnesses, μe, systematically (and undesirably) vary as the percentage changes. Here, the projected radius R5 ≡ R−2, where the logarithmic slope of the surface-brightness and intensity profiles equal 5.00 mag dex−1 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 R1/n family, $R_5=(2n/b_n)^n\, \text {$R_{\text {e}}$}$, with $R_5/\text {$R_{\text {e}}$}\rightarrow {\mathrm{e}}^{1/6}\approx 1.181$ as n → ∞. Reparameterizing the (now bn-free) R1/n model in terms of the observable pair (R5, μ5) removes the non-linear $\text {$R_{\text {e}}$}$–n coupling, and because the local slope is 5 mag dex−1 at R5, correlated measurement errors in R5 and μ5 better cancel when deriving the inferred total magnitude (∝μ5 − 5log R5). 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 R5 is shown to be connected, through a weakly n-dependent factor, g(n), to the anisotropy-insensitive internal/spatial radius r−3, yielding a refined n-dependent Wolf-type mass estimator M−3 and spatial mass-to-light ratio (Mdyn/L)−3. Specifically, $M_{-3}\equiv M( 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.