$R_{\rm e}$. I. Understanding galaxy sizes, associated luminosity densities, and the artificial division of the early-type galaxy population

Abstract: The deceptive simplicity of the radius enclosing an arbitrary 50 percent of a galaxy's light has hamstrung the understanding of early-type galaxies (ETGs). Half a century ago, using the "effective half-light radii" $R_e$ from de Vaucouleurs' $R^{1/4}$ model, S\'ersic reported that bright ETGs follow the relation $\mathfrak{M}B\propto2.5\log R_e$; and consequently one has that $\langle\mu\rangle_e\propto2.5\log R_e$ and $\mu_e\propto2.5\log R_e$, where $\mu_e$ and $\langle\mu\rangle_e$ are the effective surface brightness at $R_e$ and the mean effective surface brightness within $R_e$, respectively. S\'ersic additionally observed an apparent transition which led him to advocate for a division between what he called dwarf and giant ETGs; a belief frequently restated to occur at $\mathfrak{M}_B\approx-18$ mag or S\'ersic $n\approx 2.5$. Here, the location of this false dichotomy is shown to change by more than 3 mag simply depending on the arbitrary percentage of light used to quantify a galaxy's size, voiding claims for different formation physics operating on ETGs brighter and fainter than $\mathfrak{M}_B\approx-18$ mag. This is of further importance because quantities such as dynamical mass $\sigma^2R/G$, gravitational binding energy $GM^2/R$, acceleration $GM/R^2$, and the "Fundamental Plane" depend systematically on the arbitrary percentage of light used to define $R$, with implications for dark matter estimates, galaxy formation theories, compact massive galaxies, studies of peculiar velocity flows, and more. Finally, some of the vast literature which has advocated for segregating the ETG population at $\mathfrak{M}_B\approx-18$ mag ($M\approx1$-$2\times10^{10}\,M{\odot}$) is addressed, and it is revealed how this pervasive mindset has spilled-over to influence both the classical bulge versus pseudobulge debate and recently also correlations involving supermassive black hole masses.