The size--mass and other structural parameter ($n,{\mu }_z,R_z$) relations for local bulges/spheroids from multicomponent decompositions

We analyse the bulge/spheroid size-(stellar mass), $R_{\mathrm{e},\mathrm{S}\mathrm{p}\mathrm{h}}-M_{\ast ,\mathrm{S}\mathrm{p}\mathrm{h}}$, relation and spheroid structural parameters for 202 local (predominantly $\lesssim 110\mathrm{M}\mathrm{p}\mathrm{c}$) galaxies spanning $M_{\ast }\sim 3\times {10}^9-{10}^{12}{\mathrm{M}}_{\odot }$ and $0.1\lesssim R_{\mathrm{e},\mathrm{S}\mathrm{p}\mathrm{h}}\lesssim 32\mathrm{k}\mathrm{p}\mathrm{c}$ from multicomponent decomposition. The correlations between the spheroid Sérsic index ($n_{\mathrm{S}\mathrm{p}\mathrm{h}}$), central surface brightness (${\mu }_{0,\mathrm{S}\mathrm{p}\mathrm{h}}$), effective half-light radius ($R_{\mathrm{e},\mathrm{S}\mathrm{p}\mathrm{h}}$), absolute magnitude (${\mathfrak{M}}_{\mathrm{S}\mathrm{p}\mathrm{h}}$) and stellar mass ($M_{\ast ,\mathrm{S}\mathrm{p}\mathrm{h}}$) are explored. We also investigate the consequences of using different scale radii, $R_{z,\mathrm{S}\mathrm{p}\mathrm{h}}$, encapsulating a different fraction ($z$, from 0 to 1) of the total luminosity. The correlation strengths for projected mass densities, ${\mathrm{\Sigma }}_z$ and $⟨\mathrm{\Sigma }{⟩}_z$, vary significantly with the choice of $z$. Spheroid size ($R_{\mathrm{z},\mathrm{S}\mathrm{p}\mathrm{h}}$) and mass ($M_{\ast ,\mathrm{S}\mathrm{p}\mathrm{h}}$) are strongly correlated for all light fractions $z$. We find: $\log (R_{\mathrm{e},\mathrm{S}\mathrm{p}\mathrm{h}}\mathrm{/}\mathrm{k}\mathrm{p}\mathrm{c})=0.88\log ({\mathrm{M}}_{\ast ,\mathrm{S}\mathrm{p}\mathrm{h}}\mathrm{/}{\mathrm{M}}_{\odot })-9.15$ with a small scatter of ${\mathrm{\Delta }}_{rms}=0.24\mathrm{d}\mathrm{e}\mathrm{x}$. This result is discussed relative to the $\mathit{\text{curved}}$ size-mass relation for early-type galaxies due to their discs yielding larger galaxy radii at lower masses. Moreover, the slope of our spheroid size-mass relation is a factor of $\sim 3$, steeper than reported bulge size-mass relations, and with bulge sizes at $M_{\ast ,\mathrm{s}\mathrm{p}\mathrm{h}}\sim 3\times {10}^9{M}_{\odot }$ which are 2 to 3 times smaller. Finally, we show that the local spheroids align well with quiescent galaxies at $z\sim 1.25$–$2.25$. In essence, local spheroids and high-$z$ quiescent galaxies appear structurally similar, likely dictated by the virial theorem.