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), $R_{\rm e,Sph}-M_{\rm *,Sph}$, relation and spheroid structural parameters for 202 local (predominantly $\lesssim 110~\rm Mpc$) galaxies spanning $ M_*\sim 3\times10^{9}-10^{12}~\rm M_{\odot}$ and $ 0.1 \lesssim R_{\rm e, Sph}\lesssim32~\rm kpc$ from multicomponent decomposition. The correlations between the spheroid S\'ersic index ($n_{\rm Sph}$), central surface brightness ($\mu_{\rm 0, Sph}$), effective half-light radius ($R_{\rm e, Sph}$), absolute magnitude ($\mathfrak{M}_{\rm Sph}$) and stellar mass ($M_{\rm *,Sph}$) are explored. We also investigate the consequences of using different scale radii, $R_{z,\rm Sph}$, encapsulating a different fraction ($z$, from 0 to 1) of the total luminosity. The correlation strengths for projected mass densities, $\Sigma_z$ and $\langle \Sigma \rangle_z$, vary significantly with the choice of $z$. Spheroid size ($R_{\rm z, Sph}$) and mass ($M_{\rm *,Sph}$) are strongly correlated for all light fractions $z$. We find: $\log(R_{\rm e,Sph}/\rm kpc) = 0.88\log(M_{\rm *,Sph}/\rm M_{\odot})-9.15$ with a small scatter of $\Delta_{rms} = 0.24~\rm dex$. This result is discussed relative to the \textit{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 $\sim3$, steeper than reported bulge size-mass relations, and with bulge sizes at $M_{\rm *,sph}\sim 3\times10^9~M_\odot$ which are 2 to 3 times smaller. Finally, we show that the local spheroids align well with quiescent galaxies at $z\sim1.25$--$2.25$. In essence, local spheroids and high-$z$ quiescent galaxies appear structurally similar, likely dictated by the virial theorem.