The size–mass and other structural parameter ($n,\mu_{z},R_{z}$) relations for local bulges/spheroids from multicomponent decompositions

SG16 analyzed 66 galaxies with dynamically measured supermassive black hole (SMBH) mass, $M_{\mathrm{BH}}$, from Greenhill et al. (2003); Graham & Scott (2013); Rusli et al. (2013). It contains 47 ETGs and 19 "early-type" spiral galaxies. The analysis was performed on $3.6~{}\mu\rm m$ Spitzer/IRAC (InfraRed Array Camera) images (Fazio et al., 2004; Werner et al., 2004) (see SG16, their Section 2.1). In their subsequent paper, Savorgnan et al. (2016) convert the $3.6~{}\mu\rm m$ luminosity into stellar mass using a constant mass-to-light ($M_{*}/L_{3.6}$, or $\Upsilon_{3.6}$) ratio of $0.6$ (Meidt et al., 2014)³³3Meidt et al. (2014) assumes a Chabrier (2003) initial mass function (IMF), a Bruzual & Charlot (2003) stellar synthesised stellar population model (SSP), and an exponentially declining stellar formation history (SFH)..

D+19 performed multicomponent decompositions for 43⁴⁴4Three of which are bulgeless galaxies. late-type galaxies (LTGs). The sample was selected based on a compilation from Davis et al. (2017): a set of spiral galaxies with their SMBH mass measured via proper stellar motion, stellar dynamics, gas dynamics, and stimulated astrophysical masers (see their Table 1). The spiral galaxy data set consists of the following morphological type: 10 SA, 12 SAB, and 22 SB galaxies (defined by RC3 de Vaucouleurs et al., 1991)⁵⁵5Four of which were discarded by D+19 in the decomposition process.. D+19 analysed the galaxy structures primarily using $3.6~{}\mu\rm m$ images. D+19 analyzed the Spitzer Survey of Stellar Structure in Galaxies ($\rm S^{4}G$, Sheth et al., 2010) and the Spitzer Heritage Archive (SHA)⁶⁶6http://sha.ipac.caltech.edu. Alternative data from the $F814W$-band in Hubble Space Telescope (HST)⁷⁷7https://mast.stsci.edu/ images and $K_{\mathrm{s}}$-band images from the Two Micron All Sky Survey (2MASS) Large Galaxy Atlas (LGA, Jarrett et al., 2003) were used in the absence of Spitzer images or when greater special resolution was required (see Section 2.2 in D+19). In LTGs, because the dust in the disc glows in the infrared, D+19 used a smaller stellar mass-to-light ratio $M_{*}/L_{3.6}=0.453$, in accordance with data from Querejeta et al. (2015), to compensate for the potential overestimation of stellar mass due to non-stellar luminosity in LTGs (see Section 2.8 in D+19)⁸⁸8D+19 used the median $(L_{*}/L_{\mathrm{obs}})_{\mathrm{IRAC1}}=0.755$ from Fig. 10 of Querejeta et al. (2015) and the color-independent $M_{*}/L_{\mathrm{IRAC1}}=0.60$ from Meidt et al. (2014) to estimate the median value of $M_{*}/L_{3.6}=0.453$. See Section 2.8 in D+19 for details.. For the sake of consistency, we only use their 26 galaxies with Spitzer images.

Building upon the 47 ETGs from SG16, S+19 added 41 more ETGs, including seven remodelled ETGs (A3565 BCG, NGC 524, NGC 2787, NGC 1374, NGC 4026, NGC 5845, and NGC 7052). For these seven remodelled galaxies, we use the parameters provided by S+19. Among the newly added 41 galaxies, based on their decomposition, are 15 E, 3 ES, and 23 S0 galaxies, respectively. S+19 data sources are from: $3.6~{}\mu\rm m$-band images in Spitzer/IRAC (Sheth et al., 2010), r-band images in Sloan Digital Sky Survey (SDSS, York et al., 2000), $K_{s}$-band images from 2MASS/LGA (Jarrett et al., 2003), and $F814W$-band in SHA. As was done with the D+19 data, we only use the Spitzer galaxies from S+19, which amounts to 33 galaxies.

Similar to what was done in SG16, the spheroid $3.6~{}\mu\rm m$ luminosity from Spitzer/IRAC and SHA images are converted into stellar mass with a constant mass-to-light ratio $M_{*}/L_{3.6}=0.6$ (Meidt et al., 2014, see their Section 3.3 for details).

H+22 selected a mass- and volume-limited sample of massive ($M_{\mathrm{*,gal}}>10^{11}~{}\rm M_{\odot}$) galaxies at $\rm Distance<110~{}Mpc$ from the NASA-Sloan catalogue⁹⁹9http://www.nsatlas.org/ based on SDSS photometry (York et al., 2000; Aihara et al., 2011) in the $i$-band. The H+22 sample contains 103 massive galaxies with $M_{*}/\rm M_{\odot}>6.7\times 10^{10}$ (based on the Roediger & Courteau (2015) $M_{*}/L_{i}$ ratio) but is otherwise nondiscriminatory to galaxy morphology. According to H+22’s decomposition, the galaxies are assigned new morphologies. There are 13 ellipticals (E+ES)¹⁰¹⁰10Here, the elliptical galaxies are composed of two populations: 11 conventional ellipticals (E) and two ellicular (ES) galaxies. ES galaxies (Liller, 1966; Savorgnan & Graham, 2016b; Graham, 2019b) are ellipitcals that contain an intermediate-scale disc. The name ”ellicular” (”elliptical+lenticular”) was introduced in Graham et al. (2016)., 50 lenticular (S0) and 40 spiral (S) galaxies in this sample. H+22 use the galaxies’ $g$- and $i$-band photometry from SDSS to estimate the $M_{*}/L_{i}$ ratio. In this paper, we will use the $M_{*}/L_{i}$ ratio provided by the Roediger & Courteau (2015) prescription, which provides agreement with the Spitzer-derived masses from the three other works (Sahu et al., 2019, see their Section 3.4) ¹¹¹¹11The $M_{*}/L$ ratio utilised by the four works assume a Chabrier (2003) initial mass function and the stellar population model from Bruzual & Charlot (2003) (see Meidt et al., 2014; Roediger & Courteau, 2015).. After removing the non-Spitzer galaxies from the previous three studies and combining them with H+22, we have, in total, 202 galaxies for analysis.

In this section, we develop the spheroid $\mathfrak{M}_{\mathrm{Sph}}$–$\mu_{\mathrm{Sph}}$ and $\mathfrak{M}_{\mathrm{Sph}}$–$R_{\mathrm{Sph}}$ relations, with a different definition for the radial scale parameter, $R_{z,\rm Sph}$, and thus also the intensity scale parameter ($I_{z}:\mu_{z}\equiv-2.5\log I_{z}$). The canonical approach with the Sérsic model is to only use the effective half-light radius $R_{\mathrm{e}}$. We examine how the various relations change when a different fraction ($z$, from 0 to 1) of the total luminosity, $L_{\mathrm{tot}}$, is used to define the equally-valid alternate scale radii, $R_{\mathrm{z}}$.

In the left-hand panel of Fig. 3, we show $\mathfrak{M}_{\mathrm{Sph}}$–$\log(R_{z,\rm Sph})$ using a radius $R_{z,\rm Sph}$ encapsulating 50 percent ($z=0.5$, red points), 5 percent ($z=0.05$, black points) and 95 percent ($z=0.95$, blue points) of the light. We illustrate the trends for each relation with green dashed lines. The green points are the median value within an absolute magnitude interval of $\Delta\mathfrak{M}_{\mathrm{Sph}}=1~{}\rm mag$ from $\mathfrak{M}_{\mathrm{Sph}}=-18\rm~{}mag$ to $-25\rm~{}mag$. In the right-hand panel of Fig. 3, the $\mathfrak{M}_{\mathrm{Sph}}$–$\mu_{z,\rm Sph}$ distributions are presented, where $\mu_{z,\rm Sph}$ is the surface brightness at $R_{z,\rm Sph}$. For $z=0$ (purple points), it recreates the same plot as shown in the middle panel of Fig. 1. When $z=0$, the $\mathfrak{M}_{\mathrm{Sph}}$–$\mu_{z,\rm Sph}$ relation presents a positive trend, i.e., a brighter spheroid has a more luminous central surface brightness value. For $z=0.5$ and $z=0.95$, both relations change directions and have their absolute magnitude negatively correlating with the surface brightness, $\mu_{\mathrm{0.5,Sph}}$ (or $\mu_{\mathrm{e}}$) and $\mu_{\mathrm{0.95,Sph}}$. In both panels, the spheroids have a continuous relations between $\mathfrak{M}_{\mathrm{Sph}}\sim-18$ to $-25$ (in $i$-band AB mag). The spheroid population appears to be continuous regardless of the radius $R_{z,\rm Sph}$ used.

In an effort to extract physical meaning from these relations, as well as comparing the data from H+22 with SG16, D+19 and S+19, we converted the spheroids’ absolute magnitude, $\mathfrak{M}_{\mathrm{Sph}}$, into stellar mass, $M_{\mathrm{*,Sph}}$, and central surface brightness, $\mu_{0}$, into central projected mass density, $\Sigma_{\mathrm{0,Sph}}$. Using the data from the four sources, we plotted the spheroid stellar mass, $\log(M_{\mathrm{*,Sph}})$, versus Sérsic index, $\log(n_{\mathrm{Sph}})$, $\log(\Sigma_{\mathrm{0,Sph}})$, and $\log(R_{\mathrm{e,Sph}})$ in the left-hand, middle, and right-hand panel in Fig. 4, respectively.

Kelvin et al. (2012, their Fig.22) report how the galaxy half-light radii change with wavelength, which is related to the bulge/disc transition, such that the (small) different colours between the bulge and disc can produce a change in the galaxy size as a function of wavelength (Kennedy et al., 2016). Kelvin et al. (2012) report that the drop in the median galaxy size from the $i$-band (0.6 micron) to the $J$, $H$, and $K$-band (1-2 micron) is $\sim$4.5 kpc to 3.5 kpc, before the decline in size stabilises beyond 1 micron. This is a 22% drop, or 0.11 dex, for the galaxy half-light size. Vulcani et al. (2014) performed the same analysis, and for their ‘red galaxies’, the median galaxy size dropped from 5.5 kpc to 4 kpc (0.14 dex) when going from the $i$-band (0.6 micron) to the $H$-band (1.6 micron). However, what is required is the change in the bulge/spheroid size rather than the change in the galaxy (bulge+disc+bar) size. As discussed in Hon et al. (2022), for a uniform stellar population within the spheroidal component, there should be no systematic change in the spheroid’s size with wavelength, modulo the impact of dust. We are, however, at present unable to quantify this change from the $i$-band to 3.6 microns, and as such, have not implemented any rescaling of the spheroid sizes.

The correlation coefficients for the entire sample, $r_{\mathrm{p}}(\rm All)$ and $r_{\mathrm{s}}(\rm All)$, where "All" now includes all four galaxy samples (SG16, D+19, S+19, and H+22) in both $\log(M_{\rm*,Sph})$–$\log(n_{\mathrm{Sph}})$, $\log(M_{\rm*,Sph})$–$\log(\Sigma_{\mathrm{0,Sph}})$, and $\log(M_{\rm*,Sph})$–$\log(R_{\mathrm{e,Sph}})$ planes are comparable to that in the $\mathfrak{M}_{\rm Sph}$–$\log(n_{\mathrm{Sph}})$, $\mathfrak{M}_{\rm Sph}$–$\log(\mu_{\mathrm{0,Sph}})$, and $\mathfrak{M}_{\rm Sph}$–$\log(R_{\mathrm{e,Sph}})$ planes from Fig. 1, respectively. The inclusion of the data from SG16, D+19 and S+19 does not increase the correlation strength. In the $\log(M_{\rm*,Sph})$–$\log(\Sigma_{\mathrm{0,Sph}})$ plane, when we divide the sample into ellipticals and disc-galaxies, both subgroups return an extremely low correlation, with $r_{\mathrm{p}}(E+ES)=0.19$ and $r_{\mathrm{p}}(S0+S)=0.44$. The weak correlations between the spheroid luminosity (subsequently its stellar mass) and projected central density persists even after being converted into physical quantities, $M_{*}$ and $\Sigma_{\mathrm{0,Sph}}$. Sahu et al. (2022) reported a similar Pearson correlation coefficient, $r_{\mathrm{p}}(All)=0.57$ (their table 1), in the $M_{\mathrm{BH}}$–$\Sigma_{\mathrm{0,Sph}}$ plane. Due to the coevolution between the SMBH and the spheroid, $M_{\mathrm{BH}}$ and $M_{\mathrm{Sph}}$ are strongly correlated. Therefore, the similarity between the $\log(M_{\mathrm{BH}})$–$\log(\Sigma_{\mathrm{0,Sph}})$ and $\log(M_{\mathrm{*,Sph}})$–$\log(\Sigma_{\mathrm{0,Sph}})$ relation is somewhat expected.

In Fig. 5, we convert the $\log(M_{\mathrm{*,Sph}})$–$\log(R_{\mathrm{e,Sph}})$ and $\log(M_{\mathrm{*,Sph}})$–$\log(\Sigma_{\mathrm{0,Sph}})$ into the $\log(M_{\mathrm{*,Sph}})$–$\log(R_{z,\rm Sph})$ and $\log(M_{\mathrm{*,Sph}})$–$\log(\Sigma_{z,\rm Sph})$ distributions by using different scale radii, encompassing light fractions $z=0.05,0.5,$ and $0.95$. The behaviour is similar to that in Fig. 3. The $\log(M_{\mathrm{*,Sph}})$–$\log(R_{z,\rm Sph})$ distributions maintain a somewhat linear trend with positive slope while the $\log(M_{\mathrm{*,Sph}})$–$\log(\Sigma_{z,\rm Sph})$ distributions change slope with different values of $z$.

Some studies have suggested that the average projected mass density within a small fixed radius, say 1 kpc, $\langle\Sigma\rangle_{\mathrm{1~{}kpc}}$, or 5 kpcs, $\langle\Sigma\rangle_{\mathrm{5~{}kpc}}$, are great estimators to track the growth of SMBHs (e.g., Barro et al., 2017; Dekel et al., 2019; Ni et al., 2021; Sahu et al., 2022). By proxy, it will also pertain to the growth of spheroids, given the connection between the two. Indeed, the $M_{\mathrm{BH}}$–$\langle\Sigma\rangle_{\mathrm{5~{}kpc,Sph}}$ plane exhibits a log-linear relation ($r_{\mathrm{p}}=0.83$ and $r_{\mathrm{s}}=0.84$, see Sahu et al., 2022). However, it is noteworthy that at this fixed radius, the quantity $\langle\Sigma\rangle_{\mathrm{5~{}kpc,Sph}}$ might be sampling a different percentage of light in different spheroids. For instance, for a bulge with $R_{\mathrm{e}}=6~{}\rm kpc$, $\langle\Sigma\rangle_{\mathrm{5~{}kpc,Sph}}$ captures nearly half of the light, while for a spheroid like an E galaxy with $R_{\mathrm{e}}=12~{}\rm kpc$, it captures a small fraction. In the interest of studying how the average projected mass density¹⁴¹⁴14$\langle\Sigma\rangle_{z,\rm Sph}$ can be calculated by subsituting $\langle\mu\rangle_{z,\rm Sph}$ (see equation 6) into equation 8., $\langle\Sigma\rangle_{z,\rm Sph}$, varies as one includes a different percentage of light within the radius $R_{z,\rm Sph}$, we plot $M_{\mathrm{*,Sph}}$ versus $\langle\Sigma\rangle_{z,\rm Sph}$ in Fig. 6. The $\log(M_{\mathrm{*,Sph}})$–$\log(\langle\Sigma\rangle_{z,\rm Sph})$ relations present a negative trend for all $z$. This result is consistent with Sahu et al. (2022) who found a negative correlation in the $\log(M_{\mathrm{BH}})$–$\log(\langle\Sigma\rangle_{\mathrm{e,Sph}})$ ($=\log(\langle\Sigma\rangle_{\mathrm{0.5,Sph}})$) relation (see their Fig. 6). As the value of $z$ increases, the $\log(M_{\mathrm{*,Sph}})$–$\log(\langle\Sigma\rangle_{z,\rm Sph})$ relation appears increasingly more linear and has a steeper slope. In addition, it appears that the correlation strength of the distributions varies with $z$. At $z=0.05$, the correlation is non-existent ($r_{p},r_{s}\sim-0.08$) while at $z=0.95$, $\log(M_{\mathrm{*,Sph}})$ and $\log(\langle\Sigma\rangle_{z,\rm Sph})$ have a moderate correlation ($r_{p},r_{s}\sim-0.68$).

Since $\langle\Sigma\rangle_{z,\rm Sph}$ is calculated by substituting $\mu_{z,\rm Sph}$ with $\langle\mu\rangle_{z,\rm Sph}$ in equation 8, the varying trends in the $\log(M_{\mathrm{*,Sph}})$–$\log(\langle\Sigma\rangle_{z,\rm Sph})$ plane can be explained by looking into the formulation in equation 6. The average surface brightness, $\langle\mu\rangle_{z,\rm Sph}$, is a linear combination of the surface brightness, $\mu_{z}$, in the first term and the shape function, $B_{z}(n)$, in the second term. Because $B_{z,\rm Sph}(n)$ is purely a function of Sérsic index $n$, it inherits the moderately strong linearity in the $\log(M_{\mathrm{*,Sph}})$–$\log(n_{\mathrm{Sph}})$ plane (see Fig. 4). We demonstrate such a strong trend in Fig. 7, where the spheroid stellar mass ($M_{\mathrm{*,Sph}}$) is plotted against $-2.5~{}\log[B_{z,\rm Sph}(n_{\rm Sph})]$. Indeed, the $\log(M_{\mathrm{*,Sph}})$–$(-2.5\log[B_{z,\rm Sph}])$ relations have a rather consistent anti-correlation¹⁵¹⁵15The negative sign is due to the multiplier $-2.5$. Spheroid stellar mass, $\log(M_{\mathrm{*,Sph}})$ correlates positively with the shape function $\log(B_{z,\rm Sph})$. across $z$, with $r_{p},r_{s}\sim-0.7$. Hence, the variation in correlation strength in the $\log(M_{\mathrm{*,Sph}})$–$\log(\langle\Sigma\rangle_{z,\rm Sph})$ plane is because of the inherent non-linearity in the $\log(M_{\mathrm{*,Sph}})$–$\rm\mu_{z,\rm Sph}$ relation (see Fig. 1). This latter trend dampens the correlation strength of the $\log(M_{\mathrm{*,Sph}})$–$(-2.5\log[B_{z,\rm Sph}])$ distribution and results in an changed trend as $z$ changes. It is clear that the choice of the light fraction value $z$ significantly affects the shape and slope of the scaling relations.

Here, we explore how the correlation strength changes with different light fractions $z$. In Fig. 8, we plot the correlation strength, $r_{p}$ and $r_{s}$, in the upper and lower panel, respectively, as a function of the fraction of spheroid light, $z$, contained within the radius $R_{z,\rm Sph}$, for various scaling relations: $\log(M_{\mathrm{*,Sph}})$–$\log(R_{z,\rm Sph})$, $\log(M_{\mathrm{*,Sph}})$–$\log(\Sigma_{z,\rm Sph})$, $\log(M_{\mathrm{*,Sph}})$–$\log(\langle\Sigma\rangle_{z,\rm Sph})$, and $\log(M_{\mathrm{*,Sph}})$–$(-2.5\log[B_{z.\rm Sph}(n)])$, where all involve the stellar mass. The $\log(M_{\mathrm{*,Sph}})$–$\log(R_{z,\rm Sph})$ relations (green line) are consistently strongly coupled regardless of the value of $z$. The $\log(M_{\mathrm{*,Sph}})$–$\log(\Sigma_{z,\rm Sph})$ relation (blue line) is positively correlated at $z=0$, but it becomes negatively correlated at larger $z$, a feature that can be seen in the right-hand panel of Fig. 5. For the $\log(M_{\mathrm{*,Sph}})$–$\log(\langle\Sigma\rangle_{z,\rm Sph})$ relation at $z=0.05$ (first point on the red line), it has an extremely low correlation ($r_{\mathrm{p}},r_{\mathrm{s}}\sim 0$) but as $z$ increases, it becomes increasingly anti-correlated. Finally, the $\log(M_{\mathrm{*,Sph}})$–$(-2.5\log[B_{z,\rm Sph}(n_{\rm Sph})])$ relations (purple line) exhibit a very consistent anti-correlation trend across all $z$ with $r_{p},r_{s}\sim-0.7$.

Fig. 8 provides an important insight into spheroid scaling relations that, in general, the surface brightness, $\mu_{z,\rm Sph}$, and its derived physical quantities ($\Sigma_{z,\rm Sph}$ and $\langle\Sigma\rangle_{z,\rm Sph}$) are comparatively weak estimators for spheroid stellar mass. While the scaling relations become more reliable at higher light fraction $z$, the correlation coefficients are limited to $\sim 0.7$. The variability in correlation strength across $z$ makes these scaling relations inherently biased by choice of $R_{z,\rm Sph}$.

In Section 3.1, we found that the Sérsic index ($n_{\mathrm{Sph}}$) of a spheroid is a slightly better predictor for a spheroid stellar mass than $\mu_{0}$. Here, we show that the Sérsic shape function, $B_{z,\rm Sph}(n_{\rm Sph})$, is also a decent spheroid mass predictor across all $z$ ($r_{p},r_{s}\sim 0.7$). Similarly, $R_{z,\rm Sph}$ also has a consistently strong log-linear relation with $M_{\mathrm{*,Sph}}$ across all $z$ but with an even higher correlation strength ($r_{p},r_{s}\sim 0.9$). Hence, we advocate using $R_{z,\rm Sph}$ as the primary predictor to spheroid stellar mass ($M_{\mathrm{*,Sph}}$) and $n_{\mathrm{Sph}}$ (or $B_{z,\rm Sph}(n_{\rm Sph})$) as the secondary. Of course, $R_{z,\rm Sph}$, $n_{\mathrm{Sph}}$, and $\mu_{0,\rm Sph}$ perfectly define the stellar luminosity of the spheroid.

With the above observation that spheroid size ($R_{z,\rm Sph}$) is the most consistent and reliable predictor for spheroid stellar mass ($M_{\mathrm{*,Sph}}$) across all $z$, we proceed to investigate the spheroid size-mass relation for our expanded sample. In Fig. 9, we present the equivalent axis effective radius ($R_{\mathrm{e,Sph}}$) versus the stellar mass ($M_{\mathrm{*,Sph}}$) for the local spheroids from SG16, D+19, S+19, and H+22. In the left-hand panel of Fig. 9, the data is separated by the morphological type of the host galaxies. There is a significant overlap between S0 and S spheroids. The elliptical (E) and ellicular (ES) galaxies are the largest and most massive, occupying the region $R_{\mathrm{e,Sph}}\gtrsim 2~{}\rm kpc$ and $M_{\mathrm{*,Sph}}\gtrsim 10^{11}~{}\rm M_{\odot}$. They are also more likely to have a depleted stellar core: 19 of them are well described by the core-Sérsic model. Spheroids embedded in S0 and S galaxies reside within the range $0.2~{}\lesssim R_{\mathrm{e,Sph}}/\rm kpc\lesssim~{}5$ and $3\times 10^{9}\lesssim~{}M_{\mathrm{*,Sph}}/\rm M_{\odot}~{}\lesssim 2\times 10^{11}$.

In the right-hand panel of Fig. 9, the data points are colour-coded for their respective sources. The size–mass relation shows strong agreement between the four data sets despite having different personnel conducting the decomposition. The big spheroids follow an established trend set by elliptical galaxies (not to be confused with ETGs). Although there is an important upturn at high masses, which we speculate is due to E+E mergers rather than E built from S0+S0 mergers, we fit a single power-law to the relation using the Linmix fitting routine (Kelly, 2007):

where $S$ is the slope and $int.$ is the y-intercept of the relation.

Linmix calculates the best-fit regression line via maximising the Bayesian likelihood function (see equation 16 in Kelly, 2007). The prior distribution of the independent variable is assumed to be a Gaussian mixture. Such treatment allows greater flexibility in dealing with data heteroscedasticity, i.e., the non-uniform variance in each data point. We implemented 5000 iterations of Monte Carlo Markov Chain (MCMC) sampling to maximise the likelihood function. It is unclear in nature whether the size of a galactic structure depends on its stellar mass or vice versa. For this reason, we have performed the fitting process with $M_{\mathrm{*,Sph}}$ treated as the independent variable for the relation $R_{\mathrm{e,Sph}}(M_{\mathrm{*,Sph}})$ and $R_{\mathrm{e,Sph}}$, as the independent variable for the relation $M_{\mathrm{*,Sph}}(R_{\mathrm{e,Sph}})$. In the right-hand panel of Fig. 9, we depict the regression lines for fitting $R_{\mathrm{e,Sph}}(M_{\mathrm{*,Sph}})$ and $M_{\mathrm{*,Sph}}(R_{\mathrm{e,Sph}})$ with red and cyan shaded area, respectively. The optimal fits among these regressions are shown as a dash-dotted and dashed line, respectively. They are the following relations.

From these regression lines, we calculated the bisector line, shown as the solid black line in the panel:

with a scatter of $\Delta_{rms}=0.24~{}\rm dex$ in the vertical direction. All three scaling relations are valid but with differing assumptions. Future work may choose the appropriate relation for their purpose. However, since the spheroid size and mass are highly correlated ($r_{p}\sim 0.9$), the three scaling relations are similar.

For $R_{\mathrm{e}}\lesssim 10~{}\rm kpc$, a log-linear relation appears adequate to describe the size-mass distribution of local spheroids. An unrelated log-linear relation was also demonstrated in Sahu et al. (2020, see their Fig. 8) but is now abundantly clear with the additional 103 spheroids from H+22. As noted above, there is a slight departure from a log-linear relation at the high-mass end ($M_{\mathrm{*,Sph}}>5\times 10^{11}~{}\rm M_{\odot}$), where, at fixed mass, massive spheroids have a larger radius than the log-linear relation suggests. We speculate that this might also partly be due to the influence of intracluster light (ICL) in the brightest cluster galaxies (BCGs)¹⁶¹⁶16There are three, four, and three BCGs in SG16, S+19, and H+22, respectively. All of which have a galaxy stellar mass $M_{\mathrm{*,gal}}>4\times 10^{11}~{}\rm M_{\odot}$.. The most massive ($M_{*}\gtrsim 4\times 10^{11}~{}\rm M_{\odot}$) galaxies tend to be BCGs living in the centre of clusters with extended stellar halos. As a result, the ICL accumulates around the BCG. When modelling the galaxies with a single Sérsic model, we could be slightly biased by the ICL and overestimate their intrinsic size. For E+E mergers in which the velocity dispersion, $\sigma$, may not increase, the virial theorem ($M_{*}\propto\sigma^{2}R$) dictates $M_{*}\propto R$. For reference, we have added solid blue lines of slope $S=1$ in Fig. 9.

We present a few supplementary scaling relations: $\log(n_{\mathrm{Sph}})$–$\log(M_{\mathrm{*,Sph}})$ and $\log(B_{\mathrm{e,Sph}})$–$\log(M_{\mathrm{*,Sph}})$, $\log(\Sigma_{\mathrm{0,Sph}})$–$n_{\mathrm{Sph}}$, and $\log(R_{\mathrm{e,Sph}})$–$\log(n_{\mathrm{Sph}}$). These extra relations have a moderate to high linearity ($r_{p}\approx 0.7-0.9$) and, therefore, are suited to be secondary estimators for spheroid structures. Our intention here is to merely demonstrate and touch on the potential importance of these relations. A deeper exploration of their physical meaning shall be conducted in future works.

Sérsic index, $n$, and its derived ‘shape function’, $B_{z}(n)$, have a weaker correlation with stellar mass than the size–mass relation, with $r_{p},r_{s}\sim 0.72$ versus $\sim 0.9$. The left- and right-hand panel of Fig. 10 depict the $\log(n_{\rm Sph})$–$\log(M_{\mathrm{*,Sph}})$ plane and the $\log[B_{\mathrm{e,Sph}}(n_{\rm Sph})]$–$\log(M_{\mathrm{*,Sph}})$ plane¹⁷¹⁷17$B_{z}(n)$ (equation 7) depends on $b_{n,z}$ and thus $R_{z}$, Here, we set $R_{z}$ to $R_{\mathrm{e}}$, i.e. $z=0.5$., respectively. We performed multiple linear fits on our sample, similar to what was done for the size–mass relation in Section 3.5.1. For the $\log(n_{\mathrm{Sph}})$–$\log(M_{\mathrm{*,Sph}})$ relation, we have the following bisector relations: