2.1. de Vaucouleurs’ $R^{1/4}$ model

First in French (de Vaucouleurs, Reference de Vaucouleurs1948) then in English, de Vaucouleurs (Reference de Vaucouleurs1953) presented an empirical function that was to become known as the $R^{1/4}$ model due to how the projected (on the plane of the sky) intensity profile I(R) depends on the projected radius R raised to the 1/4 power. This mathematical model can be expressed as

(1) \begin{equation} I(R)=I_0\exp\left[ -b\left(\frac{R}{R_{\rm s}}\right) ^{1/4}\right] = \frac{I_0}{\left( {\rm e}^{b}\right)^{\left(R/R_{\rm s} \right)^{1/4}}}, \end{equation}

where $R_{\rm s}$ is a scale radius, $I_0$ is a scale intensity at $R=0$ , and b is a constant that shall be explained below. Given that the galaxies do not have clear edges — and in the middle of the 20th century it was not known how their radial profiles behaved at large radii — the practice was to extrapolate one’s adopted model to infinity in order to determine a galaxy’s total luminosity.

The projected luminosity (from three dimensions to two dimensions onto the plane of the sky) interior to a circle of radius R is determined by integrating the intensity over the enclosed area, such that

(2) \begin{equation} L({\lt}R) = \int_0^R I(R^{\prime})2\pi R^{\prime} {\rm d}R^{\prime}. \end{equation}

Using the substitution $x=b(R/R_{\rm s})^{1/4}$ in equation (1), the above integral reduces to

(3) \begin{equation} L({\lt}R)=\frac{I_0 R_{\rm s}^{2}8\pi}{b^8}\gamma (8,x), \end{equation}

where $\gamma $ (8,x) is the incomplete gamma function defined by

(4) \begin{equation} \gamma (8,x)=\int ^{x}_{0} {\rm e}^{-t}t^{8-1}{\rm d}t. \label{eqn4} \end{equation}

As noted, the total luminosity is obtained by integrating to infinity, in which case $\gamma (8,x)$ is replaced with the complete gamma function, $\Gamma (8)$ , and one has that

(5) \begin{equation} L_{\rm tot} = \frac{I_0 R_{\rm s}^{2}8\pi}{b^8}\Gamma (8). \label{eqn5} \end{equation}

Now, here is where things can, and did, become arbitrary. Gerard de Vaucouleurs elected to define the radius $R_{\rm s}$ such that it enclosed 50% of the total light $L_{\rm tot}$ . He did this by determining the value of b required to balance the equation

(6) \begin{equation} \gamma (8,b) = 0.5\,\Gamma (8). \label{eqn6} \end{equation}

With $b=-7.669$ , the projected radius $R_{\rm s}$ effectively encloses half of the model’s total light, and it was subsequently denoted $R_{\rm e}$ and referred to as the ‘effective half light radius’. The $R^{1/4}$ model’s central surface brightness, $\mu_0$ , is given by $-2.5\log I_0$ , and the projected intensity at $R=R_s \equiv R_{\rm e}$ is given by

(7) \begin{equation} I_{\rm e}=I_0{\rm e}^{-b} = I_0/2141. \label{eqn7} \end{equation}

The average intensity $\langle I \rangle_{\rm e}$ within $R_{\rm e}$ is such that

(8) \begin{equation} 0.5\,L_{\rm tot} = \pi R_{\rm e}^2 \langle I \rangle_{\rm e}, \label{eqn8} \end{equation}

and it can be shown that

(9) \begin{equation} \langle I \rangle_{\rm e} = 3.61 I_{\rm e} = I_0/594 \label{eqn9} \end{equation}

(Graham & Driver, Reference Graham and Driver2005, see their equations 7 and 9).

As alluded to above, de Vaucouleurs could have chosen a radius enclosing any fraction of the light, and his two-parameter model would still have the same functional form (equation (1)). That is, one could use a radius $R_X$ containing any percentage of the total light, and one could use an intensity $I_Y$ taken from any (similar or different) fixed radius (in units of $R_{\rm e}$ ). The homology of the $R^{1/4}$ model is such that $R_X = C_1 R_{\rm e}$ , $I_Y = C_2 I_{\rm e}$ , and $\langle I \rangle _Y = C_3 \langle I \rangle_{\rm e}$ , where $C_1$ , $C_2,$ and $C_3$ are constants. In trying to understand the behaviour of, and connections between, galaxies, astronomers could plot $\log R_X$ versus $-2.5\log I_Y$ , and versus $-2.5\log \langle I \rangle _Y$ , and the trends would be the same as obtained when using $R_{\rm e}$ , $I_{\rm e,}$ and $\langle I \rangle_{\rm e}$ , just shifted vertically or horizontally in one’s diagram. As such, the arbitrary selection of 50% by de Vaucouleurs did not appear to matter. To give a more concrete example, de Vaucouleurs could have set the scale radius $R_s = R_{10}$ , that is, enclosing 10% of the total light (e.g. Farouki, Shapiro, & Duncan, Reference Farouki, Shapiro and Duncan1983). The mean intensity $\langle I \rangle_{10}$ within this radius is related by the expression

(10) \begin{equation} 0.1\,L_{\rm tot} = \pi R_{10}^2 \langle I \rangle_{10}, \label{eqn10} \end{equation}

and the associated value of b is obtained by solving the equation

(11) \begin{equation} \gamma (8,b) = \Gamma (8)/10, \label{eqn11} \end{equation}

to give $b=4.656$ and $I_{10}=I_0{\rm e}^{-b} = I_0/105.2$ (cf. equation (7)). In this example, de Vaucouleurs’ model would then read

(12) \begin{equation} I(R)=I_0\exp\left[ \frac{-4.656}{R_{10}^{1/4}} R^{1/4}\right], \label{eqn12} \end{equation}

where $R_{10} = (4.656/7.669)^4\, R_{\rm e} = R_{\rm e}/7.361$ , and $\langle I \rangle_{10}=I_0/54.77$ .

However, and this is the crux of the matter: ETGs, and also the bulges of spiral galaxies, do not follow the $R^{1/4}$ model, that is, there is not structural homology. This has important consequences when using radii enclosing a fixed percentage of the total light, and when using the associated surface brightness terms.

It is noted that the $R^{1/4}$ model had become so entrenched during the second half of the 20th century that it was invariably referred to as the $R^{1/4}$ law. That is, this empirical model was effectively elevated to the status of a physical law because it was thought that all ETGs did have $R^{1/4}$ light profiles. Indeed, it was not uncommon for astronomers to vary the sky-background in order to make their light profiles more $R^{1/4}$ -like (e.g. Tonry et al., Reference Tonry, Blakeslee, Ajhar and Dressler1997; see also the ‘Seven Samurai’ team data from Burstein et al., Reference Burstein, Davies and Dressler1987 as presented in D’Onofrio, Capaccioli, & Caon Reference D’Onofrio, Capaccioli and Caon1994, their Figure 4). This belief was in part because of de Vaucouleurs (Reference de Vaucouleurs and Flügge1959) study that had shown that the $R^{1/4}$ model fit better than the popular Reynolds’ (Reference Reynolds1913) modelFootnote ^c, and because of de Vaucouleurs & Capaccioli’s (Reference de Vaucouleurs and Capaccioli1979) study of NGC 3379 which revealed that its light profile is remarkably well fit by the $R^{1/4}$ model over an extensive range in surface brightness (see also Fish, Reference Fish1964 in the case of M87 and M105). However, Caon, Capaccioli, & D’Onofrio (Reference Caon, Capaccioli and D’Onofrio1993, Reference Caon, Capaccioli and D’Onofrio1994, 1990) and D’Onofrio et al. (Reference D’Onofrio, Capaccioli and Caon1994), subsequently revealed that other ETGs, with different absolute magnitudes, are equally well fit down to B-band surface brightnesses of $\sim$ 28 mag arcsec^-2 when using exponents in the light profile model that are different to the value of 1/4.

2.2 Sérsic’s $R^{1/n}$ model

Today, it is widely recognisedFootnote ^d that ETGs — and the bulges of spiral galaxies — display a range of light profile shapes that are better represented by a generalised version of the $R^{1/4}$ model, referred to as the Sérsic (Reference Sérsic1963) $R^{1/n}$ model, in which the exponent $1/n$ can take on a range of values other than just 1/4. This realisation applies to not just the ordinary ETGs (e.g. Caon et al., Reference Caon, Capaccioli and D’Onofrio1993; D’Onofrio et al., Reference D’Onofrio, Capaccioli and Caon1994) but also the dwarf ETGs (e.g. Davies et al., Reference Davies, Phillipps, Cawson, Disney and Kibblewhite1988; Cellone, Forte & Geisler, 1994; James, Reference James1994; Vennik & Richter, Reference Vennik and Richter1994; Young & Currie, Reference Young and Currie1994, Reference Young and Currie1995) which had previously been fit with an exponential model (e.g. Faber & Lin, Reference Faber and Lin1983; Binggeli et al., Reference Binggeli, Sandage and Tarenghi1984). Despite this, the early assumption of structural homology for dwarf ETGs versus a different structural homology for giant ETGs had been sown into the astronomical literature and psyche. Moreover, the implications of a varying exponent upon the use of the arbitrary 50% half-light radius, and the associated surface brightness terms, remained poorly recognised.

José Sérsic’s (Reference Sérsic1963, Reference Sérsic1968a) $R^{1/n}$ model, which was introduced in Spanish, is a generalisation of de Vaucouleurs’ $R^{1/4}$ model such that

(13) \begin{eqnarray} I(R) & = & I_0\exp\left[ -b_n\left(\frac{R}{R_{\rm e}}\right) ^{1/n}\right] = \frac{I_0}{\left( {\rm e}^{b_n}\right)^{\left(R/R_{\rm e} \right)^{1/n}}} \nonumber \\ & = & I_{\rm e}\exp\left\{ -b_n\left[\left( \frac{R}{R_{\rm e}}\right) ^{1/n} -1\right]\right\}. \label{eqn13}\end{eqnarray}

The exponent $1/n,$ or its inverse n, describes the curvature of the light profile. Within $\approx 1 R_{\rm e}$ , a larger value of n results in a more centrally concentrated distribution of light, while beyond $\approx 1 R_{\rm e}$ , a larger value of n results in a less steeply declining light profile. The quantity $b_n$ was defined such that $I_{\rm e}$ is, again, the intensity at the ‘effective half light’ radius $R_{\rm e}$ that encloses half of the total light (Capaccioli, Reference Capaccioli, Corwin and Bottinelli1989; Ciotti, Reference Ciotti1991; Caon et al., Reference Caon, Capaccioli and D’Onofrio1993). The value of $b_n$ is solved via the equation

(14) \begin{equation} \gamma(2n,b_n) = 0.5\,\Gamma(2n) \label{eqn14} \end{equation}

(cf. equation (6)), and the total luminosity, giving the total magnitude, is given by

(15) \begin{equation} L_{\rm tot} = \frac{I_0 R_{\rm e}^{2}2n\pi}{(b_n)^{2n}}\Gamma (2n) \label{eqn15} \end{equation}

(cf. equation (5)). For $0.5 \lt n \lt 10$ , $b_n \approx 1.9992n-0.3271$ (Capaccioli, Reference Capaccioli, Corwin and Bottinelli1989).

However, what was initially (for the $R^{1/4}$ model) an inconsequential selection of an arbitrary scale radius enclosing 50% of the light now has considerable consequences given that galaxies do not all have the same light profile shape, that is, the same value of n. Crucially, the ratio between radii containing different fixed percentages of the projected galaxy light is no longer a constant value — as we just saw it was for the $R^{1/4}$ model — but rather changes with the Sérsic index n. Given that ETGs and bulges possess a range of light profile shapes that are described well by the $R^{1/n}$ model (e.g. Caon et al., Reference Caon, Capaccioli and D’Onofrio1993; D’Onofrio et al., Reference D’Onofrio, Capaccioli and Caon1994), this remark about the changing ratio of radii holds even if one does not fit an $R^{1/n}$ model but instead measures the radii independently of any light profile model.

What this means is that the distribution of points in scaling diagrams involving the logarithm of scale radii and scale intensities will look different depending on what scale radius is used. That is, the arbitrary choice of radius, which to date has been the 50% radius, produces a somewhat arbitrary pattern in diagrams using $\log R_{\rm e}$ , $\mu_{\rm e,}$ and $\langle \mu \rangle_{\rm e}$ . Also apparent, from equation (13), is that the scale radius no longer occurs where the intensity has declined by the same fixed amount but rather by different amounts depending on the value of ${\rm e}^{b_n}$ and thus on the value of n. To quantify this, 4.1 will explore scaling diagrams using projected radii containing fixed percentages of the total light, including 50%, revealing how the bend in scaling relations using ‘effective’ parameters changes. 4.2 will explore the use of scale radii where the intensity has dropped by the same amount, yielding monotonic size–luminosity relations without the strong bends seen in 4.1.

3.1. A representative set of ETG light profiles

Given the Sérsic function and luminosity (equations 13 and 15), and armed with the two empirical equations 16 and 17, one can readily determine not only the typical Sérsic index and central surface brightness for a given (B-band) absolute magnitude but also the typical effective surface brightness at $R_{\rm e}$ , the mean effective surface brightness within $R_{\rm e}$ , and the effective half-light radius in kpc. This information has been used here to construct a representative set of surface brightness profiles for ETGs having five different absolute magnitudes, or rather, five different Sérsic indices (Figure 2, upper panel). The associated set of mean surface brightness profiles, which display the average surface brightness enclosed within the radius R, are also shown in the lower panel of Figure 2,

Figure 2. Upper panel: Sérsic light profiles (B-band, Vega mag), for a range of Sérsic indices n, that are representative of the ETG population at large. Lower panel: Associated set of representative mean surface brightness profiles. These stem from equations 16 and 17 and the $R^{1/n}$ model.

4.1. Relations involving effective surface brightnesses and effective radii

This section reveals how the absolute magnitude associated with the bend in diagrams using effective radii, and effective surface brightnesses, changes depending on the percentage of light that these radii enclose. That is, it shows that the absolute magnitude associated with the bend does not relate to different formation processes but rather relates to the arbitrary definition of galaxy size.

4.1.1. Luminosity-(effective surface brightness) diagram

As was noted, given the absolute magnitude of an ETG, equations 16 and 17 inform one of the typical Sérsic index and central surface brightness $\mu_0$ associated with this magnitude. This is enough information to determine the surface brightness $\mu_z$ , at a radius $R_z$ , containing any fraction z (between 0 and 1, or percentage Z) of the ETG’s total light. Using $\mu(R)=-2.5\log I(R)$ , at $R=R_z$ , the Sérsic model ((equation (13)) gives

(18) \begin{equation} \mu_z = \mu_0 + 2.5b_{n,z}/\ln(10), \label{eqn18} \end{equation}

and it can be shown that the mean surface brightness is such that

(19) \begin{equation} \langle \mu \rangle_z = \mu_z - 2.5 \log[f(n)], \label{eqn19} \end{equation}

where

(20) \begin{equation} f(n)=\frac{z\,2n {\rm e}^{b_{n,z}}}{(b_{n,z})^{2n}}\Gamma(2n). \label{eqn20} \end{equation}

To date, z has invariably been set equal to 0.5, giving $R_{\rm e}$ , $\mu_{\rm e}$ , and $\langle \mu \rangle_{\rm e}$ . The quantity $b_{n,z}$ seen above is a function of both the Sérsic index n and z, and is obtained by solving

(21) \begin{equation} \gamma(2n,b_{n,z})= z\, \Gamma(2n) \label{eqn21} \end{equation}

(cf. equation (14)). Knowing $b_{n,z}$ , one can additionally calculate the radius $R_z$ containing Z percent of the total light, in terms of the effective half-light radius $R_{\rm e}$ , containing 50% of the total light:

(22) \begin{equation} R_z = \left( \frac{b_{n,z}}{b_n} \right)^n R_{\rm e}, \label{eqn22} \end{equation}

where $b_n$ is given by equation (14).

Figure 3 reveals the difference between the central surface brightness, $\mu_0$ , and both the surface brightness $\mu_z$ at the scale radius $R_z$ (left panel) and the mean surface brightness $\langle \mu \rangle_z$ within this radius (right panel). The orthogonal behaviour (at faint and bright magnitudes) seen here for any z is a consequence of the Sérsic index changing systematically and monotonically with absolute magnitude, that is, ‘structural non-homology’.

Figure 3. ETG scaling relations between absolute B-band magnitude and the B-band surface brightness at projected radii containing different percentages ( $Z=$ 2, 10, 20… 80, 90, 97) of the total light (left panel) and the mean surface brightness within these radii (right panel). The thick straight line is the relation from Figure 1 involving the central surface brightness $\mu_{0,B}$ . The curved lines corresponding to $R_{\rm e}$ , that is, the radius enclosing 50% ( $Z=50$ , $z=0.5$ ) of the total light, show the behaviour of both the effective surface brightness $\mu_{\rm e}$ and the mean effective surface brightness $\langle \mu \rangle_{\rm e}$ . As revealed in Graham (Reference Graham, Oswalt and Keel2013, see his Figures 2–8), the ETGs in Figure 1 follow the $Z=50$ curves shown here. The different absolute magnitude associated with the apparent midpoint or bend in the curves with different values of Z is not due to different formation physics at brighter or fainter magnitudes.

While the ETG population are unified by the linear $\mathfrak{M}$ – $\mu_0$ and $\mathfrak{M}$ – $\log(n)$ relations — with no evidence for a divide at $\mathfrak{M}_B \approx -18$ mag — the peak in the bend of the ( $z=0.5$ ) $\mathfrak{M}$ – $\mu_{\rm e}$ and $\mathfrak{M}$ – $\langle \mu \rangle_{\rm e}$ distribution occurs at $\mathfrak{M}_B \approx -18$ mag. This has contributed to decades of belief that different physical processes have shaped the ETGs brighter and fainter than $\mathfrak{M}_B \approx -18$ mag. However, Figure 3 reveals that had de Vaucouleurs used a radius containing 97% of the total light, then some might today be claiming that the divide between dwarf and ordinary ETGs occurs at $\mathfrak{M}_B=-17$ mag; or had de Vaucouleurs used a radius containing 2% of the galaxy’s total light, then they might be advocating for a divide at $\mathfrak{M}_B=-20.5$ mag.

The crucial point is that one should not assign a physical interpretation to the bend. Graham & Guzmán (Reference Graham and Guzmán2003), and Graham (Reference Graham, Oswalt and Keel2013), tried to make this point using only the $Z=50$ curves in Figure 3 and explaining that the bend is due to the light profile shape changing smoothly as the absolute magnitude changes. That is, it is not due to different physical processes operating at absolute magnitudes fainter and brighter than $-18$ mag (or $-17$ mag or $-20.5$ mag).

Despite the above, there has been a remarkable number of claims of supporting evidence for the false divide at $\mathfrak{M}_B \approx -18$ mag. This often pertains to observations that some quantity (e.g. Sérsic index or colour or dynamical mass-to-light ratio) is, on average, different between ETGs brighter and fainter than $\mathfrak{M}_B \approx -18$ mag. This paper has endeavoured to more fully explain the nature of ETGs by including the additional curves in Figure 3 and by revealing in the coming sections what the distribution of ETGs looks like in related diagrams involving effective radii and other measures of radii. There is much that needs addressing given the decades of literature on this subject, the engrained nature of assigning a divide between dwarf and ordinary ETGs at $\mathfrak{M}_B = -18$ mag, and the many (yet to be widely recognised and utilised) insights from understanding these curved scaling relations.

4.1.2. Luminosity-(effective radius) diagram

Due to how the light profile smoothly and systematically changes shape with absolute magnitude (e.g. Fisher & Drory, Reference Fisher and Drory2010, see their Figure 13), when using effective half-light radii ( $z=0.5$ ), it results in a distribution of ETGs — and bulges — which is curved (e.g. Lange et al., Reference Lange, Driver and Robotham2015, and references therein). Here, Graham et al. (Reference Graham, Merritt, Moore, Diemand and Terzić2006, see their Figure 1) and Graham & Worley (Reference Graham and Worley2008, see their Figure 11) are expanded upon by additionally showing what the size–luminosity relation looks like when using scale radii that effectively enclose different fractions of the total galaxy light. This also reveals how the absolute magnitude associated with the alleged dichotomy between dwarf ( $\mathfrak{M}_B > -18$ mag) and ordinary ( $\mathfrak{M}_B \lt -18$ mag) ETGs is fictitious, purely dependent on the arbitrary fraction z rather than different physical formation processes.

Building upon equation 12 from Graham & Driver (Reference Graham and Driver2005), which used $R_{\rm e}$ and thus $z=0.5$ , the generalised expression for the total absolute magnitude, in terms of the radius $R_z$ containing the fraction z of the total light, is given by

(23) \begin{equation} \mathfrak{M}_{\rm tot,B} = \langle\mu\rangle_{\rm z,B} - 2.5\log(\pi R_{\rm z,kpc}^2/z) - 36.57, \label{eqn23} \end{equation}

where $\langle\mu\rangle_{\rm z,B}$ is the mean surface brightness within $R_z$ . This can be rearranged to give the expression

(24) \begin{eqnarray} \log R_{\rm z,kpc} &=& \frac{\mu_0 - \mathfrak{M}_{\rm tot}}{5} + \frac{\log z - \log[f(n)]}{2} \nonumber \\ &+& \frac{b_{n,z}}{2\ln(10)} - 7.065, \label{eqn24}\end{eqnarray}

where f(n) is given in equation (20). Using equation (17) to replace $\mu_0$ with $\mathfrak{M}_{\rm tot}$ , this expression becomes

(25) \begin{eqnarray} \log R_{\rm z,kpc} &=& \frac{\mathfrak{M}_{\rm tot}}{10} + \frac{\log z - \log[f(n)]}{2} \nonumber \\ &+& 0.217b_{n,z} + 1.2874. \label{eqn25}\end{eqnarray}

The latter term in equation (25), involving z, cancels with the same term in f(n), and thus the dependence of $R_z$ on z occurs via the $b_{n,z}$ term (equation (21)).

Figure 4 presents the ETG luminosity–size relations for a range of fractions z, expressed there as a percentage Z. The curved behaviour is, once again, due to the ETG population smoothly changing its light profile shape — as quantified by the Sérsic index — with absolute magnitude. It can readily be appreciated that adopting some fixed fraction z, such as 0.5, and then claiming that different physical processes have shaped the luminosity–size relation on either side of the apparent bend-point would be a misleading endeavour (e.g. Fisher & Drory, Reference Fisher and Drory2010, 2016.

As can be seen in the left-hand panel of Figure 4, the bright arm of the curved $\mathfrak{M}_B$ – $\log R_{\rm e}$ relation for ETGs is approximately linear. This was noted by Fish (Reference Fish1963), who reported $\log L \propto (1\,{\rm to}\,1.5)\log R_{\rm e}$ or equivalently $\mathfrak{M}_p \propto -(2.5\,{\rm to}\,3.75)\log R_{\rm e}$ , and can be seen in Figure 1 of Sérsic (Reference Sérsic1968b, using the data from Fish, Reference Fish1964; see also Brookes and Rood, Reference Brookes and Rood1971; Gudehus & Hegyi, Reference Gudehus and Hegyi1991; Shen et al., Reference Shen, Mo and White2003; Graham & Worley, Reference Graham and Worley2008; Lange et al., Reference Lange, Driver and Robotham2015)Footnote ^hFootnote ⁱ. Sérsic (Reference Sérsic1968b) was perhaps the first to remark upon the offset nature of the faint ETGs from the bright ETGs in the $\mathfrak{M}_B$ – $\log R_{\rm e}$ diagram. Not understanding the bend in this diagram — referred to as the ‘transition region’ by Sérsic (Reference Sérsic1968b) — coupled with the inclusion of three unusually small galaxies, Sérsic attributed the bend to two populations of (dwarf and giant) elliptical galaxies, rather than one population with smoothly varying properties.

Figure 4. Left panel: relations describing the distribution of ETG B-band absolute magnitude versus the projected radii enclosing various percentages ( $Z=$ 2, 10, 20… 80, 90, 97) of their total flux (equation (25)). The relation involving the effective half-light radius corresponds to the $Z=50$ curve. Middle and right panels: It can be seen how much the scale radii vary depending on the arbitrary percentage of light used to define them.

Confounding the situation further, Sérsic added LTGs into his $\mathfrak{M}_p$ – $\log R_{\rm e}$ diagram (see his Figure 2; cf. Figures 9 and 14 from Cappellari et al., Reference Cappellari, McDermid and Alatalo2013b). Involving $R_{\rm e}$ measures from both two-dimensional spirals and three-dimensional ellipticals, Sérsic (Reference Sérsic1968b) observed a slight overlap and wrote that ‘it seems difficult to deny the existence of the sequence of irregulars and spirals joining that of the ellipticals in the transition region’. Kormendy (Reference Kormendy1985) adopted this same practice.

For $R_{\rm e} \mathbin{\lower.3ex\hbox{$\buildrel\gt\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} $ 1–2 kpc and a photographic absolute magnitude $\mathfrak{M}_p$ brighter than $-19.5$ magFootnote ^j, Sérsic (Reference Sérsic1968b) fit a line with a slope of unity to the distribution of giant elliptical galaxies in his ( $\log {\rm Mass}$ )–( $\log R_{\rm e}$ ) diagram. This distribution resembled that in his $\mathfrak{M}$ – $\log R_{\rm e}$ diagram because he claims to have used a constant mass-to-light ratio of 30. As such, Sérsic (Reference Sérsic1968b) reported a distribution in which the absolute magnitude scaled as $-2.5\log R_{\rm e}$ . Given that the magnitude of a galaxy is proportional to $\langle \mu \rangle_{\rm e} - 5\log R_{\rm e}$ (e.g. de Vaucouleurs & Page, Reference de Vaucouleurs and Page1962, see their equation 6), one immediately has the relation $\langle \mu \rangle_{\rm e} \propto 2.5\log R_{\rm e}$ for the distribution of giant elliptical galaxies. Furthermore, given that $\mu_{\rm e} - \langle \mu \rangle_{\rm e} = 1.393$ for the $R^{1/4}$ model that Sérsic (Reference Sérsic1968b) was using, one also immediately has that $\mu_{\rm e} \propto 2.5\log R_{\rm e}$ . This can be compared with Kormendy (Reference Kormendy1977)Footnote ^k who reported $\mu_{\rm e} \propto 3.02\log R_{\rm e}$ .

Somerville & Davé (Reference Somerville and Davé2015, see their Section 1.1.4) refer to the ( $\log {\rm Mass}$ )–( $\log R_{\rm e}$ ) relation as the Kormendy relation (see also Cappellari, Reference Cappellari2016, his Section 4.1.1), but it would be more appropriate if that title was assigned to the linear relation which Kormendy fit to the bright arm of what we now know is the curved $\mu_{\rm e}$ – $\log R_{\rm e}$ relation, and to instead refer to the linear ( $\log {\rm Mass}$ )–( $\log R_{\rm e}$ ) and $\mathfrak{M}$ –( $\log R_{\rm e}$ ) relations used to describe the distribution of bright elliptical galaxies as the FishFootnote ^l or Fish–Sérsic relation. The curved $\mu_{\rm e}$ – $\log R_{\rm e}$ relation is explored in 4.1.3.

Three additional insights from Figure 4 can readily be made. The first has implications for dark matter (Kent, Reference Kent1990, and references therein) if using $\sigma^2R_{\rm e}$ (e.g. Poincare & Vergne, Reference Poincare and Vergne1911; Poveda, Reference Poveda1958, Reference Poveda1961) as a proxy for massFootnote ^m in a population of ETGs with a range of absolute magnitudes and thus a range of light profile shapes. Considering how the ratio of radii ( $R_{z_1}/R_{z_2}$ ) at fixed absolute magnitude changes, for different values of $z_1$ and $z_2$ in Figure 4, one should pause for thought when using effective half-light radii ( $z=0.5$ ) to measure dynamical (stellar plus dark matter) masses via the proportionality $\sigma^2 R_{\rm e}$ (e.g. Drory, Bender, & Hopp Reference Drory, Bender and Hopp2004, see their Figure 3; Cappellari et al., Reference Cappellari, Bacon and Bureau2006 who use the luminosity-weighted $\sigma$ values within half-light radii determined from $R^{1/4}$ models; Cappellari et al., Reference Cappellari, Scott and Alatalo2013a). Using $\sigma^2R_z$ , with $z\ne0.5$ , will produce a different trend because the ratio $R_{\rm e}/R_z$ is not constant for different $\mathfrak{M}_B$ (see also the telling merger simulations by Farouki et al., Reference Farouki, Shapiro and Duncan1983 revealing how $R_{10}/R_{\rm e}$ changes with mass, and the work by Campbell et al., Reference Campbell, Frenk and Jenkins2017 and Lyskova et al., Reference Lyskova, Thomas, Churazov, Tremaine and Naab2015). Therefore, it may not be appropriate to solely invoke varying fractions of dark matter to explain the systematic differences, as a function of varying absolute magnitude, between (i) this dynamical mass estimate (based on the arbitrary radius $R_{\rm e}$ ) and (ii) the stellar mass estimate (obtained from the absolute magnitude). This will be broached in a subsequent study, covering the ‘Fundamental Plane’ (Djorgovski and Davis, Reference Djorgovski and Davis1987), improved planes, and implications for both dark matter estimates (e.g. Graves & Faber, Reference Graves and Faber2010) and ETG formation (see Cappellari, Reference Cappellari2016, his Section 4, for an overview).

Second, it is noted that the acceleration at some radius R, inside a symmetrical pressure supported system with velocity dispersion $\sigma$ , is proportional to $GM/R^2$ or $\sigma^2/R$ . Due to the structural non-homology of ETGs, this ratio will vary with M in different ways depending on what fraction z has been used to measure R. This has relevance to the critical acceleration parameter $a_0$ , or characteristic surface density $M/R^2$ , in modified Newtonian dynamics (Milgrom, Reference Milgrom1983; Sanders & McGaugh, Reference Sanders and McGaugh2002, see their Figure 7; Milgrom & Sanders, Reference Milgrom and Sanders2003; Kroupa et al., Reference Kroupa, Famaey and de Boer2010, their Figure 7; Misgeld & Hilker, Reference Misgeld and Hilker2011, their Figure 7; Famaey & McGaugh, Reference Famaey and McGaugh2012).

Third, it may also be insightful to explore the near-constant $R_{\rm e,bulge}/h_{\rm disc}$ ratio of $\sim$ 0.2 observed in spiral galaxies (e.g. Courteau et al., Reference Courteau, de Jong and Broeils1996; Graham & Worley, Reference Graham and Worley2008, and references therein), which appears irrespective of whether the bulge is considered to be a ‘classical’ bulge or a ‘pseudobulge’. For instance, the use of $z=0.1$ or $z=0.9$ , rather than $z=0.5$ , is expected to result in this ratio systematically changing, with magnitude, by a factor of $\sim$ 3 for spiral galaxies.

4.1.3. (Effective radius)-(effective surface brightness) diagram

In addition to the $R_{\rm e}$ – $\mu_{\rm e}$ and $R_{\rm e}$ – $\langle \mu \rangle_{\rm e}$ distributions (derived using $z=0.5$ ), it is instructive to show the size-(surface brightness) distributions $R_z$ – $\mu_z$ and $R_z$ – $\langle \mu \rangle_z$ that one would obtain for different values of z, corresponding to the fraction of light contained within $R_z$ . Figure 5 reveals a number of things, three of which are worth explicitly pointing out here, while many other important but less-recognised aspects will be saved for a follow-up paper pertaining to both understanding the ‘Fundamental Plane’ and constructing an improved plane/surface.

Figure 5. Size–(surface brightness) relations — describing the distribution of galaxies having a range of B-band absolute magnitudes (from $-12$ to $-23$ mag) — based on projected radii, $R_z$ , enclosing different percentages ( $Z=$ 5, 10, 20… 80, 90, 95) of the galaxy light. The innermost curve is associated with the 50% radius, known as the effective half-light radius $R_{\rm e}$ , the effective surface brightness $\mu_{\rm e}$ at this radius (left panel), and the average surface brightness $\langle \mu \rangle_{\rm e}$ within this radius (right panel). The upper envelope in the left-hand panel has a slope of $\sim$ 3.5, while the upper envelope in the right-hand panel has a slope of $\sim$ 2.8.

First, had the community been using radii enclosing 95% or 5% of the total light, then those interpreting the bend in the corresponding size-(surface brightness) diagram may likely be claiming evidence of distinctly different formation physics for galaxies brighter and fainter than $\sim -16.5$ mag or $\sim -19.5$ mag, respectively.

Second, the bunching up of tracks in the top right of Figure 5 reveals why the $R_{\rm e}$ – $\mu_{\rm e}$ and $R_{\rm e}$ – $\langle \mu \rangle_{\rm e}$ relations have a low level of scatter for ETGs with B-band absolute magnitudes brighter than $\approx -19$ mag. If one mis-measures the half-light radius and instead captures the radius enclosing 20, 30, 40, 60, 70, or 80% of the total light, the surface brightness terms associated with these radii are such that the galaxy’s location in the $R_{\rm e}$ – $\mu_{\rm e}$ diagram moves along the upper envelope seen in Figure 5 and thereby maintains a tight $R_{\rm e}$ – $\mu_{\rm e}$ relation.

Third, the inclusion of ETGs fainter than $\mathfrak{M}_B \approx -19$ mag results in a thickening of the distribution in the $R_{\rm e}$ – $\mu_{\rm e}$ diagram (e.g. Kodaira, Okamura, & Watanabe, Reference Kodaira, Okamura and Watanabe1983; Capaccioli, Caon, & D’Onofrio, Reference Capaccioli, Caon and D’Onofrio1994) as mis-measures of the half-light radius will shift galaxies perpendicular to the curved $z=0.5$ relation at faint absolute magnitudes.

Bildfell et al. (Reference Bildfell, Hoekstra, Babul and Mahdavi2008) report that ‘The Kormendy relation of our BCGs is steeper than that of the [less luminous] local ellipticals, suggesting differences in the assembly history of these types of systems’. Although the literature is full of similar claims, such interpretations are not appropriate given the curved $R_{\rm e}$ – $\mu_{\rm e}$ relation’s dependence on the arbitrary value $z=0.5$ . Countless studies which have attached a physical significance to slopes and bends in scaling diagrams involving the logarithm of $R_{\rm e}$ , $\mu_{\rm e}$ , and/or $\langle \mu \rangle _{\rm e}$ should be questioned. As already noted in Graham & Guzman (2004) and Graham (2005), this remark extends to studies of the ‘Fundamental Plane’ (Guzmán et al. 2019, in preparation).

A range of other measures for galaxy size is explored in the remainder of this section.

4.2. An alternative scheme for defining projected radii

The previous text focused on projected radii that enclosed an arbitrary fraction of light relative to the light enclosed within a radius of infinity, that is, the total light. One can, alternatively, define a radius where the intensity is an arbitrary fraction of the intensity at $R=0$ . In the case of the exponential galaxy light profile model, the parameter h denotes the scale length where the intensity has dropped by a factor of ${\rm e}\approx2.718$ . This subsection explores radii where the intensity of the $R^{1/n}$ model has dropped by fixed amounts, effectively replacing the variable ${\rm e}^{b_n}$ term in equation (13) with a constant.

For de Vaucouleurs’ $R^{1/4}$ model (equation (1)), it was noted that the intensity at $R_{\rm e}$ is e $^{7.669} \approx 2141$ times fainter than the intensity at $R=0$ . This corresponds to a surface brightness which is 8.327 mag arcsec^-2 fainter than the central surface brightness. It is informative to explore what the size–luminosity diagram looks like when using this alternative, but equally valid, measure of ETG size, that is, the radius where the surface brightness has dropped by a constant 8.327 mag arcsec^-2. This is done in Figure 6, where a few other constant values are also used.

Figure 6. Left panel: Relations between the absolute B-band magnitude, $\mathfrak{M}_B$ , and the radius where the associated light profile’s surface brightness has dropped by a fixed amount from the central $R=0$ value $\mu_0$ (equation (17)).of the Sérsic model having a Sérsic index n dictated by the value of $\mathfrak{M}_B$ (equation (16)). The leftmost curve in each panel shows the result when using $\Delta\mu=8.327$ mag arcsec^-2, which is the difference in surface brightness between $\mu_0$ and $\mu_{\rm e}$ of de Vaucouleurs’ $R^{1/4}$ model. Middle panel: Similar, except that the radii shown here denote where the surface brightness profile has dropped by the same set of constant values used in the left panel, but now starting from $R=0.01$ kpc rather than from $R=0$ . This helps to bypass the rapidly rising inner light profile of systems with high values of n, but which typically contain depleted cores. Right panel: Similar to the middle panel but starting from $R=0.1$ kpc.

The left panel of Figure 6 reveals little evidence for a divide at $\mathfrak{M}_B \approx -18$ mag between the so-called dwarf and ordinary ETGs. Had astronomers used the above system of radii, calibrated to the Sérsic model’s central surface brightness (at $R=0$ ), rather than calibrated to the Sérsic model’s total luminosity (at $R=\infty$ ), then they might well have concluded that there is a dichotomy between bright and faint ETGs at $\mathfrak{M}_B \approx -20$ mag and speculated that different physical processes must be responsible for the formation of ETGs fainter and brighter than this absolute magnitude. Some astronomers may have even heralded the observation of partially depleted cores in ETGs more luminous than $\mathfrak{M}_B \sim -20.5\pm1$ mag — thought to have formed their spheroids from major dry merger events — as the explanation for the bend seen in this alternative luminosity–size diagram.

In case some readers might be entertaining the $\Delta\mu=8.327$ mag arcsec^-2 curve in the left-hand panel of Figure 6 as evidence for a division at $\mathfrak{M}_B \approx -20$ mag, additional measures of radii based on larger differences in surface brightness from the central surface brightness have been included. One can see that the location of the bend in the scaling relations shifts from a B-band magnitude of roughly $-20$ to $-22$ mag as one samples more of the galaxy light. Once again, this demonstrates that these bends are not revealing the existence of different physical processes operating at magnitudes brighter and fainter than the location of the bend. The whip around to smaller radii seen at bright magnitudes in the left-hand panel of Figure 6 is due to the rapidly rising (with decreasing radii), inner light profile of systems with high Sérsic indices. One can devise schemes to circumvent this (see the middle and right-hand panels), which may be desirable given the partially depleted cores in these galaxies which prevent such bright $\mu_0$ values actually being realised. The monotonic size–luminosity relations in Figure 6, which do not use radii where the intensity has dropped by systematically different amounts as a function of luminosity (as occurs with $R_{\rm e}$ and $R_z$ ) reveal no grounds for segregating dwarf and giant ETGs at $\mathfrak{M}_B \approx -18$ mag.

The middle and right-hand panels of Figure 6 show the distribution of ETG sizes where their surface brightness profiles have dropped by the same values as those used in the left-hand panel, but starting the drop from a radius of 0.01 and 0.1 kpc, rather than from the central value.

4.3. Isophotal radii

Based on isophotal radii, the luminosity–size relation for ETGs was initially considered to be log-linear, that is, linear in log space, unifying dwarf and giant ETGs (e.g. Heidmann, Reference Heidmann1967, Reference Heidmann1969; Holmberg, Reference Holmberg1969; Oemler, Reference Oemler1976; Strom & Strom, Reference Strom and Strom1978), and it largely still is (e.g. Forbes et al., Reference Forbes, Lasky, Graham and Spitler2008, see their Figure 3; van den Bergh, Reference van den Bergh2008; Nair, van den Bergh, & Abraham, Reference Nair, van den Bergh and Abraham2011). This section would therefore be somewhat incomplete if it did not include isophotal radii.

Using a photographic (Pg, i.e. blue filter, Vega mag system) surface brightness of 26.5 mag arcsec^-2 to define galaxy diameters, Holmberg (Holmberg, Reference Holmberg1969, see his Figure 9) reported a linear relation, with a slope of $-6$ , between the absolute magnitude and the logarithm of the isophotal major axis diameterFootnote ⁿ. Using the major axis diameter of the isophote corresponding to a photographic surface brightness of 25 (Vega) mag arcsec^-2, Heidmann, (Reference Heidmann1967, Reference Heidmann1969: see also Fraser, Reference Fraser1977 and Bigay & Paturel, Reference Bigay and Paturel1980) obtained a less steep slope of $-4.75$ for ETGsFootnote ^o in the $\mathfrak{M}$ –( $\log R_{\rm iso}$ ) diagram, which he reported as a slope of 1.9 in the $\log L$ – $\log R_{\rm iso}$ diagram.

Some half a century later, using the semimajor axis radius of the 3.6 $\mu$ m isophote whose surface brightness equals 25.5 (AB) mag arcsec^-2, Muñoz-Mateos et al. (Reference Muñoz-Mateos, Sheth and Regan2015, see their Figure 14) presented a log-linear radius–(stellar mass) relation for different morphological types. The bright ETGs have the same slope as reported by Heidmann, with $\log M \propto (1.9\pm0.1)\log R_{3.6\mu {\rm m}=25.5}$ . Approximating the low-luminosity end of the moderately curved L– $R_{\rm iso}$ relation with a power-law, the faint ETGs in Muñoz-Mateos et al. roughly follow a relation with a slope of $2.7\pm0.2$ . Muñoz-Mateos et al. additionally show, in their Figure 15, that the use of $R_{\rm e}$ , rather than isophotal radii, results in the strongly curved size–luminosity relation seen in Figure 4.

Figure 7 reveals what the size–luminosity relation for ETGs looks like when using six different isophotal radii (specifically, those radii where the B-band surface brightness equals 25, 26, … 30 mag arcsec^-2) and using the total B-band absolute magnitude $\mathfrak{M}_B$ within a radius of infinite aperture. The smoothly changing slope is consistent with the slight curve observed for 50 years in magnitude-(isophotal radii) diagrams. For example, as noted above, a moderate change in slope is seen among the ETGs in the $\log \mathfrak{M}$ – $\log R_{3.6\mu {\rm m}=25.5}$ diagram of Muñoz-Mateos et al. (Reference Muñoz-Mateos, Sheth and Regan2015) at 4–6 kpc. This can be understood in terms of the Sérsic index varying with absolute magnitude, which gives rise to the curves in Figure 7. Although it should be noted that the mapping between Figure 14 in Muñoz-Mateos et al. (Reference Muñoz-Mateos, Sheth and Regan2015) and Figure 7 shown here is not linear because of the colour–magnitude relation for ETGs (e.g. Ferrarese et al., Reference Ferrarese, Côté and Jordán2006, see their Figure 123), in which fainter ETGs are bluer than luminous ETGs. To help anyone who may wish to explore this further, it is quickly noted that given that luminous ETGs have a ( $B-3.6$ ) colour of 4 to 5, the radius where the 3.6 $\mu$ m surface brightness equals 25.5 mag arcsec^-2 will roughly correspond to the $R_{B=30}$ isophotal radii seen in Figure 7, while ETGs with $\mathfrak{M}_B=-16$ mag have a ( $B-3.6$ ) colour of $\approx$ 2.5.

Figure 7. Six different isophotal radii are shown as a function of the B-band absolute magnitude (for which the typical Sérsic profile and Sérsic parameters are known from equations 16, 17, and 25).

Past studies which did not include ETGs fainter than $\mathfrak{M}_B\approx -16$ mag could have missed the slight curvature in the L– $R_{\rm iso}$ diagram. The horizontal flattening of the curves associated with the brighter isophotal levels, seen at small radii in Figure 7, reflects that the central surface brightnesses in galaxies with these low absolute magnitudes is close to the isophotal value. Given that the ETGs in Figure 1 have $\mathfrak{M}_B \lt -13$ mag, the curves seen in Figure 7, and elsewhere, may not be reliable at $\mathfrak{M}_B > -13$ mag. At these low magnitudes, one encounters galaxies which may be a different, more heterogeneous class of galaxy with a broad range of colours (e.g. Jerjen et al. Reference Jerjen, Binggeli and Freeman2000; Hilker, Mieske, & Infante, Reference Hilker, Mieske and Infante2003; Penny & Conselice, Reference Penny and Conselice2008).

5.1. Internal effective radii

Subsection 4.1 explored parameters arising from projected radii, R, that effectively enclosed different percentages, Z, of the total galaxy light. Here, we explore parameters arising from internal radii, r, defining spheres which effectively enclose different percentages of the total galaxy light. Trends with these internal radii $r_z$ , the average luminosity densities $\langle \nu \rangle_z$ contained within the spheres defined by these radiiFootnote ^q, and the absolute magnitude are investigated. The results can be seen in Figure 9.

Figure 9. For a range of absolute B-band magnitudes with Sérsic indices $n>1$ , the internal radius $r_z$ enclosing a sphere with Z percent of the total light is shown (left panel), as is the mean luminosity density $\langle \nu \rangle_z$ within this radius (middle panel: a somewhat similar pattern exists when using the internal luminosity density $\nu_z$ at $r_z$ ). The right-hand panel shows the $\log r_z$ – $\log \langle \nu \rangle_z$ relations for fractions $Z = 2, 10, 20, 30,... 90$ %. The over-lapping nature of the relations for the brighter galaxies is the reason behind the tight $\log r_z$ – $\log \nu_z$ relation discovered by Graham et al. (Reference Graham, Merritt, Moore, Diemand and Terzić2006).

As with the projected effective parameters, the internal effective parameters display a similar behaviour of strongly curved relations, in which the midpoint of each curve depends on Z and therefore obviously does not reflect a separation based on physically different formation processes. The midpoint of the bend shifts from roughly -19 to -16 mag as Z changes from 2 to 90%. A value of $Z=50$ corresponds to the internal half-light radius $r_{\rm e}$ and the mean luminosity density $\langle \nu \rangle_{\rm e}$ , traced by the $Z=50$ curves in Figure 9. The similarity between the $Z=50$ curve in the left-hand panel of Figure 9 and the $Z=50$ curve in the left-hand panel of Figure 4 was expected, given that $r_{\rm e} \approx 4/3 R_{\rm e}$ (Ciotti, Reference Ciotti1991). In addition, the similar patterns seen in both panels means that the different ratio of radii ( $r_{z_1}/r_{z_2}$ ) at fixed absolute magnitude, for different percentages $z_1$ and $z_2$ , will result in $\sigma^2 r/G$ mass estimates that depend on the percentage used to define r. As was seen in the middle panel of Figure 4, the ratio of radii again increasingly varies as the luminosity increases. This also coincides with an increased steepening of the velocity dispersion profile, impacting estimates of the dynamical mass (e.g. Wolf, 2011 Wolf et al., Reference Wolf, Martinez and Bullock2010; Forbes et al., Reference Forbes, Spitler and Graham2011, see their Section 9.2) and further undermining the use of $\sigma^2 r_{\rm e}/G$ in the brighter, non-dwarf, ETGs.

The right-hand panel of Figure 9 reveals that one can expect a strong $\log r_{\rm e}$ – $\log \langle \nu \rangle_{\rm e}$ relation for bright ETGs. This is because if one mis-measures the internal radius enclosing 50% of the light and obtains a radius containing say 20 or 80% of the total light, the associated mean luminosity density that one measures will largely shift one along the $\log r_{\rm e}$ – $\log \langle \nu \rangle_{\rm e}$ relation for bright ETGs (see also Trujillo et al., Reference Trujillo, Graham and Caon2001, their Section 4). At low luminosities, faint of the midpoint of the bend in these curved relations, the same such mis-measurement will move one away from the curved $\log r_{\rm e}$ – $\log \langle \nu \rangle_{\rm e}$ relation. This behaviour can be seen in the $\log r_{\rm e}$ – $\log \nu_{\rm e}$ diagram of Graham et al., (Reference Graham, Merritt, Moore, Diemand and Terzić2006, see their Figure 2b).

The collective broadening that can be seen at faint absolute magnitudes in the right-hand panel of Figure 9 can be compared with the right-hand panel of Figure 5. Note that Figure 9 only shows data for light profiles with $n>1$ ( $\mathfrak{M}_B \lt -14.3$ mag). Also bear in mind that the mean surface brightness (Figure 5) is 2.5 times the logarithm of the mean intensity, hence the greater range along the $\langle \mu \rangle_{\rm e}$ axis in Figure 5 than compared to the $\log \langle \nu \rangle_z$ axis in Figure 9.

5.2. Alternative internal radii

Similar to Subsection 4.2, we can explore the internal radii r where the internal density, $\nu$ , has dropped by a fixed amount from the value at some inner radius. Figure 8 reveals that, for Sérsic indices $n>0.5$ –1, the internal density profile rises steeply with decreasing radius. In the middle and right-hand panels of Figure 6, this rapid brightening of the projected surface brightness was circumvented by starting from the radius $R=0.01$ and 0.1 kpc. Here, we start at $r=0.01$ and 0.1 kpc to compute the internal radii where the internal luminosity density profile has dropped by a fixed amount from the density at these two inner radii. Figure 10 shows these alternative scale radii as a function of the absolute B-band magnitude (for magnitudes corresponding to $n>1$ ). The relations seen there do not support an ETG divide at $\mathfrak{M}_B \approx -18$ mag.

Figure 10. For each absolute magnitude, $\mathfrak{M}_B$ , one can see the internal radius where the logarithm of the luminosity density ( $\nu$ , in units of $L_{\odot,B}$ pc^-3) has decreased by a fixed amount from its value at $r=0.01$ kpc (left panel) and $r=0.1$ kpc (right panel).

5.3. Isodensity radii

Isodensity radii define a two-dimensional surface, such as a sphere, within a three-dimensional space. These radii are the internal analog to the projected isophotal radii seen in Subsection 4.3 and are naturally considered a better measure to define the radii of three-dimensional stellar systems. This is simply because isophotes can display an artificial (not physical) contour, arising from the projected column density through a galaxy, rather than a real boundary of equal density.

Figure 11 displays the $\mathfrak{M}_B$ – $R_{\rm isodensity}$ relations for five different luminosity densities. The trends reveal no evidence for a divide at $\mathfrak{M}_B \approx -18$ mag.

Figure 11. For each absolute magnitude, $\mathfrak{M}_B$ , one can see the internal radius where the logarithm of the luminosity density ( $\nu$ , in units of $L_{\odot,B}$ pc^-3) equals one of the five different values.

While this concludes the recapitulation of the previous section but performed using internal parameters, it would be somewhat incomplete to proceed without having used the virial radii which are popular among theorists. Therefore, the following subsection presents this, along with an observer-inspired variation.

5.4. Virial radii

The integrated luminosity, within spheres centred on a galaxy, is given by

(28) \begin{equation} L_{n}(s)=I_{\rm e}{\rm e}^{b}R_{\rm e}^{2}\, 4\pi \int _{0}^{s}\nu _{n}(s')s'^{2}ds', \label{eqn28} \end{equation}

and is shown in Figure 12 for our representative set of profiles from Figure 8.

Figure 12. Representative, cumulative luminosity profiles for different Sérsic indices n, as matched to the light profiles shown in Figure 2.

Multiplying by a stellar mass-to-light ratio gives the cumulative stellar mass profiles. The luminosity density profiles $\nu(r)$ (equation (27)) were converted into stellar mass density profiles $\rho(r)$ , using a constant B-band stellar mass-to-light ratio of $M/L_B=8$ .

Following Macciò, Murante, & Bonometto (Reference Macciò, Murante and Bonometto2003, their equation 1.1; see also Bryan & Norman, Reference Bryan and Norman1998), a proxyFootnote ^r is used for the virial radius defined as the radius of the sphere within which the average (stellar mass) density is equal to $18\, \Pi^2\, \Omega_{\rm matter}^{0.45}\, \rho_{\rm critical} \approx 177.7 \times 0.589 \times \rho_{\rm critical} \approx 104.6\, \rho_{\rm critical} \approx (339.5\,\, \Omega_{\rm matter})\, \rho_{\rm critical}$ . The Planck 2015 results (Planck et al., 2016) give $\Omega_{\rm matter}=0.308\pm0.012$ and thus $339.5\, \Omega_{\rm matter} = 104.6$ . They also report $H_0 = 67.8\pm0.9$ km s^-1 Mpc^-1 and thus $\rho_{\rm critical} \equiv 3H^2_0/(8\Pi G) = 0.864\times10^{-26}$ kg m^-3, or $39.3\, {\rm M}_{\odot}\, {\rm kpc}^{-3}$ . The popular, and smaller, $r_{200}$ radius (Carlberg, Yee, & Ellingson, Reference Carlberg, Yee and Ellingson1997), within which the average (stellar mass) density is equal to $200 \rho_{\rm critical}$ , is additionally calculated. The virial radius and the $r_{200}$ radius (associated with the stellar mass and thus ignoring any potential dark matter halo) is shown in Figure 13 as a function of the absolute magnitudeFootnote ^s.

Figure 13. The virial radius is shown as a function of the B-band absolute magnitude $\mathfrak{M}_B$ . The average stellar mass density within the virial radius equals $104.5\, \rho_{\rm critical}$ . Also shown is the radius $r_{200}$ within which the average density equals $(200/\Omega_{\rm matter}) \rho_{\rm critical}$ . The slope equals -3, except for the luminous galaxies with high Sérsic indices, and thus long tails to their light profiles, with stars beyond the virial radius.

For $\mathfrak{M}_B \mathbin{\lower.3ex\hbox{$\buildrel\gt\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} -22$ mag, the slope of the $\mathfrak{M}_B$ – $r_{\rm virial}$ and $\mathfrak{M}_B$ – $r_{200}$ relation is 7.5. Converting the magnitude axis to $\log({\rm luminosity})$ and applying a constant stellar mass-to-light ratio (as was assumed for calculating the virial radii) would give a slope of $7.5/2.5=3$ for the logarithmic mass–size relation. Thus, for $\mathfrak{M}_B \mathbin{\lower.3ex\hbox{$\buildrel\gt\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} -22$ mag, the radii are large enough that they enclose the bulk of the stellar mass M, and thus, the pursuit of a constant, average enclosed density (mass/volume) is simply giving radii that meet the condition $M \propto r^3$ . As such, the masses may as well be point masses, as the information in the density profile is effectively lost.

5.4.1. A variation

Here, a new internal galaxy radius, $r_{\rm g}$ , is introduced. It is such that the average density within this radius equals some fraction of the local density at that radius. Mathematically, this can be thought of as a variation of the virial radius, which can be expressed as

(29) \begin{equation} \log (104.6)=\log \langle \rho \rangle_{r_{\rm virial}} - \log \rho_{\rm critical}. \label{eqn29} \end{equation}

The variation introduced here can be written as

(30) \begin{equation} \log ( \mathrm{H} )=\log \langle \rho \rangle_{r_{\rm g}} - \log \rho(r=r_{\rm g}). \label{eqn30} \end{equation}

This radius is somewhat akin to Petrosian (Reference Petrosian1976) radii, used by observers, which is such that the average intensity within some projected radius $R_{\rm P}$ divided by the intensity at that radius (denoted $\eta$ ) equals some constant value, typically 5 (e.g. Bershady, Jangren, & Conselice Reference Bershady, Jangren and Conselice2000; Blanton et al., Reference Blanton, Dalcanton and Eisenstein2001). For Petrosian radii, one has the expression

(31) \begin{equation} -2.5 \log [\eta=5]=\langle\mu\rangle_{R_{\rm P}} - \mu(R=R_{\rm P}) \label{eqn31} \end{equation}

Figure 14 presents these new galaxy radii (equation (30)) for values of $\log \mathrm{H} =$ 0.5, 0.6, 0.8, and 1.0. One can see that they, unlike the virial radii, are no longer too large to be unaffected by the galaxies’ structure. They behave in a fashion somewhat similar to the internal radii containing different fractions of the galaxy light. Once again, no convincing evidence for a dichotomy at a fixed magnitude is apparent.

Figure 14. Internal radius where the mean enclosed density equals some fraction of the density at that radius.

6.1. Case study 1: Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006)

Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006) imaged 100 Virgo cluster ETGs with HST and the F475W filter (transformed to the AB photometric system’s g-band) as a part of the ‘Advanced Camera for Surveys Virgo Cluster Survey’ (ACSVCS; Côté et al., Reference Côté, Blakeslee and Ferrarese2004; Ferrarese et al., Reference Ferrarese, Côté and Jordán2006). Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006) fit seeing-convolved Sérsic and core-Sérsic models (plus optional nuclear excesses) to the (geometric mean)-axisFootnote ^t surface brightness profiles. As such, because they take the ellipticity profile into account, their models can be readily integrated to obtain the total galaxy magnitude. The (surface brightness fluctuation)-based distances from Mei et al. (Reference Mei, Blakeslee and Côté2007) have been used to convert these model magnitudes into absolute magnitudes, and they have been corrected for Galactic extinction using the values from Schlafly & Finkbeiner (Reference Schlafly and Finkbeiner2011), as tabulated in the NASA/IPAC Extragalactic Database (NED)Footnote ^u.

Of these 100 galaxies, 2 (VCC 1535; VCC 1030) could not be modelled by Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006) due to dust, and 2 (VCC 1250 and VCC 1512) have core-Sérsic fits which Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006, see their Section 4.2) discredit — and rightfully so given that the $R_{\rm e}$ values hit their limit of 490 arcsec, as was also the case for VCC 575 ( $\mathfrak{M}_B = -17.61$ mag). In addition to these five galaxies, the S0 galaxy VCC 1321 (NGC 4489, $\mathfrak{M}_B = -18.20$ mag) which was reported to have an unusually high galaxy Sérsic index of $\sim$ 6 (cf. $2.3\pm0.5$ from Table C1 of Krajnović et al., 2013) is also excluded.

Here, we will see how the linear $\mathfrak{M}$ – $\mu_0$ and $\mathfrak{M}$ –n relationsFootnote ^v, spanning the Virgo ETG sample’s full magnitude range, explain the curved trends in diagrams involving effective radii and effective surface brightnesses. The following two g-band equations approximate the distribution of data seen in the upper panels of Figure 15:

Figure 15. Sample of 94 Virgo cluster ETGs from Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006). Top row: The Sérsic parameters, including $\mathfrak{M}_g$ and $\mu_{0,g}$ , are from fits to the (geometric mean)-axis light profiles. The lines are defined in equations 32 and 33. Middle and bottom rows: The solid curves are predictions based on the linear fits in the top panels. The three equal-dashed curves in the lower panels show the $\mathfrak{M}_g = -15$ mag boundary. Looking at the two lower left-hand panels, one might be inclined to call for a divide at $\mathfrak{M} = -20.5\pm0.5$ mag, while looking at the two lower right-hand panels, one may instead be inclined to advocate for a divide at $\mathfrak{M} = -18\pm1$ mag.

(32) \begin{eqnarray} \mathfrak{M}_g &=& -10.5\log(n) - 14.0, {\rm and} \label{eqn32}\end{eqnarray}

(33) \begin{eqnarray} \mathfrak{M}_g &=& 0.63\mu_{0,g} -28.4, \label{eqn33}\end{eqnarray}

The predicted g-band $\mathfrak{M}$ – $\mu_z$ and $\mathfrak{M}$ – $R_z$ distributions are presented in the middle and lower panels of Figure 15 for $z=0.05, 0.5,$ and $0.95$ . Similar results are obtained with the $\mathfrak{M}$ – $\langle \mu \rangle$ diagram and also when using their data obtained through the F850LP filter.

The middle and lower panels of Figure 15 should be compared with Figure 117 in Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006), which used quantities at $z=0.05$ and 0.5 ( $Z=5$ % and 50%), and compared with Figure 76 in K09 which used quantities at 50%. The series of linear equations 17– 26 in Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006) — used to approximate the bright and faint ends of their $\mathfrak{M}$ – $\mu_{\rm e}$ , $\mathfrak{M}$ – $R_{\rm e}$ , $\mathfrak{M}$ – $\mu_5,$ and $\mathfrak{M}$ – $R_5$ distributions — does not adequately capture the curved nature of the scaling relations which unify the faint and bright ETGs in these diagrams. Their equations have been fit separately to the core-Sérsic and Sérsic galaxies, implying a division between these two galaxy types in these diagrams. However, their set of linear approximations are not only dependent upon the magnitude range included in the fit, but they go against the premise of a continuity in these diagrams and against the understanding that the different slopes at bright and faint magnitudes cannot be used to interpret signs of different galaxy types or formation physics in diagrams involving ‘effective’ parameters. Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006) understood that there is a continuity at $\mathfrak{M} = -18$ mag and a divide at $\mathfrak{M} \approx -20.5$ mag — as did Gavazzi et al. (Reference Gavazzi, Donati and Cucciati2005); Côté et al., Reference Côté, Ferrarese and Jordán2007, Reference Côté, Piatek and Ferrarese2006; Misgeld, Mieske, & Hilker, Reference Misgeld, Mieske and Hilker2008 2009; and Chen et al., Reference Chen, Côté, West, Peng and Ferrarese2010 — but diagrams involving effective radii and effective surface brightnesses can not be used to make this diagnosis. Similarly, the colour-coding used by K09 (see their Figure 76) is inappropriate and misleading.

6.2. Case study 2: Kormendy et al. (Reference Kormendy, Fisher, Cornell and Bender2009)

K09 acceptFootnote ^w the $\mathfrak{M}$ –n relation (see their Figure 33) but they deny the existence of a linear $\mathfrak{M}$ – $\mu_0$ relation unifying dwarf and ordinary ETGs (see their Figure 1). This follows on from Kormendy (Reference Kormendy1985, see his Figure 3), which produced an $\mathfrak{M}_B$ – $\mu_0$ diagram with a sample selection that had an absence of ETGs with magnitudes $-17 \mathbin{\lower.3ex\hbox{$\buildrel\gt\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} \mathfrak{M}_B \mathbin{\lower.3ex\hbox{$\buildrel\gt\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} -20.5$ mag ( $H_0=50$ km s-1 Mpc-1), making it difficult to know where a transition may occur.Footnote ^x Further complicating the situation was that the faint ETG data in Kormendy (Reference Kormendy1985) did not produce the known $\mathfrak{M}_B$ – $\mu_{0,B}$ trend (e.g. Binggeli et al. Reference Binggeli, Sandage and Tarenghi1984, see their Figure 8)Footnote ^y whose distribution points towards the faint-end of the $\mathfrak{M}_B$ – $\mu_{0,B}$ sequence for bright ETGs with depleted cores. That is, according to the data in Kormendy (Reference Kormendy1985), the ETGs fainter than $\mathfrak{M}_B \approx -17$ mag follow a distribution with a steeper slope in the $\mathfrak{M}_B$ – $\mu_{0,B}$ diagram than shown in the right-hand panel of Figure 1, such that the distribution of faint ETGs in Kormendy (Reference Kormendy1985) points to the bright-end of the distribution of ETGs with depleted cores. Despite the ongoing rejection by K09 for a unifying $\mathfrak{M}$ – $\mu_0$ relation across $\mathfrak{M}_B = -18$ mag ( $\mathfrak{M}_{V_T} \approx -19$ mag), K09 did not actually show the $\mathfrak{M}$ – $\mu_0$ diagram for their data set nor the $\mathfrak{M}$ – $\mu_5$ diagram used by Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006).

K09 also excluded many of the ETGs in Ferrarese et al. (Reference Ferrarese, Côté and Jordán2006) because they wanted to work with what they thought was a sample of predominantly one-component galaxies, that is, no lenticular galaxies. K09 effectively thinned-out much of the population of ETGs from $-18 > \mathfrak{M}_B > -20.5$ mag. This resulted in a sample of 42 ETGs, including 10 dwarf ETGs. However, 5 of the 32 non-dwarf galaxies were subsequently identified as S0 galaxies by K09. Furthermore, another 5 were rare compact elliptical (cE) galaxies, taken from Table XIII of Binggeli et al. (Reference Binggeli, Kunth, Thuan and Van1985; who note in Part 3 of their Appendix that the M32-like galaxies are vastly outnumbered by dwarf ETGs of similar magnitude: $-14 > \mathfrak{M}_B > -18$ mag). The cE galaxies are thought to be heavily stripped disc galaxies (e.g. Rood, Reference Rood1965; Bekki et al., Reference Bekki, Couch, Drinkwater and Gregg2001; Graham, Reference Graham2002b; Chilingarian et al., Reference Chilingarian, Cayatte and Revaz2009), while the isolated cE galaxies may have either never acquired a significant disc or may have been ejected from a cluster after losing much of their disc (e.g. Chilingarian & Zolotukhin, Reference Chilingarian and Zolotukhin2015). The cE galaxies are two-component systems, likely dominated by a remnant bulge, and are known to overlap with the bulges of spiral and S0 galaxies in the scaling diagrams (e.g. Graham Reference Graham, Oswalt and Keel2013, see his Figure 1). This over-representation of cE galaxies, relative to normal galaxies, in K09 is inappropriate for two reasons. In terms of a sample providing a balanced representation of galaxies, there should be $\sim$ 200 times (I. Chilingarian 2018, priv. comm.) fewer cE galaxies than non-cE galaxies across their co-existing range in absolute magnitude. Second, the cE galaxies are more akin to bulges, and as such, they are better compared with bulges than with parameters from single Sérsic fits to ETGs that typically contain bulges and discs.

As for the five galaxies identified by K09 as lenticular galaxies, K09 performed a bulge/disc decomposition for these. As with their fitting of a single Sérsic model, they did not convolve their models with the central image’s point spread function but excluded by eye the region they considered to be affected by either nuclear excesses or a partially depleted core. They then used the bulge parameters rather than the galaxy parameters for these 5 galaxies to compare with the galaxy parameters of the remaining galaxies which they thought were pressure-supported, single-component systems (with additional small nuclear excesses or cores). However, Emsellem et al. (Reference Emsellem, Cappellari and Krajnović2011, see their Table B1) report on the internal kinematics for the brightest 19 of the supposed 27 ( $=32-5$ ) ‘elliptical’ galaxies in K09. They reveal that 10 of these 19 are ‘fast rotators’, and Krajnović et al. (2013) provide bulge/disc decompositions for 7 of them. Furthermore, Toloba et al. (Reference Toloba, Guhathakurta and Boselli2015) contains internal kinematical information for 6 of the 10 ‘dwarf spheroidal’ galaxies in K09, reporting that 4 of these 6 are ‘fast rotators’. K09 have therefore plotted a mixture of bulge parameters (for 5 S0 galaxies) and galaxy parameters (for at least 11, and likely more, S0 galaxies). This blurs prospects for identifying connections in parameter scaling diagrams, and it explains why K09 did not find the known $\mathfrak{M}$ –n or $\mathfrak{M}$ – $\mu_0$ relations. Given that bulges and ETGs follow a different size–luminosity relation, they cannot follow the same $\mathfrak{M}$ –n and $\mathfrak{M}$ – $\mu_0$ relations (see 4.1). The $\mathfrak{M}$ –n diagram in K09 is thus a blurring of two distributions, which have been separated here in Figure 16.

Figure 16. Sample of 42 stellar systems in the Virgo cluster from K09, comprising ordinary ETGs (large red circles), 5 bulges of ETGs (light blue squares, NGC: 4570, 4660, 4564, 4489, 4318), and 6 compact elliptical galaxies (dark blue stars, VCC: 1297, 1192, 1440, 1627, 1199, 1545) which are considered to be the remnant bulges of stripped disc galaxies. An additional 128 Virgo cluster ‘dwarf’ ETGs from Binggeli & Jerjen (Reference Binggeli and Jerjen1998) show the extension to fainter magnitudes. A rough $B-V=0.8$ colour was applied uniformly to this latter sample of B-band data. Top panel: The parameters are from Table 1 in K09, where the absolute magnitudes $\mathfrak{M}_{V_T}$ were derived independently of the Sérsic model for the ETGs and cE galaxies and are from their column 11; the Sérsic indices are major-axis values; and the central surface brightnesses are the $R=0$ values from their Sérsic models fit either to the galaxy or, in 5 instances, the bulge component. Expressions for the red ETG lines are provided in equations 34 and 35. Middle and lower rows: Similar to Figure 15. The three equal dashed curves show the $\mathfrak{M}_{V_T} = -14$ mag boundary. Neither the absolute magnitudes, effective half-light (50%) radii, nor effective surface brightnesses in the lower middle panels are from the Sérsic model but were instead obtained independently from 2D profile integration by K09. The 10% and 95% radii and surface brightnesses were derived from the Sérsic model.

The upper panels in Figure 16 display the $\mathfrak{M}$ – $\mu_0$ (Vega V-band mag) diagram using the data from K09. One can see that there is an $\mathfrak{M}$ – $\mu_0$ relation for ETGs, although a couple of high $-n$ ETGs appear to have had their Sérsic index over-estimated and their total magnitudes under-estimatedFootnote ^z by the (roughly) isophotal magnitudes advocated by K09 and used here for comparative purposes. The following equations represent the lines for ETG shown in the upper row of Figure 16.

(34) \begin{eqnarray} \mathfrak{M}_{V_T} &=& -9.6\log(n) - 15.0, {\rm and} \label{eqn34}\end{eqnarray}

(35) \begin{eqnarray} \mathfrak{M}_{V_T} &=& 0.63\mu_{0,V} -29.0. \label{eqn35}\end{eqnarray}

As can be seen in Figure 16, the cE galaxies do not follow either the $\mathfrak{M}$ –n nor the $\mathfrak{M}$ – $\mu_0$ relations for ETGs. The bulges of the S0 galaxies similarly do not follow these relations. For a given central surface brightness, the bulges have fainter absolute magnitudes than the ETGs, which makes sense given that their disc light has been excluded. The offset to fainter absolute magnitudes in the $\mathfrak{M}$ –n diagram is not as great, due to the reduced Sérsic indices of these bulges relative to their galaxy Sérsic indices (which tend to be higher due to the outer disc light). K09 do not use a different colour to denote the (i) cE galaxies, (ii) bulges, and (iii) ETGs. This missing information makes it difficult to appreciate what is going on in their scaling diagrams.

K09 elected to plot the $\mathfrak{M}$ – $\mu_{10}$ , rather than the $\mathfrak{M}$ – $\mu_{0}$ or $\mathfrak{M}$ – $\mu_{5}$ , and the $\mathfrak{M}$ – $\mu_{\rm e}$ diagrams, which are shown here in Figure 16, along with the $\mathfrak{M}$ – $\mu_{95}$ diagram, with the bulge- and galaxy-type information included. The exclusion of known S0 galaxies by K09, coupled with their use of bulge rather than galaxy parameters for some S0 galaxies but not others, results in a thinning of the bridging population of ETGs around $\mathfrak{M}_B = -18$ to $-20.5$ mag in their diagrams. This practice is particularly apparent throughout Kormendy & Bender (Reference Kormendy and Bender2012), Bender et al., Reference Bender, Kormendy, Cornell and Fisher2015, and Kormendy (Reference Kormendy, Laurikainen, Peletier and Gadotti2016). Furthermore, their attempted shift of focus to dynamically hotFootnote ^aa systems, i.e., the bulge sequence (e.g. Balcells et al., Reference Balcells, Graham and Peletier2007; Graham, Reference Graham, Oswalt and Keel2013, and references therein), rather than the ETG sequence, came at the expense of realising the continuous ETG sequence, that is, the continuity between dwarf and ordinary ETGs, and contributed to their ongoing belief in the artificial divide at $\mathfrak{M} \approx -18$ mag.

Figure 17 has been included to better help one evaluate the colour-coding and information presented in the scaling diagrams of K09, Bender et al. (Reference Bender, Kormendy, Cornell and Fisher2015), Kormendy (Reference Kormendy, Laurikainen, Peletier and Gadotti2016), and elsewhere. The curved distribution for the ETGs in these scaling diagrams involving the arbitrary ‘effective’ parameters is not a sign of division but arises from the unity seen in the $\mathfrak{M}$ – $\mu_0$ and $\mathfrak{M}$ –n diagrams. As will be broached in 7.3, numerous other scaling relations also display a continuity across the alleged dwarf/ordinary ETG divide at $\mathfrak{M}_B =-18$ mag.Footnote ^bb

Figure 17. Zoom in and summary of the effective half-light parameters displayed in Figure 16. Here, the predicted relations for bulges — according to the two linear relations in Figure 16 — have also been included. This diagram facilitates comparison with, and understanding of, Figure 14 in Bender, Kormendy, & Cornell (2015) — where bright S0 galaxies were often either excluded or their bulge parameters plotted, and where dwarf S0 galaxies always have their galaxy parameters plotted.