The Correlation Luminosity-Velocity Dispersion of Galaxies and Active Galactic Nuclei

Abstract

In this work we discuss the correlation between luminosity L and velocity dispersion $\sigma$ observed in different astrophysical contexts, in particular that of early-type galaxies (ETGs; Faber–Jackson (FJ) law) and that of active galactic nuclei (AGN). Our data for the ETGs confirm the bending of the FJ at high masses and the existence of similar curvatures in the projections of the Fundamental Plane (FP) approximately at the mass scale of ∼${10}^{10}{M}_{\odot }$. We provide an explanation for such curvatures and for the presence of the Zone of Exclusion (ZoE) in these diagrams. The new prospected theory for the FJ law introduces a new framework to understand galaxy evolution in line with the hierarchical structure of the Universe. The classic analysis carried out for a class of type 1 AGN accreting gas at very high rates, confirms that a FJ law of the form $L={\mathcal{L}}_0{\sigma }^4$ is roughly consistent with the observations, with a slope quite similar to that of ETGs. We discuss the physics behind the FJ law for the AGN in different contexts and also examine the biases affecting both the luminosity and the velocity dispersion, paying particular attention to the effects induced by the spherical symmetry of the emitting sources on the accuracy of the luminosity estimates.

1. Introduction

The correlation between luminosity L and absorption/emission line broadening (closely linked to the rotational velocity V or the velocity dispersion $\sigma$ of gas and stars) is observed in different astrophysical contexts.

The most well-known is the Faber–Jackson (FJ) relation, which is the correlation observed between the luminosity L of galaxies and the velocity dispersion $\sigma$ (of stars). The history of this correlation is very long and cannot be reported here. In brief, the first papers were by Minkowski [1] (who used a sample of 13 ETGs), and Morton and Chevalier [2] a decade later. The relation took the power law form $L\propto {\sigma }^4$, that we all know only with the work of Faber and Jackson [3] (who used a sample of 25 ETGs), and since then it is commonly known as the FJ relation, customarily expressed as $L=L_0{\sigma }^{\alpha }$, where $L_0$ and $\sigma$ take suitable values that depend on the available observational data. It is worth recalling here that the usual FJ relation refers to galaxies (ETGs) of the Local Universe, therefore, old objects at redshift zero, unless otherwise specified. In other words, it is a picture of the present day situation. No temporal effects due to the ever-changing physical content of a galaxy are included. Finally, the FJ relation is a projection of the Fundamental Plane in the multidimensional space of the luminosity L, the mass M (usually the stellar mass), the radius R (usually the half-light or half-mass radius $R_e$), the specific intensity $I_e$ $(I_e\propto L/R_e)$, and the velocity dispersion $\sigma$ (see a recent discussion by [4]).

The possibility of a direct physical connection between L and $\sigma$ in a galaxy has been hastily rejected since the discovery of the correlation: why in a galaxy should the total light emitted by its stars be aware of their mean velocity? Despite this, the $L-\sigma$ correlation has been observed in different astrophysical contexts and was eventually accepted as one of the most important relationships, followed by galaxies, and it is a basic tool of analysis in many astrophysical circumstances. It suffices to mention its use of distance indicator for galaxies, because it links the distance-independent quantity $\sigma$ to an intrinsic property such as the total luminosity L of the stars (see, e.g., [5,6,7,8,9]). In recent times the classical FJ relation [3] in its modern declinations was extended by D’Onofrio et al. [10] to include evolutionary effects in the galaxy population. They adopted the same formalism of the FJ relation but supposed that $L_0$ and $\alpha$ are time and galaxy dependent. The new relation is $L=L_0^{\prime }\left(t\right){\sigma }^{beta\left(t\right)}$. This simple idea, combined with the virial theorem expressing the dynamical equilibrium of a galaxy, was very rewarding because it led to a description of the FP (inclination, thickness, and distribution of galaxies on it), together with its projection planes, as functions of the variables $L_0^{\prime }\left(t\right)$ and $\beta \left(t\right)$.

Passing to active galactic nuclei (AGN), the luminosity-$\sigma$ relation is also present in these systems. The consensus that the broadening of emission lines in quasar and Seyfert nuclei results from the Doppler effect, attributed to the rapid motion of the line-emitting gas, emerged early on; already Carl Seyfert, in his initial systematic study of galaxies named after him [11], attributed the large widths of lines to Doppler shifts, with velocities reaching up to approximately 8500 km/s for the hydrogen lines of NGC 3516 and 7469 [12]. The nature of the motions has been debated ever since (see, e.g., and references therein, [13,14,15,16]). The realization that two systems of emission lines—one dominated by virial broadening and one affected by strong outflow motions—are coexisting in AGN came slowly [17,18,19,20,21]. Since then, the broadening of the low-ionization emission lines (LILs, especially H$\beta$ and Mgii$\lambda$2800) has been used in numerous attempts to estimate the black hole mass assuming virial motions (and references therein [22,23,24,25,26]).

In this study we will focus on the formulation of updated scaling laws based on recipes to correct for viewing angle and accretion rate effects, whose importance has only very recently been appreciated in full [27,28].

So, it is perhaps not surprising that the formulation of a FJ-like law in the context of AGN has a relatively recent history. Regarding the use of a FJ law as a luminosity indicator, one has to first consider that quasars are the opposite of standard candles, since their luminosity is spread over more than five orders of magnitude (see the interviews on the exploitation of quasars as distance indicators in [29]). Attempts to define correlation between spectral parameters and luminosity have been systematically frustrated by the increasing dispersion that arose when samples of increasing size have become available. A case in point is the Baldwin effect [30,31,32], a luminosity trend that becomes shallower and statistically weaker with an increasing sample size. A connection between quasars observational and accretion parameters has become possible only after the systematic organization of the diverse spectral properties along an easily identified sequence (the so-called Eigenvector-1 sequence or quasar main sequence [14,33,34,35]). Since then, two main populations and several spectral types along the sequence have provided contextualization for most type-1 AGN [36,37], in both low and high luminosity samples [38,39]. The distinction between the two populations—conventionally named A and B—is rooted in systematic differences in terms of Eddington ratio, which in turn translates into the possibility of two different accretion modes [40,41]. Of special interest is a spectral type at one end of the quasar main sequence (MS), characterized by the most intense optical singly-ionized iron emission [9,42]. Even if the strong Feii emitters have been studied systematically since the 1990s [43,44,45], the association of Feii emissions with a high Eddington ratio is a more recent result [35,42,46,47,48]. At very high accretion rates, a geometrically thick, optically thick disk is expected to develop, with the accretion flow becoming advection-dominated [49,50]. As a consequence, the Eddington ratio tends asymptotically to values of order unity [51] for a dimensionless accretion rate $\dot{m}=\dot{M}/(L/c^2)\gg 1$, where $\dot{M}$ is the mass accretion rate and L is the quasar luminosity. This important result is at the basis of the exploitation of a quasar FJ-like relation for cosmology (as will be amply discussed below), and confirms that the luminosity follows a dependence on line broadening consistent with the virial theorem, at least for Population A sources. Finally, a luminosity versus velocity relationship is also possible for type-A AGN.

Given these introductory premises, the chief aim of this review is to report in some detail on the past history and most recent results concerning the luminosity–velocity relation in galaxies (ETGs) and AGN.

The plan of the paper is as follows. In Section 2 we briefly review the state of the art of the $L-\sigma$ relation both for ETGs and AGN. In Section 3 we deal with ETGs in some detail, present the observational data and the Illustris galaxy models we have adopted, discuss the classical FJ relation, examine the four possible projections of the FP and the distribution of galaxies in these projection planes, and report on the results obtained with the time-dependent FJ relationship. In Section 5 we address the subject of AGN, describe the two populations of AGN, the current estimates of the masses of central Black Holes, different estimates of the velocity dispersion and corrections of this by different effects, and finally derive luminosity versus velocity dispersion relationships for two groups of so-called population-A AGN. In Section 5, we apply the formalism developed by D’Onofrio et al. [10] to derive the variables $L_0^{\prime }\left(t\right)$ and $\beta \left(t\right)$ to our group of AGN, pointing out the remarkable similarity in the scaling laws for AGN and ETGs. Lastly, in Section 6 we summarize the results and present some conclusions. Finally, in Appendix A we present some details on measuring the velocity dispersion in xA sources.

All the calculations and figure are obtained using the $\mathsf{\Lambda }$-CDM cosmological parameters [52]: ${\mathsf{\Omega }}_m=0.3,{\mathsf{\Omega }}_{\mathsf{\Lambda }}=0.7\mathrm{and}{H}_0=70$ km s⁻¹ Mpc⁻¹.

2. State of the Art of the ETGs and AGN $\mathit{L}\mathbf{-}\mathit{\sigma }$ Space

2.1. The ETGs

Several works have addressed the $L-\sigma$ correlation looking at different aspects of the observed distribution of ETGs in this parameter space, taking advantage of the even larger samples of galaxies with available measurements of $\sigma$ provided by the big surveys (e.g., ATLAS, SDSS, BOSS, etc.). Many such works are concerned with the slope of the relation, the observed scatter, the evolution of the relation with redshift, and its power as a distance indicator.

Very soon it became clear that the slope $\alpha$ of the $L=L_0{\sigma }^{\alpha }$ relation depends on the range in luminosity of the ETGs under analysis. Values larger than 4 were measured for luminous ETGs (e.g., [53,54,55,56,57]) and ∼2 for faint systems (e.g., [58,59,60,61,62,63], Lauer et al. [64]) for example, using a compilation of HST observations for ∼200 ETGs, measured a slope of ∼7. Similarly the team of the ATLAS3D [65] survey, working with 260 local ETGs, measured a high-mass slope of ∼$4.7$ for the stellar mass $M-\sigma$ relation above ∼$2\times {10}^{11}{M}_{\odot }$ [55]. The analysis of Focardi and Malavasi [66] instead reported a value of ∼$5.6$, using a sample of a few hundred objects extracted from HYPERLEDA [67].

The measured slopes of the FJ therefore vary among different works, from ∼$4.5$ to ∼8 for luminous ETGs and ∼$t2$ for faint objects. Such curvature at high luminosity has been confirmed with high statistical significance with the SDSS and BOSS CMAS data [63,68,69,70,71,72]).1

Bernardi et al. [73] correctly pointed out that the slope of the correlation depends on the fitting strategy and that selection effects play a different role according to the correlation chosen. The fit of the luminosity at each velocity dispersion ($logL=alog\sigma +b$) does not give the same result of the fit of the velocity dispersion as a function of luminosity ($log\sigma =a^{\prime }logL+b^{\prime }$), nor of the joint distribution of L and $\sigma$. The first allows for the use of $\sigma$ to predict L, whereas the second allows the opposite.

For what concerns the intrinsic scatter, the conclusions are a bit contradictory: Nigoche-Netro et al. [74] measured a decrease (with mass), whereas Shu et al. [75] seems to indicate the opposite. Montero-Dorta et al. [63] measured an intrinsic scatter $s=0.047\pm 0.004$ in log units at a fixed L using the big BOSS CMAS database and found no significant evolution of the scatter up to z∼0.5. Hyde and Bernardi [70] observed instead a scatter decreasing towards high luminosities. They estimated a value of ∼0.05 dex at $M_r<-23$. At the high-mass end, Kormendy and Bender [56] derived an intrinsic scatter of 0.06 in $log\sigma$ for “core” galaxies and 0.10 for “core-less” ellipticals at a given magnitude. A mass (or luminosity) dependence of the intrinsic scatter of the $L-\sigma$ relation has been also been reported in other papers [66,69,74,76]. Nigoche-Netro et al. [71,77,78,79] also claimed that the intrinsic dispersion is affected by geometrical effects and that its value does not diminish with the magnitude range.

Being that the FJ relation is a projection of the Fundamental Plane (FP) [80,81], in this case of the $log\left(\sigma \right)-log\left(L\right)-log\left(R_e\right)$ space where the effective radius $R_e$ and the central velocity dispersion $\sigma$ are correlated with the luminosity L, the redshift evolution of the FJ relation is closely linked to the evolution of the ETG distribution in the FP. It is known that the zero-point of the FP evolves from z∼1 in a way consistent with the passive evolution of stellar populations [73,82]. Montero-Dorta et al. [63] found that the evolution of the zero-point of the FJ relation is consistent with such passive evolutions since the redshift of $z=2-3$, which is in good agreement with the FP results, and that the high-mass end of the relation does not change in the range $0.5<z<0.7$ and is consistent with that at z∼0.

The data of the hydro-dynamical numerical simulations, e.g., that of Illustris [83], analyzed by D’Onofrio et al. [10], also seem to indicate that the slope of the FJ relation was much steeper at redshift $z>1$ and that the curvature observed at high luminosity appeared only at redshifts lower than 1.5 in all projections of the FP.

In general, the indications coming from different studies suggest that several scaling relations obtained with the structural parameters of the FP change with redshift. Some works claim that only the zero-point vary with redshift (e.g., [84,85,86,87]), while others find that both zero-point and slope are variables (e.g., [82,88,89]). In this context, dwarf and bright ETGs seem to have different behaviors, possibly originating from different paths of evolution [69,90,91,92,93,94,95].

Recently D’Onofrio et al. [10] introduced a new idea concerning the origin of the FJ relation, proposing that this relation comes out of the combination of two different scaling laws, the virial theorem (VT) on one side and a new empirical relation, the $L=L_0^{\prime }{\sigma }^{\beta }$ law, that accounts for the variation in mass and luminosity due to mergers and stripping phenomena typical of the hierarchical scenario of galaxy formation and evolution. In the $L=L_0^{\prime }{\sigma }^{\beta }$ formulation the main role is played by the terms $L_0^{\prime }$ and $\beta$. The idea is that the total luminosity of galaxies is the result of star assemblies, star formation history, and stellar evolution. In this view, $L_0^{\prime }$ and $\beta$ are variables parameters that change significantly from galaxy to galaxy. The advantage of this approach is that, when the $L=L_0^{\prime }{\sigma }^{\beta }$ law is coupled with the VT, one can reproduce the whole distribution of ETGs in all the projections of the FP, the trends observed for the high mass tail of ETGs, and explain the origin of the Zone of Exclusion (ZoE) [4]. The parameter $\beta$ can assume both large positive and negative values (and $L_0^{\prime }$ consequently can spread over quite a big interval). This is not a problem for the scatter observed in the FJ, because the maximum possible variation in L is of the order of ∼$0.3$ in log units (a factor of ∼2) when two equal mass objects merge. This is a very rare phenomenon, in particular at z = 0. In practice the whole scatter of the FJ is preserved at almost all redshifts.

The extended FJ law might imply that the luminosity of galaxies L depends, although in a mild way, on the velocity dispersion $\sigma$. Such ideas can be draw back to Brosche [96], who first suggested a failing of the simple law of star formation Schmidt [97], based only on the gas density $\rho$, favoring a scenario in which star formation is a function ∼$f\left(\rho v^{\beta }\right)$, where v is the velocity of stars and $\beta$ is a variable parameter. The stars born in large gas aggregates have a characteristic velocity that depends on the physical condition of the galaxies during the star forming event (collapse, shock, merging, etc.). For this reason the global star formation and consequently the total luminosity of galaxies might keep a memory of the velocity of this gas, and later on, of the velocity dispersion of its stars. This opens the possibility of a “physical” connection between L and $\sigma$, leaving to the classical FJ relation the meaning of a “translation” of the $M-\sigma$ relation, obtained substituting the mass M (difficult to measure) with the luminosity L (easy to measure).

When we leave the domain of ETGs we discover that L and $\sigma$ correlate also in other physical contexts. It is well known that in young massive star clusters, where giant extra-galactic HII regions exist, a correlation between luminosity and width of the emission lines is observed; this is the L($H\beta$) − $\sigma$ relation [98] between the luminosity of the Hβ line and the velocity dispersion of the HII gas. The scatter in this relation is small enough that it has been used as a distance indicator [99,100,101,102,103,104]. One key property is that, when the mass of the young cluster increases, both the number of ionizing photons and the motion of the ionized gas increase. The HII feature in the optical spectrum is visible out to z∼3.5 and, according to Koo et al. [105] and Guzman et al. [106], a large fraction of compact star-forming galaxies at an intermediate redshift have a similar L($H\beta$) − $\sigma$ of local galaxies. Bordalo and Telles [102] explored the $L\left(H\alpha \right)-\sigma$ correlation and its systematic errors in the redshift range ($0\simeq z\simeq 0.08$), concluding that something close to a power law ($L\propto {\sigma }^4$) is observed when objects with Gaussian emission line profiles are considered. The root mean square (rms) scatter they measured is $\delta logL\left(H\alpha \right)$∼0.30.

It is important to emphasize that the L($H\beta$) − $\sigma$ relation observed at high redshift is that of a young burst of star formations and not of the whole parent galaxy. The size of the star-forming region enters, in fact, as a second parameter in the relation, as well as the emission-line equivalent width, the continuum colour, and the metallicity, which also contribute to produce a small rms scatter ($\delta logL\left(H\beta \right)=0.233$). The coefficients derived from the best fit of the relation are very close to the expectation of the virial theorem, as in the case of the $L-\sigma$ relation observed for ETGs. At the same time the total amount of stellar and ionized gas mass2 is similar to the dynamical mass, suggesting that gravity is the main mechanism behind the broadening of the emission lines in these young clusters.

2.2. The AGN

In the context of AGN research, a similar $L-\sigma$ relation is observed when the intensity of the Hβ line, emitted in the broad line region (BLR) around the central black hole, is correlated with the full width, half maximum, or with the velocity dispersion $\sigma$ of the gas). This correlation is observed for a particular class of quasars, the xA, that radiate at extreme $L/L_{\mathrm{Edd}}$ along the quasar main sequence [9]. Accretion disk theory predicts for these objects a low radiative efficiency at a high accretion rate and a $L/L_{\mathrm{Edd}}$ ratio converging toward a limiting value [49,107,108]. The inner part of the disk is puffed up by radiation pressure, while the outermost one remains geometrically thin. This change from the standard thin disk provides two key elements for the BLR structure: the existence of a collimated cone-like region, where the high ionization outflows might be produced, and the shadowing of the outer disk where low-ionization emission lines form [109]. The low-ionization emitting region may therefore remain shadowed from the intense radiation field that is associated with the continuum observed if the line of sight is not too far from the polar axis, and the velocity field stays unperturbed.

The self similarity of the xA spectra permits an easy identification of these sources up to z∼$3-4$, making the $L-\sigma$ relation of quasars a powerful cosmological probe [110,111,112]. xA quasars satisfy three conditions: (1) a constant Eddington ratio $L/L_{\mathrm{Edd}}$, close to the Eddington limit; (2) the assumption of virial motions for the low-ionization BLR, so that the black hole mass $M_{\mathrm{BH}}$ can be expressed by a virial relation; and (3) the spectral invariance. For xA the ionization parameter U can be written as $U=Q\left(H\right)/4\pi r_{\mathrm{BLR}}^2{n}_{\mathrm{H}}c\propto L/r_{\mathrm{BLR}}^2{n}_{\mathrm{H}}$ [15], where $Q\left(H\right)$ is the number of hydrogen ionizing photons. U has to be approximately constant, otherwise we should observe a significant change in the spectral appearance. The three constraints make it possible to derive a relation between the line width (the FWHM of the H$\beta$ broad component is expressed in units of 1000 km s⁻¹) and luminosity:

$$L\left(\mathrm{FWHM}\right)={\mathcal{L}}_0·{\left(\mathrm{FWHM}\right)}_{1000}^4{\mathrm{erg}\mathrm{s}}^{-1}$$

(1)

where ${\mathcal{L}}_0$ depends on the square of $L/L_{\mathrm{Edd}}$, the ionizing range of the spectral energy distribution, and a parameter directly derived from the UV spectra, the product density times ionization parameter that has been scaled to the typical value ${10}^{9.6}$ cm⁻³ [113,114,115]. Until now, the FWHM of the H$\beta$ broad component and of Aliii$\lambda$1860 have been adopted as virial broadening estimators [116,117,118]. Equation (1) implies that a simple measurement of the FWHM of a low-ionization line yields a z-independent estimate of the accretion luminosity (c.f. [8,9]). The virial equation has been applied to xA quasars only ($L/L_{\mathrm{Edd}}$∼ 1), although in principle it could be useful for all quasars with known $L/L_{\mathrm{Edd}}$, provided a suitable emission line broadened by virial motions is used for the luminosity computation. Potentially, the virial equation can be considered for all xA quasars distributed over a wide range of luminosity and redshift, where conventional cosmological distance indicators are not available [117,118,119].

The virial luminosity equation of AGN is therefore very similar to the FJ law of ETGs and HII regions. Why does L correlate with $\sigma$ approximately in the same way in so different a context? The aim of this work is to try to find a possible answer to such a question. In particular, we will analyze the $L-\sigma$ relation in galaxies and AGN, discussing the properties of the observed relations in both contexts and providing a uniform view of the FJ correlation. In particular, we will discuss the origin of the slope/scatter of the relation and the presence of a curvature in the observed trend. Finally, we will argue why a similar slope is observed in systems that are so different.

3. The $\mathit{L}\mathbf{-}\mathit{\sigma }$ Relation of ETGs

In this section we thoroughly discuss the case of ETGs, first presenting the observational data in use, the $L-\sigma$ relation, and the revised $L=L_0^{\prime }{\sigma }^{\beta }$ presenting the so $\beta$-$L_0^{\prime }$ theory and discussing the great advantages offered by it in the interpretation of the Fundamental Plane and all its projections on 2D planes such as $L-\sigma$, $I_e-R_e$, $I_e-\sigma$, etc.

3.1. The ETGs Data Sample in Use

The observational data for the ETGs are those extracted from the WINGS and Omega-WINGS databases [120,121,122,123,124,125,126,127,128,129]. The WINGS and Omega-WINGS datasets are two optical surveys of nearby clusters ($0<z<0.07$). The databases include a large amount of information: galaxy magnitudes, morphological types, effective radii, effective surface brightness, stellar velocity dispersions, star formation rates, etc.

The data used here are: 1. The velocity dispersion $\sigma$ of ∼1700 ETGs, already used by [130] to infer the properties of the FP; 2. A new set of $\sigma$ derived by [131] for the dwarf ETGs (∼20 objects); 3. The luminosities, the effective radii, and the effective surface brightness in the V-band of several thousand ETGs derived by [131] with the software GASPHOT [132]; 4. The distance of the galaxies derived from the redshifts measured by [122,127].

The errors on the photometric parameters are always $\simeq \mathrm{20\%}$. The values of $\sigma$ were measured within a fixed aperture of three arcsecs and come from the SDSS and NOAO databases. One might question if the results we obtain depend on such an aperture, and change using a velocity dispersion averaged within the effective radius $R_e$ (temporarily indicated as ${\sigma }_e$) instead of ${\sigma }_0$ (measured in the central region). The values of ${\sigma }_0$ and ${\sigma }_e$ are known to be related to each other by the relation $log\left({\sigma }_e\right)=0.78log\left({\sigma }_0\right)+0.46$ (see for details Busarello et al. [133], Busarello et al. [134]), so the net effect of such a change is a systematic variation of the slope $\alpha$ and the proportionality factor $L_0$ of the FJ, while the scatter of the relation remains nearly the same. In other words, such a systematic effect does not alter our conclusion about the curvature of the FJ slope between bright and faint ETGs.

In this work we have chosen to represent with different colors three different samples of objects: the black color marks the dataset of brightest (BCGs) and second brightest ETGs (2ndBCGs), both brighter than $M_V=-21.5$; the red color marks the sub-sample of the WINGS ETGs for which both stellar mass and velocity dispersion are available. These are generally objects fainter than $M_V=-21.5$. The sample also includes the dwarf ETGs analyzed by [131]; the gray color marks the data-set of ETGs used by D’Onofrio et al. [130] for the analysis of the FP properties. They span a large range of luminosities.

In addition to real data, we have used the artificial data of the Illustris simulation (see, e.g., [83,135,136]). This dataset contains ∼2400 galaxies and is marked in all plots by yellow dots. A full description of these data is given in [128,137]. We used the run with full-physics (with both baryonic and dark matter) having the highest degree of resolution, i.e., Illustris-1 (see Table 1 in [83]), extracting the $\mathrm{V}$-band luminosities of the galaxies, their mass, velocity dispersion, half-mass radii of the stellar particles, morphological types, etc. The projected light and mass profiles of ETGs have been studied in the paper of [137], who computed the effective radius (R_e) the effective surface brightness ${\langle \mu \rangle }_e$, the best-fit Sérsic index, and the line-of-sight velocity dispersion $\sigma$.

The Illustris-1 models are known to be in conflict with the real data in the case of the effective radius $R_e$ [10]. However, a careful comparison with the WINGS data showed that they nicely reproduce the tail of the $I_e-R_e$ plane and the tail of the $R_e$ − $M^{*}$ relation of the bright ETGs. They fail instead to reproduce the location of dwarf galaxies (masses in the interval ${10}^8$ to ${10}^{10}{M}_{\odot }$) on both planes. The relatively larger radii of these latter galaxies can be easily explained as an indirect effect of recent mergers and galactic winds (see for more details [138]). The velocity dispersion of the Illustris models, on the other hand, are in good agreement with the observed ones, as well as the FJ based on them (see below). For all these reasons we feel safe in using the Illustris-1 models for the purposes of the present study.

3.2. The FJ Relation for ETGs

Figure 1 shows the $L-\sigma$ relation of ETGs and simulated objects. Both samples span the same magnitude range from $M_V=-16$ to $M_V=-24$, although real data and artificial models have a different richness. This is due to observational (WINGS) and computational reasons (Illustris). At low luminosity, WINGS suffers a deficit due to the difficulty of measuring $\sigma$, whereas at high luminosity Illustris has a deficit of high mass galaxies, being that the data were extracted from a limited number of clusters. The figure clearly confirms that the slope of the FJ relation for bright objects deviates from that of faint objects.

The orthogonal fits marked by the red solid line and the black dashed line, obtained with the SLOPES software (Slope 3 Version 1) [139]), for the ETGs fainter and brighter than ${10}^{10}{L}_{\odot }$ are reported in

Table 1. The fits of the FJ relation for the bright and faint ETGs obtained using the WINGS data. The luminosity L is in solar units and the velocity dispersion $\sigma$ in km/s.

$log\left(\mathit{L}\right)=(\mathit{A}+\Delta \mathit{A})log\left(\mathit{\sigma }\right)+(\mathit{B}+\Delta \mathit{B})$
$\mathit{A}$	$\Delta \mathit{A}$	$\mathit{B}$	$\Delta \mathit{B}$	$\mathit{rms}$	$\mathit{Note}$
2.86	±0.17	3.98	±0.34	0.29	$log\left(L\right)\le 10.00$
4.24	±0.26	0.85	±0.60	0.29	$log\left(L\right)>10.00$

. The slope varies from $2.8$ to $4.2$ when we consider these two datasets. The whole rms scatter instead appears ∼$0.29$ for both samples. Considering the contribution of the measurement errors we get an intrinsic rms of ∼0.07, in agreement with the previous literature.

Notably, the data of the Illustris simulation follow the same trend of real ETGs (although with a bit smaller scatter), with the exception of the bright objects that appears to follow the same trend of fainter galaxies.

As remarked in our Introduction, the correct interpretation of such a change of slope in the FJ relation should be searched for by looking at the whole trends observed in all projections of the FP. Figure 2 shows some of these projections3.

The feature in common in all these diagrams is that all the relations are curved. The curvature occurs approximately at $L\sim {10}^{10}{L}_{\odot }$ in all diagrams. Such bending implies that the distribution of galaxies in the FP is far from uniform or that we are seeing a curvature of the FP itself. Clearly the more luminous ETGs follow a different trend with respect to the faint objects. We will try a possible explanation of this behavior in Section 3.3.

It is important to note here that the simulated data fail to reproduce correctly the effective radius of many galaxies and consequently the effective surface brightness. This was already noted and discussed by D’Onofrio et al. [10]. The most probable reason for such a discrepancy is the difficulty of taking into account the correct amount of feedback from SNe events. The strong winds of gas from these explosions can determine a puffing-up of the galaxies.

The curvature appears more pronounced in the diagrams that involve the $I_e$ and $R_e$ parameters. Indeed, when we plot both such parameters we obtain the $I_e-R_e$ plane (Figure 3). Here the two populations of bright and faint ETGs form an acute angle. In this figure we used the whole database of ETGs measured for the WINGS sample. They are shown here by the small gray dots.

The peculiar distribution of Figure 3 was first shown by Capaccioli et al. [93], who suggested the existence of two different populations of ETGs, the ‘ordinary’ and the ‘bright’. This is a long debated question concerning the existence of two distinct channels of formation for dwarf and giant ETGs (see e.g., [140,141,142,143,144,145,146,147,148,149,150]).

Of great importance in this diagram is the presence of a Zone of Exclusion (ZoE), i.e., a region completely avoided by any object. In this plane, the existence of the ZoE was first noted by [92,151] using the k-space version of the FP. The distribution of ETGs appears indeed limited in the maximum surface brightness at each $R_e$. The slope of this line of avoidance is close to $-1$ (when $I_e$ is $L_{\odot }p{c}^{-2}$ units), i.e., the slope predicted by the Virial Theorem (VT) [10].

This feature is present in every diagram involving $R_e$ and $I_e$. It is in fact easy to observe that in the $R_e-L$ and $I_e-L$ planes galaxies appear limited in the maximum luminosity that can be achieved at each radius or surface brightness. The slope of such ZoE in these cases is again the value predicted by the VT, i.e., that of the dynamical equilibrium of the stellar system.

3.3. Discussion of the Stellar FJ

There is a large consensus today that the steeper slope of the FJ relation at high masses is due to the different evolution of ETGs at different mass ranges. An attempted explanation invoked the evidence that high-mass galaxies are preferentially “core” ellipticals, whereas intermediate and lower mass objects are generally “core-less” [55,152,153]. The transition occurs at ∼${10}^{11.2}{M}^{*}$. Core ellipticals have boxy isophotes and are slow rotators, while core-less ellipticals have disky isophotes and rotate faster [55,56,152,153,154,155]. This picture is consistent with the systematic variation of the total mass profiles in the central region of ETGs [156], with more massive galaxies having shallower mass profiles. In this framework the change of slope is the result of the relative efficiency of gas cooling and feedback in ETGs at different mass scales. The cooling of gas allows the condensation of baryons in the central regions, and consequently an increase in the central concentration of mass [157,158,159,160]). Dynamical friction and SNe/AGN feedback, in contrast, are a source of heating, decreasing the central density concentration [160,161,162,163,164,165,166]).

The explanation that we propose here is instead totally different. The starting point is the $L=L_0^{\prime }{\sigma }^{\beta }$ law proposed by [4]. The interesting fact in this relation is that it is completely independent from the VT, i.e., it is not a replica of the classical FJ relation. As a consequence, one can put in a system the scalar virial law and the $L=L_0^{\prime }{\sigma }^{\beta }$ law. After some algebra, it is quite easy to get the following equations:

$$log\left(\langle I_e\rangle \right)=\frac{(2/\beta )-(1/2)}{(1/2)-(1/\beta )}log\left(R_e\right)+\mathsf{\Pi }$$

(2)

where $\mathsf{\Pi }$ is a factor that depends on $k_v$, $M/L$, $\beta$, and $L_0$ (see for details [4]. This equation give the slope in the $I_e-R_e$ plane as a function of $\beta$.

Another combination gives:

$$R_e={\left(\frac{1}{\frac{k_v}{G}{\left(\frac{2\pi \langle I_e\rangle }{L_0^{\prime }}\right)}^{2/\beta }}\right)}^{1/\left(4/\beta +1\right)}{M}^{1/\left(4/\beta +1\right)}.$$

(3)

which gives the slope in the $R_e$ − $M^{*}$ plane as a function of $\beta$.

In addition, we have:

$$R_e=\frac{k_v}{G}\frac{L}{M}{\sigma }^{\frac{2+\beta }{1+\frac{2/\beta -1/2}{1/2-1/\beta }}}$$

(4)

which gives the slope of the $R_e-\sigma$ relation, and finally

$$R_e=\frac{G}{k_v}\frac{M}{L}{L_0^{\prime }}^{2/\beta }{L}^{1-2/\beta }$$

(5)

which gives the slope of the $R_e-L$ relation.

Table 2. The slopes in the log form of the $I_e-R_e$, ${\langle \mu \rangle }_e-log\left(R_e\right)$, $R_e-M^{*}$, $R_e-L$ and $R_e-\sigma$ planes for different values of $\beta$.

$\mathit{\beta }$	${\mathit{I}}_{\mathit{e}}-{\mathit{R}}_{\mathit{e}}$	${\langle \mathit{\mu }\rangle }_{\mathit{e}}-{\mathit{R}}_{\mathit{e}}$	${\mathit{R}}_{\mathit{e}}-{\mathit{M}}^{*}$	${\mathit{R}}_{\mathit{e}}$ − L	${\mathit{R}}_{\mathit{e}}-\mathit{\sigma }$
3.0	1.0	−2.50	0.43	0.33	10.00
2.0	-	-	0.33	-	4.00
1.0	−3.00	7.50	0.20	−1.00	−6.00
0.5	−2.33	5.83	0.11	−3.00	−3.33
−0.5	−1.80	4.50	−0.14	5.00	−1.19
−1.0	−1.66	4.16	−0.33	3.00	−0.66
−1.5	−1.57	3.92	−0.66	2.33	−0.28
−2.0	−1.50	3.75	−1.00	2.00	0.00
−2.5	−1.44	3.16	−1.66	1.79	0.22
−3.0	−1.40	3.50	−3.00	1.66	0.40
−3.5	−1.36	3.41	−7.00	1.57	0.54
−4.0	−1.33	3.33	0.00	1.50	0.66
−4.5	−1.30	3.26	9.00	1.44	0.77
−5.0	−1.28	3.21	5.00	1.40	0.86
−8.0	−1.20	3.00	0.50	1.25	1.20
−11.0	−1.15	2.88	1.57	1.18	1.38
−25.0	−1.07	2.68	1.19	1.08	1.70
−50.0	−1.03	2.59	1.08	1.04	1.85
−100.0	−1.01	2.55	1.04	1.02	1.92
−1000.0	−1.00	2.50	1.00	1.00	1.99
−10,000.0	−1.00	2.50	1.00	1.00	2.00

list the observed slopes in these diagrams for different values of $\beta$. You can also appreciate the predicted slopes, marked by blue arrows in the above plots. They represent the direction of motion of a galaxy in the FP projections, when $\beta$ is equal to 3 and $-100$, two extreme cases that correspond, respectively, to an object with active star formation and an object in a passive quenched state, in which the luminosity decreases at a nearly constant $\sigma$.

Having demonstrated that the combination of the VT and $L=L_0^{\prime }{\sigma }^{\beta }$ law is the key to interpret the different aspects of the FP projection, we can also easily show that such behavior is necessarily present in the FJ plane. Starting from the VT:

$${\sigma }^2=\frac{G}{k_v}\frac{M^{*}}{R_e},$$

(6)

when the luminosity is substituted to $M^{*}$ and $R_e$, one obtains two different trends:

$$\begin{array}{cc}\hfill L=\frac{k_v}{G}{\sigma }^{2.10}& \mathrm{when}{M}^{*}\le {10}^{10}{M}_{\odot }\end{array}$$

(7)

$$\begin{array}{cc}\hfill L=\frac{k_v}{G}{\sigma }^{4.76}& \mathrm{when}{M}^{*}>{10}^{10}{M}_{\odot }\end{array}$$

(8)

This occurs because $M^{*}$ scale as $L^{1.12}$ over the whole mass range, while $R_e$ follows two different trends at the low and high mass ranges. In particular, we derive that $R_e\propto L^{0.17}$ at the low masses and $R_e\propto L^{0.70}$ at the high masses. The slopes in log units are therefore very close to that observed. Clearly a small difference in the $R_e-L$ relation can fit exactly the trends derived from the fits. The curvature of the FJ is, in conclusion, due to the role played by luminosity and the mechanism producing such curvature is the same in all the projections of the FP.

We can summarize arguing that the different condition of evolution in galaxies of different masses are the main aspect responsible for the observed distribution in all the projections of the FP. Passive and quenched objects are very close to the virial equilibrium and therefore follow a path close to the ZoE consistent with the slope predicted by the VT. In addition, the converging values of the slopes obtained in each plane, when $\beta$ diverges to extremely negative values (pure passive objects), naturally determines the existence of the ZoE, easily visible in many of these diagrams. No object can overcome the ZoE, because the coupling of the VT and the $L=L_0^{\prime }{\sigma }^{\beta }$ law constrains all passive virialized systems to move along the same direction (parallel to the ZoE).

4. The $\mathit{L}\mathbf{-}\mathit{\sigma }$ of AGN

4.1. Populations of AGN

The Eigenvector-1 correlation technique permitted the definition of two populations of luminous type-1 AGN. Population A sources have been identified as the type-1 AGN encompassing sources above a threshold value $L/L_{\mathrm{Edd}}$∼$0.1-0.2$ up to super-Eddington candidates with the maximum $L/L_{\mathrm{Edd}}$ (e.g., [9,167,168,169,170]), i.e., $L/L_{\mathrm{Edd}}$ in the range ${10}^{-2}/{10}^{-3}-\mathcal{O}\left(1\right)$. The observational selection criterion for low-z samples is FWHM H$\beta$$\lesssim 4000$ km s⁻¹, although one has to consider that the limit ≈ 4000 km s⁻¹ is luminosity dependent if it is intended to mark a fixed $L/L_{\mathrm{Edd}}$ [171]. The low-ionization emission lines (LILs) such as the Hi Balmer lines, Mgii$\lambda 2798$ are thought to be emitted by gas, mainly following a virial velocity field (and references therein [15,24,172,173]). This is not to say that the BLR of Population B—characterized by $L/L_{\mathrm{Edd}}$ $\lesssim 0.2$—is not virialized. There is strong evidence that the line broadening is predominantly virial [172], with increasing width and ionization for lines whose emission is weighted closer to the central black hole. In addition, Pop. B’s high-ionization lines are less affected by outflow motion, instead giving an overwhelming contribution to high-ionization lines such as Civ$\lambda$1549 especially in extreme Population A sources (e.g., [20,174,175,176,177]). The presence of an extended red wing in the H$\beta$ and other line profiles, even if due to gravitational redshift [178,179,180,181,182,183], is also ultimately consistent with a virial velocity field. We focus on Pop. A because these sources are believed to be more representative of high-z quasars [14]: Pop. B sources are more prone to selection effects at high-z because of the lower Eddington ratio [184]. The exploration of a FJ-like for Pop. B is deferred to future studies.

4.2. Virial $M_{\mathrm{BH}}$ Estimates

The following subsections discuss at first H$\beta$ and subsequently Aliii$\lambda$1860 scaling laws that are (1) relying on measurements of the H$\beta$ line dispersion $\sigma$ (Section 4.3), and (2) relaxing the assumption of strict equivalence with Vestergaard and Peterson [22] (Section 4.4). Both relations entail assumptions that make them more speculative than the ones presented in Marziani et al. [185], although they attempt to include corrections for the likely important effects of Eddington ratio and viewing angle that are not considered in the empirical relation entirely based on the FWHM.

The virial mass equation can be written in the following form:

$$M_{\mathrm{BH}}=f_{\mathrm{S}}\left(\theta \right)r_{\mathrm{BLR}}\left(L,\frac{L}{L_{\mathrm{Edd}}}\right)\frac{{\mathrm{FWHM}}^2}{G}$$

(9)

where the virial factor is now a function of the viewing angle (the angle between the line of sight and the accretion disk axis) and $r_{\mathrm{BLR}}$ is a function of both luminosity and $L/L_{\mathrm{Edd}}$ [27,186]. Awareness of the dependence by $L/L_{\mathrm{Edd}}$ and viewing angle has been growing in the latter years, and it is now imperative that some alternative strategies are devised to improve $M_{\mathrm{BH}}$ estimates to make them free of systematic effects as well as reduce the statistical noise down to the one expected from measurement errors of the observational parameters employed for the estimates.

4.3. $M_{\mathrm{BH}}$ Scaling Laws Based on H$\beta$

Recently, Dalla Bontà et al. [187] pointed out that the dispersion from single-epoch observations could be the best estimator of the $\sigma$ of the rms spectrum for sources studied with reverberation mapping. The $\sigma$ could be used in place of the FWHM, as recommended by Peterson and Dalla Bontà [188]. This approach is made more difficult in the context of Population A sources, where line profiles are Lorentzian-like (e.g., [36,167,189,190,191]). However, accurate measurements are possible, especially for spectra recorded with high S/N or high-quality parameter $\mathcal{Q}$. Appendix A.1 and Appendix A.2 summarize the measurements and introduce the $\mathcal{Q}$ parameter as a more proper uncertainty predictor than S/N evaluated on the continuum.

We considered a sample of Population A sources covering a wide range of redshift and luminosity ($0\lesssim z\lesssim 3.5$, $44\lesssim logL\lesssim 48$ [erg s⁻¹] for which H$\beta$, Aliii$\lambda$1860, and [Ciii]$\lambda$1909 emission lines were observed (Marziani et al. [185], hereafter [185]). For the H$\beta$ of this sample, we derive the following relation between the velocity dispersion $\sigma$ and FWHM (left panel of Figure 4):

$$\sigma \approx (0.4875\pm 0.03057)\mathrm{FWHM}+(803\pm 111)\mathrm{km}/\mathrm{s}$$

(10)

$$\mathrm{FWHM}/\sigma \approx (0.6363\pm 0.03406)log\mathrm{FWHM}+(1.1520\pm 0.1183)$$

(11)

The measurements were carried out on the Hβ profiles of our sample spectra (corresponding to ${\sigma }_{\mathrm{M}}$ in Dalla Bontà et al. [187]).

In addition, a FWHM predicted on the basis of the measured $\sigma$, FWHM_σ was obtained by isolating the Population A sources from Table 3 of Dalla Bontà et al. [187] to construct a relation yielding the FWHM_M from ${\sigma }_{\mathrm{M}}$. The procedure was similar to the one of Dalla Bontà et al. [187] and involved the exclusion of outlying points with a 2.6$\sigma$ clipping criterion4. The resulting relation applied to our data is (Figure 4, right panel):

$$log{\mathrm{FWHM}}_{\sigma }\left(\mathrm{H}\beta \right)\approx (0.701\pm 0.057)log\mathrm{FWHM}\left(\mathrm{H}\beta \right)+(1.146\pm 0.188)$$

(12)

A word of caution is that the highest luminosity covered by Dalla Bontà et al. [187] is $log\lambda L_{\lambda }\left(5100\right)\approx 45$ [erg s⁻¹]. The majority of our sources (red circles in Figure 4) are above this limit. The FWHM predicted from the $\sigma$ is larger than the observed one, especially for the narrowest H$\beta$ profiles: the constant term ensures that FWHM_σ ≳ FWHM for all reasonable FWHM values; the largest mass estimates would be unaffected since FWHM_σ ≈ FWHM if FWHM ≈ 7000 km s⁻¹, an extremely large value considering the luminosity-dependent maximum FWHM for Pop. A sources [38]. This result is potentially interesting as it is likely that the narrowest profiles are the ones most affected by an almost pole-on orientation. The simplest use of Equation (12) would be to substitute it in the $M_{\mathrm{BH}}$ H$\beta$ relation of Vestergaard and Peterson [22]. The scaling law would then become:

$$log{M}_{\mathrm{BH}}\left(\mathrm{H}\beta \right)\approx 0.91+0.5log{L}_{5100,44}+2log{\mathrm{FWHM}}_{\sigma },$$

(13)

where the luminosity at 5100 Å is in units of ${10}^{44}$ erg s⁻¹, and the FWHM is in km s⁻¹.

We now attempt to account for orientation and $L/L_{\mathrm{Edd}}$ effects. The dependence of $f_{\mathrm{S}}$ on the viewing angle has been parameterized by Mejía-Restrepo et al. [192] as $f_{\mathrm{S}}\approx$ (FWHM/4550)^−1.17, and the $r_{\mathrm{BLR}}$ dependence on the Eddington ratio by applying a correction to the standard scaling between $r_{\mathrm{BLR}}$ and optical luminosity at 5100 Å [27,193], yielding

$$log{M}_{\mathrm{BH}}\left(\mathrm{H}\beta \right)\approx 4.513+0.533log{L}_{5100,44}+0.83log\mathrm{FWHM}-0.394log\left(\frac{L}{L_{\mathrm{Edd}}}\right)\left[{\mathrm{M}}_{\odot }\right].$$

(14)

The main empirical correlate with the Eddington ratio is the $R_{\mathrm{FeII}}$ prominence parameter [14,33,35,48,194]. A tight relation between $r_{\mathrm{BLR}}$ and luminosity can be derived once $r_{\mathrm{BLR}}$ is corrected using $R_{\mathrm{FeII}}$ in place of $L/L_{\mathrm{Edd}}$. The following relation applies [27]:

$$log{r}_{\mathrm{BLR}}\approx 17.063+0.45log{L}_{5100,44}-0.35R_{\mathrm{FeII}}\left[\mathrm{cm}\right],$$

(15)

yielding:

$$log{M}_{\mathrm{BH}}\left(\mathrm{H}\beta \right)\approx 5.220+0.45log{L}_{5100,44}+0.83log\mathrm{FWHM}-0.35R_{\mathrm{FeII}}\left[{\mathrm{M}}_{\odot }\right]$$

(16)

Equation (14) implies a significant increase in the $M_{\mathrm{BH}}$ values for the sources with the narrowest profile with respect to the scaling law of Vestergaard and Peterson [22]. One of the best-studied sources in the low $M_{\mathrm{BH}}$ domain $\lesssim {10}^7$ M_⊙ is I Zw 1. A recent $M_{\mathrm{BH}}$ estimate is $9·{10}^6$ M_⊙ [195] assuming $f_{\mathrm{S}}=1$. The estimates using the Vestergaard and Peterson [22] H$\beta$ and the Aliii$\lambda$1860 and Civ$\lambda$1549 scaling laws are consistent with $log{M}_{\mathrm{BH}}\approx 7.2$. They would imply a value for $f_{\mathrm{S}}\approx 2$, as suggested by Collin et al. [196] for Population A, to be in agreement with Huang et al. [195]. Spectropolarimetric measurements are expected to provide $M_{\mathrm{BH}}$ estimates less biased from the viewing angle [197,198]. The estimate from the H$\alpha$ polarization profile is log$M_{\mathrm{BH}}$ $\approx 7.46\pm 0.30$ [197], in turn implying $f_{\mathrm{S}}=3$. In the case of Equation (14) there are two competing effects: on the one hand, the narrow H${\beta }_{\mathrm{BC}}$ requires $f_{\mathrm{S}}\approx 5$; on the other hand, the high $L/L_{\mathrm{Edd}}$ (log$L/L_{\mathrm{Edd}}$ $\approx 0.13$) implies a significant reduction in the $r_{\mathrm{BLR}}$ measurement, by $\delta log{r}_{\mathrm{BLR}}$$\approx -$0.64. The value derived from Equation (14) for I Zw 1, log$M_{\mathrm{BH}}$ $\approx 7.58$ [M_⊙], is consistent with the value derived from the spectropolarimetry.

The relation $f_{\mathrm{S}}\propto$ FWHM⁻¹ may introduce a bias if applied to a broad range of FWHM: it neglects an intrinsic dependence on luminosity (or mass) of the line width. However, the FWHM-dependent correction on $f_{\mathrm{S}}$ becomes negligible at high L, where FWHM ∼ 5000 km s⁻¹, not unlike in the case of Equation (13). At high L, the dominating effect in our sample is the correction because of the Eddington ratio that implies $\delta log{r}_{\mathrm{BLR}}\approx -0.6$. The largest $M_{\mathrm{BH}}$ estimates a decrease from log$M_{\mathrm{BH}}$ $\approx 9.8$ to 9.2 [M_⊙] (Figure 5).

4.4. Two $M_{\mathrm{BH}}$ Aliii$\lambda$1860 Relations

4.4.1. $\sigma$-Derived Relation

Contingent to the data in Ref. [185], the Aliii$\lambda$1860 is observed at a lower S/N with respect to H$\beta$. This implies that the Aliii$\lambda$1860 $\sigma$ will be measured on a line whose wings are lost in noise, creating a bias in the relation between the velocity dispersion of the two lines. On the other hand, there is an empirical consistency between the FWHM of H$\beta$ and the one of Aliii$\lambda$1860. Presently, there is evidence that H$\beta$ might be predominantly emitted in the same flattened and virialized region (e.g., [199,200]), and that Aliii$\lambda$1860 might be emitted in a region largely co-spatial with H$\beta$ [201]. These lines of evidence suggest that orientation effects might be similarly affecting the two lines.

Even if an accurate measure of Aliii$\lambda$1860 $\sigma$ is not viable, Equation (12) may still provide a way to correct for inclination effects if a correction is applied to both H$\beta$ and Aliii$\lambda$1860 FWHM and H$\beta$ and [Ciii]$\lambda$1909. The use of FWHM_σ in place of the measured FWHM yields scaling laws that are formally very similar to the ones obtained for Aliii$\lambda$1860 and [Ciii]$\lambda$1909 FWHM (Equations (17) and (18) below), with a significant change in the low-end $M_{\mathrm{BH}}$ estimates by one order of magnitude:

$$\begin{array}{ccc}\hfill log{M}_{\mathrm{BH}}\left(\mathrm{AlIII}\right)& \approx & (0.{577}_{-0.037}^{+0.033})log{L}_{1700,44}+\hfill \\ & & 2log\left({\mathrm{FWHM}}_{\sigma }\left(\mathrm{AlIII}\right)\right)+(0.{486}_{-0.056}^{+0.134}),\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill log{M}_{\mathrm{BH}}\left[\mathrm{CIII}\right]& \approx & (0.{5787}_{-0.039}^{+0.041})log{L}_{1700,44}+\hfill \\ & & 2log\left({\xi }_{\left[\mathrm{CIII}\right]}{\mathrm{FWHM}}_{\sigma }\left(\left[\mathrm{CIII}\right]\right)\right)+(0.{2876}_{-0.088}^{+0.162}).\hfill \end{array}$$

(18)

Uncertainties have been estimated as described in [185], and the subscript $\sigma$ means that the Aliii$\lambda$1860 and [Ciii]$\lambda$1909 FWHM have been “corrected” according to Equation (12). Note that for [Ciii]$\lambda$1909 the correction has to be computed first before multiplying the FWHM_σ by ${\xi }_{\left[\mathrm{CIII}\right]}$. The application of the scaling laws to our data is shown in Figure 6.

The resulting scaling laws are qualitatively consistent with the one obtained in Section 4.3 for H$\beta$, in the sense that both scaling laws predict an increase in $M_{\mathrm{BH}}$ at the low end of the $M_{\mathrm{BH}}$ range (Figure 5 and Figure 6), but differ little at the high $M_{\mathrm{BH}}$ end. The $\sigma$-based relation relies on the application of a highly uncertain correction, i.e., the relation shown in Figure 4 to the Aliii$\lambda$1860 and [Ciii]$\lambda$1909 widths. This cautions us to the use of Equations (17) and (18). A test against samples with fully independent estimates of the $M_{\mathrm{BH}}$ mass from the Vestergaard and Peterson [22] relation at low z is beyond the scope of the present work.

4.4.2. An Aliii$\lambda$1860 $M_{\mathrm{BH}}$ Scaling Law Equivalent to Equation (14)

Enforcing the equality condition (Figure 5):

$$M_{\mathrm{BH}}\left(\mathrm{AlIII}\right)\approx (1.001\pm 0.0504)M_{\mathrm{BH}}\left(\mathrm{H}\beta \right)+\left(-0.0006\pm 0.417\right)$$

(19)

between the $M_{\mathrm{BH}}$ estimated with Aliii$\lambda$1860 and with the H$\beta$ scaling law with orientation and $L/L_{\mathrm{Edd}}$ corrections (Equation (14)) implies an Aliii$\lambda$1860-based $M_{\mathrm{BH}}$ provided by:

$$log{M}_{\mathrm{BH}}\left(\mathrm{AlIII}\right)\approx (0.{344}_{-0.015}^{+0.045})log{L}_{1700,44}+2log\left(\mathrm{FWHM}\left(\mathrm{AlIII}\right)\right)+(0.{615}_{-0.725}^{+0.615}).$$

(20)

with an rms scatter $\approx 0.25$. In principle, Equation (20) takes into account two effects that are likely to affect the dispersion and bias the results of the Vestergaard and Peterson [22] relation: the increased likelihood for the narrowest profiles to be affected by an almost pole-on view [200], and the dependence on the spectral type of the $r_{\mathrm{BLR}}$ − L relation [27].

To further address this issue for the H$\beta$ and Aliii$\lambda$1860 lines would require less heterogeneous data, dedicated observations, and larger samples. Equation (20) is reported, but the introduction of a large correction associated with Equation (20) cautions about its use in actual $M_{\mathrm{BH}}$ estimates.

4.5. A FJ Law for Extreme Population A Type-1 AGN

We focus on a special sub-population involving the most extreme Eddington ratio ($L/L_{\mathrm{Edd}}$$\sim \mathcal{O}\left(1\right)$) that may involve super-Eddington accretion [9,41,110,112]. Extreme Population A is defined by the strongest Feii emission with an optical $R_{\mathrm{FeII}}\gtrsim 1$ and are at one end of the quasar Eigenvector 1/main sequence ([33,40], see [37,168] for reviews).

The xA quasars offer a potential luminosity indicator (as do spiral galaxies, following the Tully–Fisher law): the luminosity should be proportional to the 4th power of the virial FWHM: $L\propto FWH{M}^4$. A difficulty here is more to relate the accretion luminosity to a distance indicator free from anisotropic effects and uncertain corrections, starting from observed fluxes. The xA quasars, however, may provide a class of luminosity indicators (also dubbed as “Eddington standard candles) covering cosmic epochs from almost present day up to less than 1 Gyr from the Big Bang, well beyond the limits of optical cosmological distance indicators such as supernovæ [9,116,118]. The H$\beta$ observations might be difficult in correspondence of $z\sim 1$, due to strong sky emission emission in the red and near IR. At higher redshift, round based observations are possible only within the J, H, K, …bands, where atmosphere is not opaque to the frequency of the incoming radiation. Apart from these limitations, the redshift distribution could be extended up to $z\approx$ 3–4. The importance of covering the redshift $z\gtrsim 1.5$ rests in the fact that, above this limit, the effects of the energy density of matter are dominant. In addition, data in the broad range $0<z<4$ have the potential to assess the equation of state of dark energy as a function of redshift [109,110,117,202]. Preliminary applications were carried of the “virial luminosity’’ equation to the Hubble diagram have been promising [9,116,118]. The coverage of the redshift range between 1 and 3 allows for a good precision in the estimate of ${\mathsf{\Omega }}_{\mathrm{M}}$ higher than optical surveys of supernovæ [9]. Intermediate redshift quasars and supernovæ could be said to provide “ortogonal” information: tight constraints on ${\mathsf{\Omega }}_{\mathsf{\Lambda }}$ and loose constraints on ${\mathsf{\Omega }}_{\mathrm{M}}$, and the converse for quasars [118].

The virial luminosity equation derived by Marziani and Sulentic [9] can be written in the form:

$$\begin{array}{ccc}\hfill L\left(\mathrm{FWHM}\right)& =& {\mathcal{L}}_0·{\left(\mathrm{FWHM}\right)}_{1000}^4\hfill \\ & =& 7.88·{10}^{44}{\left(\frac{L}{L_{\mathrm{Edd}}}\right)}_{,1}^2·\frac{{\kappa }_{\mathrm{i},0.5}{f}_{\mathrm{S},2}^2}{h{\overline{\nu }}_{\mathrm{i},100\mathrm{eV}}}\frac{1}{{\left(n_{\mathrm{H}}U\right)}_{{10}^{9.6}}}{\left(\mathrm{FWHM}\right)}_{1000}^4{\mathrm{erg}\mathrm{s}}^{-1}\hfill \end{array}$$

(21)

where the energy value has been normalized to 100 eV (${\overline{\nu }}_{\mathrm{i},100\mathrm{eV}}\approx 2.42·{10}^{16}$ Hz), ${\kappa }_{\mathrm{i},0.5}$ is the fraction of bolometric luminosity belonging to the ionizing continuum scaled to 0.5, the product ($n_{\mathrm{H}}U$) has been scaled to the typical value ${10}^{9.6}$ cm⁻³ [113,114,115], and the FWHM of the H$\beta$ BC is expressed in units of 1000 km s⁻¹. The $f_{\mathrm{S}}$ is scaled to the value 2 following the determination of Collin et al. [196]. The FWHM of H$\beta$ broad component and of Aliii$\lambda$1860 are hereafter adopted as a virial broadening estimator. Equation (21) implies that, by a simple measurement of the FWHM of a LIL, we can derive a z-independent estimate of the accretion luminosity ([9], c.f. Teerikorpi [8]). The constant ${\mathcal{L}}_0$ can be determined from the best fit of the Hubble diagram. It is sample dependent, in the sense that different samples as considered in the following (Section 4.7) might be affected not only by slightly different intrinsic properties (right hand of Equation (21)) but also instrumental factor such as different light losses or aperture effects (for example, SDSS aperture spectroscopy vs. narrow long slit spectroscopy). The ${\mathcal{L}}_0$ has been found in the range $log{\mathcal{L}}_0\sim 44-45$ erg s⁻¹.

To achieve a calibration of Equation (21) suitable for precision cosmology, however, several issues related to the factors entering in its right end should be resolved: the foremost is the non-virial broadening of the emission lines. Continuum emission of quasars is also not isotropic, although model predicts that the dependence on the viewing angle $\theta$ should be very modest till $\theta \approx 30$ ([109], and references therein). The constant of proportionality ${\mathcal{L}}_0$ between L and FWHM depends on SED parameters [9], and it is not as yet clear if the SEDs are all consistent, and X-ray spectral slopes have a small scatter [203,204,205]. By the same token, the product Hydrogen density times ionization parameter $n_{\mathrm{H}}U$ is not expected to be exactly the same for every xA source. It would not be a surprise to find a dependence on $R_{\mathrm{FeII}}$ or chemical abundances [113,206]. Nonetheless, a large sample of inter-calibrated rest frame optical and UV ranges would make it possible to estimate the virial broadening, and hence the distance modulus $\mu$ over a wide range of redshift. In turn, an estimate of ${\mathsf{\Omega }}_{\mathrm{M}}$ independent from all other methods as yet employed would become possible.

In the following, the focus is on the possibility of using the velocity dispersion $\sigma$ as an alternative virial broadening indicator, that would be less affected than the FWHM from orientation effects [187,207,208].

4.6. Velocity Dispersion Measurements for Extreme Population-A AGN

To this aim, we measured the velocity dispersion on a sample of 133 AGN largely overlapping with the one of Negrete et al. [200]. Measurements were carried out as described in Appendix A.1. Figure 7 shows the relation between the $\sigma$ and the FWHM for the xA sources of Negrete et al. [200], computed as for the [185] sample of generic Population A quasars (Figure 4). The relation resulting from an unweighted least square fit are:

$$\sigma \approx (0.4231\pm 0.03711)\mathrm{FWHM}+(592\pm 75)\mathrm{km}/\mathrm{s}$$

(22)

$$\mathrm{FWHM}/\sigma \approx (3.114\pm 0.41427)·{10}^{-4}\mathrm{FWHM}+(0.7896\pm 0.0840)$$

(23)

roughly consistent with the ones of Population-A as a whole (Figure 4).

4.7. The Hubble Diagram

xA

We applied the measurements of the $\sigma$ to the derivation of the redshift-independent luminosity (Equation (21)). The comparison is between estimates of distance modulus from FWHM and $\sigma$ (Figure 8), taking as a reference the distance moduli computed for “concordance cosmology.” The middle panel of Figure 8 overlays the residual in distance modulus with respect to concordance, and shows that the FWHM-based measurements provide a significant bias in the estimate, with the best fitting line significantly different from 0. This is due to a luminosity dependence on the FWHM that is much shallower than the one expected for the virial relation (FWHM $\propto L$). The limited scope in luminosity of the Negrete et al. [200] sample, along with large scatter and a statistical bias (fewer data points are available in correspondence of the lowest and highest line widths; inset in the top panel of Figure 8) might be at the origin of the problem. The unbiased range in FWHM shows a trend consistent with a slope 4 in the $logL$ vs. logFWHM (steeper line in the inset). The bias visible in the middle panel is due to the H$\beta$ line profiles at higher redshift being significantly narrower than the expected values, resulting in lower luminosity estimates, and hence in lower $\mu$ values. The trend is not improved if $\sigma$ measurements are restricted to $\mathcal{Q}\gtrsim -0.2$ (bottom panel). The bias is intrinsic to the data, as it is present for both FWHM and $\sigma$ and, like the large scatter, might be associated with orientation effects: Negrete et al. [200] explained all discrepancies between the virial luminosity and the one expected from the concordance model to orientation effects due to virial factor in the form of Equation (26). The effect of orientation will be discussed in Section 4.8.

Generally speaking, the luminosity values predicted from FWHM in samples of modest size are within the scatter expected from larger samples. Indeed, a sample including 200 xA objects for which H$\beta$ FWHM measurements are available, covering redshifts from 0 to $\approx 2.6$ provides results in agreement with the concordance cosmology (Figure 9). A linear fit to the residuals yields

$$AllH\beta xA:\delta \mu \approx (-0.188\pm 0.213)log\mathrm{FWHM}+(0.175\pm 0.142),rms\approx 1.399$$

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consistent with slope 0. If the objects from [185] discussed in this paper were removed, the slope would be $\approx 0.001$, perfectly consistent with no bias with respect to concordance cosmology. The deviation introduced by the new sample is within the statistical uncertainties. Less clear if the large deviation observed from the $\sigma$ measurements can be referred to statistical fluctuations or rather to an intrinsic phenomenon, due also to the increase of the FWHM/$\sigma$ ratio as a function of FWHM. Figure 10 emphasizes the consistency between the average $\mu$(z) computed from the H$\beta$ line broadening (the data are the ones of Figure 9): it provides a sound proof that the H$\beta$ lines is emitted by a predominantly virialized system (within the framework of the assumption made in the formulation of Equation (21)). Some of the deviations from the $\mu$(z) predicted by standard cosmology are appreciable also in the averages of Figure 10. It is imperative to populate the Hubble diagram with high quality H$\beta$ observations allowing for accurate deblending of the line profiles and estimates of both FWHM and $\sigma$ to better study the evolution of the FWHM/$\sigma$ ratio as a function of luminosity, at least for xA and the non-xA Population A independently.

4.8. Orientation Dependent Luminosity-Line Broadening Relation

The effect of orientation can be quantified by assuming that the line broadening is due to an isotropic component + a flattened component whose velocity field projection along the line of sight is $\propto 1/sin\theta$:

$$\delta v_{\mathrm{obs}}^2=\frac{\delta v_{\mathrm{iso}}^2}{3}+\delta v_{\mathrm{K}}^2{sin}^2\theta .$$

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The structure factor in the Equation (9) is set to $f_{\mathrm{S}}=2$ since Collin et al. [196] derived values for $f_{\mathrm{S}}\approx$ 2.1 for Pop. A, with a substantial scatter. If we considered a flattened distribution of clouds with an isotropic $\delta v_{\mathrm{iso}}$ and a velocity component associated with a rotating flat disk $\delta v_{\mathrm{K}}$, the structure factor appearing in Equation (9) can be written as

$$f_{\mathrm{S}}=\frac{1}{4\left[\frac{1}{3}{\left(\frac{\delta v_{\mathrm{iso}}}{\delta v_{\mathrm{K}}}\right)}^2+{sin}^2\theta \right]}$$

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which can reach values $\gtrsim 1$ if $\kappa =\frac{\delta v_{\mathrm{iso}}}{\delta v_{\mathrm{K}}}\ll 1$, and if $\theta$ is also small ($\lesssim 30$ deg). The assumption $f_{\mathrm{S}}=2$ implies that we are seeing a highly flattened system (if all parameters in Equation (9) are set to their appropriate values); an isotropic velocity field would yield $f_{\mathrm{S}}=0.75$.

The virial luminosity equation may be rewritten in the form:

$$\begin{array}{ccc}\hfill L(\mathrm{FWHM},\theta |\kappa )& =& {\mathcal{L}}_0{f}_{\mathrm{S}}^{-2}{\left(\delta v_{\mathrm{K}}\right)}^4\hfill \\ & =& 4^2{\mathcal{L}}_0^{•}{\left(\delta v_{\mathrm{K}}\right)}^4{\left[\frac{{\kappa }^2}{3}+{sin}^2\theta \right]}^2=4^2{L}_{\mathrm{vir}}{\left[\frac{{\kappa }^2}{3}+{sin}^2\theta \right]}^2,\hfill \\ \end{array}$$

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where $L_{\mathrm{vir}}$ is the true virial luminosity (which implies $f_{\mathrm{S}}=1$) with ${\mathcal{L}}_0^{•}=\frac{1}{4}{\mathcal{L}}_0$, since ${\mathcal{L}}_0$ was scaled to $f_{\mathrm{S}}=2$.

An immediate application of the $L(\mathrm{FWHM},\theta )$ relation is the ability to recover the $\theta$ distribution of a sample of xA sources, assuming a particular cosmology (in practice a “standard" cosmology with ${\mathsf{\Omega }}_{\mathrm{M}}=0.3,{\mathsf{\Omega }}_{\mathsf{\Lambda }}=0.7,H_0=70$ km s⁻¹ Mpc⁻¹). The values that are obtained cannot be used for cosmology, since a given cosmology has been assumed. However, the distribution is by itself only slightly dependent on cosmology, with a modest change of the estimated viewing angles (Figure 11). This paves the road to the inclusion of correction for viewing angle by using the distribution of $\theta$ as a prior, or by applying a random correction to the FWHM value.

The resulting distribution of $\theta$ (Figure 11) is reasonable considering the constraints imposed by unification models for type-1 AGN [210,211], with $\theta \lesssim$ 40, and maximum likelihood at $\theta \approx 20$. R. If the distribution is assumed isotropic, we expect $P\left(\theta \right)\mathrm{d}\theta \propto sin\theta \mathrm{d}\theta$. A trend like this seems to be followed up to $sin\theta \approx 0.4$, meaning $\theta \approx 23$. For larger $\theta$, the detection of the sources – and their inclusion in color-based surveys—might be disfavored by the anisotropic emission of the geometrically thick torus that is expected in sources accreting at super-Eddington rates [112], and by the presence of a dusty obscuring torus that would suppress the view of the BLR at angles $\theta \gtrsim 45$, according to the unification schemes [210].

Assuming the “standard” cosmology, Equation (26) can be computed for individual objects. The missing parameter in the computation of the virial mass (Equation (9)) is then $r_{\mathrm{BLR}}$ that can be computed from a scaling law (most notably, the one of Martínez-Aldama et al. [193] that includes a correction because of the $L/L_{\mathrm{Edd}}$ dependence in the relation $r_{\mathrm{BLR}}-L$). Alternatively, $r_{\mathrm{BLR}}$ via a photoionization method, if the UV rest frame spectrum is available [115,212]. To check the reliability of the method, a comparison with reverberation mapping results should be carried out [213,214]. This comparison goes beyond the scope of the present paper and is in preparation.

Population A

The xA sources are relatively rare in optically selected samples, with a prevalence $\lesssim 10$ [173]. In view of a possible exploitation of the line width as a luminosity indicator, it is worth exploring the possibility of an Hubble diagram based on H$\beta$ for the full Population A. The Population A sample of [185] covers a much wider range in luminosity than the one of Negrete et al. [200], and the relation between L and FWHM is dominating over the large scatter of individual measurements. Figure 12 shows the relation between luminosity and $\sigma$ (top panel) and between luminosity and FWHM (bottom) for the generic Population A (less the xA, blue circles, and labeled as $\tilde{\mathrm{A}}$) and for a subsample of 16 xAs.

The resulting trends

$$\begin{array}{ccc}\hfill Pop.\tilde{A}:log{L}_{\mathrm{bol}}& \approx & (4.543\pm 0.838)log\sigma +(30.994\pm 2.844),rms\approx 0.687\hfill \\ \hfill xA:log{L}_{\mathrm{bol}}& \approx & (5.436\pm 0.914)log\sigma +(29.300\pm 3.012),rms\approx 0.543\hfill \\ \hfill Pop.\tilde{A}:log{L}_{\mathrm{bol}}& \approx & (3.510\pm 0.547)log\mathrm{FWHM}+(34.062\pm 1.927),rms\approx 0.628\hfill \\ \hfill xA:log{L}_{\mathrm{bol}}& \approx & (3.396\pm 0.558)log\mathrm{FWHM}+(34.750\pm 1.887),rms\approx 0.535\hfill \end{array}$$

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are all consistent with $L\propto FWH{M}^4$ within $\approx 1\sigma$ confidence level. The scatter is larger for Pop. Ã (32 sources, excluding xA), and for the $\sigma$-based luminosity estimates. Figure 13 shows the Hubble diagram based on the virial luminosity from $\sigma$ and FWHM. Pop. Ã data are affected by more severe biases; and the $\sigma$ estimates deviate strongly from the prediction of concordance cosmology. It is as if the line $\sigma$ fails to grow with luminosity: in this case, a lower $\sigma$ implies a lower luminosity and hence a smaller distance modulus: the source being fainter, for the same optical flux it has to be closer. The effect depends on ${\mathcal{L}}_0$, and is present also for the FWHM-derived values. However, the result is more severely biased if $\sigma$ is used to estimate the luminosity: $\delta \mu \propto 1.57z$ vs. $\delta \mu \propto 1.02z$ for $\sigma$ vs. FWHM (lower panels of Figure 13). The slope of the residuals is not changing if ${\mathcal{L}}_0$ is adjusted to reproduce the high-z range. The origin of this behavior may be residing in the line broadening mechanism: if the line wings are significantly affected by electron scattering [215], the width should be set by the electron temperature of the scattering plasma, not by the velocity field of the emitting gas. On the converse, the broadening of the core of the line might be more closely associated with the velocity and better sampled by the FWHM.

5. The $\mathit{L}\mathbf{=}{\mathit{L}}_{\mathbf{0}}^{\mathit{\prime }}{\mathit{\sigma }}^{\mathit{\beta }}$ Law for AGN

At this point an interesting issue to examine is whether the vision developed for ETGs by means of the $L=L_0^{\prime }{\sigma }^{\beta }$ theory (proportionality constant $L_0^{\prime }$ and exponent $\beta$ varying with time for any object and from object to object) can be extended to AGN, despite the completely different astrophysical context. To do this, we derived an estimate of the main astrophysical parameters involved in the scaling relations typical of ETGs (e.g., L, M, $R_e$, $I_e$, $\sigma$, etc.) also for AGN. We used the radius of the BLRs, their luminosity, their gas mass, and the measured velocities of the gas derived from the FWHM of the emission lines. With these values, the equations of the virial theorem and the variable $L=L_0^{\prime }{\sigma }^{\beta }$ were solved to derive $\beta$ and and $L_0^{\prime }$. The four panels of Figure 14 show the key scaling relations among the main parameters, that is the $L-\sigma$ plane (left upper panel), the $I_e-R_e$ plane (right upper panel), the $I_e-\sigma$ plane (lower left panel), and finally the $\beta -I_e$ plane (lower right panel) both for AGN (red dots) and ETGs (black dots). The inspection of the various panels reveals that ETGs and AGN behave in similar way but differ in some important details. While ETGs and AGN have comparable luminosity (the latter ones are indeed as luminous as the brightest ETGs or even brighter) their surface brightness is much higher than that of ETGs (by more that twelve orders of magnitude). This can be easily explained by the much smaller dimensions of the emitting regions in AGN ($R_e\le 0.1$ pc). Furthermore, both AGN and ETGs follow a $L-\sigma$ relation, the slope of which gets steeper at increasing luminosity. In addition, the $I_e-R_e$ plane (right upper panel), the $I_e-\sigma$ plane (lower left panel) show similar trends, but for the large difference in the $I_e$ of the two groups. Finally, some comments on the values of $\sigma$ in the two groups are mandatory. In ETGs the velocity dispersion $\sigma$ measures the random motions along the three coordinate directions of stars in virial equilibrium within the galaxy gravitational potential well. In AGN there are two components, namely random motions in the overall gravitational potential well combined with circular motion of gas in the central region of the disc spiraling in around the central black hole. Looking at the mean values of $log\sigma$ of ETGs and AGN brighter than about $logL/L_{\odot }>10.5$ we estimate $\Delta log({\sigma }_{AGN}-{\sigma }_{\mathrm{ETG}})$ in between 1/3 to 1/6. For the sake of consistency, when plotting some variable (e.g., L or $I_{\mathrm{e}}$) as function of $\sigma$ (bottom and top left panels of Figure 14) one should apply this correction to the data for AGN to compare them with ETGs. Finally, the bottom right panel of Figure 14 displays $\beta$ versus $I_{\mathrm{e}}$ plane for ETGs (black dots) and AGN (red dots). Apart from the effect of large difference in $I_e$, AGN yield βs spanning very large both positive and negative values. Small values of $\beta$ seem to be absent in AGN. The large values of $\beta$ seem to suggest that also AGN are in state of virial equilibrium. It is beyond the aims of this study to investigate further the physical implications of the $\beta$ parameter in the case of AGN. We limit ourselves to call attention on this important issue and leave it to future investigation.

The remarkable similarity of the scaling laws of AGN and ETGs spurred us to analyze the distribution in the $L-\sigma$ plane of different astrophysical systems. Figure 15 shows the $L-\sigma$ relation for five different kinds of objects: Galaxy Clusters, AGN, ETGs, Dwarf galaxies and Globular Clusters. The data have been extracted from our previous studies on this subject see e.g., [4,10,216]. In the figure the velocity of the AGN has been scaled to $1/3$ of measured value to correct it for the effect of the ordered circular motion around the central black hole. It is impressive to see that over an enormous range of luminosity the distribution of all systems is very similar. Apart from small deviations, that anyhow demand an explanation, a linear trend in log units between luminosity and velocity dispersion is found.

Such similarity of behavior finds its basic explanation in the virial theorem. At all scales the virialized systems vary in the same way with the mass. Since the light L is a proxy of the mass, this is not a surprise. The real interesting physics is hidden in the peculiar deviations from such a linear trend. When L varies for other causes (e.g., scattering, quenching, induced star formation, merging, etc.) that are mass independent, the variations reflect peculiar physical phenomena that are not taken into account. The $L=L_0^{\prime }{\sigma }^{\beta }$ framework is a first step toward this wider context by providing additional degrees of freedom with respect to the simple luminosity-mass dependence. It is indeed an important tool to understand the evolution of objects in the hierarchical scenario.

6. Summary and Conclusions

In this critical review, we have reported on the scaling relation between the luminosity and velocity dispersion in two groups of objects with very different properties such as ETGs and AGN. For both groups we surveyed several basic aspects of scaling relations with particular attention to recent developments, calling attention on the many point of similarity. In the case of ETGs, we have also reported on the recent revision by [10] of the classical FJ relation ($L-\sigma$), in which the proportionality factor $L_0$ and exponent $\alpha$ are let vary with time and from object to object thus taking explicitly into account the temporal evolution of the correlation between luminosity and velocity dispersion caused by many physical processes during the life of each individual galaxy (e.g., natural evolution of the stellar populations changing their luminosity, different efficiencies of star formation, both spontaneous and induced by mergers, causing sudden variation of the total luminosity, mass, radius, and velocity dispersion in turn, and so forth). The $L-\sigma$ relation is replaced by $L=L_0^{\prime }{\sigma }^{\beta }$ where ${\mathrm{Ł}}_0^{\prime }\left(t\right)$ and $\beta \left(t\right)$ vary from galaxy to galaxy and as function of time. The classical FJ relation is the mean of many different relations (one for each galaxy) at a fixed time, that is the present age (redshift z = 0). The combination of the Virial Theorem and $L=L_0^{\prime }{\sigma }^{\beta }$ allows us to determine the quantities relation ${\mathrm{Ł}}_0^{\prime }\left(t\right)$ and $\beta \left(t\right)$ at any given time; the present for the galaxies in the local Universe and back in past for those at redshift $>0$. In this context the FJ relation has to vary with time.

Similar analysis has been made for AGN showing that also these objects follow a FJ-like relationship that is fully consistent with the classical FJ for galaxies (ETGs), the slopes are nearly the same. We have extended the concept of time dependent individual $L=L_0^{\prime }{\sigma }^{\beta }$ relation to the case of AGN and found results that are fully consistent with those of classical ETGs. Since in the case of AGN the velocity contains also a component due to rotation around the central black hole, we have estimated the rotational component and corrected the observational velocity to get the component relative to motions different from rotation, therefore fully consistent with dispersion velocity of ETGs. In such a case, the position of AGN in various diagnostic planes, e.g., L − $\sigma$, $I_e$ − $R_e$, $I_e$ − $\sigma$ $\beta$ − $I_e$ closely mimics that of ETGs. Finally, objects going from Globular Clusters, to Dwarf Galaxies, ETGs, and finally AGN seem to obey the same $L-\sigma$ relation with minor deviations from it. Considering these objects span a luminosity range extending across about ten orders of magnitude, the result is very intriguing and demands further investigation.

The conclusions we derive from this analysis are as follows:

• Galaxies of all types share the FJ relation. The classical interpretation is that light L makes the role of mass M and that the correlation is generated by the virialization of the stellar systems. The different slope followed by bright and faint galaxies is likely an indication that the use of the light parameter L introduces a systematic difference for the two types of objects or that the degree of virialization is different. The scatter of the relation seems almost equivalent at all masses.
• In addition to the classical interpretation of the FJ relation, we have shown that it is possible to think at L and $\sigma$ as variable mutually connected with the evolution experienced by every galaxy. Using indeed the $L=L_0^{\prime }{\sigma }^{\beta }$ relation with two free time-dependent parameters $L_0^{\prime }$ and $\beta$, one can verify that the $\beta$ parameter enters in all the projections of the FP, providing the direction of motion of any objects forced by the evolutionary effects (e.g., mergers, accretion, star formation, etc.). Many characteristics features of the scaling relations can be understood adopting this “temporal” view of the FJ: the best example is the "zone of exclusion" (ZoE), the region empty of galaxies in the ${\langle \mu \rangle }_e-log\left(R_e\right)$ plane, that clearly have its origin in the fact that at large $\beta$’s galaxies can only move in a direction with slope $-1$.
• Regarding AGN, we have proposed several alternative scaling laws to estimate the black hole mass taking into account, at least tentatively, the effect of orientation and $L/L_{\mathrm{Edd}}$ on the line broadening and on the BLR radius, respectively. The validity of these scaling law should be tested against $M_{\mathrm{BH}}$ derived from reverberation mapping and estimates of the $r_{\mathrm{BLR}}$ obtained with photoionization method. In the near future, spectro-astrometry from GRAVITY holds the promise of estimate the BLR radius for hundreds of intermediate redshift AGN [217], much beyond the first (and already extremely valuable) estimates of the virial factor for low z AGN [218]. Simple scaling laws applicable to the huge influx of data coming from DESI, Euclid, etc. [219] will retain of course a value to permit straightforward $M_{\mathrm{BH}}$ mass estimates for samples that will be tenfold and more in size with respect to the ones available to-date. This might prove useful for cosmological applications [220], too. Figure 10 wants to emphasize the overall consistency of H$\beta$ distance moduli estimates by averaging over redshift bins.
• The virial luminosity equation for AGN is a reinstatement of a law that appears universal in the form $L\propto {\sigma }^n$, holding from stars to clusters of galaxies. In the case of AGN, the dominance of Keplerian motions and the restriction to a very limited range of $L/L_{\mathrm{Edd}}$ for the xA sources imply a well-defined scaling between luminosity and FWHM or $\sigma$ with exponent $n\approx$ 4, analogous to the 21 cm line broadening that is due to (mainly) Keplerian motion in galaxy disks [221,222]. The agreement between the magnitudes estimates is not the results of a circular argument, but rather, an indication of the fundamental correctness of the compact central object scenario and of the dominance of virial motions in at least a substantial part of the BLR. Yet, the fact that the agreement with the prediction of the virial luminosity for quasars is somewhat sample dependent means that we are not understanding in full the line broadening phenomenology of type 1 AGN.