SMBH mass and galaxy properties

On the Correlations between Galaxy Properties and Supermassive Black Hole Mass

A.  Beifiori, S. Courteau, E. M. Corsini, and Y. Zhu
Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstraße, D-85748 Garching, Germany
Institute of Cosmology and Gravitation, Dennis Sciama Building, Burnaby Road, Portsmouth, PO1 3FX, United Kingdom
Queen’s University, Department of Physics, Engineering Physics and Astronomy, Kingston, ON, K7L 3N6, Canada
Dipartimento di Astronomia, Università di Padova, vicolo dell’Osservatorio 3, I-35122 Padova, Italy
E-mail:beifiori@mpe.mpg.de
Accepted 2011 September 27. Received 2011 September 13; in original form 2011 January 5
Abstract

We use a large sample of upper limits and accurate estimates of supermassive black holes (SMBHs) masses coupled with libraries of host galaxy velocity dispersions, rotational velocities and photometric parameters extracted from Sloan Digital Sky Survey -band images to establish correlations between the SMBH and host galaxy parameters. We test whether the mass of the black hole, , is fundamentally driven by either local or global galaxy properties. We explore correlations between  and stellar velocity dispersion , -band bulge luminosity , bulge mass , bulge Sérsic index , bulge mean effective surface brightness , -band luminosity of the galaxy , galaxy stellar mass , maximum circular velocity , galaxy dynamical and effective masses  and . We verify the tightness of the  relation and find that correlations with other galaxy parameters do not yield tighter trends. We do not find differences in the  relation of barred and unbarred galaxies. The  relation of pseudo-bulges is also coarser and has a different slope than that involving classical bulges. The  is not as tight as the  relation, despite the bulge mass proving to be a better proxy of  than bulge luminosity, and despite adding the bulge effective radius as an additional fitting parameter. Contrary to various published reports, we find a rather poor correlation between  and  (or ) suggesting that  is not related to the bulge light concentration. The correlations between  and galaxy luminosity or mass are not a marked improvement over the  relation. These scaling relations depend sensitively on the host galaxy morphology: early-type galaxies follow a tighter relation than late-type galaxies. If  is a proxy for the dark matter halo mass, the large scatter of the  relation then suggests that  is more coupled to the baryonic rather than the dark matter. We have tested the need for a third parameter in the  scaling relations, through various linear correlations with bulge and galaxy parameters, only to confirm that the fundamental plane of the SMBH is mainly driven by   with a small tilt due to the effective radius. We provide a compendium of galaxy structural properties for most of the SMBH hosts known to date.

keywords:
black holes physics — galaxies: fundamental parameters — galaxies: photometry — galaxies: kinematics and dynamics — galaxies: statistics
pagerange: On the Correlations between Galaxy Properties and Supermassive Black Hole Mass6pubyear: 2011

1 Introduction

The mass  of supermassive black holes (SMBHs) is closely tied to the properties of the spheroidal component of galaxies, such as the bulge luminosity,  (Dressler, 1989; Kormendy & Richstone, 1995; Marconi & Hunt, 2003; Graham, 2007; Gültekinet al., 2009, hereafter G09), the stellar velocity dispersion,  (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002; Ferrarese & Ford 2005; G09), the mass of the bulge (Magorrian et al., 1998; Häring & Rix, 2004), the central light concentration (Graham et al., 2001), the Sérsic index (Graham & Driver, 2007), the virial mass of the galaxy (Ferrarese et al., 2006), the gravitational binding energy (Aller & Richstone, 2007), the kinetic energy of random motions of the bulge (Feoli & Mancini, 2009), and the stellar light and mass deficit associated to the core ellipticals (Lauer et al., 2007; Kormendy & Bender, 2009). Most of these relations are inter-compared in Novak et al. (2006) and in G09. Given the  relation and the correlation between  and the circular velocity, , Ferrarese (2002) and Pizzella et al. (2005) suggested a link between  and  (or equivalently, with the mass of the dark matter halo). However, Courteau et al. (2007) and Ho (2007) pointed out that the  relation actually depends on galaxy morphology (or equivalently, on its light concentration) thus precluding a simple connection between  and . Kormendy & Bender (2011) studied a sample of bulgeless galaxies and concluded that there is almost no correlation between the SMBH and dark matter halo, unless the galaxy also contains a bulge.

Several authors have noted that the residuals of the  and  relations correlate with the galaxy effective radius (e.g., Marconi & Hunt, 2003). Hopkins et al. (2007a, b) suggested the possibility of a linear combination between different galaxy properties to reduce the scatter of the  scaling laws, heralding the idea of a fundamental plane of SMBHs (BHFP). Many SMBH scaling relations could thus be seen as projections of the BHFP (Aller & Richstone, 2007; Barway & Kembhavi, 2007). A correlation of  with more than one galaxy parameter would suggest a SMBH growth sensitive to the overall structure of the host galaxy.

The local characterisation and cosmic evolution of the  scaling relations have already been examined through theoretical models for the coevolution of galaxies and SMBHs (Granato et al., 2004; Vittorini et al., 2005; Hopkins et al., 2006; Monaco et al., 2007). These studies have revealed that the observed scaling relations could be reproduced in models of SMBH growth with strong feedback from the active galactic nucleus (AGN, Silk & Rees, 1998; Cox et al., 2006; Robertson et al., 2006a, b; Di Matteo et al., 2005). In particular, these models predict the existence of the BHFP (Hopkins et al., 2007a, b; Hopkins et al., 2009). However, while the observed relations can be reproduced by the models, these still depend on the adopted slope, zero point, and scatter (Somerville, 2009) which remain ill-constrained.

In this work, we make use of a large sample of galaxies with available  estimates to improve our understanding of the known scaling laws over a wide range of , morphological type and nuclear activity, as well as to test for possible correlations of  with different combinations of spheroid and galaxy parameters.

This paper is organised as follows. The sample of SMBH hosts is described in §2. Their photometric, kinematic, and dynamical properties are presented in §3, while the correlations between  and the bulge and galaxy properties are shown in §4. We discuss our results and conclude in §5.

2 Sample Description

The  values were retrieved from two different samples: the compilation of  upper limits by Beifiori et al. (2009, hereafter B09) and the compilation of secure  by G09.

The  estimates of B09 were obtained from Hubble Space Telescope (HST) archival spectra. That sample includes nuclear spectra for 105 nearby galaxies with Mpc obtained with the Space Telescope Imaging Spectrograph (STIS) equipped with the G750M grating. The spectra cover the region of the H line and [Nii] and [Sii] doublets. The nebular-line widths were modelled in terms of gas motion in a thin disc of unknown orientation but known spatial extent following the method of Sarzi et al. (2002). B09 adopted two different inclinations for the gaseous disc with a nearly face-on disc () hosting a larger  and a nearly edge-on disc (81) harbouring a smaller . The two inclinations correspond to the 68% upper and lower confidence limits for randomly oriented discs. We augmented the B09 sample with the  upper limits of NGC 2892 and NGC 5921. The STIS/G750M spectra for these galaxies were retrieved from the HST archive and we have calculated their  upper limits from the nebular line widths following the prescription of B09. We include in Sample A the set of 105 galaxies from B09 minus 18 galaxies in common with G09 (15 of them with a secure  and 3 with a  upper limit derived from the dynamical modelling of resolved kinematics). We are left with 87  upper limits from B09 based on nebular-line widths. We also include two newly determined  upper limits and the five upper limits derived from the dynamical modelling of resolved kinematics by G09. The resulting 94 galaxies constitute our Sample A.

G09 collected  data published up to November 2008 based on the resolved kinematics of ionised gas, stars, and water maser for total sample of 49 galaxies with a secure  estimate and 18 galaxies with a  upper limit. Our Sample B is limited to the 49 definite values of . The five upper limits derived from the dynamical modelling of resolved kinematics already belong to Sample A. The remaining 13 upper limits by G09 are taken from the nebular line width measurements by Sarzi et al. (2002). They were also measured by B09 and are therefore already included in Sample A.

All the data relative to Samples A and B are listed in Table 1 and 2, respectively. The upper limits by B09 were rescaled assuming , , and to conform with G09.

Figure 1 shows the upper limits and the accurate determinations of  for the 18 galaxies in common between the B09 and G09 samples. The  estimates by B09 and G09 are consistent within of each other; no systematic offset is detected. Thus, nebular line width measurements by B09 (included in Sample A) trace well the nuclear gravitational potential dominated by the central SMBH, allowing a reliable estimate of  (as those in Sample B). Therefore, we shall use the  upper limits from Sample A to study the correlations of  versus various galaxy parameters. We adopt for our tests the case of  which maximises the upper limit on . B09 already compared their  relation with those of Ferrarese & Ford (2005) and Lauer et al. (2007) to show that their upper limits on  (included in Sample A) are a valuable proxy for the more secure determinations of  (comprised in Sample B). A Kolmogorov-Smirnov test on the distributions of  and  indicates that Samples A and B could be drawn from the same parent distribution to better than the 80% confidence level. For this reason, we merged Samples A and B into a joint AB data set. The combination of Samples A and B yields a total sample of 143  determinations. The union of those two samples proves most valuable for the proper statistical assessment of scaling relations with .

Figure 1: Comparison between the  upper limits by B09 and accurate  measurements (symbols) and upper limits (leftward arrows) by G09 and based on the resolved kinematics of gas (filled circles), stars (open circles), and water masers (open square). The upper and lower edges of the dotted lines correspond to B09’s  values estimated assuming an inclination of and for the unresolved Keplerian disc, respectively.

3 Galaxy Properties

The aim of this study is to determine the strength of the correlations, if any, between  and the properties of their host galaxies. For the latter, we have compiled as large a collection as possible of homogeneous measurements of photometric parameters (effective radius, effective surface brightness, Sérsic index, and luminosity of the bulge, effective radius, effective surface brightness, concentration, and total luminosity of the galaxy), kinematic properties (stellar velocity dispersion and circular velocity), and masses (bulge mass, galaxy stellar mass, virial and dynamical mass of the galaxy).

The sample demographics are as follows: 29% of the host galaxies are ellipticals, 27% lenticulars, and 44% are spirals (de Vaucouleurs et al., 1991, hereafter RC3). Regarding nuclear activity, 23% of the sample galaxies are Low-Ionisation Nuclear Emission-line Regions (LINERs), 11% host Hii nuclei, 25% are Seyferts, and 8% are classified as transition objects according to Ho et al. (1997). The remaining 33% do not show central emission.

We describe in the sub-sections below the extraction of all the galaxy structural parameters.

3.1 Galaxy Photometric Parameters

We retrieved and -band images from the seventh data release (DR7, Abazajian et al., 2009) of the Sloan Digital Sky Survey (SDSS, York et al., 2000) for as many Sample A and B galaxies as possible. The total SDSS sample includes 90 galaxies, 62 from Sample A and 28 from Sample B, from which to derive structural parameters.

The SDSS images are already bias subtracted, flat-fielded and cleaned from bright stars; however, we performed our own sky subtraction since the SDSS pipeline sky levels may be flawed for extended galaxies (Bernardi et al., 2007; Lauer et al., 2007). This issue has been addressed in the SDSS-III Data Release 8 (Aihara et al., 2011; Blanton et al., 2011). For our purposes, we estimated the sky level of each galaxy image by isolating five regions away from the galaxy, free of any contaminant, and calculating the mode of the sky intensities per pixel within each sky region. The average and standard deviation of the five sky values was then computed. The difference between our measured sky values and those provided by SDSS can be as large as . The SDSS sky level is always biased high, likely due to the inclusion of bright, extended sources while our interactive technique ensures a contaminant-free selection of the sky fields. We find that the typical surface brightness error in the and bands is 0.1 mag arcsec at mag arcsec and mag arcsec, respectively (see McDonald et al., 2011, for more details). The large angular extent of NGC 224 and NGC 4594 relative to the field of view thwarted their proper sky subtraction; these two galaxies were therefore excluded from our SDSS sample. NGC 221 was also discarded due to improper positioning of the image on the detector. The remaining images were flux-calibrated based on the SDSS photometric zero-point, corrected for Galactic extinction (Schlegel et al., 1998) as well as for internal extinction and -correction following Shao et al. (2007).

The galaxy surface brightness profiles were extracted using the isophotal fitting methods outlined in Courteau et al. (1996) and McDonald et al. (2011). These make use of the astronomical data reduction package XVISTA111See http://astronomy.nmsu.edu/holtz/xvista/index.html.. The azimuthally-averaged surface brightness profiles, projected onto the major axis of each galaxy, are shown in Figure 2 for four sample galaxies. These are the elliptical galaxy NGC 5127, two high- (NGC 4036) and low- (NGC 4477) inclination lenticular galaxies, and the spiral galaxy NGC 3675 which boasts a remarkably high dust content.

Figure 2: Typical (red points) and -band (blue points) surface brightness profiles extracted along the major axis for a few sample galaxies.

The and -band total magnitude of each galaxy is then determined by summing the flux at each isophote and extrapolating the light profile to infinity. The colour of each galaxy was calculated from the difference of the fully corrected and -band magnitudes. The remaining structural parameters were measured from the -band light profiles since, of all the SDSS band passes, the -band suffers least dust extinction. We extracted the isophotal radius, , corresponding to the surface brightness of 24.5 mag arcsec, the half-light (or effective) radius of the galaxy, , the effective surface brightness of the galaxy, , and the galaxy concentration , where and are the radii which enclose and of the total luminosity, respectively. These -band structural parameters are listed in Table 3. The -band magnitude is also listed. Based on simulated models of spiral galaxies (MacArthur et al., 2003), the typical error per galaxy structural parameter is roughly 10-20%.

3.2 Photometric Parameters of Bulges and Discs

The structural parameters for elliptical galaxies, modelled typically as a single spheroid, and for spiral galaxies, modelled as the sum of a spheroid and a disc, were derived by applying the two-dimensional photometric decomposition algorithm GASP2D (Méndez-Abreu et al., 2008) to the SDSS band images.

The surface brightness of the spheroid component (typically the entire elliptical galaxy or the bulge component for a disc galaxy) is modelled using a Sérsic function (Sérsic 1968; see also Graham & Driver 2005)

(1)

where is the effective radius, is the surface brightness at , and is a shape parameter that describes the curvature of the radial profile. With or , the Sérsic function reduces to the exponential or de Vaucouleurs function, respectively. The coefficient (Caon et al., 1993) is a normalisation term. The spheroid model elliptical isophotes have constant position angle PA and constant axial ratio .

The surface brightness distribution of the disc component is assumed to follow an exponential law (Freeman, 1970)

(2)

where and are the scale length and central surface brightness of the disc, respectively. The disc model elliptical isophotes have constant position angle PA and constant axial ratio . The fitting algorithm GASP2D relies on a minimisation of the intensities in counts, for which we must adopt initial trial parameters that are as close as possible to their final values. The latter were estimated from our ellipse-averaged light profiles from which basic fits to the bulge and disc were applied to estimate structural parameters (see Méndez-Abreu et al., 2008, for details). The minimisation is based on the robust Levenberg-Marquardt method by Moré et al. (1980). The actual computation has been done using the MPFIT algorithm (Markwardt, 2009) under the IDL222Interactive Data Language is distributed by ITT Visual Information Solutions. It is available from http://www.ittvis.com/. environment.

The GASP2D software yields structural parameters for the bulge (, , , PA, and ) and disc (, , PA, and ) and the position of the galaxy centre . In GASP2D, each image pixel intensity is weighted according to the variance of its total observed photon counts due to the contribution of both galaxy and sky, and accounting for photon and detector read-out noise. Seeing effects were taken into account by convolving the model image with a circular Moffat point spread function (Moffat, 1969, hereafter PSF) with shape parameters measured from the stars in the galaxy image. Only the image pixels with an intensity larger than 1.5 times the sky standard deviation were included in the fit. Foreground stars were masked and excluded from the fit. The initial guesses were adopted to initialise the non-linear least-squares fit to galaxy image, where the parameters were all allowed to vary. A model of the galaxy surface brightness distribution was built using the fitted parameters. It was convolved with the adopted circular two-dimensional Moffat PSF and subtracted from the observed image to obtain a residual image. In order to confirm the minimum in the -space found in this first pass, two more iterations were performed. In these iterations, all the pixels and/or regions of the residual image with values greater or less than a fixed threshold, controlled by the user, were rejected. Those regions were masked out and the fit was repeated assuming, as initial trials for the free parameters, the values obtained in the previous iteration. These masks are useful when galaxies have spiral arms and dust lanes, which can affect the fitted parameters. We found that our algorithm converges after three iterations.

The model decompositions for few elliptical galaxies (NGC 4473, NGC 4636, NGC 4649) were significantly improved by including a disc component, as found in a few other elliptical galaxies (Kormendy et al., 2009; McDonald et al., 2009). For the other ellipticals, the values of obtained either directly (empirically) from the light profile (in §3.1) or by using a Sérsic fitting function (from the two-dimensional photometric decomposition) agree within their respective errors.

We eliminated 33 galaxies from our sample because of poor decompositions due to either a strong central bar, a Freeman II profile (Freeman, 1970), or just the overall inadequacy of our single or double-component modelling (e.g., due to the presence of strong dust lanes and/or spiral arms). We successfully performed photometric decompositions, as judged by a global figure-of-merit, for the 57 galaxies ( 38 from Sample A, 19 from Sample B) listed in Table 4. The latter includes the resulting bulge and disc structural parameters. Some examples illustrating the various fitting strategies are shown in Figure 3. These are the same galaxies, whose azimuthally-averaged surface brightness profiles are shown in Figure 2. For all of them, the ellipse-averaged radial profiles of surface brightness, ellipticity, and position angle of the model image are consistent with those measured on the galaxy image. The differences between model and data found in the ellipticity and position angle of NGC 4477 and in the ellipticity of NGC 3675 are due to the presence of a weak bar and strong dust lanes, respectively. These features are clearly seen in the galaxies’ residual images. Nevertheless, the surface brightness residuals are remarkably small; mag arcsec for all the galaxies shown in Figure 3, except for some portions of the dust lanes of NGC 3675 where the residuals increase to mag arcsec. Although part of the NGC 3675 disc is missing, this does not affect the fit result. GASP2D performs a reliable fit as soon as the observed galaxy can be modelled by the sum of two axisymmetric components and the field of view covers at least half of the galaxy (Méndez-Abreu, 2008).

Figure 3: Two-dimensional photometric decomposition of the sample galaxies shown in Figure 2 illustrating the various fitting strategies adopted with GASP2D. For each galaxy, we show in the upper panels the SDSS -band image (left), the best-fit image (middle), and the residual (i.e., observed-model) image (right). The lower panels show the ellipse-averaged radial profiles of the surface brightness (left), ellipticity (middle), and position angle (right) measured in the SDSS (dots) and model image (green continuous line). The difference between the ellipse-averaged radial profiles from the observed and model images are also shown. The dashed blue and dotted red lines represent the intrinsic surface brightness radial profiles of the bulge and disc, respectively. No disc contribution is assumed for the elliptical galaxy NGC 5127.

The GASP2D formal errors obtained from the minimisation method are not representative of the real errors in the structural parameters (Méndez-Abreu et al., 2008). Instead, the estimated errors given in Table 4 were obtained through a series of Monte Carlo simulations. To this end, we generated a set of 400 images of elliptical galaxies with a Sérsic spheroid and 400 images of disc galaxies with a Sérsic bulge and an exponential disc; each with a different PA and ellipticity. The range of tested parameters for the simulated images was taken from the photometric analysis of nearby elliptical galaxies by Kormendy et al. (2009) and face-on disc galaxies by Gadotti (2009) (but assuming a wider range of axial ratios than the latter). Our tests include treatment for resolution effects (galaxy distance, pixel scale, and seeing), colour effects (accounting for and bands), inclination, and more. The two-dimensional parametric decomposition was applied to analyse the images of the artificial galaxies as if they were real. The artificial and observed galaxies were divided in bins of 1 mag. The relative errors on the fitted parameters of the artificial galaxies were estimated by comparing the input and output values and were assumed to be normally distributed. In each magnitude bin, the mean and standard deviation of relative errors of artificial galaxies were adopted as the systematic and typical error on the relevant parameter for the observed galaxies. Overall, we find that GASP2D recovers the galaxy structural parameters with an uncertainty ranging from 1%, to 10%, and to 20% for brighter (), intermediate (), and fainter () sample galaxies, respectively.

The inclination of disc galaxies was calculated from the fitted disc axial ratio in the -band as

(3)

where is the intrinsic disc axial ratio. We obtain the latter from Paturel et al. (1997):

(4)

where is the galaxy Hubble type from RC3.

The fitted -band disc axial ratios agree well with the isophotal axial ratios at a surface brightness level mag arcsec reported in the RC3. Our disc axial ratios are on average lower than RC3 with a standard deviation of . We adopted the RC3 axial ratios for the lenticular and spiral galaxies whose GASP2D solution for the disc was too uncertain; this amounts to 28 galaxies in Sample A and another 8 in Sample B.

3.3 Stellar Velocity Dispersion

The measured stellar velocity dispersions, , for galaxies in Sample A were taken from the same sources as B09. We applied the aperture correction of Jørgensen et al. (1995) to transform the  into the equivalent of an effective stellar dispersions, , measured within a circular aperture of radius . The effective radii were also taken from the same sources as B09, except for the 35 galaxies for which  was obtained from our own decomposition of the SDSS images (Tables 3 and 4). The maximum difference between our and literature values of  is about , though the comparison often involves different band passes which broadens the discrepancy.

The aperture correction was also applied to the  measured for NGC 2892 and NGC 5921 by Ho et al. (2009) and Wegner et al. (2003), respectively.

For Sample B galaxies, we adopted the values of  given by G09. These were derived as the luminosity-weighted mean of the stellar velocity dispersion within .

3.4 Circular Velocity

The values for the galaxy circular velocity, , for both elliptical and disc galaxies were collected from different sources.

We retrieved  from the compilation of Ho (2007) for 40 disc galaxies (31 from Sample A, 9 from Sample B). These were derived from the Hi line widths available in the HyperLeda catalogue (Paturel et al., 2003). The and line widths are measured at and the of the total Hi line profile flux. For galaxies missing in Ho (2007), line widths were found in HyperLeda for an additional 27 galaxies (22 from the Sample A, 5 from Sample B). Given multiple sources in HyperLeda, we favoured the larger survey source for each galaxy in order maximise the homogeneity of the data base333We selected the most recent and highest resolution observations available in the on-line HyperLeda catalogue (http://leda.univ-lyon1.fr) up to September 2009.. All line widths were already corrected for instrumental resolution. We further applied a correction for cosmological stretching and broadening by gas turbulence following Bottinelli et al. (1983) and Verheijen & Sancisi (2001). Finally, the corrected line widths and were deprojected using the prescription of Paturel et al. (1997)

(5)
(6)

where the inclination, , is listed in Table 1 and 2. We take the final circular velocity as the average of and ,

(7)

Following Ho (2007), we adopted a 5% error on  for the maximum velocity listed in Hyperleda.

For 14 early-type galaxies (7 in Sample A, 7 in Sample B), we adopted the value of  derived from dynamical modelling (IC 1459, Samurović & Danziger 2005; NGC 1052, Binney et al. 1990; NGC 3115, Bender et al. 1994; NGC 3608, Coccato et al. 2009; NGC 4314 Quillen et al. 1994; and 9 ellipticals in Kronawitter et al. 2000). For a few other galaxies (five in Sample A, one in Sample B), either the ionised-gas (NGC 2911, Sil’chenko & Afanasiev 2004; NGC 5252, Morse et al. 1998), the Hi (IC 342, Pizzella et al. 2005; NGC 3801, Hota et al. 2009) or the CO kinematics (NGC 4526, Young et al. 2008), Milky Way (Baes et al. 2003) rotational velocity at large radii are assumed to represent . We adopted a 10% error when the  uncertainty from models or observations was not given.

Finally, we eliminated 7 galaxies (4 from Sample A, 3 from Sample B) since their quoted circular velocities are unrealistically low (NGC 1497, NGC 3384, and NGC 5576, Paturel et al. 2003; NGC 3642, NGC 4429, NGC 5347, and NGC 7052, Ho 2007).

The final circular velocities of 88 galaxies from Sample A (65) and Sample B (23) are listed in Tables 1 and 2.

3.5 Masses

We estimated the bulge mass as , where is the gravitational constant and (Cappellari et al., 2006) is a dimensionless constant that depends on galaxy structure. The effective radius, , for elliptical galaxies is extracted from their azimuthally-averaged light profile (Table 3), whereas the bulge effective radius of lenticular and spiral galaxies is obtained from two-dimensional photometric decomposition (Table 4). In deriving , we implicitly assumed that the measured value of  is dominated by the bulge component. For late-type galaxies we expect that the disc contribution to  results in an increase of the scatter of  as a function of the morphological type. This does not affect our analysis since we could measure (and therefore ) for only a few Sb–Sbc galaxies. Moreover, the ratio except for three galaxies.

Ferrarese et al. (2006) suggested a connection between  and the mass of early-type galaxies calculated as

(8)

Note that such a mass estimate is indicative of the galaxy mass within and is thus an incomplete representation of the total galaxy mass. We calculated  for the elliptical and lenticular galaxies in our sample by adopting calculated from the azimuthally-averaged light profiles (Table 3). For elliptical galaxies, . The virial estimator by Cappellari et al. (2006) captures the entire dynamical mass so long as total mass traces light (Thomas et al., 2011). Wolf et al. (2010) found a different coefficient () for their derivation of a mass estimator for stellar systems supported by velocity dispersion. The latter is a good estimate of the total dynamical mass inside a radius which is just larger than the effective radius (Thomas et al., 2011). The appropriate value of is actually a function of the Sérsic shape index  (Trujillo et al., 2004; Cappellari et al., 2006). However the exact application of the relation, which results in a zero-point offset for , does not affect our conclusions. We thus make use of the Cappellari et al. (2006) mass estimator. Error estimates on  and  account for the uncertainty on .

We derived the dynamical mass of the disc galaxies from

(9)

where (Table 3). Most of the circular velocities that we derived from Hi data yield no information about their radial coverage. Nevertheless,  corresponds to the galaxy mass within the optical radius, because is roughly indicative of the galaxy optical radius and the observations of spatially-resolved Hi kinematics in spirals show that the size of Hi discs closely matches that of a galaxy’s optical disc (Ho et al., 2008a).

The masses computed from Eq. (8) and Eq. (9) are both available only for the 10 lenticular galaxies in our sample, since they share dynamical properties of both elliptical and spiral galaxies. For these galaxies, /.

The galaxy stellar masses, , were derived from  under the assumption of constant mass-to-light ratio . estimates were inferred from our colours following Bell et al. (2003). Their mass-to-light ratios were derived from the stellar population models of Bruzual & Charlot (2003), which are tuned to reproduce the ages and metallicities of the local spiral galaxies.

4 Analysis

4.1 Correlations Between  and Bulge and Galaxy Parameters

We used the data obtained in §3 to build scaling relations between  and the bulge (i.e., the velocity dispersion, luminosity, virial mass, Sérsic index, and mean effective surface brightness) and galaxy (luminosity, circular velocity, stellar, virial, and dynamical mass) properties. In addition to the Sample AB, we also built a comparison sample with the 30 galaxies for which all the desired physical parameters could be measured. For each of the above parameters , we assume that there exists a relation of the form

(10)

where is a normalisation value chosen near the mean of the distribution of values.

The best-fit values of and , their associated uncertainties, the total scatter and intrinsic scatter of the relation, the Spearman rank correlation coefficient , and the level of significance of the correlation were obtained as a function of the sample at hand. The results are given in Table 5. The quoted errors are all derived through a bootstrap technique.

Samples AB and the comparison sample include both accurate determinations and upper limits of . The proper handling of these heterogeneous data sets requires that we perform a censored regression analysis (Feigelson & Nelson, 1985; Isobe et al., 1986) with the ASURV444The FORTRAN source code of the Astronomy Survival Analysis Package (v.1.3) is available at http://www2.astro.psu.edu/statcodes/asurv. package (Lavalley et al., 1992). Linear regressions were calculated with the EM maximum-likelihood algorithm implementing the technique of expectation (E-step) and maximization (M-step) by Dempster et al. (1977) which assumes normal residuals. For each relation, we checked that the distribution of residuals about the fitted line was normal. With uncensored data, this analysis would be equivalent to that of a standard least-squares linear regression. ASURV gives as correlation coefficient the generalised Spearman rank correlation coefficient (Akritas, 1989) and computes the level of significance of the correlation . We derived the scatter of the relation as the root-mean square (rms) deviation in from the fitted relation assuming no measurement errors. This assumption does not affect our results, since we wish to compare the relative tightness of the different scaling relations for the comparison sample.

G09 focused their analysis on  and . We have extended their analysis to a broader range of scaling relations by performing least-square linear regressions with the secure values of  and the bulge and galaxy parameters measured for Sample B. We used the MPFITEXY555The IDL source code of MPFITEXY is available at http://purl.org/mike/mpfitexy/. algorithm (Williams et al., 2010), which accounts for measurement errors in both variables and the intrinsic scatter of the relation. We used the IDL routine R_CORRELATE to compute the standard Spearman rank correlation coefficient and the level of significance of the correlation. The intrinsic scatter of each relation was assessed by varying in the fitting process to ensure that the reduced . The total scatter was derived as the rms deviation in from the fitted relation weighted by measurement errors.

In addition to fitting the –bulge and galaxy scaling relations as outlined above, we also look in the subsections below for correlations against morphological type or nuclear activity as listed in Tables 1 and 2. We will see that the latter (morphology or nuclear activity) do not play a strong rôle in any of the –bulge and galaxy scaling relations. Morphology only plays a small rôle in the  and in the  relations.

4.1.1  versus

We have plotted the  distribution for the Sample AB in Figure 4. The slope of our  relation is consistent within the errors with those by Ferrarese & Ford (2005) and Lauer et al. (2007) as discussed by B09, and with that by G09 by default. On the other hand, the inclusion of the upper limits measured by B09 and G09 slightly changes the zero point of the relation but does not appreciably affect its scatter. The  relation is indeed the tightest correlation that we measure; its scatter is consistent with G09 ( dex) and slightly larger than Ferrarese & Ford (2005, dex). According to G09, the increased fraction of spirals may be a source of the increased scatter with respect to previous estimates based mostly on SMBH measurements in early-type galaxies. Consistently, our analysis reveals that the  scatter for late-type galaxies ( dex) alone is slightly larger than that of early-type galaxies ( dex). Nevertheless, at small , some upper limits exceed the expected  as the line-widths for such low- outliers are most likely affected by the mass contribution of a conspicuous nuclear cluster (B09). The scatter for the late-type galaxies reduces to ( dex) by excluding the nucleated galaxies from the fit. Therefore, they are the cause for the observed difference between the scatter of early and late-type galaxies. We conclude that the  relation is the same for both early and late-type galaxies. S0 galaxies overlap ellipticals at high  values and spirals at low  values.

Barred and unbarred galaxies follow the same  relation within the errors. The same is not true for galaxies with classical and pseudo-bulge. We have classified our bulges according to their measured Sérsic index ( for classical and for pseudo-bulges; Kormendy & Kennicutt, 2004). We find slope differences in the  relations of our 46 classical () and 11 pseudo-bulges (), whereas their zero points ( and , respectively) are in closer agreement within the errors. Conversely, Hu (2008) found similar slopes and different zero points. At face values, our and Hu et al.’s correlation coefficients are in agreement. This is mostly due to the large error bars on the  coefficients for pseudo-bulges, which translate into a lower level of significance of the  relation for pseudo-bulges () with respect to that for classical bulges (). This results supports the recent findings by Kormendy et al. (2011) that there is little or no correlation between  and pseudo-bulges.

Figure 4:  as a function of  for 94 Sample A galaxies (89 upper limits from nebular line widths, circles; 5 upper limits from resolved kinematics, arrows) and 49 Sample B galaxies (squares). The total number of galaxies is . The error bars for  are shown only in the upper panel for clarity. The lower and upper ends of the dotted lines correspond to  upper limits estimated assuming an inclination of  and 81 for the unresolved Keplerian disc, respectively. Galaxies are plotted according to morphological type (upper panel) and nuclear activity (lower panel). The dashed line is the G09  relation. The Sbc bin includes all galaxies classified as Sbc or later.

At high  values, the  of the Sample AB shows a weak steepening with respect to  confirming previous results by Wyithe (2006), Lauer et al. (2007), and Dalla Bontà et al. (2009).

Finally, Figure 4 shows the absence of trends of the  relation with nuclear activity. The different types of nuclear activity cover all the  range, with most LINERs at high  and most Seyferts at low . This is expected considering the strong dependence of nuclear spectral class on Hubble type (and therefore on ), with LINERs and Seyfert nuclei being more frequent in ellipticals and spirals, respectively (see Ho, 2008, for a review).

4.1.2  versus

The correlation between  and the luminosity of the spheroidal component of a galaxy is also fairly tight (see Graham 2007, and references therein). Graham (2007) compared the different versions of the  by Kormendy & Gebhardt (2001), McLure & Dunlop (2002), Marconi & Hunt (2003), and Erwin et al. (2004). Recently, Sani et al. (2011) derived the  relation by measuring the 3.6 m bulge luminosity. Since the differing relations do not predict the same SMBH mass, Graham (2007) investigated the effects of possible biases on the  relation including any dependency on the Hubble constant, the correction for dust attenuation in the bulges of disc galaxies, the mis-classification of lenticular galaxies as elliptical galaxies and the adopted regression analysis. These adjustments resulted in relations which are consistent with each other and suitable for predicting similar . In particular, a detailed photometric decomposition of the galaxy light profile is crucial to obtaining a representative bulge luminosity and therefore a reliable .

G09 derived an updated value of the intrinsic scatter of the  relation, which is comparable to that of the  relation in early-type galaxies, confirming the results of Marconi & Hunt (2003). According to them, the scatter of the  relation is significantly reduced when the bulge effective radius is extracted from a careful two-dimensional image decomposition.

Figure 5 shows our  relation. The best-fit coefficients are in agreement within the errors with Graham (2007) and Sani et al. (2011). The scatter is slightly higher than G09 who fit only early-type galaxies, but larger than that of the  relation. All galaxies show a similar  distribution with no dependence on morphological type or nuclear activity. We cannot test the claim that barred and unbarred galaxies follow different  relations (Graham, 2008; Graham & Li, 2009; Gadotti & Kauffmann, 2009; Hu, 2009), since the sample galaxies with a strong bar were excluded for photometric decomposition and thus lack a  measurement. For all the remaining galaxies, we did not model any other components than the bulge and disc (see Table 4).

Figure 5:  as a function of  for 38 Sample A galaxies (35 upper limits from nebular line widths, circles; 3 upper limits from resolved kinematics, arrows) and 19 Sample B galaxies (squares) for which two-dimensional bulge-to-disc decompositions of the SDSS -band images were performed. Symbols and panels are as in Figure 4.

4.1.3  versus

The connection between the  and  suggests a linear correlation with mass of the spheroidal component of the galaxy (Magorrian et al., 1998; McLure & Dunlop, 2002; Marconi & Hunt, 2003; Häring & Rix, 2004).

The slope of our  relation (Fig. 6) is consistent within the errors with McLure & Dunlop (2002), Marconi & Hunt (2003), Aller & Richstone (2007) and Sani et al. (2011) and slightly shallower than Häring & Rix (2004). This regression is not a marked improvement over the  relation, as one might expect considering the additional fitting parameter in the mass measurement (namely the radius). Indeed, the  is worse than the  relation. However,  is still a better proxy for  than . Different Hubble types follow the same  relation, with the lenticular galaxies covering the whole range of masses. There is also no dependence of this relation on nuclear activity, in agreement with McLure & Dunlop (2002).

Figure 6:  as a function of  for the same sample as in Figure 5. Symbols and panels are as in Figure 4. The dashed line is the Häring & Rix (2004)  relation.

4.1.4  versus Sérsic n and

The two-dimensional photometric decompositions yield a panoply of galaxy structural parameters, including the Sérsic shape index  and mean effective surface brightness , both used as measure of the concentration of the bulge light. The Sérsic index  has indeed been adopted by some as a good tracer for  (Graham et al., 2001, 2003; Graham & Driver, 2007).

Our  relation (Figure 7) differs significantly from the linear and quadratic relations with   and   by Graham et al. (2001, 2003) and Graham & Driver (2007), respectively. In particular, the values of for galaxies with high  are not as large as those in Graham & Driver (2007). The relation is characterised by a large scatter, small Spearman correlation coefficient, and low level of significance. Therefore, the correlation between  and is poor and the  relation is not reliable for predicting . Our findings are in agreement with Hopkins et al. (2007b) who found no correlation between  and Sérsic index with both observations and hydrodynamical simulations.

The correlation between  and  (Figure 8) is also poor, lending further support to the idea that  is unrelated to the light concentration of the bulge, regardless of galaxy morphology and nuclear activity.

Figure 7:  as a function of  for the same sample as in Figure 5 for which two-dimensional bulge-to-disc decompositions of the SDSS -band images were performed. Symbols and panels are as in Figure 4. The  relations by Graham et al. (2001, 2003) and Graham & Driver (2007) are shown as the dashed and continuous lines, respectively.
Figure 8:  as a function of  for the same sample as in Figure 7. Symbols and panels are as in Figure 4.

4.1.5  versus

Kormendy (2001) compared the -band total magnitude with dynamically secure  estimates in spheroids and few bulgeless disc galaxies. The poor correlation between  and  led him to conclude that the evolution of SMBHs is linked to bulges rather than discs.

We compared our own  with  as a function of the morphological type in Figure 9. The Spearman rank correlation coefficient suggests a correlation at 17% significance level, which is tighter than Kormendy (2001) but not as tight as that for the  relation. Our sample is short of bulgeless galaxies which may explain the better agreement for our  and  with respect to Kormendy (2001). On the other hand, total and bulge luminosity differ for disc galaxies. The later the morphological type, the larger the discrepancy resulting in both a larger slope and scatter of the  relation with respect to . The scatter is larger and the correlation weaker when the band total luminosity is considered. Our findings are consistent with Hu (2009, see their Fig. 6) and confirm that bulge luminosity is a better tracer of the mass of SMBHs than total light of the galaxy.

Figure 9:  as a function of  for 62 Sample A galaxies (57 upper limits from nebular line widths, circles; 5 upper limits from resolved kinematics, arrows) and 28 Sample B galaxies (squares) for which azimuthally-averaged luminosity profiles from SDSS -band images were extracted. Symbols and panels are as in Figure 4.

4.1.6  versus , , and

We have tested whether the scatter in predicting  could be reduced by adopting  rather than , since the galaxy luminosity is a proxy for its stellar mass. The  relation shown in Figure 10 is a slight improvement over the  relation. The slopes are comparable but the Spearman rank correlation coefficient of the former is higher with a 3% significance level. Ellipticals and lenticulars follow a tighter  relation than late-type galaxies; the disc light clearly plays an anti-correlating rôle, indicating once more that the bulge parameters drive SMBH correlations.

Figure 10:  as a function of  for the same sample as in Figure 9 for which colours from SDSS and -band images were derived. Symbols and panels are as in Figure 4.

We adopted the mass estimator from Cappellari et al. (2006) (i.e., the mass within the effective radius) for our ellipticals and lenticulars. The distribution of  as a function of  for the early-type galaxies is plotted in Figure 11. They follow the same trend as the  relation by Ferrarese et al. (2006). This is especially true for M where the fit slope and normalisation are consistent within the errors with Ferrarese et al. (2006). Less-massive galaxies seem to follow a steeper  relation with a smaller normalisation. More data are however needed in this mass range to carefully address any trend differences. Introducing a new parameter  and a specific exponent for  ought to considerably reduce the scatter of the  relation but, in fact, it does not. In particular, the scatter of the  relation is smaller than that of  if the same sample of early-type galaxies is considered.

The dynamical mass  (i.e., the mass within the optical radius) was taken as the galaxy mass estimator for lenticulars and spirals. It does not correlate with  (Figure 12), irrespective of Hubble type and nuclear activity. Our analysis reveals that the coarse  relation found by Ho et al. (2008b, dex) does not hold when galaxies with M (i.e., at the lower mass end of the  range) are taken into account. The scatter of the –galaxy mass relation is larger and the correlation weaker when  and  are considered. Since the stellar and dynamical masses include the mass contribution of the disc component, we conclude that the bulge mass is a better tracer of . Nevertheless, the  is tighter and yet again more fundamental than the  relation.

Figure 11:  as a function of  for 27 Sample A galaxies (24 upper limits from nebular line widths, circles; 3 upper limits from resolved kinematics, arrows) and 24 Sample B galaxies (squares) ranging from E to S0a. Symbols and panels are as in Figure 4. The dashed line is the Ferrarese et al. (2006)  relation.
Figure 12:  as a function of  for 41 Sample A disc galaxies (38 upper limits from nebular line widths, circles; 3 upper limits from resolved kinematics, arrows) and 6 Sample B (squares). Symbols and panels are as in Figure 4. The dashed line is the Ho et al. (2008b)  relation.

4.1.7  and  relations

Assuming that  at large radii trace the dark matter halo, we study the possible link between SMBHs and dark matter halos by comparing  with . A possible relation between  and the dark matter depends clearly on the radius at which  is measured. Theoretical models that reproduce the observed luminosity function of quasars (Cattaneo, 2001; Adams et al., 2003; Volonteri et al., 2003; Hopkins et al., 2005a; Springel et al., 2005) suggest that  scales as a power law of the virial velocity (i.e., the circular velocity of the galactic halo at the virial radius) of the galactic halo of the SMBH host. However, the conversion between  measured at large radii and virial velocity, depends on the assumed model.

We show  against  in Figure 13. There is a weak correlation between these quantities, which is mostly driven by the late-type galaxies.

Figure 13:  as a function of  for 65 Sample A disc galaxies (61 upper limits from nebular line widths, circles; 4 upper limits from resolved kinematics, arrows) and 23 Sample B galaxies (squares). Symbols and panels are as in Figure 4. The dashed line is the Ho (2007)  relation.

This is in agreement with previous studies by Zasov et al. (2005) and Ho et al. (2008b). Zasov et al. (2005) used a collection of 41 galaxies with , , and  from the literature to conclude that there is a coarse  relation and that, for a given , early-type galaxies have larger  than late-type galaxies. Ho et al. (2008b) studied the  relation for a sample of 154 nearby galaxies comprising both early and late-type systems for which  was estimated from the mass-luminosity-line width relation (Kaspi et al., 2000; Greene & Ho, 2005; Peterson & Bentz, 2006) and  from Hi and optical data (Ho et al., 2008a). They found that the correlation between  and  improves if only the galaxies with the most reliable  measurements are considered. The distribution of our spiral galaxies is consistent with the  relation by Ho et al. (2008b), whereas the value of  for elliptical and lenticular galaxies is almost constant over the full observed  range (Figure 13).

Ho et al. (2008b) suggested that the main source of scatter in the  relation is related to the dependence of / on the bulge-to-disc ratio (Courteau et al., 2007; Ho, 2007). Since the bulge-to-disc ratio scales with the light concentration (Doi et al., 1993), which is another way to estimate the degree of bulge dominance, we derived / as a function of . We found a general agreement with the relation found by Courteau et al. (2007). The Jeans equation evokes a relation between the circular velocity of galaxy and its velocity dispersion for a pressure-supported system. The / relation was first reported by Whitmore et al. (1979) and more recently by Ferrarese (2002), Buyle et al. (2004), and Pizzella et al. (2005). Courteau et al. (2007) and Ho (2007) further demonstrated that the  relation must depend on a third parameter which depends on the galaxy structure.

The existence of the  relation and the absence of a single universal  for all the morphological types is in contrast with the hypothesis that  is more fundamentally connected to halo than to bulge, as suggested by Ferrarese (2002) and Pizzella et al. (2005). We conclude that the SMBH mass is associated with the bulge and not the halo (see also Peng, 2010) by analysing the  relation as a function of  for our sample galaxies (Figure 14). For a given , the range of  is on average 1.6 times larger than the range of  demonstrating that  is driven by  and not by . The few bulgeless galaxies known to host a SMBH (NGC 4395, Filippenko & Ho 2003; POX 52, Barth et al. 2004; NGC 1042, Shields et al. 2008) are an exception to this scenario, which is supported by cases like M33. The latter is a pure disc galaxy, which does not show any evidence of a SMBH (Gebhardt et al., 2001) but has a massive dark matter halo (Corbelli, 2003).

More recently, Kormendy & Bender (2011) confirmed our early findings (Beifiori, 2010) by extending the analysis of the  relation to bulgeless galaxies. They reported an absence of correlation between  and  unless the galaxy also contains a bulge, suggesting that the fundamental driver for  is  for all Hubble types and that  plays only a small rôle in galaxies with a bulge. Although at this later cosmic time the  is coarser than the  relation, Volonteri et al. (2011) remind us that a tighter connection in the past is not precluded. At earlier epochs, the SMBH assembly was more likely coupled to the dark matter halo properties based on a merger-driven hierarchical formation scenario. Still, the scatter we measure for the  relation is a factor larger than that of the  relation, which is significantly more than argued by Volonteri et al. (2011) in their analysis of Kormendy & Bender (2011)’s data set.

Figure 14:  relation as a function of  for the same sample as in Figure 13. The dashed line is the Ho (2007)  relation.

4.2 Correlations Between  and Linear Combinations of Bulge and Galaxy Parameters

We wish to understand whether the relations between  and bulge and galaxy properties studied in §4.1 can be improved by the addition of a third parameter. For the bulge or the galaxy parameters and in Table LABEL:tab:fit_combination, a correlation of the form

(11)

is assumed, where and are normalisation values chosen near the mean of the distribution of and values respectively. The fit parameters are the offset , and the logarithmic slopes and . The best-fit values of , , and , their associated uncertainties, the total scatter and intrinsic scatter of the relation are given in Table LABEL:tab:fit_combination. The Spearman rank test cannot be evaluated in the same way as described in §4.1 as it applies to two variables. To estimate the degree of correlations between the parameters in use, we applied the Spearman rank statistic using the projections of the planes once the combination of the two independent variables has been fixed to its best-fit values. This provides an indication of the degree of correlation between several linear combinations. The Spearman rank correlation coefficient , and the significance of the correlation derived for the projections are also given in Table LABEL:tab:fit_combination. The quoted errors are all computed through a bootstrap technique.

For both Samples AB and the comparison sample, we performed a censored regression analysis with two independent variables assuming normal residuals. The best-fit parameters were computed with the EM algorithm implemented in the ASURV package which yields maximum likelihood estimates from a censored data set allowing, in the general form, N observables distributed via a multivariate Gaussian distribution. We applied this technique to the special case of two independent variables and another dependent variable containing the censored data. The regression problem solves like a least-square fit but accounting for censored data. It uses initial guesses from an ordinary linear regression of the non-censored data and then converges on the new censored values.

The standard deviation is estimated in the standard fashion, weighted for a factor which accounts for the censored data (see Isobe et al., 1986, for details). The generalised Spearman rank correlation coefficient, and the significance of the correlation for the projections of planes, are based on the technique described in Akritas (1989). The total scatter was computed as the rms deviation in from the fitted relation assuming no measurement errors.

We used our own modified version of the MPFITEXY algorithm to perform a least-squares linear regression of Sample B’s data with two independent variables. Measurement errors were invoked in the derivation of best-fit parameters, total, and intrinsic scatter. The Spearman rank correlation coefficient and level of significance of the correlation were also derived by using the projections of the best-fit plane given all possible parameter combinations. The actual computation was done using the IDL routine R_CORRELATE.

4.2.1  versus Bulge Parameters

So far we have found that  is more strongly related with  than  and . Nevertheless, the correlation between the  and  suggests that a possible combination of the bulge properties can result in a tighter correlation with . Note that the BHFP derives from the FP which connects , , and  together (Dressler et al., 1987; Djorgovski & Davis, 1987; Jørgensen et al., 2007, and reference therein). Therefore, we have analysed the residuals of , , and  relations to test whether there is the evidence for an additional dependence with an additional parameter. The residuals were obtained as the difference between the observed data and their expected values from the fitting relations in Table 5. We compared the residuals in , , and  with the bulge photometric and kinematic properties (i.e., , , , and ; Figure 15). There is no trend between  and other bulge properties (Figure 15, left column), whereas  and  show a very weak dependence on  according to the the Spearman rank correlation coefficient and their significance intervals % and %, respectively (Figure 15, middle and left columns, respectively). This was confirmed by Sani et al. (2011), who found no correlation of the residual with .

Figure 15: Residuals of the , ,  relations versus , , ,  for 38 Sample A galaxies (35 upper limits from nebular line widths, filled circles; 3 upper limits from resolved kinematics, open circles) and 19 Sample B galaxies (squares). The total number of galaxies is . The Spearman rank correlation coefficient is given in each panel.

We have tested for the more fundamental driver of . To this aim we considered different linear combinations of the bulge parameters with  (Figure 16). The correlation between , , and  is as tight as the  (Figure 16a). Taking into account  does not significantly improve the  fit. The relation between , , and  (Figure 16b) is not as strong as that between , , and , but it is a slight improvement over . The relation between , , and  is as tight as that between , , and  (Figure 16c). The relation between , , and  (Figure 16d) is expected since  is known to correlate with  and it depends on both  and . For this reason the two correlations with  and either  and  or  and  are different (but not independent) expressions of the same relationship. Thus, larger SMBHs are associated with more massive and larger bulges, as understood in the framework of coevolution of spheroids and SMBHs. Hopkins et al. (2007a, b) first studied the residuals of the  and  relations with bulge properties, finding tighter correlations between the combination of the bulge parameters and . This bolstered the notion of a BHFP. The bulge parameters  and  are indeed physically related to each other through the fundamental plane relation (FP) and virial theorem. The existence of such BHFP has important implications for the largest  and resolves the apparent conflict between expected and measured values of  for the outliers in both  and  relations (e.g. Bernardi et al., 2007; Lauer et al., 2007). A similar correlation, but between , , and , was reported by Barway & Kembhavi (2007) for nearby ellipticals with measured . This correlation has a smaller scatter than the  and  relations and gives further support to the existence of a FP-type relation for SMBHs.

Figure 16:  as a function of  and  (a),  and  (b),  and  (c),  and  (d) for the same sample as in Figure 15. The logarithmic slopes and and offset of the fitted relation are given in each panel.

Furthermore, by comparing the tightness of the correlations between  and linear combination of the bulge parameters, we argue that  is the parameter that drives the connection with  in the BHFP. A small contribution is due to  or . Similar results were obtained by Aller & Richstone (2007) who compared  with , and . Thus, including an additional parameter to the relation does not improve the quality of the fit, with  always being the dominant parameter.

4.2.2  versus Galaxy Parameters

We also wish to test if the shallow  and  relations can be improved by the addition of another galaxy parameter. Therefore, we calculated the residuals of , , and  relations to look for the signature of an additional galaxy parameter. The residuals were obtained as the difference between the data points and the relation fits listed in Table 5. We compared the residuals in , , and  with the galaxy photometric and kinematic properties (i.e., , , , and ; Figure 17). For , no trends with other galaxy properties are found whereas the  and  relations show a weak dependence on  according to the Spearman rank correlation coefficient with a significance levels of 0.3% and 0.9%, respectively. We also examined the residuals of the relation to check that our results are not affected by the propagated errors in the determination of . Figure 18 shows the different linear combinations of two galaxy parameters with . A few tight correlations are found when considering the total galaxy rather than bulge parameters. The tightness always depends on having  as a fitting parameter (Table LABEL:tab:fit_combination), i.e., the Spearman rank correlation coefficient is higher when  is taken into account. The constancy of the Spearman rank correlation coefficient despite the addition of other fitting parameters, argues that the main acting parameter is .

Figure 17: Residuals of the , ,  relations versus , , ,  for 62 Sample A galaxies (57 upper limits from nebular line widths, filled circles; 5 upper limits from resolved kinematics, open circles) and 28 Sample B galaxies (squares). The total number of galaxies is . The Spearman rank correlation coefficient is given in each panel.

We conclude that the addition of bulge (e.g., , , ) or galaxy (e.g., , , ) structural parameters does not appreciably modify and improve the  relation. We exclude that parameter covariances would produce the same scatter in both the  and  relations. Such a conspiracy would require an anti-correlation between  and  (or ) which is not observed.

Figure 18:  as a function of  and  (a), and  (b),  and  (c),  and  (d), and  and  (e) for the same sample as in Figure 17. The logarithmic slopes and and offset of the fitted relation are given in each panel.

5 Discussion

The existence of the  scaling relations implies that SMBH and galaxy formation processes are closely linked. Some of the challenges of the current models of SMBH formation and evolution include reproducing and maintaining the relations regardless of the sequence of galaxy evolution during the hierarchical mass assembly (Robertson et al., 2006a; Schawinski et al., 2006; Croton, 2009). A more comprehensive assessment the  scaling relations scatter thus enables a better characterisation of the different SMBH/galaxy formation models.

Our large sample of galaxies with secure determination or upper limits of  has enabled a thorough investigation of the SMBH demography over a wide range of , morphological type, and nuclear activity. After establishing the unbiased mapping of  upper limits against that of secure , we tested whether  is more fundamentally driven by one of the several bulge (i.e., the velocity dispersion, luminosity, virial mass, Sérsic index, and mean effective surface brightness) and galaxy (luminosity, circular velocity, stellar, virial, and dynamical mass) parameters known to correlate with the SMBH mass, and if the known scaling relations can be improved with the addition of a third parameter.

  • We argue that  is fundamentally driven by , considering that the  relation has the tightest scatter with respect to all other scaling relations. The scatter of our  relation is comparable to G09 but slightly larger than that of Ferrarese & Ford (2005). At small , some  upper limits exceed the expectation value as they likely account for the mass of a conspicuous nuclear cluster (B09). Excluding these galaxies from the fit reduces the scatter to Ferrarese & Ford (2005)’s value. Barred and unbarred galaxies as in B09 and G09 follow the same  relation within the errors, contrary to Graham (2008)’s findings. Classical and pseudo-bulges were identified according to their Sérsic index (Kormendy & Kennicutt, 2004). The  relations of classical and pseudo-bulges have a different slope. The low significance of the  relation for pseudo-bulges is in agreement with more recent findings by Kormendy et al. (2011) and suggests that the formation and growth histories of SMBHs depend on the host bulge type.

  • The  relation is clearly not as tight as the  one. However, the fact that G09 found a  relation as tight as the  one is not in contradiction with our results, since G09 accounted only for early-type galaxies. The same is true for the  correlation, although the bulge mass proved to be a better proxy of  than  and it includes  as an additional fitting parameter.

  • Contrary to previous findings (Graham et al., 2001, 2003; Graham & Driver, 2007), we find little or no correlation between  and Sérsic  in agreement with Hopkins et al. (2007b). The latter stated that  is unrelated with the light concentration of the bulge based on observations and simulations. Consistently, we confirm that  and  are poorly correlated.

  • The correlations between  and galaxy luminosity or mass are not a marked improvement over the  relation. These scaling relations are strongly sensitive to the morphology of the host galaxies, with the presence of a disc playing an anti-correlating rôle, as first pointed out by (Kormendy, 2001). This is a further indication that bulges are driving the SMBH correlations and SMBH evolution.

  • We found that the  relation is significantly coarser than that of the  relation, with a scatter about twice larger, suggesting that  is more strongly controlled by the baryons than the dark matter.

  • To assess the need for an additional parameter in the -bulge/galaxy scaling relations, we performed both a residual analysis and third-parameter fits as in Hopkins et al. (2007b), Barway & Kembhavi (2007), and Aller & Richstone (2007). To this aim we considered different linear combinations of bulge or galaxy parameters with . The strongest correlations always include  as a fundamental structural parameter. The tightest relation is found between , , and  (also known as the BHFP, Hopkins et al. 2007a, b). Since its scatter does not change appreciably compared to that of the  relation, the addition of  is barely an improvement. This is a further confirmation that  is the fundamental parameter which drives also the BHFP.

Our findings about the tightness of scaling relations such as the  may be interpreted in the framework of self-regulating feedback in galaxies Hopkins et al. (2009). These authors argued that the energy released from the accretion of gas onto a SMBH is enough to stop further gas accretion, drive away the gas, and quench star formation. Hopkins et al. (2009) studied the relation between  and the mass of the gas accreting onto the SMBH as a function of the distance from the central engine. The scatter of the -stellar mass relation increases at smaller radii (i.e., approaching the radius of influence of the SMBH) and decreases at larger radii (i.e., where the SMBH scaling relations are defined). This has been interpreted in terms of self-regulated growth of SMBHs, where the SMBH accretes and regulates itself without accounting for the gas supply but just being controlled by the bulge properties.

In this scenario,  is the property which gives the tightest connection with MBH since it determines the depth of the local potential well from which the gas must be expelled. Younger et al. (2008) studied self-regulated models of SMBHs growth in different scenarios of major mergers, minor mergers, and disc instabilities, to find that SMBHs depend on the scale at which self-regulation occurs. They compared the bulge binding energy and total binding energy with , finding that the total binding energy is not a good indicator of  mass in disc-dominated systems. This agrees with our findings that the late-type systems deviate most significantly from the -galaxy and BHFP scaling relations.

Several SMBH formation models predict a connection between  and the total mass of the galaxy (Haehnelt et al., 1998; Silk & Rees, 1998; Adams et al., 2001; Croton et al., 2006; Croton, 2009) such that if the dark and baryonic matter act to form the bulge and SMBH, the dark halo determines the bulge and SMBH properties. Therefore, the mass of the SMBH and dark matter halo should be connected (Cattaneo, 2001; Hopkins et al., 2005a, b). Numerous recent studies have erred along those lines. For instance, Bandara et al. (2009) studied the correlation between  and total (luminous+dark) mass of the galaxy estimated from numerical simulations of gravitational lensing. Their relation suggests that the more massive halos are more efficient at forming SMBHs than the less massive ones; the slope of their relation suggests merger-driven, feedback-regulated processes of SMBH growth. Likewise, Booth & Schaye (2010) used self-consistent simulations of the coevolution of SMBHs and galaxies to confirm the relation by Bandara et al. (2009) and to prove that self-regulation of the SMBH growth occurs on dark matter halo scales. Volonteri & Natarajan (2009) investigated through numerical simulations the observational signature of the self-regulated SMBH growth by analysing the mass assembly history of black hole seeds. They found that the  relation stems from the merging history of massive dark halos, and that its slope and scatter depend on the halo seed and specific SMBH self-regulation process.

Our observational results are not supportive of such results. We found only a weak correlation of  with . This agrees with our expectations based on the observed tightness of the  relation and given the fact that the scatter of the  relation is large and morphologically-dependent (Courteau et al., 2007; Ho, 2007).

The - relation may be improved with homogeneous measurements of the  data base. Indeed, while our collection of observed velocities is as current and reliable as possible, especially for spiral galaxies, we still lack a fully homogeneous data base. Many of us are working on improvements of this nefarious situation. However it is clear, as this work demonstrates, that the - is the definitive fundamental relation of the SMBH-bulge connection.


Acknowledgements

We are indebted to Ralf Bender, Michele Cappellari, Lodovico Coccato, Elena Dalla Bontà, Victor Debattista, John Kormendy, Tod Lauer, Lorenzo Morelli, Alessandro Pizzella, Marc Sarzi, and Roberto Saglia for many useful discussions and suggestions. We also thank Jairo Méndez Abreu for his GASP2D package which we used to measure the photometric parameters of our sample galaxies. We acknowledge the anonymous referee for valuable comments that led to an improved presentation. AB is grateful to Queen’s University for its hospitality while part of this paper was being written. AB was also supported by a grant from Accademia dei Lincei and Royal Society as well as STFC rolling grant ST/I001204/1 “Survey Cosmology and Astrophysics”. EMC was supported by Padua University through grants CPDA089220/08 and CPDR095001/09 and by Italian Space Agency through grant ASI-INAF I/009/10/0. SC and YZ acknowledge support from the Natural Science and Engineering Science Council of Canada. This research has made use of the HyperLeda database, NASA/IPAC Extragalactic Database (NED), and Sloan Digital Sky Survey (SDSS).

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