iint TXFiint

# On the scaling relations of bulges and early type galaxies

Bogdan A. Pastrav1
1Cosmology and Astroparticle Physics Laboratory, Institute of Space Science,
Atomistilor 409, 077125, Bucharest-Magurele, Romania
<EMAIL_ADDRESS>
(Accepted 2021 June 15. Received 2021 June 11; in original form 2020 December
21.)

###### Abstract

Following from our recent work, we present here a detailed structural analysis
of a representative sample of nearby spiral and early-type galaxies taken from
the KINGFISH/ SINGS survey. The photometric parameters of bulges are obtained
from bulge-disc decompositions using GALFIT data analysis algorithm. The
method and corrections for projection and dust effects previously obtained are
used to derive intrinsic photometric and structural bulge parameters. We show
the main bulge scaling relations and the black hole relations, both observed
and intrinsic ones, in B band. We find dust and inclination effects to produce
more important changes in the parameters of the Kormendy relation for spiral
galaxies, with the respective bulges of late-type galaxies residing on a
steeper slope relation that the early-type galaxies. We observe that the
Kormendy relation in combination with a bulge Sérsic index ($n_{b}$)
threshold, does not produce a conclusive morphological separation of bulges.
The $n_{b}$\- bulge-to-total flux ratio ($B/T$) and $B/T$-stellar mass could
be used to discriminate between late-type and early-type galaxies, while a
further use of these parameters to divide bulges with different morphologies
is problematic due to overlaps in the two distributions or large spread in
values. We confirm the existence of two distinct intrinsic relations between
the bulge luminosity (or absolute magnitude) and Sérsic index for late- and
early-type galaxies, while the relations between the black-hole mass
($M_{BH}$) and bulge luminosity are not found to be statistically different at
p<0.05. Within errors, we find statistically similar intrinsic $M_{BH}-n_{b}$
relations for all bulges.

###### keywords:

galaxies: bulges – galaxies: elliptical and lenticular, cD – galaxies:
structure – ISM: dust, extinction – galaxies: evolution – galaxies:
fundamental parameters

††pubyear: 2020††pagerange: On the scaling relations of bulges and early type
galaxies–References

##  1 Introduction

Bulges, traditionally defined as the excess of light above the inner
extrapolation of discs and/or bars, were for a long time thought to be
elliptical-like structures, predominantly pressure-supported, embedded in the
galaxy disc. These components - the classical bulges - have similar
observational properties as elliptical galaxies (Faber 1977, Gott 1977,
Renzini 1999) and form out of major mergers (Brooks & Christensen 2016, Tonini
et al. 2016, Rodriguez-Gomez et al. 2017) or coalescence of giant clumps
(Bournaud et al. 2007, Bournaud et al. 2009, Elmegreen et al. 2008, Bournaud
et al. 2016). However, many studies have shown the existence of a dichotomy of
bulges (Kormendy 2003, Kormendy & Kennicutt 2004, Athanassoula 2005, Fisher &
Drory 2016), with the second category of bulges - named pseudo-bulges, being
flat, disc-like stuctures, rotation-supported and with younger stellar
populations than classical bulges. It was also observed a coexistence of the
two types of bulges within the same galaxy (Fisher & Drory 2008, Erwin et al.
2015).
Understanding the formation scenarios of bulges and their subsequent evolution
is of critical importance in the galaxy formation and evolution models. The
scaling relations of these components are also essential. Thus, the classical
bulges follow the same scaling relations as ellipticals, such as Faber-Jackson
(Faber & Jackson 1976) relation, the Fundamental Plane (Djorkovsky & Davis
1987) and its projection - the Kormendy (Kormendy 1977) relation. It has also
been shown the connection of bulges with the central massive black holes from
the centre of galaxies, their masses being correlated (Gebhardt et al. 2000,
Ferrarese & Merritt 2000, Kormendy & Gebhardt 2001). The associated scaling
relations, the black-hole scaling relations, are also very important for the
understanding of the co-evolution of galaxies and the central black holes.

Some of these relations and/or bulge structural and photometric parameters
were used alone or in combination to separate bulges of smaller or larger
samples of galaxies into classical or pseudo-bulges. Thus, Fisher & Drory
(2008) used the bulge Sérsic index ($n_{b}$) for this division, while Gadotti
(2009) and Gao et al. (2020) argued that a classification using this parameter
alone, produces an important degree of contamination in the two categories of
bulges. Allen et al. (2006) used a combination of bulge-to-total flux ratios
($B/T$) and the ratio between bulge and disc sizes. Neumann et al. (2017) on
the other hand, used a combination of four parameters to classify bulges and
found the Kormendy relation together with their custom concentration index to
provide the most accurate classification. Gadotti (2009), Sachdeva et al.
(2017), Sachdeva et al. (2019) and Gao et al. (2020) used the Kormendy
relation of the eliptical galaxies to classify bulges and found with a high
degree of accuracy that pseudo-bulges are the ones which are situated far
(more than two or three standard deviations) from the aforementioned relation.
Sachdeva et al. (2020) defined a new morphology indicator from the Kormendy
relation - the relative distance from the $\pm 1\sigma$ boundary of the
Kormendy relation, and used it to investigate the degree of accuracy of other
bulge morphology classifiers (e.g. $n_{b}$, $B/T$, concentration index or
central velocity dispersion). They found this to be the most efficient
indicator of bulge morphology and stellar activity. Despite all this, as
pointed out by Neumann et al. (2017), there is not a single criterion that
should be used when trying to distinguish between classical and pseudo-bulges,
but a combination of them for a more robust classification, as one type of
bulges may not follow all the criteria or have all the
morphological/structural properties usually characteristic for them (see
Fisher & Drory 2016, Kormendy 2016).
Added to all this, it has been shown by Tuffs et al. (2004), Möllenhoff et al.
(2006), Gadotti et al. (2010), Pastrav et al. (2013a) and Pastrav et al.
(2013b) that dust (which pervades the disc) and projection (inclination)
effects produce biases in the parameters involved in the scaling relations of
bulges and early-type galaxies, or in those employed when trying to separate
bulges with different morphologies. These biases are more sigmificant at
shorter wavelengths and higher inclinations. Not taking into account these
effects may produce biased scaling relations, leading to misleading
conclusions, misclassifications of bulges, or validation/invalidation of
certain criteria for bulge morphology classification.

Thus, in this paper we continue the study presented in Pastrav (2020)
(hereafter, Paper I) and show the results of a detailed analysis of a sample
of low-redhsift spiral and early-type galaxies taken from the KINGFISH survey
(Kennicutt et al. 2011). Here we focus on the scaling relations of bulges of
spirals and early-type galaxies, showing how dust, inclination and
decomposition effects bias the specific parameters of the relations, such as
slope, zero point or scatter. Special attention is dedicated to a set of
scaling relations, important for understanding the connection between bulges
and black holes. For certain relations, we investigate the degree of
correlation between the parameters involved. We also investigate, in relation
with previous works, which relations or combination of parameters can produce
a more accurate classification of galaxy bulges into classical or pseudo-
bulges, even though our sample is rather small. We then compare our main
results with other similar works in the literature. As before, we decompose
each galaxy into its main components (bulge+disc). Then, we use the method of
Pastrav et al. (2013a) and Pastrav et al. (2013b) and their numerical
corrections for projection (inclination), dust and decomposition effects, to
recover the intrinsic photometric and structural paraneters involved in the
scaling relations. The numerical corrections were derived by analysing and
fitting simulated images of galaxies produced by means of radiative transfer
calculations and the model of Popescu et al. (2011). The empirical relation
found by Grootes et al. (2013) is used to determine the central face-on dust
opacity, a parameter which is essential when applying the corrections for dust
effects, in the case of bulges and lenticular galaxies.

The paper is organised as follows. In Sect. 2 we present the galaxy sample
used in this study, while in Sect. 3 we describe the method used for the
overall procedure (photometry, fitting algorithm, derive dust opacity and dust
mass, derive intrinsic parameteres needed in the scaling relations). In Sect.
4 we present the main results - the galaxy scaling relations for bulges of
spiral galaxies and the early-type ones, both observed and intrinsic (dust-
free) ones, together with all the numerical results. The black-hole scaling
relations are presented and analysed in Sect. 4.3. In Sect. 5 we discuss upon
the main results and possible sources of errors and differences with other
studies, while in Sect. 6 we summarise the results obtained in this study and
draw conclusions.
Throughout this paper, where necessary, a Hubble constant of $H_{0}=67.8$
km/s/Mpc (Planck Collaboration 2016) was used.

##  2 Sample

Our sample consists of the original sample of 18 late-type galaxies analysed
in Paper I, to which another 11 early-type galaxies (6 ellipticals and 5
lenticulars) were added, the latter also included in the SINGS (Spitzer
Infrared Nearby Galaxies Survey; Kennicutt et al. 2003) survey and the
KINGFISH project (Key Insights on Nearby Galaxies: a Far-Infrared Survey with
Herschel; Kennicutt et al. 2011). Thus, in total our sample consists of 29
galaxies, which is used to study the scaling relations of both bulges and
early-type galaxies at low redshift. The KINGFISH project is an imaging and
spectroscopic survey, consisting of 61 nearby (d<30 Mpc) galaxies, chosen to
cover a wide range of galaxy properties (morphologies, luminosities, SFR,
etc.) and local ISM environments characteristic for the nearby universe. The
optical images were extracted from the NASA/IPAC Infrared Science Archive
(IRSA) and NASA IPAC Extragalactic Database (NED). The images were taken with
the KPNO (Kitt Peak National Observatory) and CTIO (Cerro Tololo Inter-
American Observatory) telescopes (see Kennicutt et al. (2003)).
As in Pastrav (2020), barred galaxies present in the KINGFISH survey were
excluded from our selected samples because at the moment we cannot properly
account and correct for the effects of dust on the photometric and structural
parameters of bars. The images analysed here are in optical regime, in B band,
for a number of reasons. Thus, the galaxy images in the short optical bands
(e.g B, r) present a much higher resolution, and are better resolved overall
and up to larger galactocentric radii than at longer wavelengths. Thus, on one
hand, one can analyse and disentangle the galaxy components with a higher
degree of accuracy. On the other hand, as short optical wavelength images can
be obtained up to a considerably deeper (lower) surface brightness limit, one
can go deep into the backround when analysing images at these wavelengths and
have a more correct picture of the galaxy outskirts and possible truncations
in the galaxy surface brightness profiles at large galactocentric radii.
Moreover, as dust and inclination effects are stronger at shorter wavelengths
(as shown in Pastrav et al. (2013a), Pastrav et al. (2013b)), and because our
method is suitable for cases where optical data are available, we have chosen
the B band images.

##  3 Method

The method used in this study is in general the same as in Paper I when it
comes to the fitting procedure, the sky determination and subtraction, the
photometry and the dust opacity and dust mass derivation, with a few
differences. Therefore, for a more detailed description, we refer the reader
to Paper I, where the whole procedure is presented in great detail. Here, a
shorter and concise version is given below.

###  3.1 Fitting procedure

For the fitting procedure of the early-type galaxies (ETGs) we used the GALFIT
(version 3.0.2) data analysis algorithm (Peng et al. 2002, Peng et al. 2010),
as we did for the case of spirals in our sample. The analysis of the bulges of
spiral galaxies was done in Paper I when analysing the sample of late-type
galaxies. Thus, in this study we just analysed the ellipticals and lenticulars
added here for the purpose of studying their scaling relations. For the bulge-
to-disc decomposition and to fit the observed surface brightness of ETGs we
used the exponential (“expdisc”) and the Sérsic (“sersic”) functions available
in GALFIT, for the disc and bulge surface brightness profiles, while the "sky"
function was used for an estimation of the background in each image.

In most studies from the literature, elliptical galaxies are usually
considered as a single component, a spheroid, and thus fitted with a single
Sérsic function only. This approach may be accurate, especially at lower
resolution and for higher redshift galaxy images. Hovewer, it has also been
shown in quite a few studies, such as the ones of Capaccioli (1987), Bender
(1990), Rix & White (1990), Rix & White (1992) or Nieto et al. (1991) that
ellipticals can present a stellar disc in their structure or intermediate-
scale discs. Huang et al. (2013a), Huang et al. (2013a), Huang et al. (2016)
found the structure of local, bright elliptical galaxies to be formed by three
sub-components, with the surface brightness of each being described by a
Sérsic function. The works of Bellstedt et al. (2017), Oh et al. (2017) and
Zhu et al. (2021) came to emphasize that ellipticals are not just simple one-
component structures and the need to consider extra-components in their
structural analysis when trying to explain their evolutionary history. Oh et
al. (2017) found that using a single-component to model the surface brightness
of the ellipticals cannot always match the ellipticity curve, or
alternatively, changes in the isophotal shape. Thus, in this study, as our
images are well resolved and at high resolution and signal-to-noise ratio, we
decided to use the same combination of functions as for spirals and
lenticulars.

As in our previous work, the free parameters of the fits are: the X and Y
coordinates of the centre of the galaxy in pixels, the bulge and disc
integrated magnitudes, the disc scale-length / bulge effective radius (for
exponential/Sérsic function), axis-ratios of discs and bulges, bulge Sérsic
index (for Sérsic function), the sky background (only in the preliminary fit -
Step 1, see Paper I) and the sky gradients in X and Y. The input values for
the coordinates of galaxy centre were determined after a careful inspection of
each image. Initial values for the position angles (PA) and axis-ratios were
taken from NED. Although the central coordinates are free parameters, we
imposed a constraint on the fitting procedure, ensuring that the bulge and
disc components were centred on the same position. The axis-ratio is defined
as the ratio between the semi-minor and semi-major axis of the model fit (for
each component). The position angle is the angle between the semi-major axis
and the Y axis (increasing counter clock-wise). To mask the pixels
corresponding to the additional light coming from neighboring galaxies, stars,
compact sources, AGN or image artifacts, for each galaxy image we used a
complex star-masking routine to create a bad pixel mask. This was used as
input in GALFIT.

###  3.2 Sky determination and subtraction. Photometry

We follow the procedure with the steps described in Paper I (Step 1, 2 and,
where necessary - Step 3) to estimate the background level, derive the
structural and photometric parameters, and calculate the integrated fluxes for
each galaxy, together with the bulge-to-disc ratios. The integrated (total)
flux of the galaxy is calculated from the maximum curve-of-growth (CoG) value
(in counts), at the $R_{max}$ radius (this is defined as the radius beyond
which there is no galaxy emission and, therefore, the CoG is essentially flat
towards larger radii). The bulge-to-disc ratio ($B/D$) is estimated from the
disc and bulge CoGs and compared with the one determined by the ratio of the
total counts of the decomposed disc and bulge images, as it has to be
consistent, within errors. The uncertainties in the fluxes are estimated from
the root mean square of the CoG values from the first 10 elliptical anulli
beyond $R_{max}$. As before, we have used the positive sky residuals in the
outer parts of galaxies (towards $R_{max}$ and beyond) to estimate the
systematic errors in bulge-to-disc ratios. This is important because the sky
level errors have an important contribution to the systematic errors in bulge-
to-disc decompositions, as shown by Simard et al. (2002). In addition, in
nearby, well resolved, high resolution galaxy images, the choice of modelling
functions for the surface distribution can produce a more significant
contribution to the error budget and the best-fit bulge parameters (including
the bulge-to-total flux ratio), like the studies of Laurikainen et al. (2005),
Kim et al. (2014) and Gao & Ho (2017) have determined.
The derived integrated fluxes and bulge-to-disc ratios for the ETGs of our
sample are given in Table 1, together with the distances to each galaxy used
in this study, taken from NED.

With the best-fitting structural/photometric parameters and the integrated
fluxes for the galaxies and their main constituents already determined, we
then calculated the central, effective and mean effective surface brightness
for bulges, together with the corresponding apparent and absolute bulge
magnitudes. For the bulges of the spiral galaxies, these were already
calculated in our previous paper (see Table 4 in Pastrav (2020)). Thus,
following for example Graham & Driver (2005) or Graham & Worley (2008), we can
write the equations for the bulge effective surface brightness, central and
mean effective surface brightness, bulge apparent and absolute magnitudes as:

$\displaystyle\mu_{e,b}=-2.5\log[\frac{F_{b}}{2\pi(R_{e,b})^{2}\exp(\kappa_{n})n\kappa_{n}^{-2n}\Gamma(2n)Q_{b}}/F_{0}]$
(1) $\displaystyle\mu_{0,b}=\mu_{e,b}-2.5\times(\kappa_{n})/log(10)$ (2)
$\displaystyle<\mu_{e,b}>=\mu_{e,b}-2.5\log([\frac{n\exp(\kappa_{n})\Gamma(2n)}{\kappa_{n}^{2n}}]$
(3)

where $F_{b}$ is the integrated flux of the bulge, $R_{e,b}$ is the effective
radius (in arcsecs), $\Gamma(2n)=2\gamma(2n,\kappa_{n})$ (Ciotti 1991), with
$\Gamma$ and $\gamma$ the complete and incomplete gamma functions, $n$ is the
Sérsic index, while $\kappa_{n}$ is a variable coupled with $n$ (see Ciotti &
Bertin 1999 and Graham & Driver 2005)

$\displaystyle
m_{b}=\mu_{e,b}-2.5\log(2\pi(R_{e,b})^{2}Q_{b})-2.5\log[\frac{n\exp(\kappa_{n})\Gamma(2n)}{\kappa_{n}^{2n}}]$
(4) $\displaystyle M_{b}=m_{b}-25-5\log(d_{gal}/Mpc)$ (5)

The observed axis-ratio of the bulge ($Q_{b}$) is included in Eqs. 2&4 because
bulges are seen in projection. Thus, this has to be taken into consideration
later when deriving the intrinsic parameters, and correct for projection
effects.

Table 1:  The calculated fluxes for the early-type galaxies (B band). The columns represent: (1) - galaxy name; (2) - morphological type; (3) - distance to each galaxy, taken from NASA Extragalactic Database (NED), as derived in: $a$ \- Tully et al. (2013), $b$ \- Kennicutt et al. (2011), $c$ \- Blakeslee et al. (2001), $d$ \- Theureau et al. (2007), $e$ \- Tully & Fisher (1988), $f$ \- Sabbi et al. (2018); (4) bulge-to-disc ratios ($B/D$) derived from the decomposed images, with systematic uncertainties, derived as described in Sec. 3.2; (5) - the integrated flux for each galaxy, in $Jy$; (6) - the error for the galaxy flux; (7) - the integrated flux of the bulge, in $Jy$. $Galaxy$ | $Type$ | $d_{gal}$ | $B/D$ | $F_{gal}$ | $\sigma_{F_{gal}}$ | $F_{b}$
---|---|---|---|---|---|---
|  | [Mpc] |  | [Jy] | [Jy] | [Jy]
NGC0584 | E4 | $20.00^{a}$ | $2.02_{-0.00}^{+0.00}$ | 0.310 | 0.006 | 0.190
NGC0855 | E | $8.83^{a}$ | $0.37_{-0.00}^{+0.00}$ | 0.025 | 0.001 | 0.007
NGC1404 | E1(2) | $19.50^{a}$ | $5.52_{-0.03}^{+0.05}$ | 0.630 | 0.004 | 0.530
NGC3265 | E | $19.60^{b}$ | $0.72_{-0.00}^{+0.00}$ | 0.025 | 0.009 | 0.013
NGC4125 | E6pec | $23.90^{c,a}$ | $3.40_{-0.05}^{+0.06}$ | 0.300 | 0.003 | 0.230
NGC4552 | E0-1 | $15.90^{a}$ | $0.93_{-0.08}^{+0.02}$ | 0.400 | 0.003 | 0.190
NGC1377 | S0 | $21.00^{d}$ | $1.63_{-0.01}^{+0.02}$ | 0.023 | 0.001 | 0.014
NGC1482 | SA0(p) | $19.60^{e}$ | $1.11_{-0.00}^{+0.00}$ | 0.172 | 0.002 | 0.094
NGC1705 | SA0p | $5.22^{f,a}$ | $1.58_{-0.00}^{+0.00}$ | 0.105 | 0.004 | 0.064
NGC3773 | SA0 | $17.00^{e}$ | $0.54_{-0.08}^{+0.08}$ | 0.030 | 0.001 | 0.011
NGC5866 | S0 | $14.70^{a}$ | $0.36_{-0.05}^{+0.06}$ | 0.231 | 0.001 | 0.060

###  3.3 Correcting for dust, projection and decomposition effects

In Paper I (see section 3.3), we showed the procedure and equations used to
derive the dust opacity and dust mass for the analysed sample of spiral
galaxies. Thus, we have used the correlation between the central face-on B
band dust opacity, $\tau_{B}^{f}$, and the stellar mass surface density of
nearby galaxies ($\mu_{*}$, derived using the scale-length found in the bulge-
disc decomposition), found by Grootes et al. (2013). This relation was
obtained by analysing a sample of spiral galaxies taken from the Galaxy and
Mass Assembly (GAMA) survey.

$\displaystyle\log(\tau_{B}^{f})=1.12(\pm
0.11)\cdot\log(\mu_{*}/M_{\odot}kpc^{-2})-8.6(\pm 0.8)$ (6)

The dust masses for each galaxy were derived using Eq. (2) from Grootes et al.
(2013) (see also Eqs. (A1-A5) from Appendix A of the same paper). To derive
this relation, the dust geometry of the Popescu et al. (2011) model was
considered.

$\displaystyle\tau_{B}^{f}=K\frac{M_{dust}}{R_{s,d}}$ (7)

where $K=1.0089pc^{2}/kg$ is a constant containing the details of the dust
geometry and the spectral emissivity of the Weingartner & Draine (2001) model,
while $R_{s,d}$ is the scale-length of the disk, expressed in kpc.

However, one of the differences with respect to Paper I approach is that, for
the elliptical galaxies, the dust opacity and dust mass were not derived
because the relations used previously for spiral galaxies were not calibrated
for the specific geometry of this type of galaxies. Thus, the correlation from
Eq. 6 was discovered by analysing a sample of nearby spiral galaxies, while
the relation in Eq. 7 relies on the dust geometry of the Popescu et al. (2011)
model (which considers that the diffuse dust in the disc is distributed
axisymetrically in two exponential discs), also calibrated on a small sample
of nearby spiral galaxies. Taking into account these considerations, we only
used the aforementioned relations (Eqs. (10)-(11)) for the spirals and
lenticulars in our enlarged sample.

To correct all the parameters involved in the bulge and ETG scaling relations,
we used again the method developed and presented in Pastrav et al. (2013a) and
Pastrav et al. (2013b). More specifically, we used the whole chain of
corrections presented in Eqs. (4-13) from Pastrav et al. (2013a) and Eqs.
(3-13) from Pastrav et al. (2013b) for the bulges and lenticular galaxies,
together with all the numerical results (given in electronic form as data
tables at CDS) to correct the measured parameters for projection
(inclination), dust and decomposition effects, in order to obtain their dust-
free, intrinsic values. For the elliptical galaxies however, we used just the
projection corrections together with the ones for projection effects in the
decomposition process (a part of the third term in the chain of corrections).
The reason for this is that, as we mentioned earlier, the dust opacity was not
calculated with Grootes et al. (2013) relation and thus, dust effects on the
structural and photometric parameters of the ellipticals could not be
estimated.
However, as it was shown in Pastrav et al. (2013a) and Pastrav et al. (2013b)
(see Figs. 5,6, 19 & 22 in Pastrav et al. (2013a)), the projection effects on
bulge parameters are far more significant that those for the disc, having a
stronger influence than the dust effects over the measured structural
parameters of bulges. Therefore, these effects should be taking into account
for elliptical galaxies, even tough the dust effects are not quantified or
considered.
Due to the fact that the numerical corrections are a function of wavelength,
dust opacity and/or bulge-to-disc ratio, we used the values of $\tau_{B}^{f}$
(where available), and $B/D$ already derived individually for each galaxy and
did all the needed interpolations to obtain the final values for the
structural and photometric parameters. As a result, $B/D$, $n$, $R_{e,b}$,
$\mu_{0,b}$, $\mu_{e,b}$, $<\mu_{e,b}>$, $m_{app,b}$, $M_{abs,b}$ were fully
corrected. Thus, we rewrite Eqs. 1-5 to determine the intrinsic (corrected)
values for these parameters:

$\displaystyle\mu_{0,b}^{i}=\mu_{0,b}-A_{ext}+A_{dim}$ (8)

where $A_{ext}$ is the attenuation due to the foreground galactic extinction
(taken from NED, as in Schlafly & Finkbeiner 2011 recalibration of the
Schlegel et al. 1998 infrared based dust map), and $A_{dim}=-2.5\log(1+z)^{3}$
is the attenuation due to cosmological redshift dimming, per unit frequency
interval,

$\displaystyle\mu_{e,b}^{i}=\mu_{0_{b}}^{i}+2.5\times\kappa_{n}/log(10)$ (9)
$\displaystyle<\mu_{e,b}^{i}>=\mu_{e,b}^{i}-2.5\log([\frac{n^{i}\exp(\kappa_{n})\Gamma(2n^{i})}{\kappa_{n}^{2n^{i}}}]$
(10) $\displaystyle
m_{b}^{i}=<\mu_{e,b}^{i}>-2.5\log(2\pi(R_{e,b}^{i})^{2}Q_{b})$ (11)
$\displaystyle M_{b}^{i}=m_{b}^{i}-25-5\log(d_{gal}/Mpc)$ (12)

All the galaxies in our enlarged sample are at low redshift and therefore we
did not apply K-corrections or evolutionary ones. Correspondingly, the
correction due to cosmological redshift dimming is also quite small, in the
range $0.01-0.05$ mag. The values for all the previously mentioned parameters
are shown in Tables 2 and 3, for the whole sample of spirals and early-type
galaxies.

Table 2:  The photometric and structural parameters for the bulges of late-type (LTGs) and early-type (ETGs) galaxies. The columns represent: (1) - galaxy name; (2) - the intrinsic bulge-to-disc ratio; (3) - the observed bulge axis-ratio; (4), (5) - the observed and intrinsic bulge effective radii; (6-8) - the observed, the intrinsic bulge effective surface brightness, and the standard deviation; (9-11) - the observed, the intrinsic mean effective surface brightness, and the standard deviation; (12-14)- the observed, the intrinsic bulge absolute magnitude, and the standard deviation: (15), (16) - observed and intrinsic bulge Sérsic index. In square brackets we have the units in which these quantities are expressed. $Galaxy$ | $(B/D)^{i}$ | $Q_{b}$ | $R_{e,b}$ | $R_{e,b}^{i}$ | $\mu_{e,b}$ | $\mu_{e,b}^{i}$ | $\sigma_{\mu_{e,b}}$ | $<\mu_{e,b}>$ | $<\mu_{e,b}^{i}>$ | $\sigma_{<\mu_{e,b}>}$ | $M_{b}$ | $M_{b}^{i}$ | $\sigma_{M_{b}}$ | $n$ | $n^{i}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | [kpc] | [kpc] | $[\frac{mag.}{arcsec^{2}}]$ | $[\frac{mag.}{arcsec^{2}}]$ |  | $[\frac{mag.}{arcsec^{2}}]$ | $[\frac{mag.}{arcsec^{2}}]$ |  | [mag.] | [mag.] |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16)
LTGs |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
NGC0024 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
NGC0628 | 0.03 | 0.92 | 0.45 | 0.38 | 20.19 | 19.92 | 0.24 | 19.42 | 18.89 | 0.46 | -17.34 | -17.49 | 0.58 | 1.17 | 1.45
NGC2841 | 0.25 | 0.66 | 0.69 | 0.63 | 19.06 | 18.98 | 0.23 | 18.17 | 17.65 | 0.45 | -19.13 | -19.47 | 0.61 | 1.50 | 2.01
NGC2976 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00
NGC3031 | 0.97 | 0.70 | 1.47 | 0.91 | 20.84 | 20.54 | 0.20 | 19.56 | 18.37 | 0.41 | -19.45 | -19.63 | 0.51 | 3.20 | 4.56
NGC3190 | 0.31 | 0.27 | 1.53 | 1.26 | 18.54 | 18.40 | 0.27 | 18.11 | 17.83 | 0.52 | -19.96 | -19.82 | 0.61 | 0.56 | 0.66
NGC3621 | 0.03 | 0.53 | 1.15 | 0.40 | 21.29 | 20.98 | 0.31 | 21.19 | 20.88 | 0.62 | -17.01 | -15.04 | 0.70 | 0.15 | 0.21
NGC3938 | 0.03 | 0.94 | 0.45 | 0.35 | 20.22 | 20.11 | 0.29 | 19.60 | 19.23 | 0.52 | -17.19 | -16.99 | 0.82 | 0.86 | 1.08
NGC4254 | 0.21 | 0.79 | 1.65 | 1.54 | 21.97 | 21.80 | 0.21 | 20.93 | 20.32 | 0.42 | -18.46 | -18.93 | 0.75 | 2.02 | 2.64
NGC4450 | 1.01 | 0.62 | 2.18 | 3.86 | 21.75 | 21.62 | 0.21 | 20.40 | 19.17 | 0.41 | -19.33 | -21.82 | 0.64 | 3.63 | 5.31
NGC4594 | 4.81 | 0.76 | 21.31 | 6.48 | 25.87 | 25.66 | 0.20 | 24.38 | 22.62 | 0.40 | -20.53 | -19.71 | 0.44 | 4.76 | 7.18
NGC4736 | 1.08 | 0.90 | 0.27 | 0.20 | 17.89 | 17.82 | 0.23 | 16.97 | 16.47 | 0.44 | -18.66 | -18.50 | 0.59 | 1.60 | 2.11
NGC4826 | 0.04 | 0.66 | 0.12 | 0.07 | 18.28 | 18.12 | 0.32 | 17.70 | 17.27 | 0.55 | -15.82 | -15.13 | 0.83 | 0.78 | 0.99
NGC5033 | 0.35 | 0.40 | 1.76 | 0.74 | 20.18 | 20.10 | 0.23 | 19.34 | 18.85 | 0.44 | -19.46 | -18.06 | 0.69 | 1.34 | 1.78
NGC5055 | 0.15 | 0.63 | 1.32 | 0.59 | 21.21 | 21.13 | 0.23 | 20.47 | 20.08 | 0.45 | -18.19 | -16.81 | 0.78 | 1.10 | 1.40
NGC5474 | 0.19 | 0.77 | 1.04 | 1.08 | 23.82 | 23.77 | 0.22 | 22.86 | 22.42 | 0.43 | -15.51 | -16.03 | 0.62 | 1.72 | 2.24
NGC7331 | 0.25 | 0.40 | 1.25 | 1.07 | 19.35 | 18.99 | 0.25 | 18.81 | 18.26 | 0.49 | -19.24 | -19.44 | 0.62 | 0.71 | 0.86
NGC7793 | 0.01 | 0.85 | 0.03 | 0.02 | 17.73 | 17.66 | 0.60 | 16.90 | 16.58 | 0.81 | -13.76 | -12.87 | 1.44 | 1.33 | 1.62
ETGs |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
NGC0584 | 1.86 | 0.66 | 1.40 | 1.15 | 19.37 | 19.19 | 0.22 | 18.23 | 18.04 | 0.42 | -20.63 | -20.38 | 0.62 | 2.50 | 2.49
NGC0855 | 0.38 | 0.51 | 0.36 | 0.30 | 21.11 | 20.84 | 0.25 | 20.30 | 19.92 | 0.47 | -15.31 | -15.31 | 0.72 | 1.27 | 1.41
NGC1404 | 5.21 | 0.75 | 4.87 | 3.99 | 21.39 | 21.31 | 0.20 | 20.00 | 19.94 | 0.40 | -21.68 | -21.32 | 0.54 | 3.92 | 3.87
NGC3265 | 0.73 | 0.68 | 0.83 | 0.70 | 20.61 | 20.49 | 0.38 | 19.81 | 19.57 | 0.60 | -17.94 | -17.80 | 0.86 | 1.26 | 1.41
NGC4125 | 3.20 | 0.50 | 4.74 | 3.90 | 21.22 | 21.11 | 0.21 | 19.98 | 19.88 | 0.41 | -21.22 | -20.90 | 0.68 | 2.95 | 2.93
NGC4552 | 0.69 | 0.90 | 1.79 | 1.47 | 20.87 | 20.70 | 0.21 | 19.59 | 19.42 | 0.41 | -20.13 | -19.87 | 0.59 | 3.25 | 3.21
NGC1377 | 1.62 | 0.49 | 1.95 | 1.62 | 22.22 | 22.08 | 0.23 | 21.35 | 21.13 | 0.44 | -17.90 | -17.71 | 0.94 | 1.44 | 1.55
NGC1482 | 1.08 | 0.55 | 3.67 | 3.06 | 21.88 | 21.71 | 0.21 | 20.92 | 20.69 | 0.42 | -19.82 | -19.65 | 1.25 | 1.74 | 1.83
NGC1705 | 1.29 | 0.85 | 0.12 | 0.10 | 18.60 | 18.56 | 0.27 | 17.28 | 17.26 | 0.48 | -16.53 | -16.04 | 0.98 | 3.46 | 3.37
NGC3773 | 0.62 | 0.57 | 0.22 | 0.19 | 17.97 | 17.84 | 0.41 | 17.47 | 17.02 | 0.65 | -17.18 | -17.28 | 1.78 | 0.66 | 0.88
NGC5866 | 0.46 | 0.67 | 8.44 | 7.55 | 24.13 | 24.05 | 0.26 | 24.06 | 23.32 | 0.53 | -18.71 | -19.21 | 0.68 | 0.27 | 0.57
Table 3:  Dust masses and dust opacities, derived using Eqs. (1) and (5) from Grootes et al. (2013), for the early-type galaxies in our sample. The different columns represent: (1) - galaxy name; (2) - B band face-on dust optical depth; (3) - stellar mass surface densities; (4) - corrected stellar mass surface densities; (5) - stellar masses taken from: $a$ \- Rémy-Ruyer et al. (2015), $b$ \- Skibba et al. (2011), $c$ \- Wilson et al. (2013), $d$ \- Zibetti & Groves (2011), $e$ \- Grossi et al. (2015); (6) - dust masses; (7) - corrected dust masses; (8)-(11) - standard deviation for $\tau_{B}^{f}$, $\mu_{*}$, $M_{*}$ and $M_{dust}$. The errors for the corrected quantities are the same and thus not given here. In square brackets we have the units in which these quantities are expressed. All quantities except dust optical depth are given in decimal logarithm unit scale. $Galaxy$ | $\tau_{B}^{f}$ | $log(\mu_{*})$ | $log(\mu_{*}^{i})$ | $log(M_{*})$ | $log(M_{dust})$ | $log(M_{dust}^{i})$ | $\sigma_{\tau_{B}^{f}}$ | $\sigma_{\mu_{*}}$ | $\sigma_{M_{*}}$ | $\sigma_{M_{dust}}$
---|---|---|---|---|---|---|---|---|---|---
|  | $[M_{\odot}/kpc^{2}]$ | $[M_{\odot}/kpc^{2}]$ | $[M_{\odot}]$ | $[M_{\odot}]$ | $[M_{\odot}]$ |  | $[M_{\odot}/kpc^{2}]$ | $[M_{\odot}]$ | $[M_{\odot}]$
NGC0584 | – | 8.50 | 9.02 | $10.87^{a}$ | – | – | – | 0.20 | 0.20 | –
NGC0855 | – | 7.75 | 8.19 | $9.20^{a}$ | – | – | – | 0.21 | 0.21 | –
NGC1404 | – | 10.21 | 10.80 | $10.85^{b}$ | – | – | – | 0.14 | 0.13 | –
NGC3265 | – | 6.36 | 6.80 | $9.55^{a}$ | – | – | – | 0.21 | 0.21 | –
NGC4125 | – | 8.73 | 9.27 | $11.38^{c}$ | – | – | – | 0.11 | 0.11 | –
NGC4552 | – | 7.54 | 8.10 | $10.97^{d}$ | – | – | – | 0.10 | 0.10 | –
NGC1377 | 0.01 | 5.98 | 6.43 | $9.47^{a}$ | 6.34 | 6.34 | 0.01 | 0.21 | 0.21 | 0.24
NGC1482 | 0.08 | 6.69 | 7.16 | $9.99^{b}$ | 6.94 | 6.92 | 0.02 | 0.11 | 0.11 | 0.12
NGC1705 | 0.90 | 7.64 | 8.22 | $8.19^{e}$ | 5.26 | 5.13 | 0.16 | 0.07 | 0.04 | 0.12
NGC3773 | 0.21 | 7.08 | 7.48 | $8.31^{b}$ | 5.31 | 5.36 | 0.09 | 0.16 | 0.16 | 0.19
NGC5866 | 5.07 | 8.31 | 8.70 | $10.02^{b}$ | 7.17 | 7.22 | 1.19 | 0.09 | 0.09 | 0.11

###  3.4 Estimation of errors

As previously mentioned in Pastrav (2020), the best-fit parameters given by
GALFIT suffer from an underestimation of uncertainties, which was shown in
Häussler et al. (2007). To estimate the systematic errors on the main
photometric parameters involved in this study, namely the ones that
characterise the bulges of spiral galaxies and the ETG galaxies, we ran a new
set of fits, for a few sampled galaxies (both late-type and early-type). In
this process, we fixed the sky value to the one found initially by GALFIT and
added $\pm 1\sigma$, or $\pm 3\sigma$ ($\sigma$ being the uncertainty in the
sky level), leaving free the parameter of interest (either bulge effective
radius, axis-ratio or Sérsic index) while all other parameters were also fixed
to the values found by GALFIT. This approach of error estimation of bulge
parameters was also used previously by Gao et al. (2020) and Gao et al.
(2019), following from Gao & Ho (2017) and Huang et al. (2013a). The
systematic errors in the bulge effective radii were within the range 1-10
pixels (1-3 arcsecs). For the bulge axis-ratios, the uncertainties were less
significant, of up to $0.01$. For the Sérsic index, the uncertainties observed
were more significant as this parameter is quite sensitive to the choice of
the sky or innacurare star masking, and in general is determined with less
precision compared with other photometric parameters. Thus, the uncertainties
were in the range $0.1-0.75$ which corresponds to 0.04-0.2 dex for
log$(n_{b})$. The error over $d_{gal}$ (measured distance to the galaxy) was
taken from NED. Having the flux uncertainties already shown in Table 1, we
performed propagation of errors in Eqs. 1-12 to calculate the standard
deviations ($\sigma$) for all the required parameters. The uncertainties are
shown in Table 2.

Figure 1:  Left panel The Kormendy relation for the ETGs and LTGs in our
sample. The observed average bulge surface brightness ($<\mu_{e,b}>$) values
for the bulges of spirals are represented with black diamonds, while the
corrected ones are plotted with red stars. The solid black and red lines are
the best-fit to the data points obtained through a linear regression.
Similarly, for the early-type galaxies, the blue diamonds show the observed
(measured) $<\mu_{e,b}>$ values, while green stars correspond to the corrected
(intrinsic) ones. Similar to the spirals, green and blue dashed lines are the
best-fit Kormendy relations for the ETG galaxies. Additionally, we plot with
yellow and cyan dash-dotted lines the best-fit to all the data points. The
error bars represent the standard deviations. The average uncertainty of
$R_{e,b}$ is shown in the lower left side. Right panel The same relation, but
with the sample of spiral galaxies now divided according with the bulge Sérsic
index - black diamonds and stars show the observed and intrinsic values for
spiral galaxies with $n_{b}<1.5$, while red diamonds and stars correspond to
the ones with $n_{b}\geq 1.5$. We overplot the best-fits relation for early-
type galaxies with blue and green solid lines (observed and intrinsic) and the
corresponding $\pm 3\sigma$ boundaries with dashed blue and green lines. For
comparing purposes, we additionally plot the data points for ETGs, just as in
the left panel (same symbols, not divided by $n_{b}$). The legends overplotted
in both panels illustrate the correspondence of each symbol and line to the
data.

##  4 Results

In this section, the main results of this study are presented, the scaling
relations obtained for the bulges of the spiral galaxies and the early type
galaxies in our sample.

###  4.1 The Kormendy relation

We present first, the relation between the mean bulge surface brightness
($<\mu_{e,b}>$) and bulge effective radius, the photometric projection of the
Fundamental Plane of elliptical galaxies (Djorkovsky & Davis 1987, Dressler et
al. 1987), also known as the Kormendy relation (Kormendy 1977), in the left-
hand panel of Fig. 1. We derive the general relation, for the whole sample of
late-type and early type galaxies (dash-dotted cyan and yellow lines for the
measured and corrected relation), but also separately for bulges of the spiral
galaxies and the early-type ones (ellipticals+lenticulars). We do this as a
means to verify if there are significant differences between the
characteristics of the relation for bulges of spirals and the ones
corresponding to early-type galaxies and if these two morphologically distinct
categories occupy a different locus on the $<\mu_{e,b}>-R_{e,b}$ relation.
Moreover, both the observed (black / blue diamonds for LTG/ETG) and intrinsic
(red / green stars for LTG/ETG) Kormendy relation are shown.
The best-fit for all cases is obtained through a linear regression procedure,
having the slope and the intercept as coefficients of the relation
$<\mu_{e,b}>=a+b\times\log(R_{e,b})$, and given in Table 4.
Looking at the observed best-fit Kormendy relation for bulges of spiral
galaxies compared with corresponding one for the early-type galaxies, one can
notice differences in the slope of the relation, with bulges residing on a
slightly steeper slope relation than ETGs. This could be explained in part by
the changes induced by dust and projection effects, which have a larger extent
for bulges in spirals. We remind the reader that dust effects were not
quantified for the ellipticals in our sample. This can also be checked if we
compare the pair of observed to intrinsic relations for bulges (solid black
and red lines) and early-type galaxies (dashed blue and green lines). Thus, we
notice a more significant change in the slope and intercept for the former
ones (see also Table 4). The steepest relation is the general one, for the
whole sample, with an observed slope of $2.94\pm 0.21$, consistent with other
recent studies (e.g. Sachdeva et al. 2020, Gao et al. 2020), while the
intrinsic relation suffering a more important decrease in slope due to the
aforementioned biases compared with the separated relations.
Gadotti (2009) and Gao et al. (2020) performed a multi-component structural
analysis on a large sample of elliptical and disc galaxies, using the Kormendy
relation to separate classical and pseudo-bulges. Thus, they considered the
best-fit Kormendy relation of ellipticals and classified pseudo-bulges as
those bulges that are situated below the $3\sigma$ boundary of the
aforementioned relation. They showed that this is more precise means for bulge
classification than Sérsic indices. Moreover, in Fig. 8 of Gadotti (2009) and
Figs. 5&6 of Gao et al. (2020), the authors showed that when the bulges are
divided on the basis of their Sérsic indices in the Kormendy relation, there
is a pronounced overlap of the distribution of bulges and pseudo-bulges. We
have not explicitely separated our sample into galaxies with classical or
pseudo-bulges. Nevertheless, we tested this result by dividing the spiral
galaxies sample based on our derived Sérsic indices, into galaxies with
$n_{b}<1.5$ bulges, and the rest with $n_{b}\geq 1.5$ (as we would try to
classify the sample into galaxies containing a classical or a pseudo-bulge),
to check how the bulges classified in this way would be situated with respect
to the Kormendy relation of the ETGs. We show this in the right panel of Fig.
1, where the black diamonds correspond to the observed values for spiral
galaxies with low Sérsic index $n_{b}>1.5$, while the red diamonds show the
other category of spiral galaxies (with $n_{b}>1.5$). In a similar way, we
show the intrinsic values, with black and red stars, respectively. We overplot
the best-fit observed and intrinsic Kormendy relation for ETGs from the left-
hand panel in solid blue and green lines, while the corresponding dashed lines
represent the $\pm 3\sigma$ boundaries over the zero-point (the slope
remaining fixed). Additionally, for comparing purposes, we overplot the data
points corresponding to the sample of early-type galaxies, just as in the left
panel - not divided by a $n_{b}$ value (same symbols and color scheme). One
can see that most of the high Sérsic index LTGs (with classical bulges) are
situated in the region where most of the early-type galaxies reside.
Nevertheless, comparing the two plots, using the available morphological
classification and taking into account that our sample is rather small (but
representative), we cannot clearly observe a similar separation as in Fig. 1
of Gao et al. (2020), where most of their classified pseudo-bulges lie beyond
the 3$\sigma$ boundary of their ellipticals Kormendy relation. However, we do
notice, as in Fig. 8 in Gadotti (2009) and Fig. 5 in Gao et al. (2020), that
high- and low-Sérsic index bulges significantly overlap in the
$<\mu_{e,b}>-R_{e,b}$ relation. This is also the case when the sample of LTGs
is divided using a $n_{b}=2.0$ threshold. This result is in contrast with that
found by Neumann et al. (2017), who analysed a small sample of unbarred disc
galaxies and found that using the Kormendy relation of ETGs and separating
their bulges using a $n_{b}=1.5$ threshold produces a clear separation of
ellipticals, low- and high-Sérsic index bulges, with almost no overlap. The
authors also found pseudo-bulges to be located below the 2$\sigma$ boundary of
the relation. In contrast with Gadotti (2009) and Gao et al. (2020), Costantin
et al. (2017) found the Kormendy relation not reliable for separating bulge
populations due to the large scatter and the overall dependence of the best-
fit coefficients of this relation on the absolute magnitude range considered.
Moreover, Fisher & Drory (2016) found that selecting bulges solely by their
position in the $<\mu_{e,b}>-R_{e,b}$ space does not clearly separate bulges,
while adding a $n_{b}=2.0$ selection threshold to the Kormendy relation
produces a cleaner separation of classical and pseudo-bulges.
Thus, according with our findings, using the Kormendy relation of early-type
galaxies in combination with the Sérsic index of bulges does not provide a
relevant and conclusive classification of bulge morphology. This happens
because the Sérsic index is not a good bulge morphology discriminator, and
using it in combination with an apparently more accurate criterium (the
Kormendy relation for ETGs) does not produce conclusive results. Thus, we want
to emphasize that a threshold in the bulge / galaxy Sérsic alone is not
sufficient for an accurate separation between classical and pseudo-bulges, as
it may lead to errors and misclassifications. Using $B/T$ additionally would
give us a more clear picture, as one can see that bulges with higher Sérsic
index are situated in general at higher effective radii and $<\mu_{e,b}>$ on
the Kormendy relation (right panel of Fig. 1).
In Table 4 we also show the derived scatter (rms in the $<\mu_{e,b}$
direction) for all the plotted relations and the corresponding correlation
coefficients. The scatter for the general relation and the one for the spiral
galaxies is comparable with small differences between the values for the
measured and intrinsic relations. In the case of early-type galaxies, the
scatter is larger, mainly due to the reduced number of galaxies analysed,
compared with the overall sample and the number of spirals. As expected for a
well-established relation, the Pearson coefficients are close to 1.0,
indicating a near perfect correlation. The coefficents are lower for the ETGs,
which may also be explained by the above argument for the rms.

Figure 2:  Upper panel Bulge absolute magnitudes as a function of bulge
central surface brightness, $\mu_{0,b}$. The average uncertainty of
$\mu_{0,b}$ is shown in the upper left corner. Lower panel Bulge absolute
magnitudes as a function of bulge effective radii. The average uncertainty of
$R_{e,b}$ is shown in the lower right corner. The symbols and color scheme
(common for both panels and illustrated in the legend overplotted in the
bottom panel) are as follows: black and blue diamonds represent the observed
values for bulges of spiral galaxies and the early-type ones; red and green
stars correspond to the intrinsic values. The error bars represent the
standard deviations.

###  4.2 Other bulge scaling relations

The bulge absolute magnitudes are plotted against the bulge central surface
brightness in the upper panel of Fig. 2. The linear continuous trend is
noticed for both the observed and corrected data points, without any real,
noticeable separation for spirals from early-type galaxies, as also shown in
Graham (2013) and Graham & Guzman (2003). This relation is in close connection
with another linear relation, namely the $M_{b}-n_{b}$ one, which will be
shown in the next section, dedicated to the black-hole scaling relations. The
apparent small gaps in the data at $\mu_{0,b}=16.0$ and $13.0$ are mostly due
to our sample size. In the lower panel of the same figure, the bulge absolute
magnitude is plotted as a function of the effective radius, with the symbols
and colors being the same as in the previous plot. Although we do not attempt
to fit the data, we can see that the relation $M_{b}-R_{e,b}$ is curved,
unlike the previous one, as are all relations which involve effective
quantities (e.g. $R_{e},\mu_{e}$ or $<\mu_{e}>$), a fact pointed out by Graham
(2013).

Figure 3:  The Sérsic index of the bulges of spirals and early-type galaxies vs. $B/T$ (bulge-to-total flux ratio). The average uncertainty of $n_{b}$ is shown in the upper left corner, while the average uncertainty of $B/T$ is $0.025$. The symbols and color scheme are the same as in Fig. 2. Table 4:  The linear regression best-fit parameters for the Kormendy relation, for the late type and early type galaxies shown in the left panel of Fig. 1,for both the measured (observed) and corrected (intrinsic) ones: $a$ \- the intercept, $b$ \- the slope. We also present here in the last two columns, the scatter of the relations, $\sigma$, and Pearson correlation coefficient, $r^{corr}$.  Kormendy relation | $a$ | $b$ |  $\sigma$ (rms) | $r^{corr}$
---|---|---|---|---
obs., LTG+ETG | $19.70\pm 0.12$ | $2.94\pm 0.21$ | 0.59 | 0.90
intrin., LTG+ETG | $19.57\pm 0.13$ | $2.69\pm 0.21$ | 0.63 | 0.87
obs., LTG | $19.59\pm 0.18$ | $2.45\pm 0.33$ | 0.58 | 0.98
intrin., LTG | $19.39\pm 0.20$ | $2.29\pm 0.32$ | 0.56 | 0.91
obs., ETG | $19.19\pm 0.31$ | $2.03\pm 0.56$ | 0.80 | 0.83
intrin., ETG | $19.16\pm 0.30$ | $2.11\pm 0.56$ | 0.79 | 0.84

In figure 3, the bulge Sérsic index is plotted against the bulge-to-total flux
ratio ($B/T$). The black and blue diamomds represent the observed values of
the spiral bulges and those of the early type galaxies, while red and green
stars show the corresponding intrinsic values. One can see a quite clear
separation in the $n_{b}-B/T$ plane for bulges of spiral galaxies and those of
ETG, irrespective of a few outliers. Thus, the early-type ones occupy the area
of high $B/T$ and moderate to high values of the Sérsic index, as expected.
This holds true for both the measured and intrinisc parameters. A similar
result was observed by Gadotti (2009) by doing a structural analysis of a much
larger sample of galaxies from the SDSS survey. Both Gadotti (2009) and Fisher
& Drory (2008) have also shown that a correlation exists between $n_{b}$ and
$B/T$ for classical bulges, while this is not present in the case of pseudo-
bulges. Although our sample is smaller, we can confirm this correlation, even
though we did not divide our sample into pseudo- and classical bulges.
Laurikainen et al. (2010) observed a similar separation in the $n_{b}-B/T$
plot between the regions ocupied by lenticulars and Sa-Sc galaxies, and
determined that a correlation exists between these two measured quantities.
The fact that late-type and early-type galaxies reside in distinct areas in
the $n_{b}-B/T$ plane confirms this relation as a useful and accurate
criterion to morphologically classify galaxies, as long as $n_{b}$ and $B/T$
are used in combination, and with the inherent biases taken out from
measurements. On a different note, Costantin et al. (2017) and Costantin et
al. (2018) found that bulges with low Sérsic index found in galaxies with low
$B/T$ are not naturally disc-like, rotationally supported components (as
pseudo-bulges), and cand have a variety of intrinsic shapes, unlike classical,
spheroid-like bulges.
Here we also note the average bulge Sérsic index (measured and intrinsic) for
the spirals - $<n_{b}>=1.47$ & $<n_{b}^{i}>=2.01$, while for ETGs the
respective values are $<n_{b}>=2.06$ & $<n_{b}^{i}>=2.14$. These values are
consistent with those found in other studies such as those of Laurikainen et
al. (2007) or Graham (2001).

Figure 4:  Upper panel Bulge-to-total ($B/T$) ratio of the sample, as a
function of stellar mass. The average uncertainty of $M_{\star}$ is shown in
the lower right corner. Lower panel Bulge-to-disc ratio versus morphological
type, $T$. The symbols and color legend are the same as in Fig. 2. The error
bars represent the standard deviations.

Another relation that can be used when trying to separate galaxies with
different morphologies is the one displayed in the upper panel of Fig. 4,
namely the bulge-to-total flux ratio versus the galaxy stellar mass,
$M_{\star}$. The $B/T$ values have been corrected for inclination and dust
effects for the late-type and lenticulars, while for those corresponding to
ETGs, only the inclination (projection) corrections were applied. The black
and blue triangles are the observed values for the late-type and early type
galaxies, while the red and green stars represent the corrected values. It can
be seen the clear difference in the trends followed by the two classes of
galaxies. The late type bulge-to-total ratios show an increase with stellar
mass from very low, almost bulgeless galaxies, to values similar to early-type
galaxies. In contrast, the early-type (ellipticals and lenticulars) show less
of a trend, being almost constant with stellar mass, with values fluctuating
around $B/T=0.5$. The corrections introduced by the aforementioned effects do
not produce important changes in the trend seen for the observed versus the
one corresponding to the intrinsic values. Overall, we consider
$B/T-M_{\star}$ relevant when trying to divide galaxies into late-type and
early-type galaxies.
In the lower panel of Fig. 4 we show $B/D$ as a function of the morphological
type ($T$) for the spiral galaxies. This relation, sometimes in the form
$B/T-T$, has been shown in similar studies by Dong & De Robertis (2006),
Laurikainen et al. (2007), Graham & Worley (2008), Weinzirl et al. (2009),
Laurikainen et al. (2010), Méndez-Abreu et al. (2017) and Gao et al. (2019).
As expected, one can notice an important decrease in the bulge-to-disc ratio
when going from early to later morphologies, as the bulge influence on the
surface brightness profile of the galaxy starts to gradually decrease. The
trend seen here is similar with that shown in the previous works mentioned
above. One can also notice a certain degree of scatter in the values of $B/T$.
A large scatter has been reported before in studies on larger samples than
ours, with better statistics, such as the ones of Weinzirl et al. (2009),
Laurikainen et al. (2010) and Gao et al. (2019). This is the reason other
authors concluded that due to this systematic scatter in $B/T$ at any Hubble
type, one should not use Hubble type to quantitavely infer $B/D$ values and/or
bulge proeminence. Due to the low statistics for each morphological type in
our sample, we cannot examine this issue in more detail or make a fair
comparison of the average value found for bulge-to-total flux ratios or
$n_{b}$, for different morpholgies, like S0 or ellipticals. We just note there
the respective values for the average intrinsic $B/T=0.48$ for lenticular
galaxies and 0.56 for the ellipticals.
As in the upper plot, we can see that dust and inclination effects do not
produce great changes in the $B/D$ values for the spiral galaxies, unlike for
the rest of the disc and bulge structural and photmetric parameters.

###  4.3 Black-hole scaling relations

In this section, we present an important set of relations, the black-hole
scaling relations, essential for understanding the co-evolution of galaxies
(more specific their bulge component) and their central black holes, which
reside at their centre. These relations are likewise important as they can
infer predictions on the black hole masses of galaxies using their bulge
structural or photometric parameters. Since dust and inclination can bias some
of the parameters involved in these relations, the bulge photometric and
structural parameters, it’s important to see if there are any important
changes introduced by these effects in the main characteristics of these
relations (e.g. the slope, the scatter). We have not derived here the black-
hole masses. These were compiled from the relevant papers in the literature,
and are given in Table 5.

The relation between the mass of the central black hole ($M_{BH}$) and the
stellar light concentration of the galaxy’s bulge/spheroidal component, given
by the Sérsic index, was first observed by Graham (2001). This was thought to
exist as there were already proven correlations which involved the stellar
luminosity of the bulge / spheroid $L_{b}$ (or alternatively, the bulge
absolute magnitude, $M_{b}$), such as the $M_{BH}-L_{b}$ correlation (e.g.
Kormendy & Richstone 1995, Magorian et al. 1998, Marconi & Hunt 2003), or the
$L_{b}-n_{b}$ one (e.g. Young & Currie 1994, Jerjen et al. 2000, Graham &
Guzmán 2003). Thus, Graham & Driver (2007) showed for the first time the
$M_{BH}-n_{b}$ correlation by doing a structural analysis of a small sample of
spiral and elliptical galaxies.
We present the $M_{b}-\log(n_{b})$ relation in the left-hand panel of Fig. 5,
with the symbols and the colors the same as in Fig. 1. This is a linear
relation, as is the $M_{b}-\mu_{0,b}$ one, presented in the previous section,
irrespective of a few outliers. The trend of the bulge absolute magnitude (or
alternatively bulge luminosity, $L_{b}$) is to linearly increase with the
concentration of the bulge / spheroid stellar light, for both late-type galaxy
bulges and early type ones. We do a linear regression procedure to derive the
specific parameters of the relation (slope and zero-point). The best-fit
measured and corrected relations are plotted with black and red solid lines
for the spiral galaxies, while the corresponding ones for the ETGs are plotted
with blue and green lines. Comparing the observed or the corrected relations
between bulges of spirals and early-type galaxies, one immediately notices a
difference in both the slope and the zero-point of the relations, with the
ETGs showing a steeper slope relation: $2.44\pm 0.39$ vs. $1.97\pm 0.16$ for
the observed relation, and $2.68\pm 0.45$ vs. $2.36\pm 0.18$ for the correted
one (see Table 6). These values were obtained from the initial best-fit
parameters of the $M_{b}-n_{b}$ relation, knowing that
$M_{b}=-2.5\times\log(L_{b})$, and are thus the best-fit parameters for the
$L_{b}-n_{b}$ relation.
Different slopes for the ETGs and LTGs were also reported previously in Graham
(2001), Savorgnan et al. (2013) and Savorgnan (2016). Our results for the
slope of the corrected relation show a shallower slope than the one found by
Savorgnan (2016) \- $3.60\pm 0.19$, for early-type galaxies
(ellipticals+lenticulars), while for the late-types it is steeper than the
$1.44\pm 0.52$ value derived by the same authors. One can see that dust and
inclination effects produce a significant change in the slope and intercept of
this relation, especially for the bulges of late-type galaxies. However, as
Savorgnan (2016) performed their multicomponent structural analysis on
$3.6\mu$m band images, which are not much affected by dust extinction, dust
and inclination effects alone cannot account for this difference when compared
with our intrinsic values. In Section 5, we are coming back to this issue.
In the last two columns of Table 6 we also show the scatter of this relation
($\sigma$) in the $M_{b}$ direction, and the Pearson correlation coefficients.
The minimum number of outliers, situated far-off from the average value, were
given minimum weight in the linear regression and thus were practically
excluded from these calculations. However, all the data points were considered
in the Pearson correlation coefficients calculation, even though it is known
these coefficents are quite sensitive to far-off values which are prone to
high uncertainties, and including them would lead to misinterpretations of the
overall trend observed and of the results obtained. Comparing the respective
values, there are small differences in the scatter, with the intrinsic
relation for the early-type galaxies having a slightly less scatter than the
bulges of spiral galaxies. The values are comparable, though smaller, with
those found by Savorgnan (2016). Similar behaviour can be observed for the
correlation coefficients, with very small differences between the coefficients
for the observed vs. intrinsic relation. It is important to note however the
high values, close to $1.0$ of these coefficients for the LTGs, which indicate
the already established, strong correlation between the luminosity of the
bulge / spheroidal component of a galaxy and $n_{b}$. The coefficents for ETGs
are smaller but still indicate a strong correlation. Excluding the outliers
has the result of increasing these coefficents to values similar to the ones
corresponding to LTGs - $0.88$ and $0.89$ respectively.

Table 5:  Black-hole masses for galaxies in our sample, as compiled from the literature: $a$ \- Savorgnan & Graham (2015), $b$ \- Beifiori et al. (2012), $c$ \- Graham & Scott (2013) (predicted), $d$ \- Do et al. (2014) (estimated), $e$ \- Caramete & Biermann (2010) catalog (estimated); $f$ \- van den Bosch (2016); $g$ \- Dullo et al. (2020) (predicted). In Beifiori et al. (2012) and Do et al. (2014) there were no estimated/derived uncertainties for the black hole masses, while in Dullo et al. (2020), an error of 0.85 dex on $\log(M_{BH})$ was considered for all their estimated masses. Caramete & Biermann (2010) have a rough uncertainty of 50% for all their derived masses. $Galaxy$ | $M_{BH}(x10^{8}M_{\odot})$ | $\sigma_{M_{BH}}(x10^{8}M_{\odot})$
---|---|---
| $[M_{\odot}]$ | $[M_{\odot}]$
NGC3031 | $0.74^{a}$ | 0.21
NGC4594 | $6.40^{a}$ | 0.40
NGC4736 | $0.06^{b}$ | 0.014
NGC4826 | $0.016^{a}$ | 0.004
NGC0628 | $4.90^{c}$ | 1.00
NGC3938 | $5.00^{c}$ | 1.00
NGC2841 | $1.10^{d}$ | 0.00
NGC3190 | $2.14^{e}$ | 1.07
NGC4450 | $2.20^{b}$ | 0.28
NGC7331 | $1.04^{f}$ | 0.43
NGC3621 | $0.01^{f}$ | 0.008
NGC5055 | $8.32^{f}$ | 1.91
NGC4254 | $0.097^{g}$ | 0.85
NGC5033 | $1.047^{g}$ | 0.85
NGC5474 | $0.0022^{g}$ | 0.85
NGC7793 | $0.0046^{g}$ | 0.85
NGC0024 | $0.026^{g}$ | 0.85
NGC2976 | $0.126^{g}$ | 0.85
NGC0584 | $2.60^{e}$ | 0.00
NGC0855 | $0.042^{e}$ | 0.021
NGC1404 | $1.97^{d}$ | 0.98
NGC3265 | $0.083^{e}$ | 0.041
NGC4125 | $3.95^{e}$ | 1.97
NGC4552 | $4.70^{a}$ | 0.50
NGC1377 | $0.29^{e}$ | 0.14
NGC1482 | $1.06^{e}$ | 0.53
NGC1705 | $0.006^{e}$ | 0.003
NGC3773 | $0.036^{e}$ | 0.018
NGC5866 | $1.33^{e}$ | 0.66
Figure 5:  Black-hole scaling relations for the ETGs and LTGs in our sample.
The panels are as follows: Left Bulge absolute magnitude (luminosity) vs.
bulge Sérsic index; Middle Black-hole mass as a function of bulge absolute
magnitude; Right Black-hole mass plotted against the bulge Sérsic index. The
solid lines represent the best-fit relation to the data points from a linear
regression procedure. The symbols and color legend, common for all panels, is
shown on the left hand panel. Additionally, in the right-hand panel, the $\pm
1\sigma$ limits just for the corrected $M_{BH}-n_{b}$ relations of the LTG and
ETG galaxies are overplotted in red and green dashed lines, with the
corresponding color and line legend displayed. The error bars represent the
standard deviations. The average uncertainties of $log(n_{b})$ and $M_{b}$ are
overplotted in the upper left and lower right corners.

To check if this correlation is significant and if the two different slope
relations found for the two samples of LTG and ETG are in fact statistically
different, we perform two statistical tests. First, we calculate a t-value
based on the correlation coefficients previously derived for this relation,
using the formula:

$\displaystyle t(r^{corr})=r^{corr}\sqrt{n-2}/\sqrt{1-(r^{corr})^{2}}$ (13)

where $n$ is the number of galaxies (early-type or late-type) and $r^{corr}$
is the Pearson correlation coefficient. We do this for both observed and
intrinsic relations. The values are displayed in Table 7, together with their
associated $p$-values. As one can see, in all cases, the $p$-values for this
relation are lower or much lower than the standard $p=0.05$ value
(alternatively $95\%$ confidence level), which is generally accepted as being
a reliable indicator of the statistic significance of a result - in this case,
of the correlation. Then, working in the null hypothesis that the two samples
of galaxies have the same mean and should be described by one single
$M_{b}-n_{b}$ correlation, we perform an independent two sample t-test to
calculate the t-statistics (or t score), for the pair of observed (for LTG and
ETG) relations using all the data in the samples (including outliers) and then
for the other pair, corresponding to the intrinsic relations. The t-values are
also shown in Table 7. Based on this, and taking into account the number of
degrees of freedom for our sample, we then derive the associated $p$-values
for both situations. The corresponding $p$-values (one-tail probability) are
0.085 for the pair of observed correlations and 0.064 for the intrinsic pair.
A confidence level of at least $95\%$ (and thus p<0.05) would correspond to a
t-statistics higher than 1.706. We can thus say that the two relations for ETG
and LTG are statistically different only at a $p=0.1$ significance level. If
one repeats the test and excludes the extreme outliers in the calculation of
t-score, this results in $p$-values of 0.025 and 0.033, which means the two
relations are statistically different at $95\%$ confidence level. Better
statistics (a larger sample) would most likely produce an improvement in the
statistical significance of this conclusion, that there are two different
$M_{b}-n_{b}$ relations for LTG and ETGs.

The second relation that we present and analyse in this section is the
$M_{BH}-M_{b}$ relation (or alternatively $M_{BH}-L_{b}$). This is displayed
in the central panel of Fig. 5, the symbols and color scheme being the same as
for the previous relation. One can notice from this plot that galaxies with
more luminous bulges host more massive black-holes, as expected, with the
increasing trend being linear. This is valid for all the galaxies in our
sample, both late and early-type ones. We derive the best-fit parameters of
this relation by linear regression fitting. Here, the three spiral galaxies in
the upper middle part of the plot, way far-off from the general trend and with
their black-hole mass estimated, together with the early-type galaxy with the
lowest black-hole mass were excluded from the fit and from the calculation of
the correlation coefficients as the values are deemed highly uncertain. While
we plot $M_{BH}$ vs $M_{b}$, taking into account that
$M_{b}=-2.5\times\log(L_{b})$, the best-fit parameters in Table 6 correspond
to the $M_{BH}-\log(L_{b})$ correlation.
It can be seen that the bulges of spiral galaxies reside on a slightly steeper
relation than the ones of early-type galaxies. This result was also found by
Savorgnan et al. (2016) and Graham & Scott (2013). In contrast, Beifiori et
al. (2012) found that there is no significant variation of this relation for
different morphological types. Comparing the slope of our best-fit relation
for the ETGs, $0.91\pm 0.16$ (observed) and $0.87\pm 0.17$ (corrected), with
other studies in the literature, we see that these are consistent within
errors with those in Savorgnan et al. (2016), who found $\alpha=1.03\pm 0.11$
on average, from their multiple regression methods. However, their relation
for bulges of late-type galaxies, with $\alpha=2.88\pm 0.68$, is considerably
steeper than either our observed or intrinsic relation - $1.32\pm 0.15$ &
$1.00\pm 0.23$. Considering that their analysed images are at $3.6\mu$m, where
dust influence is not significant, the diffence in the slopes cannot be
attributed solely to dust and inclination effects. Possible explanations are
discussed in the next section. Dullo et al. (2020) fitted a single relation
for their whole sample of ETGs and LTGs, considering as well the bulge
luminosity at $3.6\mu$m, and found a slope of $1.23\pm 0.15$, consistent
within errors with our best-fit parameters. In another study, Beifiori et al.
(2012) found considerably shallower slopes for their samples of galaxies
(early and late types) than our results.
Zhu et al. (2021) determined a similar slope value for the relation between
$M_{BH}$ and the stellar mass of classical bulges and of the elliptical
galaxies in their entirety (they consider ellipticals as multi-component
galaxies, with a core and two extra components - extended envelopes). Taking
into account only the core of the ellipticals, which the authors conclude that
should be considered in the coevolution of black-holes and their elliptical
hosts, they find a $M_{BH}-M_{core}$ relation with a slope of $1.22\pm 0.11$
and a scatter of 0.41 dex. With a proper transformation of our $L_{b}$ into
the stellar bulge masses, considering the mass-to-light ratio in B band, the
result of Zhu et al. (2021) could be comparable and consistent within errors
with our values for the slope and scatter for this relation. Kormendy & Ho
(2013) excluded pseudo-bulges from the the $M_{BH}-M_{b}$ relation because of
their different formation scenario, and because the black hole mass correlates
more tightly with the mass of classical bulges and elliptical galaxies, and
not with the ones of pseudo-bulges and discs.
The changes in the zero-point and slope when correcting for all the
aforementioned effects are much more significant for the late-type bulges than
for the early-types, as expected, with the corrected magnitudes shifting to
higher values, which decrease the slope of the best-fit relation. The change
is much less significant in the case of early-type galaxies.
As in the case of the first relation, since the slopes for the two samples of
late-type galaxies have close enough values, we check again the statistical
significance of the $M_{BH}-L_{b}$ relations derived, and if the two relations
are statistically different from each other (or in other words, if the means
of the two samples are different). We use the same formula from Eq. 13 and the
t-score to investigate this. The respective values are presented in the same
table. One can see from the first two columns that this relation is also
statistically significant, having the $p$-value associated with the
$t(r^{corr})$ value much lower than the 0.05 standard. Testing the second
issue, we see that the pair of observed relations for late-type and early-type
galaxies are in this case significantly different only at $p<0.1$ level
(p=0.086), while for the intrinsic pair, this is not the case, with the
associated p-value being larger than 0.10. We can thus conclude that the two
relations are not statistically different at p<0.05 level.

Table 6:  The linear regression best-fit parameters for the black-hole scaling relations, for the late type and early type galaxies shown in Fig. 5, both the measured (observed) and corrected (intrinsic) ones: $\alpha$ \- the intercept, $\beta$ \- the slope. We also present here in the last two columns, the scatter of the relations, $\sigma$, and Pearson correlation coefficient, $r^{corr}$.  BH relation | $\alpha$ | $\beta$ |  $\sigma$ (rms) | $r^{corr}$
---|---|---|---|---
$L_{b}-n_{b}$ (obs., LTG) | $7.02\pm 0.06$ | $1.97\pm 0.16$ | 0.52 | 0.90
$L_{b}-n_{b}$ (intrin., LTG) | $6.62\pm 0.08$ | $2.36\pm 0.18$ | 0.61 | 0.92
$L_{b}-n_{b}$ (obs., ETG) | $7.12\pm 0.14$ | $2.44\pm 0.39$ | 0.58 | 0.67
$L_{b}-n_{b}$ (intrin., ETG) | $6.91\pm 0.17$ | $2.68\pm 0.45$ | 0.56 | 0.66
$M_{BH}-L_{b}$ (obs., LTG) | $6.33\pm 0.26$ | $1.32\pm 0.15$ | 0.39 | 0.84
$M_{BH}-L_{b}$ (intrin., LTG) | $7.01\pm 0.33$ | $1.00\pm 0.17$ | 0.53 | 0.80
$M_{BH}-L_{b}$ (obs., ETG) | $7.36\pm 0.27$ | $0.91\pm 0.16$ | 0.35 | 0.91
$M_{BH}-L_{b}$ (intrin., ETG) | $7.33\pm 0.25$ | $0.87\pm 0.17$ | 0.38 | 0.90
$M_{BH}-n_{b}$ (obs., LTG) | $7.32\pm 0.09$ | $3.42\pm 0.37$ | 0.49 | 0.80
$M_{BH}-n_{b}$ (intrin., LTG) | $7.02\pm 0.12$ | $3.08\pm 0.35$ | 0.43 | 0.83
$M_{BH}-n_{b}$ (obs., ETG) | $6.89\pm 0.19$ | $3.14\pm 0.54$ | 0.35 | 0.81
$M_{BH}-n_{b}$ (intrin., ETG) | $6.57\pm 0.22$ | $3.79\pm 0.59$ | 0.33 | 0.76
Table 7:  The statistical tests parameteres for the black-hole scaling relations, for the late type and early type galaxies, corresponding to Table 6, for both the measured (observed) and corrected (intrinsic) ones: columns 1-2 - $t(r^{corr})$-test value, derived from the formula presented in Eq. 13 and the corresponding p-value; columns 3-4 - the t-test score (independent two-sample t-test) and its corresponding p-value. BH relation | $t(r^{corr})$ |  $p$-value |  $t$-test |  $p$-value
---|---|---|---|---
$L_{b}-n_{b}$ (obs., LTG) | 7.84 | <<0.05 | 1.41 | 0.085
$L_{b}-n_{b}$ (intrin., LTG) | 8.83 | <<0.05 | 1.57 | 0.064
$L_{b}-n_{b}$ (obs., ETG) | 2.73 | 0.011 | 1.41 | 0.085
$L_{b}-n_{b}$ (intrin., ETG) | 2.63 | 0.013 | 1.57 | 0.064
$M_{BH}-L_{b}$ (obs., LTG) | 5.75 | <<0.05 | 1.40 | 0.086
$M_{BH}-L_{b}$ (intrin., LTG) | 1.91 | <<0.05 | 1.09 | 0.143
$M_{BH}-L_{b}$ (obs., ETG) | 2.44 | <<0.05 | 1.40 | 0.086
$M_{BH}-L_{b}$ (intrin., ETG) | 2.61 | <<0.05 | 1.09 | 0.143
$M_{BH}-n_{b}$ (obs., LTG) | 5.08 | <<0.05 | 1.25 | 0.111
$M_{BH}-n_{b}$ (intrin., LTG) | 5.69 | <<0.05 | 1.27 | 0.107
$M_{BH}-n_{b}$ (obs., ETG) | 4.09 | 0.004 | 1.25 | 0.111
$M_{BH}-n_{b}$ (intrin., ETG) | 3.48 | 0.009 | 1.27 | 0.107

Coming back to the first relation that we mentioned in this section, we show
it in the right-hand panel of Fig. 5, plotted as $M_{BH}$ vs $\log(n_{b})$. We
have seen previously that early-type and late-type galaxies present two
different slope relations in the $M_{b}-n_{b}$ relation and similar ones in
the $M_{BH}-M_{b}$ one. Here we check what is the case for this last relation.
Previous recent work by Savorgnan (2016) showed that the values for the best-
fit linear regression parameters for the two morphological classes of galaxies
are consistent with each other, within errors, meaning that only one relation
exists for both LTGs and ETGs.
We overplot the best-fit relation with solid lines, in similar colors as in
the other two panels, for both spirals and early-type galaxies. The best-fit
linear regression parameters are presented in Table 6, as for the previous two
relations. Here, a few points were excluded from the fit (namely the three
LTGs in the upper middle part of the plot and the two ETG in the opposite
corners, far-off from the average trend), as the measurements of the black
hole mass or Sérsic index were unreliable and highly uncertain. This was done
as before, to mitigate the effects of these outliers on the correlation
coefficients or the best-fit parameters. At a first look it is apparent that
there are two similar, but slightly different trends between the black-hole
mass and the Sérsic index of the bulge, for both the observed and corrected
case - a linear increase. To investigate this, we overplot the $\pm 1\sigma$
limits just for the corrected relations (to avoid an unnecessary clutter of
the plot) in dashed lines. One can see a partial overlap of the two regions
delimited by the standard deviations, especially at the higher-end of the
relations. Keeping in mind the uncertainties of $\log(n_{b})$ from Sect. 3.4,
which are $\pm 0.2$ dex at most (with the average being shown on the plot),
and the ones associated with $M_{BH}$, but also the slopes of the two
relations, we could draw the same conclusion as Savorgnan (2016). However,
considering the large scatter in the data for late-type galaxies and the
number of ETGs analysed, it is certain that more data would be beneficial for
a more definitive conclusion.
Still, to investigate further this issue, and the statistical significance of
this correlation, we perform similar statistical tests as in the case of the
first two black hole relations. The respective parameters can be found in the
same Table 7. One can see from the $t,p$ values that the relations are
statistically significant. Then, working as before in the null hypothesis that
the two relations are not different (and have the same mean), the low t-score
and high associated $p$-value leads us to the conclusion that at least at
p<0.1 ($<90\%$ CL) level, the two relations are not different.
Here we would like to note the consistency (within errors) of our best-fit
parameters for the slope and intercept of the relations with those derived by
Savorgnan (2016) and Sahu et al. (2020). Moreover, the scatter in the $M_{BH}$
direction for the last two relations is also comparable, though in some cases
slightly larger, than the values found by Savorgnan (2016) or Sahu et al.
(2020) for their larger sample of early and late-type galaxies. The Pearson
coefficents indicate a lower degree of correlation, not as strong as for the
$M_{b}-n_{b}$ relation. The scatter is also quite large here, but also in the
previous relation. The main reason for this is that the black hole masses in
Table 5 are measured with a variety of techniques and instrumentation, some
are predicted while others estimated, and thus the measurements are not
heterogeneous and uniform in quality. The large scatter in this relation was
also observed by Beifiori et al. (2012), who also note that this correlation
is poor, with a low level of significance and a poor predictor for $M_{BH}$.
Still, our findings suggests that this correlation exists despite the large
scatter, attributed mainly to the uncertainties associated with the black-hole
masses and in some cases, the bulge Sérsic indices.
Overall, the last two relations are important in predicting black-hole masses
in large sample of galaxies, then deriving the black hole / supermassive black
hole mass functions, and further, the black hole mass density for early-type
and late-type galaxies. Therefore, to clarrify the issue of having two or one
single $M_{BH}-n_{b}$ / $M_{BH}-L_{b}$ relations is essential. However, all
existent studies rely on small catalogues of black-hole masses, determined, as
in the case of our sample, with different methods, and at various wavelengths,
affected by the effects of dust attenuation. As shown here, accurate bulge
parameters and correcting the relations for these effects is important, and
can add information to the issue of having two separate relations or a single
one for all morphologies. However, better statistics and more robust
measurements are needed for a definitive answer.

Figure 6:  Upper pannel The bulge-to-disc ratios of spirals and early-type
galaxies vs. observed ellipticity of the bulges, $\epsilon$. Black diamonds
and stars represent the observed and intrinsic values for galaxies with
$n_{b}<2.0$, while red diamonds and stars are for galaxies with $n_{b}\geq
2.0$ (see the overplotted legend). The average uncertainty of $n_{b}$ is
overplotted in the upper left corner, while the ones for ellipticity are not
significant enough to be distinguished. The average uncertainty for $B/D$ is
$0.025$. The error bars show the standard deviations, which are mostly of the
size of the data points. Lower pannel Bulge Sérsic indices vs. observed
ellipticities of bulges. The symbols and color legend are similar as for the
upper pannel plot.

##  5 Discussion

We are coming back in this section to a few issues observed when analysing the
scaling relations presented in the previous section.

###  5.1 Testing B/D vs ellipticity and $n_{b}$ vs ellipticity as viable bulge
classification criteria

We saw previously in the right-hand panel of Fig. 1 that using the Kormendy
relation for the early-type galaxies in combination with a division of our
sample using a Sérsic index threshold did not produce a clear separation
between spiral galaxies with classical and pseudo-bulges in the
$<\mu_{e,b}>-R_{e,b}$ plane. The main reason is that Sérsic index is not
sufficient for this separation, apparently even when it is corrected for dust
and inclination effects. We test here a different approach to check if we can
distinguish between classical and pseudo-bulges considering their different
geometry (e.g. the intrinsic shape) as classical bulges are rounder,
spheroidal shape, while pseudo-bulges are flatter, disc-like structures. Thus,
we plot in upper pannel of Fig. 6 the $B/D$ ratios as a function of the
observed (measured) ellipticity of the bulges, $\epsilon=1-Q_{b}$, while in
the lower pannel, we plot $n_{b}$ vs. $\epsilon$. The bulge axis-ratio $Q_{b}$
is affected by projection effects, which are larger for bulges than for discs,
as shown in Pastrav et al. (2013a). Still, we only plot the observed values as
we do not have neccesary corrections for this parameter. We note that if
available, applying these corrections could produce significant changes,
comparable with those induced by projection effects on the bulge effective
radius or Sérsic index. Our sample is again divided (in both plots) using the
$n_{b}=2.0$ threshold in bulge Sérsic index. One can see in the upper plot
that low $n_{b}$ galaxies show a large spread in ellipticity, from those
characterstic to discs / pseudo-bulges to lower ones, overlapping with the
values characteristic for elliptical galaxies. Galaxies with high $n_{b}$ have
ellipticity values similar with elliptical galaxies, with a smaller dispersion
in values, occupying a separate locus in the $B/D-\epsilon$ plot. By analysing
the intrinsic shape of bulges in a sample of nearby galaxies, Costantin et al.
(2018) found a similar result. Since low Sérsic index galaxies show a
diversity in ellipticity values, and because of the relative overlap with the
high $n_{b}$ bulges, we caution that using $B/D$ and $\epsilon$, with a
threshold in $n_{b}$ to classify bulges will not produce an accurate, real
division, as among the $n_{b}<2.0$ bulges can hide galaxies with classical
bulges. Looking at the lower plot in Fig. 6, one can see that low Sérsic index
bulges show the whole range of ellipticities, as disc-like feature would
present, while high $n_{b}$ bulges have lower ellipticities as classical
bulges and elliptical bulges. This is also in agreement with the studies of
Costantin et al. (2017), Costantin et al. (2018) and Gao et al. (2020). There
is also a noticeable separation between the two populations of bulges in the
$n_{b}-\epsilon$ plot so one could conclude that this combination of
parameters would be suitable to classify bulges. However, as mentioned
earlier, we need to keep in mind the projection effects on $Q_{b}$, which
could bias this result. This is the reason why Gao et al. (2020) cautioned
that measured ellipticities of bulges should not be used when trying to
separate classical and pseudo-bulges.

###  5.2 $n_{b}$-$B/T$ and $B/T$-$M_{\star}$ as proxies for bulge
classification?

We saw earlier in Fig. 3 and upper panel of Fig. 4 from section 4 how the
combinations $n_{b}$-$B/T$, and $B/T$-$M_{\star}$ can be used to
morphologically separate galaxies into early- and late-type ones. One may ask
if these criteria can be extended further, to be used to classify bulges,
namely into classical and pseudo-bulges (disc-like).
It has been reported in the literature by Kormendy (1993), Carollo et al.
(1997) and Carollo et al. (1998) that most early-type galaxies, from S0 to Sab
contain classical bulges, while pseudo- bulges are very common starting with
type Sb galaxies. Using the results for galaxy central components
classification in HST (Hubble Space Telescope) V and H bands from Carollo et
al. (2002) (which at that time had the best statistics), complemented with
their classifications for the galaxies not fitted in Carollo et al. (2002)
sample, Kormendy & Kennicutt (2004) find the proportion of classical and
pseudo-bulges to be as high as $50\%,60\%$, and $44\%$ in S0-Sa, Sab and Sb
galaxies. Starting with Hubble type Sb on, the overwhelming majority of late-
type galaxies have a pseudo-bulge. Kormendy & Kennicutt (2004) also state in
their criteria for recognising pseudo-bulges that bulges with small $B/T$ are
not guaranteed to be disc-like bulges, but if $B/T$ is larger than $0.33\div
0.5$ it can be concluded that the respective galaxy has a classical bulge,
while a bulge with $B/T<1/3$ is likely to be a disc-like bulge. Gao et al.
(2020) find in their study that pseudo-bulges have in general a small bulge-
to-total ratio (smaller than 0.1), while classical bulges show a large spread
in $B/T$, 26% of their sample having also $B/T<0.1$. Because of the strong
overlap at low $B/T$, the authors stress that bulge-to-total flux ratio alone
is not a reliable parameter for separating bulges. A significant overlap in
the distribution of bulge Sérsic indices for classical and pseudo-bulges is
also found in the work of Gao et al. (2020). Gao et al. (2018) and Gao et al.
(2020) report that up to $95\%$ of the bulges found in S0-Sab galaxies are
classical, while the percentage of pseudo- bulges is between $62\%$ for Sc
galaxies and up to $90-100\%$ for late-type spirals.
Taking into account all this, and looking at Fig. 3, imposing a combination of
$n_{b}<2.0$ and $B/T<1/3$ selection criteria to distinguish the pseudo-bulges
does separate most of the bulges found in the spiral galaxies of our sample,
with two ETG bulges overlapping into this space. However, there are a few
other bulges of late-type galaxies which do not fit in this region. Imposing a
more restrictive limit in bulge-to-total flux ratio (e.g. $B/T<0.1$, as
previously mentioned studies found) produces a less clean separation, now with
more bulges of late-type galaxies outside the range and into the classical
bulge zone. We appreciate this second combination of selection criteria not to
be accurate ($n_{b}<2.0$ & $B/T<0.1$) in separating the distribution of
pseudo-bulges in our sample. This could be due to the small size of our sample
and could potentially change in case of better statistics.
As Gao et al. (2020) found that only 10% of their classified pseudo-bulges
reside in galaxies with stellar masses higher than $10.5M_{\odot}$, if one
uses a combination of selection thresholds of $B/T>1/3$ and
$\log(M_{\star})>10.5$ in Fig. 4 to separate classical bulges, an overlap with
the more massive spirals in our sample is noticeable. Considering a less
restrictive limit for $B/T$ (e.g. $B/T>0.1$) will produce more overlap between
bulges in spirals and ETG. In any case, one notices that a significant
proportion of ETG bulges will be classified as pseudo-bulges, as the
respective galaxy stellar masses are inferior to the imposed threshold. Again,
a larger sample size and with better statistics would be beneficial to
validate this combination of criteria to discriminate between bulges, but so
as adding another threshold in $n_{b}$ to the two selection thresholds.
Overall, detailed multi-component decomposition of galaxy images remain the
most accurate method for determination of bulge types in galaxies. However, in
case of large samples, when this process can be extremley time-consuming,
and/or when the galaxy images do not have high-enough resolution and signal-
to-noise ratio, proxies for bulge classification such as combination of
parameters (e.g. $n_{b}-B/T$, with the associated thresholds) - could be used,
albeit the uncertainties can be non-negligible. As pointed out by Neumann et
al. (2017), the more criteria are used together, the more accurate is the
classification obtained.

###  5.3 Potential sources of systematic errors and comparison with other
studies

Comparing with other similar studies, the small number of galaxies analysed
here (especially the ETGs) could explain the larger scatter in the
$M_{BH}-n_{b}$ or $M_{BH}-L_{b}$ relations and the differences in the best-fit
parameters compared with the works of Savorgnan et al. (2013), Savorgnan et
al. (2016), Savorgnan (2016) or Sahu et al. (2020), for example. Another
aspect is a different method of calculation for the black hole masses in Table
5, where some galaxies have directly measured $M_{BH}$, some are estimated
while others are predicted, with a high degree of uncertainty associated for
the latter. This contributes as well to best-fit linear regression parameter
values, their uncertainties and the scatter of the respective relations.

Another aspect that could produce some differences in relation with other
studies, is, as mentioned in Sect. 3.1, the consideration of ellipticals as
multicomponent objects, that contain a bulge and disc. While in previoulsy
mentioned similar studies the ellipticals were treated as a single spheroidal
component, practically considering as galaxy only the cores of the elliptical
galaxies, for our low-redhsift, high-resolution sample of elliptical galaxies,
we consider our approach to be more suitable and accurate. However, we do
recognize that this can produce some differences in the results, compared with
those in other works.

An important issue that may influence the parameters involved in the relations
presented (black-hole relations included), and a potential source of
systematic erorrs is the fact that we only considered a bulge and a disc in
our decomposition process, without any other extra-components, such as a
nucleus, nuclear rings, lens/ovals, disk breaks or spiral arms. There are a
number of previous works which underlined and shown the importance of
inclusion of such components in the structural analysis of galaxies, of
course, if their presence is verified observationally, from an analysis of the
residual images and/or surface brigthness or ellipticity profiles.
Laurikainen et al. (2009) revelead that lens/ovals are present in their
analysed sample of early-type disc galaxies (S0-Sa) in large fractions, up to
$86\pm 6\%$ for unbarred lenticulars and Sa0-Sa/Sab galaxies, with higher
fractions for lenticulars. By doing multi-component decomposition on mid-
infrared images of a medium sample of nearby barred spiral galaxies, Kim et
al. (2014) included models with a disc break. They showed that in Type II disc
galaxies (discs with a normal break/truncation in the outer parts), not
including a break can produce a derived disc scale-length with a intermediate
value between the one of the inner and outer disc. At the same time, a part of
the disc flux will be embedded in the bulge or bar flux for the same type of
disc galaxies, with $B/T$ suffering a $10\%$ decrease on average. On a similar
note, Laurikainen et al. (2006) emphasized the importance of considering more
components, especially in barred galaxies, otherwise the bulge mass and flux
will be biased, with an overestimation of $B/T$.
Gao & Ho (2017) perform a detailed study of a small sample of representative,
nearby, resolved disc galaxies, spanning all morphological types, with the
purpose of investigating the impact of choosing different combination of model
functions for the various morphological features in galaxies, on the measured
bulge parameters. The authors also quantify the errors induced by specific
model choices and establish that outer lenses, outer rings and spiral arms can
be safely ignored from a structural analysis, as their effect on bulge
parameters is at most $10\%$. Nuclear and inner lenses/rings on the other
hand, together with the disk breaks and bars, are found to have a more
significant effect on $B/T$ and $n_{b}$ and need to be considered.
Nevertheless, the authors mention that is quite time-consuming to do such a
thorough analysis on large samples of galaxies.

In light of this, we do recognize our simplified approach to the structural
analysis of the galaxies in our sample. As mentioned in Paper I, we considered
only a two-component fit first because our method is suitable for this
combination of fitting functions, while the dust and projection effects were
quantified for the parameters of these components only. Another reason is that
for the purpose of our study, we needed the structural and morphological
parameters of bulges as a whole, even though they might have some extra
components embedded in them. Here we have only analysed unbarred galaxies (due
to the lack of proper projection and dust corrections for this component), and
thus the systematic errors which arise when fitting only a bulge and a disk in
a barred galaxy are not present in our results. Taking into account the
available classifications, galaxies with outer rings / lenses are not present
in the analysed sample, although we cannot completely exclude them. In any
case, as found by Gao & Ho (2017), these features can be ignored. Thus the
main source of systematic errors on bulge parameters in our analysis will be
from not taking into account the potential presence of disk
breaks/truncations, nuclear and inner rings/lenses.
We conclude that our results for the galaxies which were analysed in similar
studies might differ to a certain extent. However, this issue does not alter
significantly the main results and conclusions of this study.

##  6 Summary and conclusions

We presented here a detailed analysis of the scaling relations of low-redshift
bulges and early-type (ellipticals+lenticulars) galaxies and the biases
introduced by the combined effects of dust, inclination (projection) and
decomposition on the main parameters involved in these relations. Our study
was performed on a representative sample of 29 spiral and early-type galaxies
(18 spirals and 11 ETGs) taken from the SINGS/KINGFISH survey, in B band. A
thorough surface photometry and sky determination and subtraction was done to
derive the integrated fluxes of each galaxy and its main components - discs
and bulges. We derived the measured (observed) photometric and structural
parameters of bulges (the spiral galaxies were analysed in Paper I) by doing a
2-component bulge-disc decomposition of galaxies, using GALFIT data analysis
algorithm. Together with the integrated fluxes and the structural parameters,
we calculated the central, effective and mean effective bulge surface
brightnesses, the absolute magnitudes and the bulge-to-disc ratios.

The central face-on dust opacities in B band ($\tau_{B}^{f}$) were derived
using the same method as in Paper I, for lenticular galaxies only.
Subsequently, we have used the numerical results for projection, dust and
decomposition effects derived in Pastrav et al. (2013a) and Pastrav et al.
(2013b) to correct all the necessary photometric and strucutural parameters to
obtain their intrinsic values. We then derived bulge and early-type scaling
relations, such as the Kormendy relation, central bulge surface brightness -
absolute magnitude, or the black-hole scaling relations, among others. Both
the observed (measured) and the intrinsic (corrected) relations were
presented, corrected for all the aforementioned effects, in order to visualise
the differences in the overall trend, their slopes and zero points. The
scatter and the Pearson correlation coefficents were also determined, where
possible, to analyse the strength of the most import correlations, and a
comparison with the values found in similar studies were presented. We
performed statistical tests to investigate the significance of the
correlations, and if the relations found are statistically different for
early- and late-type galaxies. By analysing these relations, our main
conclusions are:

  * •

bulges of spiral galaxies reside on a steeper slope Kormendy relation that the
early-type galaxies

  * •

dust and inclination effects produce more important changes in the slope and
zero-point of the Kormendy relation for spiral galaxies than those for ETGs

  * •

if we divide our sample into galaxies with pseudo- or classical bulges on the
basis of a Sérsic index threshold ($n_{b}=1.5$ or $2.0$), we cannot clearly
notice pseudo-bulges residing far away (more than $3\sigma$) from the
$<\mu_{e,b}>-R_{e,b}$ relation of the ETGs, as it was observed by other
authors

  * •

$n_{b}-B/T$ and $B/T-M_{\star}$ can be used when trying to classify galaxies
into early and late types, as the two classes occupy different areas
($n_{b}-B/T$) or show different trends ($B/T-M_{\star}$); extending the use of
these combinations of parameters (with specific thresholds) further to divide
bulges into pseudo- or classical bulges, is problematic due to overlaps in the
two distributions or large spread in values

  * •

we confirm two diferent intrinsic $M_{b}-n_{b}$ (or alternatively
$L_{b}-n_{b}$) relations for late-type and early-type galaxies, with the
latter showing a steeper slope relation; the dust and inclination effects bias
more the parameters of this relation (slope, intercept, scatter) for bulges of
spiral galaxies than in the case of early-type ones;

  * •

the black hole mass vs bulge luminosity - $M_{BH}-M_{b}$ (or $M_{BH}-L_{b}$)
relation shows different best-fit parameters for bulges of LTGs and ETGs, with
the corresponding parameters for ETGs being consistent with other studies;
while the slope is steeper for the bulges of late-type galaxies, the two
relations are not statistically different at p<0.05 level, with only the
observed relations are found to be different at p<0.1 probability;

  * •

we can confirm within errors and taking into account the size of our sample,
statistically similar $M_{BH}-n_{b}$ relations for both the early and late-
type galaxies, with more data needed for a decrease in the uncertainty of this
result

  * •

overall, our set of best-fit parameters (slope, zero-point, scatter) for the
intrinsic set of black hole associated scaling relations (Fig. 5) are
consistent within errors with values found in other studies, while the Pearson
coefficients derived indicate the strongest correlation for the $M_{b}-n_{b}$
relation, while the lowest degree of correlation was found for the
$M_{BH}-n_{b}$ relation

The results presented here were obtained on a rather small sample of nearby
spiral and early-type galaxies. Even so, we managed to draw some important
conclusions and confirm a few of the previous results obtained from analysing
larger samples. More data, especially for the black holes, will be beneficial
for deriving more tight, less scattered, intrinsic bulge scaling relations.

## Acknowledgements

The author would like to thank the referee for a careful reading of the
manuscript and for the useful suggestions which improved the quality and
clarity of this paper.
This research made use of the NASA/IPAC Extragalactic Database (NED), which is
operated by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration.
The author would like to acknowledge financial support from the Research and
Inovation Ministery (MCI), project Nucleu- LAPLAS VI/2019.

## Data Availability

The data underlying this article are available in the article and in its
online supplementary material.

## References

  * Allen et al. (2006) Allen, P. D., Driver, S. P., Graham, A. W., et al. 2006, MNRAS, 371, 2
  * Athanassoula (2005) Athanassoula, E. 2005, MNRAS, 358, 1477
  * Beifiori et al. (2012) Beifiori, A., Courteau, S., Corsini, E. M., Zhu, Y. 2012, MNRAS, 419, 2497
  * Bellstedt et al. (2017) Bellstedt, S., Graham, A. W., Forbes, D. A., et al. 2017, MNRAS, 470, 1321
  * Bender (1990) Bender, R. 1990, A&A, 229, 441
  * Blakeslee et al. (2001) Blakeslee, John P., Lucey, John R., Barris, Brian J., Hudson, Michael J., Tonry, John L. 2001, MNRAS, 327, 1004
  * Bournaud et al. (2007) Bournaud, F., Elmegreen, B. G., & Elmegreen, D. M. 2007, ApJ, 670, 237
  * Bournaud et al. (2009) Bournaud, F., Elmegreen, B. G., & Martig, M. 2009, ApJL, 707, L1
  * Bournaud (2016) Bournaud, F. 2016, in "Galactic Bulges", ed. E. Laurikainen, R. Peletier & D. Gadotti, 355
  * Brooks & Christensen (2016) Brooks, A. & Christensen, C. 2016, in "Galactic Bulges", ed. E. Laurikainen, R. Peletier & D. Gadotti, 317
  * Capaccioli (1987) Capaccioli, M. 1987, IAU Symp. 127, “Structure and Dynamics of Elliptical Galaxies”, ed. P. T. de Zeeuw, 47
  * Caramete & Biermann (2010) Caramete, L. I. & Biermann, P. L. 2010, A&A, 521, A55
  * Carollo et al. (1997) Carollo, C. M., Stiavelli, M., de Zeeuw, P. T., Mack, J. 1997, AJ, 114, 2366
  * Carollo et al. (1998) Carollo, C. M., Stiavelli, M., Mack, J. 1998, AJ, 116, 68
  * Carollo et al. (2002) Carollo, C. M., Stiavelli, M., Seigar, M., de Zeeuw, P. T., Dejonghe, H. 1998, AJ, 123, 159
  * Ciotti (1991) Ciotti, L. 1991, A&A, 249, 99
  * Ciotti & Bertin (1999) Ciotti, L. & Bertin, G. 1999, A&A, 352, 447
  * Costantin et al. (2017) Costantin, L., Méndez-Abreu, J., Corsini, E. M. et al. 2017, A&A, 601, A84
  * Costantin et al. (2018) Costantin, L., Méndez-Abreu, J., Corsini, E. M. et al. 2018, A&A, 609, A132
  * Djorgovski & Davis (1987) Djorgovski, S. & Davis, M. 1987, ApJ, 313, 59
  * Do et al. (2014) Do T., Wright S.A., Barth A.J. et al. 2014, AJ, 147, 93
  * Dong & De Robertis (2006) Dong X. Y., De Robertis M. M., 2006, AJ, 131, 1236
  * Dressler et al. (1987) Dressler, A., Lynden-Bell, D., Burstein, D., et al. 1987, ApJ, 313, 42
  * Dullo et al. (2020) Dullo, B. T., Bouquin, A. Y. K., Gil de Paz, A., Knapen, J. H., Gorgas, J. 2020, ApJ, 898, 83
  * Elmegreen et al. (2008) Elmegreen, B. G., Bournaud, F., Elmegreen, D. M. 2008, ApJ, 688, 67
  * Erwin et al. (2015) Erwin, P., Saglia, R. P., Fabricius, M. et al. 2015, MNRAS, 446, 4039
  * Faber & Jackson (1976) Faber, S. M. & Jackson, R. E. 1976, ApJ, 204, 668
  * Faber (1977) Faber, S. M. 1977, in "Evolution of Galaxies and Stellar Populations", ed. B. M. Tinsley & R. B. Larson, 157
  * Ferrarese & Merritt (2000) Ferrarese, L., Merritt, D. 2000, ApJ, 539, L9
  * Fisher & Drory (2008) Fisher, D. B., Drory, N. 2008, AJ, 136, 773
  * Fisher & Drory (2016) Fisher, D. B., Drory, N. 2016, in "Galactic Bulges", ed. E. Laurikainen, R. Peletier & D. Gadotti, 418, 41
  * Gadotti (2009) Gadotti, D. 2009, MNRAS, 393, 1531
  * Gadotti et al. (2010) Gadotti A. D., Baes M., Falony S. 2010, MNRAS, 403, 2053
  * Gao & Ho (2017) Gao, H., & Ho, L. C. 2017, ApJ, 845, 114
  * Gao et al. (2018) Gao, H., Ho, L. C., Barth, A. J., Li, Z.-Y. 2018, ApJ, 862, 100
  * Gao et al. (2019) Gao, H., Ho, L. C., Barth, A. J., Li, Z.-Y. 2019, ApJS, 244, 34
  * Gao et al. (2020) Gao, Hua, Ho, L. C., Barth, A. J., Li, Z.-Y. 2020, ApJS, 247, 20
  * Gebhardt et al. (2000) Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13
  * Gott (1977) Gott, III, J. R. 1977, ARA&A, 15, 235
  * Graham (2001) Graham, A. W. 2001, AJ, 121, 820
  * Graham & Guzman (2003) Graham A.W., Guzmán R. 2003, AJ, 125, 2936
  * Graham & Driver (2005) Graham, A.W. & Driver, S.P. 2005, PASA, 22, 118
  * Graham & Driver (2007) Graham, A. W., & Driver, S. P. 2007, ApJ, 655, 77
  * Graham (2013) Graham, A. W. 2013, in "Planets, Stars and Stellar Systems", ed. T.D.Oswalt & W.C.Keel, Vol. 6, 91
  * Graham & Scott (2013) Graham, A. W. & Scott, N. 2013, ApJ, 764, 151)
  * Graham & Worley (2008) Graham, A. W. & Worley, C. C. 2008, MNRAS, 388, 1708
  * Grossi et al. (2015) Grossi, M., Hunt, L. K., Madden, S. C. et al. 2015, A&A, 574, A126
  * Grootes et al. (2013) Grootes, M., Tuffs, R.J., Popescu, C.C. et al. 2013, ApJ, 766, 59
  * Häussler et al. (2007) Häussler, B., McIntosh, D. H., Barden, M. et al. 2007, ApJS, 172, 615
  * Huang et al. (2013a) Huang, S., Ho, L. C., Peng, C. Y., Li, Z.-Y., Barth, A. J. 2013a, ApJ, 766, 47
  * Huang et al. (2013a) Huang, S., Ho, L. C., Peng, C. Y., Li, Z.-Y., Barth, A. J. 2013b, ApJL, 768, L28
  * Huang et al. (2016) Huang, S., Ho, L. C., Peng, C. Y., Li, Z.-Y., Barth, A. J. 2016, ApJ, 821, 114
  * Jerjen et al. (2000) Jerjen, H., Binggeli, B. & Freeman, K. C. 2000, AJ, 119, 593
  * Kennicutt et al. (2011) Kennicutt, R. C., Calzetti, D., Aniano, G. et al. 2011, PASP, 123, 1347
  * Kennicutt et al. (2003) Kennicutt, R. C., Armus, L., Bendo, G. et al. 2003, PASP, 115, 928
  * Kim et al. (2014) Kim, T., Gadotti, D. A., Sheth, K., et al. 2014, ApJ, 782, 6
  * Kormendy (1977) Kormendy, J. 1977, ApJ, 218, 333
  * Kormendy (1993) Kormendy, J. 1993, IAU Symposium 153, 209
  * Kormendy & Richstone (1995) Kormendy, J., & Richstone, D. 1995, ARA&A, 33, 581
  * Kormendy & Gebhardt (2001) Kormendy, J., & Gebhardt, K. 2001, in 20th Texas Symposium on relativistic astrophysics, eds. J. C. Wheeler, & H. Martel, AIPC, 586, 363
  * Kormendy & Kennicutt (2004) Kormendy, J., Kennicutt, Jr., R. C. 2004, ARA&A, 42, 603
  * Kormendy & Ho (2013) Kormendy, J., Ho, L. C. 2013, ARA&A, 51, 511
  * Kormendy (2016) Kormendy, J. 2016, ASSL, 418, 431
  * Laurikainen et al. (2005) Laurikainen, E., Salo, H., Buta, R. 2005, MNRAS, 362, 1319
  * Laurikainen et al. (2006) Laurikainen E., Salo H., Buta R. et al. 2006, AJ, 132, 2634
  * Laurikainen et al. (2007) Laurikainen E., Salo H., Buta R., Knapen J., 2007, MNRAS, 381, 401
  * Laurikainen et al. (2009) Laurikainen, E., Salo, H., Buta, R., Knapen, J. H. 2009, ApJL, 692, L34
  * Laurikainen et al. (2010) Laurikainen, E., Salo, H., Buta, R., Knapen, J. H., Comerón, S. 2010, MNRAS, 405, 1089
  * Magorrian et al. (1998) Magorrian, J., Tremaine, S., Richstone, D., et al. 1998, AJ, 115, 2285
  * Marconi & Hunt (2003) Marconi, A., & Hunt, L. K. 2003, ApJL, 589, L21
  * Méndez-Abreu et al. (2017) Méndez-Abreu, J., Ruiz-Lara, T., Sánchez-Menguiano, L., et al. 2017, A&A, 598, A32
  * Möllenhoff et al. (2006) Möllenhoff, C., Popescu, C. C., Tuffs, R. J. 2006, A&A, 456, 941
  * Neumann et al. (2017) Neumann, J., Wisotzki, L., Choudhury, O. S., et al. 2017, A&A, 604, A30
  * Nieto et al. (1991) Nieto, J.-L., Bender, R., Arnaud, J., & Surma, P. 1991, A&A, 244, L25
  * Oh et al. (2017) Oh, S., Greene, J. E., & Lackner, C. N. 2017, ApJ, 836, 115
  * Pastrav (2020) Pastrav, B. A. 2020, MNRAS, 493, 3580
  * Pastrav et al. (2013a) Pastrav, B. A., Popescu, C. C., Tuffs, R. J., Sansom, A. E., 2013a, A&A, 553, A80
  * Pastrav et al. (2013b) Pastrav, B. A., Popescu, C. C., Tuffs, R. J., Sansom, A. E. 2013b, A&A, 557, A137
  * Peng et al. (2002) Peng, C. Y., Ho, L. C., Impey, C. D., Rix, H.-W. 2002, AJ, 124, 266
  * Peng et al. (2010) Peng, C. Y., Ho, L. C., Impey, C. D., Rix, H.-W. 2010, AJ, 139, 2097
  * Planck Collaboration (2016) Planck Colaboration 2016, A&A, 594, A13
  * Popescu et al. (2011) Popescu, C. C., Tuffs, R. J., Dopita, M. A. et al. 2011, A&A, 527, A109
  * Rémy-Ruyer et al. (2015) Rémy-Ruyer, A., Madden, S. C., Galliano, F. et al. 2015, A&A, 582, A121
  * Renzini (1999) Renzini, A. 1999, in "The Formation of Galactic Bulges", ed. C. M. Carollo, H. C. Ferguson & R. F. G. Wyse, Cambridge Univ. Press, 9
  * Rix & White (1990) Rix, H.-W., White, S. D. M. 1990, ApJ, 362, 52
  * Rix & White (1992) Rix, H.-W., White, S. D. M. 1992, MNRAS, 254, 389
  * Rodriguez-Gomez et al. (2017) Rodriguez-Gomez, V., Sales, L. V., Genel, S. et al. 2017, MNRAS, 467, 3083
  * Sabbi et al. (2018) Sabbi, E., Calzetti, D., Ubeda, L. et al. 2018, ApJS, 235, 23
  * Sachdeva et al. (2017) Sachdeva, S., Saha, K., Singh, H. P. 2017, ApJ, 840, 79
  * Sachdeva et al. (2019) Sachdeva, S., Gogoi, R., Saha, K., Kembhavi, A., Raychaudhury, S. 2019, MNRAS, 487, 1795
  * Sachdeva et al. (2020) Sachdeva, S., Ho, L. C., Li, Y. A., Shankar, F. 2020, ApJ, 899, 89
  * Sahu et al. (2020) Sahu, N., Graham, A. W., Davis, B. L. 2020, ApJ , 903, 97
  * Savorgnan et al. (2013) Savorgnan, G., Graham, A. W., Marconi, A. et al. 2013, MNRAS, 434, 387
  * Savorgnan & Graham (2015) Savorgnan G.A.D., Graham A.W. 2015, MNRAS, 446, 2330
  * Savorgnan et al. (2016) Savorgnan, G., Graham, A. W., Marconi, A., Sani, E. 2016, ApJ, 817, 21
  * Savorgnan (2016) Savorgnan G.A.D. 2016, ApJ, 821, 88
  * Schlafly & Finkbeiner (2011) Schlafly, E. F. & Finkbeiner, D. P. 2011, ApJ, 737, 103
  * Schegel et al. (1998) Schlegel D. J., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525
  * Simard et al. (2002) Simard, L., Willmer, C. N. A., Vogt, N. P. et al. 2002, ApJS, 142, 1
  * Skibba et al. (2011) Skibba, R. A., Engelbracht, C. W., Dale, D. . et al. 2011, ApJ, 738, 89
  * Theureau et al. (2007) Theureau, G., Hanski, M. O., Coudreau, N., Hallet, N., Martin, J. -M 2007, A&A, 465, 71
  * Tonini et al. (2016) Tonini, C., Mutch, S. J., Croton, D. J., Wyithe, J. S. B. 2016, MNRAS, 459, 4109
  * Tully & Fisher (1988) Tully, R. Brent & Fisher, J. Richard 1988, "Catalog of Nearby Galaxies", Cambridge Univ. Press, ISBN 0521352991
  * Tully et al. (2013) Tully, R. B., Courtois, H. M., Dolphin, A. E. et al. 2013, AJ, 146, 86
  * Tuffs et al. (2004) Tuffs, R. J., Popescu, C. C., Völk, H. J., Kylafis, N. D., Dopita, M. A. 2004, A&A, 419, 821
  * van den Bosch (2016) van den Bosch, Remco C. E 2016, ApJ, 831, 134
  * Weingartner & Draine (2001) Weingartner, J.C. & Draine, B.T. 2001, ApJ, 548, 296
  * Weinzirl et al. (2009) Weinzirl, T., Jogee, S., Khochfar, S., Burkert, A., & Kormendy, J. 2009, ApJ, 696, 411
  * Wilson et al. (2013) Wilson, C. D., Cridland, A., Foyle, K. et al. 2013, ApJL 776, 30
  * Young & Currie (1994) Young, C. K. & Currie, M. J. 1994, MNRAS, 268, L11
  * Zhu et al. (2021) Zhu, P., Ho, L. C. & Gao, H. 2021, ApJ, 907, 6
  * Zibetti & Groves (2011) Zibetti, S. & Groves, B. 2011, MNRAS, 417, 812