Galaxy luminosities, stellar masses, sizes, velocity dispersions as a function of morphological type

Abstract

We provide fits to the distribution of galaxy luminosity, size, velocity dispersion and stellar mass as a function of concentration index C_r and morphological type in the SDSS. We also quantify how estimates of the fraction of `early' or `late' type galaxies depend on whether the samples were cut in color, concentration or light profile shape, and compare with similar estimates based on morphology. Our fits show that Es account for about 20% of the r-band luminosity density, rho_Lr, and 25% of the stellar mass density, rho_*; including S0s and Sas increases these numbers to 33% and 40%, and 50% and 60%, respectively. Summed over all galaxy types, we find rho_* ~ 3 * 10^8 M_Sun Mpc^{-3} at z ~ 0. This is in good agreement with expectations based on integrating the star formation history. However, compared to most previous work, we find an excess of objects at large masses, up to a factor of ~ 10 at M_* ~ 5*10^{11} M_Sun. The stellar mass density further increases at large masses if we assume different IMFs for Es and spiral galaxies, as suggested by some recent chemical evolution models, and results in a better agreement with the dynamical mass function. We also show that the trend for ellipticity to decrease with luminosity is primarily because the E/S0 ratio increases at large L. However, the most massive galaxies, M_* > 5 * 10^{11} M_Sun, are less concentrated and not as round as expected if one extrapolates from lower L, and they are not well-fit by pure deVaucouleur laws. This suggests formation histories with recent radial mergers. Finally, we show that the age-size relation is flat for Es of fixed dynamical mass, but, at fixed M_dyn, S0s and Sas with large sizes tend to be younger. Explaining this difference between E and S0 formation is a new challenge for models of early-type galaxy formation. Comment: 42 pages, 34 figures, 9 tables. Accepted for publication in MNRAS

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arXiv:0910.1093v2 [astro-ph.CO] 29 Jan 2010
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 29 January 2010(MN LATEX style file v2.2)
Galaxy luminosities, stellar masses, sizes, velocity
dispersions as a function of morphological type
M. Bernardi1⋆, F. Shankar2, J. B. Hyde1, S. Mei3,4, F. Marulli5& R. K. Sheth1,6
1Department of Physics & Astronomy, University of Pennsylvania, 209 S. 33rd St., Philadelphia, PA 19104, USA
2Max-Planck-Instit¨ ut f¨ ur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748, Garching, Germany
3University of Paris Denis Diderot, 75205 Paris Cedex 13, France
4GEPI, Observatoire de Paris, Section de Meudon, 5 Place J. Janssen, 92195 Meudon Cedex, France
5Dipartimento di Astronomia, Universit´ a degli Studi di Bologna, via Ranzani 1, I-40127 Bologna, Italy
6Center for Particle Cosmology, University of Pennsylvania, 209 S. 33rd St., Philadelphia, PA 19104, USA
29 January 2010
ABSTRACT
We provide fits to the distribution of galaxy luminosity, size, velocity dispersion and
stellar mass as a function of concentration index Cr and morphological type in the
SDSS. (Our size estimate, a simple analog of the SDSS cmodel magnitude, is new: it
is computed using a combination of seeing-corrected quantities in the SDSS database,
and is in substantially better agreement with results from more detailed bulge/disk
decompositions.) We also quantify how estimates of the fraction of ‘early’ or ‘late’
type galaxies depend on whether the samples were cut in color, concentration or light
profile shape, and compare with similar estimates based on morphology. Our fits show
that ellipticals account for about 20% of the r-band luminosity density, ρLr, and 25%
of the stellar mass density, ρ∗; including S0s and Sas increases these numbers to 33%
and 40%, and 50% and 60%, respectively. The values of ρLrand ρ∗, and the mean
sizes, of E, E+S0 and E+S0+Sa samples are within 10% of those in the Hyde &
Bernardi (2009), Cr ≥ 2.86 and Cr ≥ 2.6 samples, respectively. Summed over all
galaxy types, we find ρ∗∼ 3×108M⊙Mpc−3at z ∼ 0. This is in good agreement with
expectations based on integrating the star formation history. However, compared to
most previous work, we find an excess of objects at large masses, up to a factor of ∼ 10
at M∗∼ 5× 1011M⊙. The stellar mass density further increases at large masses if we
assume different IMFs for elliptical and spiral galaxies, as suggested by some recent
chemical evolution models, and results in a better agreement with the dynamical mass
function.
We also show that the trend for ellipticity to decrease with luminosity is primarily
because the E/S0 ratio increases at large L. However, the most massive galaxies, M∗≥
5×1011M⊙, are less concentrated and not as round as expected if one extrapolates from
lower L, and they are not well-fit by pure deVaucouleur laws. This suggests formation
histories with recent radial mergers. Finally, we show that the age-size relation is flat
for ellipticals of fixed dynamical mass, but, at fixed Mdyn, S0s and Sas with large
sizes tend to be younger. Hence, samples selected on the basis of color or Crwill yield
different scalings. Explaining this difference between E and S0 formation is a new
challenge for models of early-type galaxy formation.
Key words: galaxies: formation - galaxies: haloes - dark matter - large scale structure
of the universe
1 INTRODUCTION
Each galaxy has its own peculiarities. Nevertheless, even
to the untrained eye, sufficiently well-resolved galaxies can
⋆E-mail: bernardm@physics.upenn.edu
be separated into three morphological types: disky spirals,
bulgy ellipticals, and others which are neither. The mor-
phological classification of galaxies is a field that is nearly
one hundred years old, and sample sizes of a few thousand
morphologically classified galaxies are now available (e.g.
Fukugita et al. 2007; Lintott et al. 2008). However, such

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M. Bernardi et. al.
eyeball classifications are prohibitively expensive in the era
of large scale sky surveys, which image upwards of a few
million galaxies. Moreover, the morphological classification
of even relatively low redshift objects from ground-based
data is difficult. Thus, a number of groups have devised au-
tomated algorithms for discerning morphologies from such
data (e.g. Ball et al. 2004 and references therein).
In parallel, it has been recognized that relatively simple
criteria, using either crude measures of the light profile (e.g.
Strateva et al. 2001), the colors (e.g. Baldry et al. 2004), or
some combination of photometric and spectroscopic infor-
mation (Bernardi et al. 2003; Bernardi & Hyde 2009) allow
one to separate early-type galaxies from the rest. Because
they are so simple, these tend to be more widely used. The
main goal of this paper is to show how samples based on
such crude cuts compare with those which are based on
the eyeball morphological classifications of Fukugita et al.
(2007). We do so by comparing the luminosity, stellar mass,
size and velocity dispersion distributions for cuts based on
photometric parameters with those based on morphology.
These were chosen because the luminosity function is stan-
dard, although it is becoming increasingly common to com-
pare models with φ(M∗) rather than φ(L) (e.g. Cole et al.
2001; Bell et al. 2003; Panter et al. 2007; Li & White 2009);
the size distribution φ(R) has also begun to receive con-
siderable attention recently (Shankar et al. 2009b); and the
distribution of velocity dispersions φ(σ) (Sheth et al. 2003) is
useful, amongst other things, to reconstruct the mass distri-
bution of super-massive black holes (e.g. Shankar et al. 2004;
Tundo et al. 2007; Bernardi et al. 2007; Shankar, Weinberg
& Miralda-Escud´ e 2009; Shankar et al. 2009a) and in studies
of gravitational lensing (Mitchell et al. 2005).
Section 2 describes the dataset, the photometric and
spectroscopic parameters derived from it, and the subsam-
ple defined by Fukugita et al. (2007). This section shows
how we use quantities output from the SDSS database to
define seeing-corrected half-light radii which closely approx-
imate the result of bulge + disk decompositions. We describe
our stellar mass estimator in this section as well; a detailed
comparison of it with stellar mass estimates computed by
three different groups is presented in Appendix A. The re-
sult of classifying objects into two classes, on the basis of
color, concentration index, or morphology are compared in
Section 3. Luminosity, stellar mass, size and velocity dis-
persion distributions, for the Fukugita et al. morphological
types are presented in Section 4, where they are compared
with those based on the other simpler selection cuts. This
section includes a discussion of the functional form, a gen-
eralization of the Schechter function, which we use to fit
our measurements. We find more objects with large stellar
masses than in previous work (e.g. Cole et al. 2001; Bell
et al. 2003; Panter et al. 2007; Li & White 2009); this is
the subject of Section 5, where implications for the match
with the integrated star formation rate, and the question of
how the most massive galaxies have evolved since z ∼ 2 are
discussed.
While we believe these distributions to be interesting in
their own right, we also study a specific example in which
correlations between quantities, rather than the distribu-
tions themselves, depend on morphology. This is the cor-
relation between the half-light radius of a galaxy and the
luminosity weighted age of its stellar population. Section 6
shows that the morphological dependence of this relation
means it is sensitive to how the ‘early-type’ sample was se-
lected, potentially resolving a discrepancy in the recent lit-
erature (Shankar & Bernardi 2009; van der Wel et al. 2009;
Shankar et al. 2009c). A final section summarizes our results,
many of which are provided in tabular form in Appendix B.
Except when stated otherwise, we assume a spatially
flat background cosmology dominated by a cosmological
constant, with parameters (Ωm,ΩΛ) = (0.3,0.7), and a Hub-
ble constant at the present time of H0 = 70 km s−1Mpc−1.
When we assume a different value for H0, we write it as
H0 = 100h km s−1Mpc−1.
2 THE SDSS DATASET
2.1 The full sample
In what follows, we will study the luminosities, sizes, velocity
dispersions and stellar masses of a magnitude limited sample
of ∼ 250,000 SDSS galaxies with 14.5 < mPetrosian < 17.5
in the r−band, selected from 4681 deg2of sky. In this band,
the absolute magnitude of the Sun is Mr,⊙ = 4.67.
The SDSS provides a variety of measures of the light
profile of a galaxy. Of these the Petrosian magnitudes and
sizes are the most commonly used, because they do not de-
pend on fits to models. However, for some of what is to fol-
low, the Petrosian magnitude is not ideal, since it captures
a type-dependent fraction of the total light of a galaxy. In
addition, seeing compromises use of the Petrosian sizes for
almost all the distant lower luminosity objects, leading to
systematic biases (see Hyde & Bernardi 2009 for examples).
Before we discuss the alternatives, we note that there
is one Petrosian based quantity which will play an impor-
tant role in what follows. This is the concentration index
Cr, which is the ratio of the scale which contains 90% of
the Petrosian light in the r-band, to that which contains
50%. Early-type galaxies, which are more centrally concen-
trated, are expected to have larger values of Cr, and two
values are in common use: a more conservative Cr≥ 2.86
(e.g. Nakamura et al. 2003; Shen et al. 2003) and a more
cavalier Cr≥ 2.6 (e.g. Strateva et al. 2001; Kauffmann et al.
2003; Bell et al. 2003). We show below that from the first
approximately two-thirds of the sample comes from E+S0
types, whereas the second selects a mix in which E+S0+Sa’s
account for about two-thirds of the objects.
The SDSS also outputs deV or exp magnitudes and
sizes which result from fitting to a deVaucouleur or expo-
nential profile, and fracDev, a quantity which takes values
between 0 and 1, which is a measure of how well the deVau-
couleur profile actually fit the profile (1 being an excellent
fit). In addition, it outputs cmodel magnitudes; this is a
very crude disk+bulge magnitude which has been seeing-
corrected. Rather than resulting from the best-fitting lin-
ear combination of an exponential disk and a deVaucouleur
bulge, the cmodel magnitude comes from separately fitting
exponential and deVaucouleur profiles to the image, and
then combining these fits by finding that linear combina-
tion of them which best-fits the image. Thus, if mexp and
mdeVare the magnitudes returned by fitting the two models,

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Galaxy distributions per morphological type
3
Figure 1. Comparison between apparent magnitudes and sizes obtained from performing full bulge+disk decompositions (denoted
deV+exp), with those output by or constructed from parameters in the SDSS database. In all cases, symbols with error bars show the
mean relation and the error on the mean, and dashed lines show the actual rms scatter. Filled circles in left panels show results for the
SDSS cmodel magnitudes and effective radii (see text for details) and right panels are for the SDSS deV (red stars; the effective radius
is the value in the SDSS database multiplied by
?b/a, i.e.
database a) quantities. Cyan triangles in bottom left panel show the result of picking either the deV or exp size, based on which of the
corresponding magnitudes were closer to the cmodel magnitude. Although these triangles are almost indistinguishable from the filled
circles, the rms scatter is substantially larger, particularly at small fracdev.
√ba) or exp (blue open squares; the effective radius is the value in the SDSS
then
10−0.4msdss−cmodel
=(1 − fracdeV)10−0.4mExp
+
fracdeV10−0.4mdeV. (1)
Later in this paper, we will be interested in seeing-
corrected half-light radii. We use the cmodel fits to define
these sizes by finding that scale re,cmodel where
10−0.4msdss−cmodel
2
=(1 − fracdeV)2π
?rsdss−cmodel
0
?rsdss−cmodel
0
dθθ Iexp(θ)
+ fracdeV2π dθθ IdeV(θ),(2)
where I is the surface brightness associated with the two fits.
Note that the SDSS actually performs a two dimensional fit
to the image, and it outputs the half-light radius of the long
axis of the image a, and the axis ratio b/a. The expression
above assumes one dimensional profiles, so we use the half-
light radius a of the exponential fit, and
for the deVaucouleur fit. We describe some tests of these
cmodel quantities shortly.
?b/a × a =√ba
We would also like to study the velocity dispersions of
these objects. One of the important differences between the
SDSS-DR6 and previous releases is that the low velocity dis-
persions (σ < 150 km s−1) were biased high; this has been
corrected in the DR6 release (see DR6 documentation, or
discussion in Bernardi 2007). The SDSS-DR6 only reports
velocity dispersions if the S/N in the spectrum in the rest-
frame 4000 − 5700˚ A is larger than 10 or with the status
flag equal to 4 (i.e. this tends to exclude galaxies with emis-
sion lines). To avoid introducing a bias from these cuts, we
have also estimated velocity dispersions for all the remain-
ing objects (see Hyde & Bernardi 2009 for more discussion).
These velocity dispersions are based on spectra measured
through a fiber of radius 1.5 arcsec; they are then corrected

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4
M. Bernardi et. al.
Figure 2. Top panels: Comparison of cmodel magnitudes (right) and sizes (left) with those obtained from performing full bulge+disk
decompositions as function of cmodel sizes. The sample was divided in three bins based on the shape of the light profile (as indicated in
the panels). Thin solid curves (blue, green and red) show fits to equations (3) and (4). Except for the sample with fracDev > 0.8, the
coefficients of these fits are given in Table 1. For fracDev > 0.8, the coefficients in Table 1 are based on the larger sample of Hyde &
Bernardi (2009) (see text for the origin of the small offsets); this results in the thick (magenta) solid curve shown. Bottom panels: Similar
to panels on left of Figure 1, but with cmodel magnitudes and sizes corrected following equations (3) and (4). In all panels, symbols show
the mean relation, error bars show the error on the mean and dashed lines (bottom) show the rms scatter.
to re/8, as is standard practice. (This is a small correction.)
The velocity dispersion estimate for emission line galaxies
can be compromised by rotation. In addition, the disper-
sion limit of the SDSS spectrograph is 69 km s−1, so at
small σ the estimated velocity dispersion may both noisy
and biased. We will see later that this affects the velocity
dispersion function. The size and velocity dispersion can be
combined to estimate a dynamical mass; we do this by set-
ting Mdyn= 5Reσ2/G.
2.2 A morphologically selected subsample
Recently, Fukugita et al. (2007) have provided morphologi-
cal classifications (Hubble type T) for a subset of 2253 SDSS
galaxies brighter than mPet = 16 in the r−band, selected
from 230 deg2of sky. Of these, 1866 have spectroscopic in-
formation. Since our goal is to compare these morphological
selected subsamples with those selected based on relatively
simple criteria (e.g. concentration index), we group galaxies
classified with half-integer T into the smaller adjoining in-
teger bin (except for the E class; see also Huang & Gu 2009
and Oohama et al. 2009). In the following, we set E (T = 0
and 0.5), S0 (T = 1), Sa (T = 1.5 and 2), Sb (T = 2.5 and 3),
and Scd (T = 3.5, 4, 4.5, 5, and 5.5). This gives a fractional
morphological mix of (E, S0, Sa, Sb, Scd) = (0.269, 0.235,
0.177, 0.19, 0.098). Note that this is the mix in a magnitude
limited catalog – meaning that brighter galaxies (typically
earlier-types) are over-represented.
2.3
cmodel magnitudes and sizes
As a check of our cmodel sizes, we have performed deVau-
couleurs bulge + exponential disk fits to light profiles of a
subset of the objects; see Hyde & Bernardi (2009) for a de-
tailed description and tests of the fitting procedure. If we
view these as the correct answer, then the top left panel

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Galaxy distributions per morphological type
5
Table 1. Coefficients used in equations (3) and (4) to correct
sizes and magnitudes for sky subtraction problems.
SampleCr0
Cr1
Cr2
fracDev > 0.8
& re,sdss−cmodel > 1.5 arcsec
& re,sdss−cmodel < 1.5 arcsec
0.3 < fracDev < 0.8
& re,sdss−cmodel > 1.5 arcsec
& re,sdss−cmodel < 1.5 arcsec
fracDev < 0.3
& re,sdss−cmodel > 1.5 arcsec
& re,sdss−cmodel < 1.5 arcsec
0.582
0.249
−0.221
0
0.065
0
0.201
0.182
−0.034
0
0.015
0
0.368
0.231
−0.110
0
0.021
0
SampleCm0
Cm1
Cm2
fracDev > 0.8
0.3 < fracDev < 0.8
& re,sdss−cmodel > 6 arcsec
& re,sdss−cmodel < 6 arcsec
fracDev < 0.3
0−0.014−0.001
0.147
0
0
−0.023
0
0.001
0
0
−0.001
of Figure 1 shows that the cmodel magnitudes are in good
agreement with those from the full bulge+disk fit, except
at fracDev≈ 0 and fracDev≈ 1 where cmodel is fainter by
0.05 mags (top left). This is precisely the regime where the
agreement should have been best. As discussed shortly, the
discrepancy arises mainly because the SDSS reductions suf-
fer from sky subtraction problems (see, e.g., SDSS DR7 doc-
umentation), whereas our bulge-disk fits do not (see Hyde
& Bernardi 2009 for details). Comparison with the top right
panel shows that cmodel is a significant improvement on
either the deV or the exp magnitudes alone.
The bottom panels show a similar comparison of the
sizes. At intermediate values of fracDev, neither the pure
deVaucouleur nor the pure exponential fits are a good de-
scription of the light profile, so the sizes are also biased
(bottom right). However, at fracDev=1, where the deVau-
couleurs model should be a good fit, the deV sizes returned
by the SDSS are about 0.07 dex smaller than those from
the bulge+disk decomposition. There is a similar discrep-
ancy of about 0.05 dex with the SDSS Exponential sizes at
fracDev=0. We argue below that these offsets are related
to those in the magnitudes, and are primarily due to sky
subtraction problems.
The filled circles in the bottom left panel show that our
cmodel sizes (from equation 2) are in substantially better
agreement with those from the bulge+disk decomposition
over the entire range of fracDev, with a typical scatter of
about 0.05 dex (inner set of dashed lines). For comparison,
the triangles show the result of picking either the deVau-
couleur or exponential size, based on which of these mag-
nitudes were closer to the cmodel magnitude (this is essen-
tially the scale that the SDSS uses to compute model colors).
Note that while this too removes most of the bias (except at
small fracDev), it is a substantially noisier estimate of the
true size (outer set of dashed lines). This suggests that our
cmodel sizes, which are seeing corrected, represent a signif-
icant improvement on what has been used in the past.
The SDSS reductions are known to suffer from sky sub-
traction problems which are most dramatic for large objects
or objects in crowded fields (see DR7 documentation). The
top panels in Figure 2 show this explicitly: while there is lit-
tle effect at small size, the SDSS underestimates the bright-
nesses and sizes when the half-light radius is larger than
about 5 arcsec. Note that this is actually a small fraction
of the objects: 6% of the objects have sdss-cmodel sizes
larger than 5 arcsec; 13% are larger than 4 arcsec. Whereas
previous work has concentrated on mean trends for the full
sample, Figure 2 shows that, in fact, the difference depends
on the type of light profile – galaxies with fracDev > 0.8
(i.e. close to deVaucouleur profiles) are more sensitive to
sky-subtraction problems than later-type galaxies. Some of
this is due to the fact that such galaxies tend to populate
more crowded fields.
To correct for this effect, we have fit low order poly-
nomials to the trends; the solid curves in the top panels
of Figure 2 show these fits. Except for the sample with
fracDev > 0.8, we use these fits to define our final corrected
cmodel sizes by:
re,cmodel
=re,sdss−cmodel + Cr0+ Cr1re,sdss−cmodel
+ Cr2r2
e,sdss−cmodel
(3)
and
mcmodel
=msdss−cmodel + Cm0+ Cm1re,sdss−cmodel
+ Cm2r2
e,sdss−cmodel, (4)
where the coefficients Cm0, Cm1, Cm2, Cr0, Cr1 and Cr2
are reported in Table 1.
For objects with fracDev > 0.8, the trends we see are
similar to those shown in Fig. 5 of Hyde & Bernardi (2009),
which were based on a (larger) sample of about 6000 early-
type galaxies. The thick solid (magenta) line in the top pan-
els of Figure 2 show the Hyde-Bernardi trends, with a small
offset to account for the fact that they did not integrate the
fitted profiles to infinity (because the SDSS, to which they
were comparing, does not), whereas we do. The thick solid
curve differs from the thin one at sizes larger than about
5 arcsec. Since the thick curve is based on a larger sample,
we use it to define our final corrected cmodel sizes. The cor-
rections are again described by equations (3) and (4), with
coefficients that are reported in Table 1.
The bottom panels of Figure 2 show that these cor-
rected quantities agree quite well with those from the full
bulge+disk fit, even at small and large fracDev.
2.4 Stellar Masses
Stellar masses M∗ are typically obtained by estimating
M∗/L (in solar units), and then multiplying by the rest-
frame L (which is not evolution corrected). In the following
we compare three different estimates of M∗. All these esti-
mates depend on the assumed IMF: Table 2 shows how we
transform between different IMFs.
The first comes from Bell et al. (2003), who report
that, at z = 0, log10(M∗/Lr)0 = 1.097(g − r)0 + zp,
where the zero-point zp depends on the IMF (see their Ap-
pendix 2 and Table 7). Their standard diet-Salpeter IMF
has zp = −0.306, which they state has 70% smaller M∗/Lr
at a given color than a Salpeter IMF. In turn, a Salpeter
IMF has 0.25 dex more M∗/Lr at a given color than the
Chabrier (2003) IMF used by the other two groups whose