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 Sloan Digital Sky Survey (SDSS). (Our size estimate, a simple analogue of the SDSS cmodel magnitude, is new: it is computed using a combination of seeing-corrected quantities in the SDSS data base, and is in substantially better agreement with results from more detailed bulge/disc decompositions.) We also quantify how estimates of the fraction of ‘early’- or ‘late’-type galaxies depend on whether the samples were cut in colour, concentration or light profile shape, and compare with similar estimates based on morphology. Our fits show that ellipticals account for about 20 per cent of the r-band luminosity density, forumla, and 25 per cent of the stellar mass density, ρ*; including S0s and Sas increases these numbers to 33 per cent and 40 per cent, and 50 per cent and 60 per cent, respectively. The values of forumla and ρ*, and the mean sizes, of E, E+S0 and E+S0+Sa samples are within 10 per cent of those in the Hyde & Bernardi, Cr≥ 2.86 and Cr≥ 2.6 samples, respectively. Summed over all galaxy types, we find ρ*∼ 3 × 108 M 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 × 1011 M. The stellar mass density further increases at large masses if we assume different initial mass functions 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 × 1011 M, 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 colour or Cr will yield different scalings. Explaining this difference between E and S0 formation is a new challenge for models of early-type galaxy formation.

1 INTRODUCTION

Each galaxy has its own peculiarities. Nevertheless, even to the untrained eye, sufficiently well-resolved galaxies can be separated into three morphological types: discy spirals, bulgy ellipticals and others which are neither. The morphological 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 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 automated 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 colours (e.g. Baldry et al. 2004) or some combination of photometric and spectroscopic information (Bernardi et al. 2003; Hyde & Bernardi 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 standard, although it is becoming increasingly common to compare 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 considerable attention recently (Shankar et al. 2009c); and the distribution of velocity dispersions φ(σ) (Sheth et al. 2003) is useful, amongst other things, to reconstruct the mass distribution of super-massive black holes (e.g. Shankar et al. 2004; Bernardi et al. 2007; Tundo et al. 2007; Shankar, Bernardi & Haiman 2009b; Shankar, Weinberg & Miralda-Escudé 2009a) and in studies of gravitational lensing (Mitchell et al. 2005).

Section 2 describes the data set, the photometric and spectroscopic parameters derived from it, and the subsample defined by Fukugita et al. (2007). This section shows how we use quantities output from the Sloan Digital Sky Survey (SDSS) data base to define seeing-corrected half-light radii which closely approximate the result of bulge + disc 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 result of classifying objects into two classes, on the basis of colour, concentration index, or morphology are compared in Section 3. Luminosity, stellar mass, size and velocity dispersion 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 generalization 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 (SFR), 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 distributions themselves, depend on morphology. This is the correlation 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 selected, potentially resolving a discrepancy in the recent literature (Shankar & Bernardi 2009; van der Wel et al. 2009; Shankar et al. 2009d). 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 Hubble constant at the present time of H0= 70 km s−1 Mpc−1. When we assume a different value for H0, we write it as H0= 100 h km s−1 Mpc−1.

2 THE SDSS DATA SET

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 deg2 of 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 depend on fits to models. However, for some of what is to follow, 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 important role in what follows. This is the concentration index Cr, which is the ratio of the scale which contains 90 per cent of the Petrosian light in the r band, to that which contains 50 per cent. Early-type galaxies, which are more centrally concentrated, 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; Bell et al. 2003; Kauffmann 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 exponential profile, and fracDev, a quantity which takes values between 0 and 1, which is a measure of how well the deVaucouleur profile actually fit the profile (1 being an excellent fit). In addition, it outputs cmodel magnitudes; this is a very crude disc+bulge magnitude which has been seeing-corrected. Rather than resulting from the best-fitting linear combination of an exponential disc 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 combination of them which best fits the image. Thus, if mexp and mdeV are the magnitudes returned by fitting the two models, then  

1
formula

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  

2
formula
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 forumla for the deVaucouleur fit. We describe some tests of these cmodel quantities shortly.

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 dispersions (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 signal-to-noise ratio (S/N) in the spectrum in the rest-frame 4000–5700 Å is larger than 10 or with the status flag equal to 4 (i.e. this tends to exclude galaxies with emission lines). To avoid introducing a bias from these cuts, we have also estimated velocity dispersions for all the remaining objects (see Hyde & Bernardi 2009 for more discussion). These velocity dispersions are based on spectra measured through a fibre of radius 1.5 arcsec; they are then corrected 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 dispersion 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 setting Mdyn= 5Reσ2/G.

2.2 A morphologically selected subsample

Recently, Fukugita et al. (2007) have provided morphological classifications (Hubble-type T) for a subset of 2253 SDSS galaxies brighter than mPet= 16 in the r band, selected from 230 deg2 of sky. Of these, 1866 have spectroscopic information. 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 integer 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 catalogue – 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 deVaucouleurs bulge + exponential disc fits to light profiles of a subset of the objects; see Hyde & Bernardi (2009) for a detailed description and tests of the fitting procedure. If we view these as the correct answer, then the top-left panel of Fig. 1 shows that the cmodel magnitudes are in good agreement with those from the full bulge+disc fit, except at fracDev ≈0 and fracDev ≈1 where cmodel is fainter by 0.05 mag (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 suffer from sky subtraction problems (see, e.g. SDSS DR7 documentation), whereas our bulge-disc 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.

Figure 1

Comparison between apparent magnitudes and sizes obtained from performing full bulge+disc decompositions (denoted deV+exp), with those output by or constructed from parameters in the SDSS data base. 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-hand panels show results for the SDSS cmodel magnitudes and effective radii (see text for details) and right-hand panels are for the SDSS deV (red stars; the effective radius is the value in the SDSS data base multiplied by forumla, i.e. forumla) or exp (blue open squares; the effective radius is the value in the SDSS data base 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.

Figure 1

Comparison between apparent magnitudes and sizes obtained from performing full bulge+disc decompositions (denoted deV+exp), with those output by or constructed from parameters in the SDSS data base. 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-hand panels show results for the SDSS cmodel magnitudes and effective radii (see text for details) and right-hand panels are for the SDSS deV (red stars; the effective radius is the value in the SDSS data base multiplied by forumla, i.e. forumla) or exp (blue open squares; the effective radius is the value in the SDSS data base 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.

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 description of the light profile, so the sizes are also biased (bottom right). However, at fracDev = 1, where the deVaucouleurs model should be a good fit, the deV sizes returned by the SDSS are about 0.07 dex smaller than those from the bulge+disc decomposition. There is a similar discrepancy 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+disc 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 deVaucouleur or exponential size, based on which of these magnitudes were closer to the cmodel magnitude (this is essentially the scale that the SDSS uses to compute model colours). 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 significant improvement on what has been used in the past.

The SDSS reductions are known to suffer from sky subtraction problems which are most dramatic for large objects or objects in crowded fields (see DR7 documentation). The top panels in Fig. 2 show this explicitly: while there is little effect at small size, the SDSS underestimates the brightnesses 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 per cent of the objects have SDSS-cmodel sizes larger than 5 arcsec; 13 per cent are larger than 4 arcsec. Whereas previous work has concentrated on mean trends for the full sample, Fig. 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.

Figure 2

Top panels: comparison of cmodel magnitudes (right) and sizes (left) with those obtained from performing full bulge+disc 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 Fig. 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.

Figure 2

Top panels: comparison of cmodel magnitudes (right) and sizes (left) with those obtained from performing full bulge+disc 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 Fig. 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 correct for this effect, we have fit low-order polynomials to the trends; the solid curves in the top panels of Fig. 2 show these fits. Except for the sample with fracDev > 0.8, we use these fits to define our final corrected cmodel sizes by  

3
formula
and  
4
formula
where the coefficients Cm0, Cm1, Cm2, Cr0, Cr1 and Cr2 are reported in Table 1.

Table 1

Coefficients used in equations (3) and (4) to correct sizes and magnitudes for sky subtraction problems.

Sample Cr0 Cr1 Cr2 
fracDev > 0.8 
& re,SDSS-cmodel > 1.5 arcsec 0.582 −0.221 0.065 
& re,SDSS-cmodel < 1.5 arcsec 0.249 
0.3 < fracDev < 0.8 
& re,SDSS-cmodel > 1.5 arcsec 0.201 −0.034 0.015 
& re,SDSS-cmodel < 1.5 arcsec 0.182 
fracDev < 0.3 
& re,SDSS-cmodel > 1.5 arcsec 0.368 −0.110 0.021 
& re,SDSS-cmodel < 1.5 arcsec 0.231 
Sample Cm0 Cm1 Cm2 
fracDev > 0.8 −0.014 −0.001 
0.3 < fracDev < 0.8 
& re,SDSS-cmodel > 6 arcsec 0.147 −0.023 
& re,SDSS-cmodel < 6 arcsec 
fracDev < 0.3 0.001 −0.001 
Sample Cr0 Cr1 Cr2 
fracDev > 0.8 
& re,SDSS-cmodel > 1.5 arcsec 0.582 −0.221 0.065 
& re,SDSS-cmodel < 1.5 arcsec 0.249 
0.3 < fracDev < 0.8 
& re,SDSS-cmodel > 1.5 arcsec 0.201 −0.034 0.015 
& re,SDSS-cmodel < 1.5 arcsec 0.182 
fracDev < 0.3 
& re,SDSS-cmodel > 1.5 arcsec 0.368 −0.110 0.021 
& re,SDSS-cmodel < 1.5 arcsec 0.231 
Sample Cm0 Cm1 Cm2 
fracDev > 0.8 −0.014 −0.001 
0.3 < fracDev < 0.8 
& re,SDSS-cmodel > 6 arcsec 0.147 −0.023 
& re,SDSS-cmodel < 6 arcsec 
fracDev < 0.3 0.001 −0.001 

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 panels of Fig. 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 corrections are again described by equations (3) and (4), with coefficients that are reported in Table 1.

The bottom panels of Fig. 2 show that these corrected quantities agree quite well with those from the full bulge+disc 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 estimates depend on the assumed initial mass function (IMF): Table 2 shows how we transform between different IMFs.

Table 2

M*/L IMF offsets used in this work. Offset is with respect to the Salpeter (1955) IMF: log10M*/L (IMF Salpeter) =log10M*/L (IMF reference) + offset.

IMF Offset (dex) Reference 
Kennicut 0.30 Kennicut (1983) 
Scalo 0.25 Scalo (1986) 
Diet-Salpeter 0.15 Bell & de Jong (2001) 
Pseudo-Kroupa 0.20 Kroupa (2001) 
Kroupa 0.30 Kroupa (2002) 
Chabrier 0.25 Chabrier (2003) 
Baldry & Glazbrook 0.305 Baldry & Glazebrook (2003) 
IMF Offset (dex) Reference 
Kennicut 0.30 Kennicut (1983) 
Scalo 0.25 Scalo (1986) 
Diet-Salpeter 0.15 Bell & de Jong (2001) 
Pseudo-Kroupa 0.20 Kroupa (2001) 
Kroupa 0.30 Kroupa (2002) 
Chabrier 0.25 Chabrier (2003) 
Baldry & Glazbrook 0.305 Baldry & Glazebrook (2003) 

The first comes from Bell et al. (2003), who report that, at z= 0, log10(M*/Lr)0= 1.097(gr)0+zp, where the zero-point zp depends on the IMF (see their appendix 2 and table 7). Their standard diet-Salpeter IMF has zp=−0.306, which they state has 70 per cent smaller M*/Lr at a given colour than a Salpeter IMF. In turn, a Salpeter IMF has 0.25 dex more M*/Lr at a given colour than the Chabrier (2003) IMF used by the other two groups whose mass estimates we use (see Table 2 for conversion of different IMFs). Therefore, we set zp=−0.306 + 0.15 − 0.25 =−0.406, making  

5
formula
We then obtain M*Bell by multiplying M*Bell/Lr by the SDSS r-band luminosity. When comparing with previous work, we usually use Petrosian magnitudes, although our final results are based on the cmodel magnitudes which we believe are superior.

Note that this expression requires luminosities and colours that have been k- and evolution-corrected to z= 0 (E. Bell, private communication). Unfortunately, these corrections are not available on an object-by-object basis. Bell et al. (2003) report that the absolute magnitudes brighten as 1.3z and gr colour becomes bluer as 0.3z. Although these estimates differ slightly from independent measurements of evolution by Bernardi et al. (2003) and Blanton et al. (2003), and more significantly from more recent determinations (Roche, Bernardi & Hyde 2009a), we use them, because they are the ones from which equation (5) was derived. Thus, in terms of rest-frame quantities,  

6
formula
If we use the rest-frame ri colour and Lr luminosity instead, then  
7
formula

These two estimates of M* will differ because there is scatter in the (gr) − (ri) colour plane. Unfortunately, Bell et al. do not provide a prescription which combines different colours. Although we could perform a straight average of these two estimates, this is less than ideal because the value of ri at fixed gr may provide additional information about M*/L. In practice, we will use the gr estimate as our standard, and ri to illustrate and quantify intrinsic uncertainties with the current approach.

The second estimate of M* is from Gallazzi et al. (2005). This is based on a likelihood analysis of the spectra, assumes the Chabrier (2003) IMF, and returns M*/Lz. The stellar mass M*Gallazzi is then computed using SDSS Petrosian z-band rest-frame magnitudes (i.e. they were k-corrected, but no evolution correction was applied). In this respect, they differ in philosophy from M*,Bell, in that the M* estimate is not corrected to z= 0. In practice, since we are mainly interested in small look-back times from z= 0, for which the expected mass-loss to the intergalactic medium is almost negligible, this almost certainly makes little difference for the most massive galaxies. These estimates are only available for 205 510 of the objects in our sample (∼82 per cent). The objects for which Gallazzi et al. do not provide stellar mass estimates are lower luminosity, typically lower mass objects; we show this explicitly in Fig. 3.

Figure 3

Stellar mass functions estimated from Petrosian magnitudes. Top: green open diamonds show the distribution of M* estimated from gr colours (equation 6) for the full SDSS sample; filled black circles show the same after removing outliers at L≥ 1011 L or M*≥ 1011.5 M (see Fig. 21), and filled grey circles use ri (equation 7) instead. Grey open squares use M* estimates from Gallazzi et al. (2005). Open red triangles show M*Petro from Blanton & Roweis (2007). Solid cyan line shows our fit to equation (9) with parameters reported in Table 4 for log10M*/M > 10.5. Solid grey curve shows the fit reported by Bell et al. (2003), dash–dotted blue curve that of Cole et al. (2001), dash–dot–dotted green curve that of Panter et al. (2007) and dotted red curve the fit from Li & White (2009), all transformed to H0= 70 km s−1 Mpc−1 and Chabrier (2003) IMF. Bottom: same as top panel, but now all quantities have been normalized by the Bell et al. (2003) fit (and the results corresponding to the green open diamonds are not shown).

Figure 3

Stellar mass functions estimated from Petrosian magnitudes. Top: green open diamonds show the distribution of M* estimated from gr colours (equation 6) for the full SDSS sample; filled black circles show the same after removing outliers at L≥ 1011 L or M*≥ 1011.5 M (see Fig. 21), and filled grey circles use ri (equation 7) instead. Grey open squares use M* estimates from Gallazzi et al. (2005). Open red triangles show M*Petro from Blanton & Roweis (2007). Solid cyan line shows our fit to equation (9) with parameters reported in Table 4 for log10M*/M > 10.5. Solid grey curve shows the fit reported by Bell et al. (2003), dash–dotted blue curve that of Cole et al. (2001), dash–dot–dotted green curve that of Panter et al. (2007) and dotted red curve the fit from Li & White (2009), all transformed to H0= 70 km s−1 Mpc−1 and Chabrier (2003) IMF. Bottom: same as top panel, but now all quantities have been normalized by the Bell et al. (2003) fit (and the results corresponding to the green open diamonds are not shown).

A final estimate of M*/Lr comes from Blanton & Roweis (2007), and is based on fitting the observed colours in all the SDSS bands to templates of a variety of star formation histories and metallicities, assuming the Chabrier (2003) IMF. In the following, we use the Blanton & Roweis stellar masses computed by applying the SDSS Petrosian and model (rest frame) r magnitudes to these mass-to-light estimates: M*Petro and M*Model. Blanton and Roweis also provide mass estimates from a template which is designed to match luminous red galaxies (LRGs); we call these M*LRG. Note that in this case M*/L is converted to M* using (rest-frame) model magnitudes only, since the Petrosian magnitude is well known to underestimate (by about 0.1 mag) the magnitudes of LRG-like objects (Blanton et al. 2001, DR7 documentation). To appreciate how different the LRG template is from the others, note that it allows ages of up to 10 Gyr, whereas that for the others, the age is more like 7 Gyr.

A detailed comparison of the different mass estimates is presented in Appendix A. This shows that to use the Blanton & Roweis (2007) masses, one must devise an algorithm for choosing between M*Model and M*LRG. In principle, some of the results to follow allow one to do this, but exploring this further is beyond the scope of the present paper. On the other hand, to use the Gallazzi et al. (2005) estimates, one must be wary of aperture effects. Finally, stellar masses based on the k+e corrected ri colour show stronger systematics than do the gr based estimates of M*. Therefore, in what follows, our preferred mass estimate will be that based on k+e corrected cmodel r magnitudes and gr colours (i.e. equation 6 with rest frame and evolution-corrected magnitudes and colours). Note that we believe the cmodel magnitudes to be far superior to the Petrosian ones. Of course, when we compare our results with previous work which used Petrosian magnitudes (Section 5), we do so too.

3 MORPHOLOGY AND SAMPLE SELECTION

This section compares a number of ways in which early-type samples have been defined in the recent literature, with the morphological classifications of Fukugita et al., and discusses what this implies for the ‘red’ fraction. When we show luminosities, they have been corrected for evolution by assuming that the magnitudes brighten with redshift as 1.3z.

3.1 Simple measures of the light profile

Concentration index, axis ratio, and fracDev have all been used as proxies for selecting red, massive, early-type galaxies. So it is interesting to see how these quantities correlate with morphological type.

The bottom-right panel of Fig. 4 shows the distribution of all objects in the Fukugita et al. sample in the space of concentration versus luminosity. The two horizontal lines show Cr= 2.6 and Cr= 2.86, the two most popular choices for selecting early-type samples. The symbols with error bars show the mean concentration index at each L if the sample is selected following Hyde & Bernardi (2009a): i.e. fracDev = 1 in g- and r-band, and r-band b/a > 0.6. To this we add the condition log10(re,g/re,r) < 0.15, which is essentially a cut on colour gradient (Roche, Bernardi & Hyde 2009b). This removes a small fraction (<2 per cent) of late-type galaxies which survive the other cuts. Dashed curves show the scatter around the Cr–luminosity relation in the Hyde & Bernardi sample.

Figure 4

Distribution of morphological types from Fukugita et al. (2007) in the space of Cr versus luminosity. Bottom-right panel shows the full sample. Horizontal lines (same in all panels) show two popular cuts for selecting early types. Text in each panel shows the fraction of objects in the panel which lie above these lines. Cyan symbols with error bars and flanked by dashed curves (same in each panel) show the median relation and the rms scatter defined by a sample selected following Hyde & Bernardi (2009).

Figure 4

Distribution of morphological types from Fukugita et al. (2007) in the space of Cr versus luminosity. Bottom-right panel shows the full sample. Horizontal lines (same in all panels) show two popular cuts for selecting early types. Text in each panel shows the fraction of objects in the panel which lie above these lines. Cyan symbols with error bars and flanked by dashed curves (same in each panel) show the median relation and the rms scatter defined by a sample selected following Hyde & Bernardi (2009).

The other panels show the result of separating out the various morphological types. Whereas Es and S0s occupy approximately the same region in this space, most Es (93 per cent) have Cr≥ 2.86, whereas the distribution of Cr for S0s is somewhat less peaked. (Here, the percentages we quote are per morphological type, in the Fukugita et. al. sample – meaning a sample that is magnitude limited to mr,Pet < 16, with no 1/Vmax weighting applied.) Samples restricted to Cr≥ 2.6 have a substantial contribution from both Sas (74 per cent of which satisfy this cut) and S0s (for which this fraction is 93 per cent), and this remains true even if Cr≥ 2.86 (50 per cent of Sas and 77 per cent of S0s). Thus, it is difficult to select a sample of Es on the basis of concentration index alone. On the other hand, the larger mean concentration of the Hyde & Bernardi selection cuts suggest that they produce a sample that is dominated by ellipticals/S0s, and less contaminated by Sas (see Section 3.5 and Table 3). Fig. 5 shows that replacing luminosity with stellar mass leads to similar conclusions.

Table 3

The morphological mix in differently selected samples, from V−1max weighted counts in the Fukugita et al. (2007) sample restricted to Mr < −19; numbers in brackets are from the raw counts.

Type HB09 Cr > 2.86 Cr > 2.6 
0.69 (0.73) 0.38 (0.51) 0.26 (0.43) 
S0 0.23 (0.20) 0.22 (0.23) 0.20 (0.22) 
Sa 0.07 (0.06) 0.25 (0.17) 0.30 (0.20) 
Sb 0.01 (0.01) 0.12 (0.07) 0.19 (0.12) 
Scd – 0.03 (0.02) 0.05 (0.03) 
Type HB09 Cr > 2.86 Cr > 2.6 
0.69 (0.73) 0.38 (0.51) 0.26 (0.43) 
S0 0.23 (0.20) 0.22 (0.23) 0.20 (0.22) 
Sa 0.07 (0.06) 0.25 (0.17) 0.30 (0.20) 
Sb 0.01 (0.01) 0.12 (0.07) 0.19 (0.12) 
Scd – 0.03 (0.02) 0.05 (0.03) 
Figure 5

Same as previous figure, but as a function of stellar mass rather than luminosity.

Figure 5

Same as previous figure, but as a function of stellar mass rather than luminosity.

Before moving on, note that although the mean concentration increases with luminosity and stellar mass in the Hyde & Bernardi sample, this is no longer the case at the highest Lr or M*: we will have more to say about this shortly.

Figs 6 and 7 show a similar analysis of the axis ratio b/a. The different symbols with error bars (same in all panels) show samples selected to have Cr≥ 2.6, Cr≥ 2.86, and following Hyde & Bernardi. At Mr≤−19 or so, the mean b/a in the first two cases increases with luminosity up to Mr=−22.7 or so; it decreases for the brightest objects. At low L, the sample with Cr≥ 2.6 has smaller values of b/a on average, though the scatter around the mean is large. Moreover, while there are essentially no Es with b/a < 0.6 about 26 per cent of S0s have b/a < 0.6. On the other hand, a little less than half the Sa and many Scd galaxies also have b/a > 0.6 (because they are face on). Evidently, just as Cr alone is not a good way to select a pure sample of ellipticals, selecting on b/a alone is not good either.

Figure 6

Similar to Fig. 4, but now in the space of b/a versus luminosity. Magenta open squares and green diamonds, each with error bars and flanked by dashed curves (same in all panels), show the samples selected using cuts in Cr larger of 2.6 and 2.86, respectively.

Figure 6

Similar to Fig. 4, but now in the space of b/a versus luminosity. Magenta open squares and green diamonds, each with error bars and flanked by dashed curves (same in all panels), show the samples selected using cuts in Cr larger of 2.6 and 2.86, respectively.

Figure 7

Same as previous figure, but as a function of stellar mass rather than luminosity.

Figure 7

Same as previous figure, but as a function of stellar mass rather than luminosity.

The filled red circles (with error bars) in the top-left panel show that, for Es, b/a≈ 0.85 independent of Mr, except at the highest luminosities where b/a decreases. This independence of Mr differs markedly from that in either of the Cr samples, but is reproduced by the Hyde & Bernardi sample, for which b/a= 0.8 except at Mr≤−22.7 where it decreases. The difference of about 0.05 in b/a arises because the Hyde & Bernardi sample includes some S0s (we quantify this in Table 3), for which b/a∼ 0.7 (filled red circles in top-right panel). This leads to an important point. While it has long been known that b/a tends to increase with luminosity, even in ‘early-type’ samples, our Fig. 6 shows that this increase is driven by the changing morphological mix – the change from S0s to Es – at Mr > −23. Whether this is due to environmental or pure secular evolution effects is an open question.

On the other hand, there is a plausible, environmentally driven model for the decrease in b/a at the highest L. This decrease has been expected for some time (see González-García & van Albada 2005; Boylan-Kolchin, Ma & Quataert 2006 and references therein) – it was first found by Bernardi et al. (2008). This is thought to indicate an increasing incidence of radial mergers, since these would tend to result in more prolate objects. The decrease in concentrations at these high luminosities (Fig. 4) is consistent with this picture, as is the fact that most of these high-luminosity objects are found in clusters. All of the preceding discussion remains true if one replaces luminosity with stellar mass.

Before concluding this section, we note that we have also considered the quantity fracDev which plays an important role in the selection cuts used by Hyde & Bernardi (2009). The vast majority of ellipticals (85 per cent) have fracDev = 1, with only a per cent or so having fracDev ≤ 0.8. The distribution of fracDev has a broader peak for S0s, but they otherwise cover the same range as Es: only 37 per cent have fracDev < 1. However, 70 per cent of Sas have fracDev < 1, whereas for Scs's, only 10 per cent have fracDev ≥ 0.4. (Note that the above percentages were computed in the magnitude-limited catalogue, i.e. were not weighted by 1/Vmax.)

3.2 Colours

In addition to simple measures based on the light profile, or more commonly, as an alternative to such methods, colour is sometimes used as a way to selecting early types. This is typically done by noting that the colour–magnitude distribution is bimodal (e.g. Baldry et al. 2004), and then adopting a crude approximation to this bimodality (e.g. Zehavi et al. 2005; Blanton & Berlind 2007; Skibba & Sheth 2009). Fig. 8 shows this bimodal distribution in the Fukugita et al. sample (Fig. 9 shows the corresponding colour–M* relation). The dotted green line in Fig. 8 shows  

8
formula
It runs approximately parallel to the ‘red’ sequence, and is similar to that obtained by subtracting −0.17 mag from equation (4) in Skibba & Sheth (2009); it is shallower than equation (7) in Skibba & Sheth (2009) or equation (1) of Young, Mo & van den Bosh (2009) (note that we k-correct to z= 0 and we use h= 0.7). ‘Red’ galaxies are those which lie above this line; ‘blue’ lie below it.

Figure 8

Colour–magnitude relation for the different morphological types in the Fukugita et al. sample. Solid and dashed curves (same in each panel) show the mean of the ‘red’ and ‘blue’ sequences, and their thickness, which result from performing double-Gaussian fits to the colour distribution at fixed magnitude, of the full SDSS galaxy sample (from Bernardi et al., in preparation). Dotted green line shows the luminosity-dependent threshold used to separate ‘red’ from ‘blue’ galaxies (equation 8).

Figure 8

Colour–magnitude relation for the different morphological types in the Fukugita et al. sample. Solid and dashed curves (same in each panel) show the mean of the ‘red’ and ‘blue’ sequences, and their thickness, which result from performing double-Gaussian fits to the colour distribution at fixed magnitude, of the full SDSS galaxy sample (from Bernardi et al., in preparation). Dotted green line shows the luminosity-dependent threshold used to separate ‘red’ from ‘blue’ galaxies (equation 8).

Figure 9

Same as previous figure, but now for stellar mass in place of luminosity.

Figure 9

Same as previous figure, but now for stellar mass in place of luminosity.

However, note that many late types (Sb and later) lie above this line – these tend to be edge-on discs. In addition, some Es lie below it. (See Huang & Gu 2009 for a more detailed analysis of such objects, which show either a star-forming active galactic nucleus or post-starburst spectrum.) We intend to present a more detailed study of the morphological dependence of the colour–magnitude, stellar mass and velocity dispersion relations in a future paper (Bernardi et al., in preparation). For our purposes here, we simply note that Figs 8 and 9 illustrate that cuts in colour are not a good way to select early-type galaxies.

3.3 Ages

Later in this paper, we will also study correlations between stellar age and galaxy mass, size and morphology. The age estimates we use, from Gallazzi et al. (2005), are based on a detailed analysis of spectral features. Since this is the same analysis that provided M*Gallazzi, errors in age and stellar mass are correlated (see Bernardi 2009 for a detailed discussion): erroneously large M* will tend to have erroneously large age as well.

Fig. 10 shows the age–M* correlation for the objects in the Fukugita et al. sample for which age estimates are available. This shows that, for any given morphological type, massive galaxies tend to be older (this correlation is not due to correlated errors). However, as expected, the later types tend to be substantially younger: Whereas two-thirds of the Es in this sample are older than 8 Gyr, only half of S0s, one-quarter of Sas, and fewer than 10 per cent of the later types (Sb, Sc, etc.) are this old. The bottom-right panel suggests that the age–M* distribution separates into two populations – one which is younger than about 7 Gyr and another which is older. However, this is not simply correlated with morphological type: the top panels show that this bimodality is also present in the E and S0s samples.

Figure 10

Joint distribution of age and stellar mass in the Fukugita et al. (2007) sample. Cyan-filled circles, green diamonds and magenta squares show the median age at fixed stellar mass for a subsample selected following Hyde & Bernardi (2009), a subsample with Cr > 2.86, and a subsample with Cr > 2.6, respectively. The dashed lines show the 1σ range around the median.

Figure 10

Joint distribution of age and stellar mass in the Fukugita et al. (2007) sample. Cyan-filled circles, green diamonds and magenta squares show the median age at fixed stellar mass for a subsample selected following Hyde & Bernardi (2009), a subsample with Cr > 2.86, and a subsample with Cr > 2.6, respectively. The dashed lines show the 1σ range around the median.

To study this further, Fig. 11 shows the age–M* distribution in a random subsample of the galaxies selected following Hyde & Bernardi (2009) from the full SDSS catalogue. Note that although 90 per cent of the objects are older than 6 Gyr, this selection clearly includes a population of younger objects. This population of ‘rejuvenated’ early-type galaxies has been the subject of some recent interest (e.g. Huang & Gu 2009; Thomas et al. 2010). Cyan-filled circles show that the median age increases with stellar mass and dashed lines show the 1σ range around the median. Green diamonds and magenta squares show the median age at fixed stellar mass for objects with Cr > 2.86 and Cr > 2.6, respectively. At large M*, both these samples produce similar age–M* relations to the Hyde–Bernardi sample; at smaller M*, allowing smaller Cr includes younger galaxies. These median relations are superimposed on the panels of Fig. 10.

Figure 11

Joint distribution of age and stellar mass in the full sample. Small dots show a random subsample of the galaxies selected following Hyde & Bernardi (2009). Cyan-filled circles, green diamonds and magenta squares show the median age at fixed stellar mass for a subsample selected following Hyde & Bernardi (2009), a subsample with Cr > 2.86 and a subsample with Cr > 2.6, respectively. The dashed lines show the 1σ range around the median. The fraction of galaxies with luminosity weighted ages older than 8 Gyr and younger than 6 Gyr, for each of the selection methods, are shown.

Figure 11

Joint distribution of age and stellar mass in the full sample. Small dots show a random subsample of the galaxies selected following Hyde & Bernardi (2009). Cyan-filled circles, green diamonds and magenta squares show the median age at fixed stellar mass for a subsample selected following Hyde & Bernardi (2009), a subsample with Cr > 2.86 and a subsample with Cr > 2.6, respectively. The dashed lines show the 1σ range around the median. The fraction of galaxies with luminosity weighted ages older than 8 Gyr and younger than 6 Gyr, for each of the selection methods, are shown.

3.4 The red and blue fractions

There is considerable interest in the ‘build-up’ of the red sequence, and the possibility that some of the objects in the blue cloud were ‘transformed’ into redder objects. We now compare estimates of the red or blue fraction that are based on colour and concentration, with ones based on morphology. In particular, we show how these fractions vary as a function of L, M* and σ.

All the results which follow are based on samples which are Petrosian magnitude limited, so in all the statistics we present, each object is weighted by V−1max (L), the inverse of the maximum volume to which it could have been seen. This magnitude limit is fainter for the full sample (mPet= 17.75) than it is for the Fukugita et al. subsample (mPet= 16), and note that Vmax depends on our model for how the luminosities evolve (absolute magnitudes brighten as 1.3z).

The panels on the left of Fig. 12 show how the mix of objects changes as a function of luminosity, stellar mass and velocity dispersion, for the crude but popular hard cuts in concentration and colour (as described in the previous sections).

Figure 12

Left-hand panels: the ‘red’ fraction in the full sample (mr,Pet < 17.5), where being red means the object lies redward of a luminosity-dependent threshold which runs parallel to the ‘red’ sequence (orange stars); or has concentration greater than 2.6 (magenta squares); or greater than 2.86 (green diamonds); or satisfied the Hyde & Bernardi selection cuts (red crosses). Also shown is the result of combining the colour cut with one on b/a (brown triangles). Right-hand panels: the fraction of objects of a given L, M*, σ and Re, as later and later morphological types are added to the Fukugita et al. sample (mr,Pet < 16). In all cases, each object was weighted by V−1max (L), the maximum volume to which it could have been seen, given the apparent magnitude limit.

Figure 12

Left-hand panels: the ‘red’ fraction in the full sample (mr,Pet < 17.5), where being red means the object lies redward of a luminosity-dependent threshold which runs parallel to the ‘red’ sequence (orange stars); or has concentration greater than 2.6 (magenta squares); or greater than 2.86 (green diamonds); or satisfied the Hyde & Bernardi selection cuts (red crosses). Also shown is the result of combining the colour cut with one on b/a (brown triangles). Right-hand panels: the fraction of objects of a given L, M*, σ and Re, as later and later morphological types are added to the Fukugita et al. sample (mr,Pet < 16). In all cases, each object was weighted by V−1max (L), the maximum volume to which it could have been seen, given the apparent magnitude limit.

Fig. 12 shows that the fraction of objects which satisfies the criteria used by Hyde & Bernardi (2009) increases with increasing L, M* and σ, except at the largest values (about which, more later). Requiring Cr≥ 2.86 instead results in approximately 5 to 10 per cent more objects (compared to the Hyde & Bernardi cuts) at each L or M*; although, in the case of σ∼ 300 km s−1, this cut allows about 20 per cent more objects. Relaxing the cut to Cr≥ 2.6 allows an additional 15 per cent, with slightly more at intermediate L and M*.

Selecting objects redder than a luminosity-dependent threshold (equation 8) which runs parallel to the ‘red’ sequence allows even more objects into the sample, but combining the colour cut with one on b/a reduces the sample to one which resembles Cr≤ 2.86 rather well. The cut in b/a is easy to understand, since edge-on discs will lie redward of the colour cut even though they are not early types – the additional cut on b/a is an easy (but rarely used!) way to remove them.

It is interesting to compare these panels with their counterparts on the right of Fig. 12, in which later and later morphological types are added to the Fukugita et al. subsample which initially only contains Es. This suggests that the Hyde & Bernardi selection will be dominated by E, Cr≥ 2.86 will be dominated by E+S0s, and Cr≥ 2.6 will be dominated by E+S0+Sas. We quantify this in the next subsection. Fig. 13 shows a similar comparison with the blue fraction. Note that the contamination of the red fraction by edge-on discs is a large effect: 60 per cent of the objects at log10(M*/M) = 10.5 are classified as being red, when E+S0+Sas sum to only 40 per cent. Fig. 14 shows this more directly: the reddest objects at intermediate luminosities are late-, not early-type galaxies.

Figure 13

Left-hand panels: same as left-hand panel of previous figure, but now showing the ‘blue’ fraction – the objects which did not qualify as being ‘red’. Right-hand panels: similar to right-hand panel of previous figure, but now starting with later types and adding successively more early types.

Figure 13

Left-hand panels: same as left-hand panel of previous figure, but now showing the ‘blue’ fraction – the objects which did not qualify as being ‘red’. Right-hand panels: similar to right-hand panel of previous figure, but now starting with later types and adding successively more early types.

Figure 14

Dependence of the bimodality in the colour distribution on morphological type, for a few bins in luminosity. Dashed histogram shows the distribution of the SDSS sample (mr,Pet < 17.5), while solid histograms show the distribution in the Fukugita et al. sample (mr,Pet < 16) as later and later morphological types are added. Note that the reddest objects at intermediate luminosity are late-type galaxies.

Figure 14

Dependence of the bimodality in the colour distribution on morphological type, for a few bins in luminosity. Dashed histogram shows the distribution of the SDSS sample (mr,Pet < 17.5), while solid histograms show the distribution in the Fukugita et al. sample (mr,Pet < 16) as later and later morphological types are added. Note that the reddest objects at intermediate luminosity are late-type galaxies.

Before concluding this section, we note that both concentration cuts greatly underpredict the red fraction of the most luminous or massive objects, as does the application of a b/a cut to the straight colour selection or the Hyde & Bernardi selection. The most luminous or massive elliptical galaxies in the Fukugita et al. sample show the same behaviour. That is, the most massive objects are less concentrated for their luminosities than one might have expected by extrapolation from lower luminosities and their light profile is not well represented by a pure deVaucoleur law. This is consistent with results in the previous section where, at the highest luminosities, b/a tends to decrease with luminosity (Fig. 6). These trends suggest an increasing incidence of recent radial mergers for the most luminous and massive galaxies.

3.5 Distribution of morphological types in differently selected samples

Much of the previous analysis suggests that the Hyde & Bernardi selection will produce a sample that is dominated by Es, Cr≥ 2.86 will include more S0s and Sas, and Cr≥ 2.6 will include Sas and later types. Table 3 shows the distribution of types in subsamples selected from the Fukugita et al. (2007) catalogue to have Cr≥ 2.6, Cr≥ 2.86 and following Hyde & Bernardi (2009). Of the 1596 objects in the magnitude-limited catalogue, 1009, 802 and 470 satisfy these cuts. The table shows that, in samples where Cr≥ 2.6, 54 per cent of the objects are Sa or later. This fraction falls to 40 per cent for Cr≥ 2.86 and to less than 10 per cent for the Hyde–Bernardi cuts (these numbers are obtained after weighting by V−1max, so they do not depend on the selection effect associated with the apparent magnitude limit of the catalogue).

These numbers indicate that Es comprise at least two-thirds of the Hyde–Bernardi sample, but in a sample where Cr≥ 2.86, to reach this fraction one must include S0s, and if Cr≥ 2.6, then reaching this fraction requires adding Sas as well. Stated differently, Es comprise more than two-thirds of a Hyde–Bernardi sample, but about one-third of a sample with Cr≥ 2.86 and one-quarter of a sample with Cr≥ 2.6. If we weight each object by its stellar mass, then (E+S0)s account for (72+21) per cent of the total stellar mass in a Hyde & Bernardi sample, (47+23) per cent in a sample with Cr≥ 2.86 and (39+23) per cent if Cr≥ 2.6. These differences will be important in Section 6.

4 DISTRIBUTIONS FOR SAMPLES CUT BY MORPHOLOGY OR CONCENTRATION

We now show how the luminosity, stellar mass, size and velocity dispersion distributions –φ(L), φ(M*), φ(Re) and φ(σ)– depend on how the sample was defined. We use the same popular cuts in concentration as in the previous sections, Cr≥ 2.86 and Cr≥ 2.6, which we suggested might be similar to selecting early-type samples which, in addition to Es, include S0s + Sas, and S0s + Sas + Sbs, respectively. We then make similar measurements in the Fukugita et al. subsample, to see if this correspondence is indeed good.

In this section, we use cmodel rather than Petrosian quantities, for the reasons stated earlier. The only place where we continue to use a Petrosian-based quantity is when we define subsamples based on concentration, since Cr is the ratio of two Petrosian-based sizes, or for comparison with results from previous work.

The results which follow are based on samples which are Petrosian magnitude limited, so in all the statistics we present, each object is weighted by V−1max (L), the inverse of the maximum volume to which it could have been seen. In addition to depending on the magnitude limit (mPet≤ 17.5 for the full sample, and mPet≤ 16 for the Fukugita subsample), the weight V−1max (L) also depends on our model for how the luminosities have evolved. A common test of the accuracy of the evolution model is to see how 〈V/Vmax〉, the ratio of the volume to which an object was seen to that which it could have been seen, averaged over all objects, differs from 0.5. In the full sample, it is 0.506 for our assumption that the absolute magnitudes evolve as 1.3z; had we used 1.62z (Blanton et al. 2003), it would have been 0.509. On the other hand, if we had ignored evolution entirely, it would have been 0.527.

In addition, SDSS fibre collisions mean that spectra were not taken for about 7 per cent of the objects which satisfy mPet≤ 17.5. We account for this by dividing our V−1max weighted counts by a factor of 0.93. This ignores the fact that fibre collisions matter more in crowded fields (such as clusters); so in principle, this correction factor has some scatter, which may depend on morphological type. We show below that, when we ignore this scatter, then our analysis of the full sample produces results that are in good agreement with those of Blanton et al. (2003), who account for the fact that this factor varies spatially. This suggests that the spatial dependence is small, so, in what follows, we ignore the fact that it (almost certainly) depends on morphological type.

4.1 Parametric form for the intrinsic distribution

We will summarize the shapes of the distributions we find by using the functional form  

9
formula
This is the form used by Sheth et al. (2003) to fit the distribution of velocity dispersions; it is a generalization of the Schechter function commonly fit to the luminosity function (which has β= 1, a slightly different definition of α). We have found that the increased flexibility which β≠ 1 allows is necessary for most of the distributions which follow. This is not unexpected: at fixed luminosity, most of the observables we study below scatter around a mean value which scales as a power law in luminosity (e.g. Bernardi et al. 2003; Hyde & Bernardi 2009). Because this mean does not scale linearly with L, and because the scatter around the mean can be significant, then if φ(L) is well fit by a Schechter function, it makes little physical or statistical sense to fit the other observables with a Schechter function as well.

4.2 Effect of measurement errors

In practice, we will also be interested in the effect of measurement errors on the shape of the distribution. If the errors are lognormal (Gaussian in ln X) with a small dispersion, then the observed distribution is related to the intrinsic one by  

10
formula
 
11
formula
The peak of Xφ(X) occurs at Xmax=X*(α/β)1/β, where Cmax=−αβ. Since α and β are usually positive, errors typically act to decrease the height of the peak. Since the net effect of errors is to broaden the distribution, hence extending the tails, errors also tend to decrease β. The expression above shows that, in the O/O*≫ 1 tail, errors matter more when β is large.

Fitting to equation (10) rather than to equation (9) is a crude but effective way to estimate the intrinsic shape (i.e. to remove the effect of measurement errors on the fitted parameters), provided the measurement errors are small. The rms errors on (ln Lr, ln M*, ln Re, ln σ) are indeed small: σerr= (0.05, 0.25, 0.15, 0.15), and so it is the results of these fits which we report in what follows. However, to illustrate which distributions are most affected by measurement error, we also show results from fitting to equation (9); in most cases, the differences between the returned parameters are small, except when β > 1.

In practice, the fitting was done by minimizing  

12
formula
where yi was log10 of the V−1max weighted count in the ith logarithmically spaced bin (and recall that, because of fibre collisions, the weight is actually V−1max/0.93).

4.3 Covariances between fitted parameters

When fitting to equation (9), a reasonable understanding of the covariance between the fitted parameters can be got by asking that all parameter combinations give the same mean density (Sheth et al. 2003):  

13
formula
In practice, φ* is determined essentially independently of the other parameters, so it is the other three parameters which are covariant. Hence, it is convenient to define  
14
formula
If β is fixed to unity, then this becomes 〈X〉=X*α. But if β is not fixed, then a further constraint equation can be got by asking that all fits return the same peak position or height. For distributions with broad peaks, it may be better to instead require that the second central moment,  
15
formula
be well reproduced. Thus, the covariance between α and β is given by requiring that σX/〈X〉 equals the measured value for this ratio. The changes to these correlations are sufficiently small if we instead fit to equation (10), so we have not presented the algebra here.

4.4 Distributions for samples cut by concentration

Fig. 15 shows φ(L) and φ(M*) in the full sample (top curve in top panel), when one removes objects with Cr < 2.6 (second from top), objects with Cr < 2.86 (third from top), and when one selects early types on the basis of a number of other criteria (bottom, following Hyde & Bernardi 2009). For comparison, the dotted curves show the measurement in the full sample when Petrosian quantities are used (i.e. from Figs 3 and 16).

Figure 15

The effect of different selection cuts on the cmodel luminosity and stellar mass functions: black crosses, magenta triangles, green diamonds and blue squares show the full sample, a subsample with Cr > 2.6, a subsample with Cr > 2.86, and a subsample selected following Hyde & Bernardi (2009). Top panels: smooth solid curves show the fits to the observed distributions (equation 9) while dashed curves (almost indistinguishable from solid curves) show the intrinsic distributions (equation 10). Cyan solid line shows our fit to the full sample for Mr < −20 and M* > 1010 M. The parameters of these fits are reported in Tables B1–B4. Red dotted line shows the measurement associated with Petrosian quantities (i.e. from Figs 3 and 16). Bottom panels: same as top panel, but now each set of data points and dashed curve are shown after dividing by their associated solid curve.

Figure 15

The effect of different selection cuts on the cmodel luminosity and stellar mass functions: black crosses, magenta triangles, green diamonds and blue squares show the full sample, a subsample with Cr > 2.6, a subsample with Cr > 2.86, and a subsample selected following Hyde & Bernardi (2009). Top panels: smooth solid curves show the fits to the observed distributions (equation 9) while dashed curves (almost indistinguishable from solid curves) show the intrinsic distributions (equation 10). Cyan solid line shows our fit to the full sample for Mr < −20 and M* > 1010 M. The parameters of these fits are reported in Tables B1–B4. Red dotted line shows the measurement associated with Petrosian quantities (i.e. from Figs 3 and 16). Bottom panels: same as top panel, but now each set of data points and dashed curve are shown after dividing by their associated solid curve.

Figure 16

Luminosity function in the r band determined from Petrosian apparent magnitudes in the range 14.5 ≤mr,Pet≤ 17.5. Red line shows the fit reported by Bell et al. (2003). Cyan line shows our fit to equation (9) for log10M*/M > 10.5, with parameters reported in Table 4.

Figure 16

Luminosity function in the r band determined from Petrosian apparent magnitudes in the range 14.5 ≤mr,Pet≤ 17.5. Red line shows the fit reported by Bell et al. (2003). Cyan line shows our fit to equation (9) for log10M*/M > 10.5, with parameters reported in Table 4.

The solid curves show the result of fitting to equation (9), and the dashed curves (almost indistinguishable from the solid ones, except at the most massive end) result from fitting to equation (10) instead, so as to remove the effects of measurement error on our estimate of the shape of the intrinsic distribution. To reduce the dynamic range, bottom panels show each set of curves divided by the associated solid curve (i.e. by the fit to the observed sample). The dotted lines in these bottom panels show that Petrosian-based counts lie well below those based on cmodel quantities at Mr≤−20 or log10M*/M≥ 10.6. The dashed lines in the bottom-right panel show that the intrinsic distribution has been notably broadened by errors above log10M*/M≥ 11.

Fig. 17 shows a similar analysis of φ(Re) and φ(σ). For φ(σ) we also compare our results with those of Sheth et al. (2003). The Sheth et al. analysis was based on a sample selected by Bernardi et al. (2003), which was more like Cr > 2.86 at large masses, but because of cuts on emission lines and S/N in the spectra, had few low-mass objects. Indeed, at large σ, our measured φ(σ) is similar to theirs.

Figure 17

Same as previous figure, but now for the size and velocity dispersion. Cyan solid line shows our fit to the full sample for 1.5 < Re/kpc < 20 and σ > 125 km s−1. The top-right panel also shows the fits obtained by Sheth et al. (2003) to the observed (solid grey line) and intrinsic (dashed grey) φ(σ) distribution.

Figure 17

Same as previous figure, but now for the size and velocity dispersion. Cyan solid line shows our fit to the full sample for 1.5 < Re/kpc < 20 and σ > 125 km s−1. The top-right panel also shows the fits obtained by Sheth et al. (2003) to the observed (solid grey line) and intrinsic (dashed grey) φ(σ) distribution.

Tables B1–B4 report the parameters of the fits (to equations 9 and 10) shown in Figs 15 and 17. It is worth noting that, for a given sample, the fits return essentially the same value of φ* in all the tables, even though this was not explicitly required. And note that φ(M*) and φ(σ) are the distributions that are most sensitive to errors; the former because the errors are large, and the latter because β is large.

Table B1

Best-fitting parameters of equation (10) to the measured r-band cmodel luminosity function φ(L). Values between round brackets show the parameters obtained fitting equation (9) to the data ignoring measurement errors.

Sample φ*/10−2 Mpc−3 L*/109 L α β ρL/109 L Mpc−3 
HB09 (0.095) 0.095 ± 0.005 (10.69) 10.99 ± 4.06 (1.38) 1.37 ± 0.18 (0.769) 0.776 ± 0.071 0.024 
CI > 2.86 (0.174) 0.174 ± 0.010 (6.96) 7.21 ± 2.52 (1.38) 1.37 ± 0.16 (0.692) 0.698 ± 0.051 0.038 
CI > 2.6 (0.382) 0.382 ± 0.022 (3.16) 3.28 ± 1.30 (1.32) 1.31 ± 0.16 (0.583) 0.588 ± 0.039 0.061 
All (4.123) 4.196 ± 1.693 (12.21) 12.48 ± 3.42 (0.20) 0.19 ± 0.10 (0.728) 0.734 ± 0.051 0.135 
All (Mr < −20) (1.227) 1.197 ± 0.240 (2.08) 1.88 ± 0.86 (1.07) 1.12 ± 0.24 (0.540) 0.533 ± 0.035 0.125 
F07-E 0.13 ± 0.11 72.10 ± 34.38 0.320 ± 0.444 1.752 ± 0.750 0.022 
F07-S0 1.05 ± 0.65 64.05 ± 10.15 0.065 ± 0.046 1.462 ± 0.203 0.038 
F07-Sa 0.80 ± 0.07 27.19 ± 1.15 0.254 ± 0.020 0.944 ± 0.023 0.058 
F07-Sb 17.22 ± 1.46 36.01 ± 1.35 0.014 ± 0.001 1.079 ± 0.030 0.084 
F07-Scd 23.48 ± 1.93 28.37 ± 1.04 0.017 ± 0.001 0.980 ± 0.023 0.112 
F07-All 16.00 ± 8.98 25.67 ± 4.87 0.027 ± 0.016 0.934 ± 0.086 0.116 
Sample φ*/10−2 Mpc−3 L*/109 L α β ρL/109 L Mpc−3 
HB09 (0.095) 0.095 ± 0.005 (10.69) 10.99 ± 4.06 (1.38) 1.37 ± 0.18 (0.769) 0.776 ± 0.071 0.024 
CI > 2.86 (0.174) 0.174 ± 0.010 (6.96) 7.21 ± 2.52 (1.38) 1.37 ± 0.16 (0.692) 0.698 ± 0.051 0.038 
CI > 2.6 (0.382) 0.382 ± 0.022 (3.16) 3.28 ± 1.30 (1.32) 1.31 ± 0.16 (0.583) 0.588 ± 0.039 0.061 
All (4.123) 4.196 ± 1.693 (12.21) 12.48 ± 3.42 (0.20) 0.19 ± 0.10 (0.728) 0.734 ± 0.051 0.135 
All (Mr < −20) (1.227) 1.197 ± 0.240 (2.08) 1.88 ± 0.86 (1.07) 1.12 ± 0.24 (0.540) 0.533 ± 0.035 0.125 
F07-E 0.13 ± 0.11 72.10 ± 34.38 0.320 ± 0.444 1.752 ± 0.750 0.022 
F07-S0 1.05 ± 0.65 64.05 ± 10.15 0.065 ± 0.046 1.462 ± 0.203 0.038 
F07-Sa 0.80 ± 0.07 27.19 ± 1.15 0.254 ± 0.020 0.944 ± 0.023 0.058 
F07-Sb 17.22 ± 1.46 36.01 ± 1.35 0.014 ± 0.001 1.079 ± 0.030 0.084 
F07-Scd 23.48 ± 1.93 28.37 ± 1.04 0.017 ± 0.001 0.980 ± 0.023 0.112 
F07-All 16.00 ± 8.98 25.67 ± 4.87 0.027 ± 0.016 0.934 ± 0.086 0.116 
Table B2

Best-fitting parameters of equation (10) to the measured stellar mass function φ(M*).

Sample φ*/10−2 Mpc−3 M*/109 M α β ρ*/109 M Mpc−3 
HB09 (0.095) 0.095 ± 0.005 (20.25) 25.24 ± 8.70 (1.29) 1.28 ± 0.14 (0.641) 0.696 ± 0.051 0.066 
CI > 2.86 (0.174) 0.174 ± 0.009 (13.62) 17.62 ± 6.14 (1.26) 1.24 ± 0.13 (0.590) 0.640 ± 0.043 0.105 
CI > 2.6 (0.388) 0.389 ± 0.020 (7.49) 10.07 ± 3.89 (1.10) 1.07 ± 0.12 (0.522) 0.563 ± 0.036 0.166 
All (5.285) 5.850 ± 3.054 (34.88) 39.06 ± 8.81 (0.11) 0.09 ± 0.06 (0.650) 0.694 ± 0.040 0.289 
All (M* > 3 × 1010 M(0.761) 0.672 ± 0.123 (0.14) 0.02 ± 0.01 (1.95) 2.68 ± 0.30 (0.342) 0.308 ± 0.010 0.261 
F07-E 0.09 ± 0.04 158.43 ± 115.34 0.54 ± 0.54 1.31 ± 0.53 0.062 
F07-S0 1.11 ± 0.66 206.53 ± 35.47 0.05 ± 0.04 1.44 ± 0.25 0.104 
F07-Sa 1.70 ± 2.52 144.73 ± 45.50 0.07 ± 0.11 1.18 ± 0.23 0.151 
F07-Sb 1.46 ± 1.31 90.24 ± 46.21 0.15 ± 0.18 0.96 ± 0.21 0.203 
F07-Scd 4.09 ± 5.46 81.34 ± 30.52 0.07 ± 0.11 0.92 ± 0.15 0.248 
F07-All 6.76 ± 9.92 84.20 ± 24.84 0.04 ± 0.07 0.93 ± 0.13 0.251 
Sample φ*/10−2 Mpc−3 M*/109 M α β ρ*/109 M Mpc−3 
HB09 (0.095) 0.095 ± 0.005 (20.25) 25.24 ± 8.70 (1.29) 1.28 ± 0.14 (0.641) 0.696 ± 0.051 0.066 
CI > 2.86 (0.174) 0.174 ± 0.009 (13.62) 17.62 ± 6.14 (1.26) 1.24 ± 0.13 (0.590) 0.640 ± 0.043 0.105 
CI > 2.6 (0.388) 0.389 ± 0.020 (7.49) 10.07 ± 3.89 (1.10) 1.07 ± 0.12 (0.522) 0.563 ± 0.036 0.166 
All (5.285) 5.850 ± 3.054 (34.88) 39.06 ± 8.81 (0.11) 0.09 ± 0.06 (0.650) 0.694 ± 0.040 0.289 
All (M* > 3 × 1010 M(0.761) 0.672 ± 0.123 (0.14) 0.02 ± 0.01 (1.95) 2.68 ± 0.30 (0.342) 0.308 ± 0.010 0.261 
F07-E 0.09 ± 0.04 158.43 ± 115.34 0.54 ± 0.54 1.31 ± 0.53 0.062 
F07-S0 1.11 ± 0.66 206.53 ± 35.47 0.05 ± 0.04 1.44 ± 0.25 0.104 
F07-Sa 1.70 ± 2.52 144.73 ± 45.50 0.07 ± 0.11 1.18 ± 0.23 0.151 
F07-Sb 1.46 ± 1.31 90.24 ± 46.21 0.15 ± 0.18 0.96 ± 0.21 0.203 
F07-Scd 4.09 ± 5.46 81.34 ± 30.52 0.07 ± 0.11 0.92 ± 0.15 0.248 
F07-All 6.76 ± 9.92 84.20 ± 24.84 0.04 ± 0.07 0.93 ± 0.13 0.251 
Table B3

Best-fitting parameters of equation (10) to the measured r-band cmodel size function φ(Re).

Sample φ*/10−2 Mpc−3 R*/kpc α β R〉/kpc 
HB09 (0.093) 0.093 ± 0.006 (0.0001) 0.0003 ± 0.0002 (8.70) 9.29 ± 0.45 (0.293) 0.294 ± 0.007 3.25 
CI > 2.86 (0.173) 0.173 ± 0.011 (0.0001) 0.0002 ± 0.0002 (9.82) 10.73 ± 0.53 (0.294) 0.290 ± 0.007 3.19 
CI > 2.6 (0.400) 0.400 ± 0.024 (0.0001) 0.0002 ± 0.0002 (7.58) 8.08 ± 0.38 (0.295) 0.295 ± 0.006 2.74 
All (6.009) 6.040 ± 0.766 (0.4838) 0.5809 ± 0.1773 (1.31) 1.27 ± 0.23 (0.688) 0.729 ± 0.045 1.41 
All (1.5 < Re < 20 kpc) (2.951) 2.858 ± 0.420 (0.5523) 0.5612 ± 0.2033 (2.22) 2.39 ± 0.44 (0.765) 0.790 ± 0.058 2.38 
F07-E 0.10 ± 0.02 0.0007 ± 0.0009 6.44 ± 1.55 0.352 ± 0.032 3.13 
F07-S0 0.19 ± 0.03 0.0008 ± 0.0008 6.07 ± 1.05 0.356 ± 0.025 2.78 
F07-Sa 0.37 ± 0.06 0.0004 ± 0.0004 6.08 ± 1.01 0.337 ± 0.021 2.46 
F07-Sb 0.83 ± 0.15 0.0123 ± 0.0104 4.24 ± 0.82 0.444 ± 0.037 2.30 
F07-Scd 1.55 ± 0.21 0.0062 ± 0.0042 5.72 ± 0.76 0.431 ± 0.026 2.80 
F07-All 3.92 ± 0.77 0.0049 ± 0.0033 4.02 ± 0.64 0.402 ± 0.024 1.81 
Sample φ*/10−2 Mpc−3 R*/kpc α β R〉/kpc 
HB09 (0.093) 0.093 ± 0.006 (0.0001) 0.0003 ± 0.0002 (8.70) 9.29 ± 0.45 (0.293) 0.294 ± 0.007 3.25 
CI > 2.86 (0.173) 0.173 ± 0.011 (0.0001) 0.0002 ± 0.0002 (9.82) 10.73 ± 0.53 (0.294) 0.290 ± 0.007 3.19 
CI > 2.6 (0.400) 0.400 ± 0.024 (0.0001) 0.0002 ± 0.0002 (7.58) 8.08 ± 0.38 (0.295) 0.295 ± 0.006 2.74 
All (6.009) 6.040 ± 0.766 (0.4838) 0.5809 ± 0.1773 (1.31) 1.27 ± 0.23 (0.688) 0.729 ± 0.045 1.41 
All (1.5 < Re < 20 kpc) (2.951) 2.858 ± 0.420 (0.5523) 0.5612 ± 0.2033 (2.22) 2.39 ± 0.44 (0.765) 0.790 ± 0.058 2.38 
F07-E 0.10 ± 0.02 0.0007 ± 0.0009 6.44 ± 1.55 0.352 ± 0.032 3.13 
F07-S0 0.19 ± 0.03 0.0008 ± 0.0008 6.07 ± 1.05 0.356 ± 0.025 2.78 
F07-Sa 0.37 ± 0.06 0.0004 ± 0.0004 6.08 ± 1.01 0.337 ± 0.021 2.46 
F07-Sb 0.83 ± 0.15 0.0123 ± 0.0104 4.24 ± 0.82 0.444 ± 0.037 2.30 
F07-Scd 1.55 ± 0.21 0.0062 ± 0.0042 5.72 ± 0.76 0.431 ± 0.026 2.80 
F07-All 3.92 ± 0.77 0.0049 ± 0.0033 4.02 ± 0.64 0.402 ± 0.024 1.81 
Table B4

Best-fitting parameters of equation (10) to the measured velocity dispersion function φ(σ).

Sample φ*/10−2 Mpc−3 σ*/km s−1 α β 〈σ〉/km s−1 
HB09 (0.097) 0.096 ± 0.006 (184.08) 173.41 ± 16.85 (2.44) 2.83 ± 0.41 (2.91) 3.10 ± 0.35 149.15 
CI > 2.86 (0.182) 0.179 ± 0.010 (177.34) 166.50 ± 17.02 (2.17) 2.54 ± 0.41 (2.76) 2.93 ± 0.33 139.14 
CI > 2.6 (0.663) 0.590 ± 0.088 (190.57) 175.96 ± 16.88 (0.80) 1.06 ± 0.34 (2.86) 2.99 ± 0.35 92.40 
All (2.099) 24.736 ± 8.376 (113.78) 140.96 ± 7.09 (0.94) 0.07 ± 0.02 (1.85) 2.22 ± 0.13  7.91 
All (σ > 125 km s−1(8.133) 2.611 ± 0.161 (176.99) 159.57 ± 1.48 (0.11) 0.41 ± 0.02 (2.54) 2.59 ± 0.04 44.02 
F07-E 0.11 ± 0.04 218.27 ± 43.06 1.53 ± 1.21 4.47 ± 1.90 131.56 
F07-S0 0.19 ± 0.05 197.35 ± 58.97 1.66 ± 1.25 3.49 ± 1.75 128.40 
F07-Sa 0.41 ± 0.13 193.07 ± 44.05 1.06 ± 0.77 3.35 ± 1.29 99.32 
F07-Sb 0.61 ± 0.25 158.18 ± 59.10 1.22 ± 1.14 2.57 ± 1.05 93.12 
F07-Scd 2.47 ± 3.96 180.45 ± 33.69 0.29 ± 0.54 2.96 ± 0.85 37.48 
F07-All 1.23 ± 0.31 59.04 ± 31.48 2.44 ± 1.17 1.35 ± 0.30 86.74 
Sample φ*/10−2 Mpc−3 σ*/km s−1 α β 〈σ〉/km s−1 
HB09 (0.097) 0.096 ± 0.006 (184.08) 173.41 ± 16.85 (2.44) 2.83 ± 0.41 (2.91) 3.10 ± 0.35 149.15 
CI > 2.86 (0.182) 0.179 ± 0.010 (177.34) 166.50 ± 17.02 (2.17) 2.54 ± 0.41 (2.76) 2.93 ± 0.33 139.14 
CI > 2.6 (0.663) 0.590 ± 0.088 (190.57) 175.96 ± 16.88 (0.80) 1.06 ± 0.34 (2.86) 2.99 ± 0.35 92.40 
All (2.099) 24.736 ± 8.376 (113.78) 140.96 ± 7.09 (0.94) 0.07 ± 0.02 (1.85) 2.22 ± 0.13  7.91 
All (σ > 125 km s−1(8.133) 2.611 ± 0.161 (176.99) 159.57 ± 1.48 (0.11) 0.41 ± 0.02 (2.54) 2.59 ± 0.04 44.02 
F07-E 0.11 ± 0.04 218.27 ± 43.06 1.53 ± 1.21 4.47 ± 1.90 131.56 
F07-S0 0.19 ± 0.05 197.35 ± 58.97 1.66 ± 1.25 3.49 ± 1.75 128.40 
F07-Sa 0.41 ± 0.13 193.07 ± 44.05 1.06 ± 0.77 3.35 ± 1.29 99.32 
F07-Sb 0.61 ± 0.25 158.18 ± 59.10 1.22 ± 1.14 2.57 ± 1.05 93.12 
F07-Scd 2.47 ± 3.96 180.45 ± 33.69 0.29 ± 0.54 2.96 ± 0.85 37.48 
F07-All 1.23 ± 0.31 59.04 ± 31.48 2.44 ± 1.17 1.35 ± 0.30 86.74 

4.5 Comparison with morphological selection

Figs 18 and 19 show φ(L), φ(M*), φ(Re) and φ(σ) for the Fukugita et al. sample as one adds more and more morphological types. The smooth curves (same in each panel) show the fits to the samples shown in Figs 15 and 17. In the top panels of Fig. 18 only, we also show the result of removing objects with Cr < 2.6 before making the measurements. This allows a direct comparison with one of the curves from the previous figure. Note that this gives results which are very similar to those from the larger (fainter, deeper) full sample; the different magnitude limits do not matter very much.

Figure 18

Distributions of luminosity, stellar mass, size and velocity dispersion for objects of different morphological types in the Fukugita et al. (2007) sample (cyan filled circles with error bars): E galaxies (top four panels) and E+S0 galaxies (bottom four panels). A subsample with Cr≥ 2.6 is also shown in the ‘E’ panels (red triangles). Smooth solid curves are same as in Figs 15 and 17: from top to bottom in each panel, they show the full SDSS sample (black), a subsample with Cr > 2.6 (magenta), a subsample with Cr > 2.86 (green), and a subsample selected following Hyde & Bernardi (2009) (blue).

Figure 18

Distributions of luminosity, stellar mass, size and velocity dispersion for objects of different morphological types in the Fukugita et al. (2007) sample (cyan filled circles with error bars): E galaxies (top four panels) and E+S0 galaxies (bottom four panels). A subsample with Cr≥ 2.6 is also shown in the ‘E’ panels (red triangles). Smooth solid curves are same as in Figs 15 and 17: from top to bottom in each panel, they show the full SDSS sample (black), a subsample with Cr > 2.6 (magenta), a subsample with Cr > 2.86 (green), and a subsample selected following Hyde & Bernardi (2009) (blue).

Figure 19

Same as Fig. 18 but for E+S0+Sa galaxies (top four panels) and for all galaxies (bottom four panels).

Figure 19

Same as Fig. 18 but for E+S0+Sa galaxies (top four panels) and for all galaxies (bottom four panels).

The results of fitting equation (10) to the Fukugita subsamples are provided in Tables B1–B4, which also show the associated luminosity and stellar mass densities, and the mean sizes. Ellipticals account for about 19 per cent of the r-band cmodel luminosity density and 25 per cent of the stellar mass density; including S0s increases these numbers to 33 per cent and 41 per cent, and adding Sas brings the contributions to 50 and 60 per cent, respectively.

Note that the number, luminosity and stellar mass densities of Es – about 10−3 Mpc−3, 0.2 × 108 L Mpc−3 and 0.6 × 108 M Mpc−3, respectively – are very similar to those of the early-type sample selected following Hyde & Bernardi (2009), as is the mean half-light radius of 3.2 kpc. Some of this match is fortuitous – we showed before that E's account for about 70 per cent of this sample, not 100 per cent. However, this lack of purity is balanced by the fact that Hyde and Bernardi select about 75 per cent of the Es, not 100 per cent: the purity and completeness effects approximately cancel. Similarly, although requiring Cr≥ 2.86 produces counts which are similar to those of E+S0s, about 40 per cent of the sample is made of Sas and later types, but the purity again approximately cancels the incompleteness. Finally, the counts when Cr≥ 2.6 are similar to those in the E+S0+Sa sample, although 25 per cent of the objects are of later type.

5 THE STELLAR MASS FUNCTION IN THE FULL SAMPLE

Our stellar mass function has considerably more objects at large M* than reported in previous work. Before we quantify this, we show the results of a variety of tests we performed to check that the discrepancy with the literature is real. This is important, since the high-mass end has been the subject of much recent attention (e.g. in the context of the build-up of the red sequence).

5.1 Consistency checks

We first checked that we were able to reproduce previous results for the luminosity function. These have typically used Petrosian rather than cmodel magnitudes, and different H0 conventions. The top panel in Fig. 20 shows the result of estimating the luminosity function using Petrosian magnitudes (using the Vmax method and code as in the previous sections). The curve in this panel shows the Schechter function fit reported by Blanton et al. (2003). The agreement between the curve and the symbols shows that our algorithms correctly transform between different H0 conventions, and between different definitions of k-corrections. It also shows that varying the evolution correction between the value reported by Blanton et al. (1.6z) and that from Bell et al. (1.3z), which we use when estimating stellar masses, makes little difference.

Figure 20

Luminosity function in the 0.1r and r bands (top and bottom panels), determined from r-band Petrosian apparent magnitudes in the range 14.5–17.5, for the two choices of the pure luminosity evolution parameter advocated by Blanton et al. (2003) and Bell et al. (2003): 1.6z and 1.3z, respectively. Dashed line in top panel shows the fit reported by Blanton et al. (2003); solid line in bottom panel shows that reported by Bell et al. (2003).

Figure 20

Luminosity function in the 0.1r and r bands (top and bottom panels), determined from r-band Petrosian apparent magnitudes in the range 14.5–17.5, for the two choices of the pure luminosity evolution parameter advocated by Blanton et al. (2003) and Bell et al. (2003): 1.6z and 1.3z, respectively. Dashed line in top panel shows the fit reported by Blanton et al. (2003); solid line in bottom panel shows that reported by Bell et al. (2003).

The bottom panel shows the Schechter function fit reported by Bell et al. (2003); there is good agreement. However, note that here we have not shifted the Petrosian magnitudes brightwards by 0.1 mag for galaxies with Cr > 2.6 (Bell et al. did this to account crudely for the fact that Petrosian magnitudes underestimate the luminosity of early types). At faint luminosities, the measurements oscillate around the fits, suggesting that fits to the sum of two Schechter functions will provide better agreement, but we do not pursue this further here. At the bright end, we find slightly more objects than either of the Bell et al. (top) or Blanton et al. (bottom) fits, but a glance at fig. 5 in Blanton et al. shows that the fit they report slightly underestimates the counts in the high-luminosity tail. Fig. 16 shows the luminosity function using our H0 convention and k-correction. It also shows our fit to equation 9 for log 10M*/M > 10.5 (the parameters are reported in Table 4.

Table 4

Top two rows: parameters of φ(Lr) (fit to Mr < −17.5) and φ(M*) (fit to log10M*/M > 8.5) derived from fitting equations (9) (in brackets) and (10) to the observed counts based on Petrosian magnitudes. Bottom two rows: parameters of φ(Lr) (fit to Mr < −20) and φ(M*) (fit to log10M*/M > 10.5). These second set of fits better reproduce the high luminosity and mass end.

Sample φ*/10−2 Mpc−3 X* α β ρX 
All L/109 L (8.427) 8.749 ± 4.228 (15.77) 16.04 ± 2.18 (0.08) 0.08 ± 0.05 (0.827) 0.833 ± 0.036 0.128 
All M*/109 M (5.620) 5.886 ± 1.839 (23.20) 25.86 ± 5.85 (0.14) 0.13 ± 0.05 (0.616) 0.654 ± 0.034 0.289 
All L/109 L (1.693) 1.707 ± 0.432 (7.32) 7.56 ± 2.05 (0.55) 0.54 ± 0.16 (0.698) 0.705 ± 0.040 0.117 
All M*/109 M (0.888) 0.857 ± 0.097 (0.94) 0.95 ± 0.35 (1.39) 1.50 ± 0.16 (0.410) 0.421 ± 0.016 0.247 
Sample φ*/10−2 Mpc−3 X* α β ρX 
All L/109 L (8.427) 8.749 ± 4.228 (15.77) 16.04 ± 2.18 (0.08) 0.08 ± 0.05 (0.827) 0.833 ± 0.036 0.128 
All M*/109 M (5.620) 5.886 ± 1.839 (23.20) 25.86 ± 5.85 (0.14) 0.13 ± 0.05 (0.616) 0.654 ± 0.034 0.289 
All L/109 L (1.693) 1.707 ± 0.432 (7.32) 7.56 ± 2.05 (0.55) 0.54 ± 0.16 (0.698) 0.705 ± 0.040 0.117 
All M*/109 M (0.888) 0.857 ± 0.097 (0.94) 0.95 ± 0.35 (1.39) 1.50 ± 0.16 (0.410) 0.421 ± 0.016 0.247 

In contrast to the good agreement for the luminosity function, our estimates of the stellar mass function (Fig. 3) show a significant excess relative to previous work at M*≥ 1011.5 M. (Note that, to compare with previous work, we convert from M*/Lr to M* by using the Petrosian magnitude. However, we do not shift the Petrosian magnitudes brightwards by 0.1 mag for galaxies with Cr > 2.6– a shift that was made by Bell et al. (2003). If we had done so the excess would be even larger.) To ensure that the discrepancy with previous fits is not caused by outliers, we removed galaxies with L≥ 1011 L or M*≥ 1011.5 M which differ by more than 0.3 dex from the linear fit (solid line in Fig. 21), in luminosity or M*. Except for the gr based M*, where we show results before and after removal of these outliers, all the other measurements shown in Fig. 3 are from the sample in which these outliers have been removed. Note that while removing outliers makes the plot slightly cleaner (see Fig. 3), the discrepancy at high M* remains.

Figure 21

Petrosian Lr versus M* (M* computed using equation 6). Small dots show a representative subsample of the galaxies. Solid green line shows the linear fit measured for M*≥ 1011 M. Dashed and dotted green lines show the 1 and 2 −σ range around the median, respectively. Red filled circles show outliers that were removed when determining φ(L) and φ(M*).

Figure 21

Petrosian Lr versus M* (M* computed using equation 6). Small dots show a representative subsample of the galaxies. Solid green line shows the linear fit measured for M*≥ 1011 M. Dashed and dotted green lines show the 1 and 2 −σ range around the median, respectively. Red filled circles show outliers that were removed when determining φ(L) and φ(M*).

The discrepancy is most severe if we use stellar masses from Gallazzi et al. (2005). (Note that they do not provide stellar mass estimates for fainter, typically lower mass objects.) The discrepancy is slightly smaller if we use equation (6) to translate gr colour into M*/Lr (for which both gr and Lr are evolution corrected; if we had used rest-frame quantities without correcting for evolution the discrepancy would be even worse). Using ri instead (equation 7) yields results which are more similar to the original Bell et al. (2003) fit. And finally, φ(M*) based on M*Petro of Blanton & Roweis (2007) has the lowest abundances of all.

The dotted line shows that our measurement of the distribution of M*Petro is well fit (except for a small offset) by the formula reported by Li & White (2009), which was based on their own estimate of φ(M*) from Blanton & Roweis M*Petro. The fact that we find good agreement with their fit suggests that our algorithm for estimating φ(M*) from a given list of M* values is accurate.

The dash–dot–dotted green line shows the fit reported by Panter et al. (2007) who computed stellar masses for a sample of 3 × 105 SDSS galaxies based on the analysis of the spectral energy distribution of the SDSS spectra. While this fit lies slightly below our data at high M*, the discrepancy is smaller than it is for most of the other fits we show.

We argued previously that aperture effects may have inflated the Gallazzi et al. masses slightly. As a check, we recomputed masses from the gr colour using equation (6), but now, using the fibre colour output by the SDSS pipeline. In contrast to the model colour, which measures the light on a scale which is proportional to the half-light radius, this measures the light in an aperture which has the same size as the SDSS fibre. The spectrophotometry of the survey is sufficiently accurate that this is a meaningful comparison (e.g. Roche et al. 2009a). Note that this gives abundances which are larger than those based on the model colours. Moreover, they are almost indistinguishable from those of M*Gallazzi.

The cyan solid curve shows the result of fitting equation (9) to our measurements of φ(M*) based on gr colour (we do not show fits to equation 10, because none of the other fits in the literature account for errors in the stellar mass estimate) when log10M*/M > 10.5. The best-fitting parameters are reported in Table 4. The table also reports the fit to the full sample (i.e. log10M*/M > 8.5). (These differ from those reported in the previous section, because here they are based on Petrosian rather than cmodel magnitudes.) The figures show the results for Mr < −20 and log10M*/M > 10.5 because this is the regime of most interest here.

Our estimates depart from the Bell et al. fit at densities of the order of 10−4 Mpc−3. However, their analysis was based on only 412 deg2 of sky, for which the expected number of objects on the mass scale where we begin to see a discrepancy (M*∼ 1011.5 M) is of the order of 10s. This, we suspect, is the origin of the discrepancy between our results and theirs – we are extrapolating their fit beyond its regime of validity.

Bell et al. (2003) did not account for errors, so the most straightforward way to quantify the increase we find is to compare our measured counts with their fit. If we do this for our Petrosian-based counts, then the stellar mass density in objects more massive than (1, 2, 3) × 1011 M is ∼(3, 24, 86) per cent larger than one would infer from the Bell et al. fit. However, Bell et al. attempted to account for the fact that Petrosian magnitudes underestimate the total light by shifting galaxies with Cr > 2.6 brightwards by 0.1 mag. Since we have not performed such a shift, the appropriate comparison with their fit is really to use our measured cmodel-based counts, so the difference between our counts and Bell et al. are actually larger. We do this in the next section.

If we compare our estimate of the stellar mass density in objects more massive than (1, 2, 3) × 1011 M with those from the Li & White (2009) fit, then our values are ∼(109, 202, 352) per cent larger.

5.2 Towards greater accuracy at large M*

It is well known that the cmodel luminosities are more reliable at the large masses where the discrepancy in φ(M*) is largest. Therefore, Fig. 22 compares various estimates of φ(M*) based on cmodel magnitudes. In this case, the estimates based on M*Model of Blanton & Roweis (2007) produce the lowest abundances, (but note they are larger than those based on M*Petro in Fig. 3), whereas those based on M*LRG are substantially larger, as one might expect (cf. Fig. A3). The M*LRG abundances are also in good agreement (slightly smaller) with those based on gr colour (equation 6, with cmodel magnitudes), except at smaller masses. (Although it is not apparent because of how we have chosen to plot our measurements, above 1011 M, the abundances based on M*Gallazzi and M*LRG are in good agreement.)

Figure 22

Stellar mass functions estimated from cmodel magnitudes. Top: Filled black circles show M* estimated from gr colours (equation 6). Open magenta and green triangles show M*Model and M*LRG from Blanton & Roweis (2007). Solid and dashed red lines show our fits to equations 9 and 10 with parameters reported in Table 4 for log10M*/M > 10.5 estimated from cmodel magnitudes. For comparison, solid cyan line shows our fit to equation (9) with parameters reported in Table 4 for log10M*/M > 10.5 estimated from Petrosian magnitudes (as in Fig. 3). Solid grey curve shows the fit reported by Bell et al. (2003) (transformed to H0= 70 km s−1 Mpc−1). Bottom: same as top panel, but now all quantities have been normalized by the Bell et al. (2003) fit.

Figure 22

Stellar mass functions estimated from cmodel magnitudes. Top: Filled black circles show M* estimated from gr colours (equation 6). Open magenta and green triangles show M*Model and M*LRG from Blanton & Roweis (2007). Solid and dashed red lines show our fits to equations 9 and 10 with parameters reported in Table 4 for log10M*/M > 10.5 estimated from cmodel magnitudes. For comparison, solid cyan line shows our fit to equation (9) with parameters reported in Table 4 for log10M*/M > 10.5 estimated from Petrosian magnitudes (as in Fig. 3). Solid grey curve shows the fit reported by Bell et al. (2003) (transformed to H0= 70 km s−1 Mpc−1). Bottom: same as top panel, but now all quantities have been normalized by the Bell et al. (2003) fit.

Figure A3

Comparison of various stellar mass estimates from Blanton & Roweis (2007), with M*Bell and M*Gallazzi for objects with Cr > 2.86.

Figure A3

Comparison of various stellar mass estimates from Blanton & Roweis (2007), with M*Bell and M*Gallazzi for objects with Cr > 2.86.

We argued previously that we believe the gr masses are more reliable than those based on ri. Therefore, we only show fits to the distribution of gr derived masses: solid and dashed curves show fits to equations (9) and (10), respectively (the latter account for broadening of the distribution due to errors in the determination of M*). Both result in larger abundances than the observed abundances based on Petrosian quantities – here represented by the fit shown in the previous figure – although the intrinsic distribution we determine for the cmodel-based masses is similar to the observed distribution of Petrosian-based masses.

If we sum up the observed counts to estimate the stellar mass density (M* from equation 6) in objects more massive than (1, 2, 3) × 1011 M, then the result is ∼(30, 68, 170) per cent larger than that one infers from the Bell et al. (2003) fit. Using our fit to the observed distribution (values between round brackets in Table B2, for log10M*/M > 10.5) gives similar results: stellar mass densities ∼(21, 60, 160) per cent larger than those from the Bell et al. (2003) fit. In practice, however, the Bell et al. fit tends to be used as though it were the intrinsic quantity, rather than the one that has been broadened by measurement error. Our fit to the intrinsic distribution, based on cmodel magnitudes (from Table B2, for log10M*/M > 10.5), gives stellar mass densities in objects more massive than (1, 2, 3) × 1011 M of (8.84, 3.00, 1.03) × 107 M Mpc−3. This corresponds to an extra ∼(15, 40, 109) per cent more than from the Bell et al. fit.

Another way to express this difference is in terms of the mass scale at which the integrated comoving number density of objects is 10−5 Mpc−3. For our fits to the intrinsic cmodel-based counts, this scale is 3.98 × 1011 M (4.27 × 1011 M for the observed fit), whereas for Bell et al. (2003) it is 3.31 × 1011 M. For 10−5.5 Mpc−3, these scales are 5.25 × 1011 M (5.75 × 1011 M for the observed fit) and 4.07 × 1011 M, respectively. Fig. 23 illustrates these differences graphically. Note that the stellar masses used by Panter et al. and Li & White were obtained using Petrosian magnitudes. To account for the difference between Petrosian and model luminosities one could add ∼0.05 dex to their values of M* (strictly speaking, to those objects with Cr > 2.86) – we have not applied such a shift.

Figure 23

Comparison of various estimates of the cumulative number density of objects more massive than M* (top), and the stellar mass density in these objects (bottom). All measurements were transformed to H0= 70 km s−1 Mpc−1 and Chabrier (2003) IMF.

Figure 23

Comparison of various estimates of the cumulative number density of objects more massive than M* (top), and the stellar mass density in these objects (bottom). All measurements were transformed to H0= 70 km s−1 Mpc−1 and Chabrier (2003) IMF.

5.3 On major dry mergers at high masses

The increase in z∼ 0 counts at high masses matters greatly in studies which seek to constrain the growth histories of massive galaxies by comparing with counts at z∼ 2. Fig. 24 shows a closer-up view of the top panel of Fig. 23. The plot is in the same format as fig. 3 in Bezanson et al. (2009). We have added a (solid magenta) curve showing the cumulative counts at 2 < z≤ 3 from Marchesini et al. (2009), shifted to our Chabrier IMF by subtracting 0.05 dex from their M* values. The dashed curve below it shows the same counts shifted downwards by a factor of 2, to reflect the fact that only perhaps half of the galaxies at z∼ 2.5 are quiescent.

Figure 24

Cumulative number density for comparison with fig. 3 in Bezanson et al. (2009). Most of the curves are as labelled in top panel of Fig. 23. The solid magenta line shows the cumulative number density from Marchesini et al. (2009) for galaxies at 2 < z < 3. The magenta dashed line shows the same result but assuming that the quiescent fraction of galaxies at z∼ 2.5 is 0.5. Dashed black line shows the number density of galaxies with M* > 1011 M from the above fit as in Bezanson et al. (2009). Solid black line shows the result of having equal mass mergers.

Figure 24

Cumulative number density for comparison with fig. 3 in Bezanson et al. (2009). Most of the curves are as labelled in top panel of Fig. 23. The solid magenta line shows the cumulative number density from Marchesini et al. (2009) for galaxies at 2 < z < 3. The magenta dashed line shows the same result but assuming that the quiescent fraction of galaxies at z∼ 2.5 is 0.5. Dashed black line shows the number density of galaxies with M* > 1011 M from the above fit as in Bezanson et al. (2009). Solid black line shows the result of having equal mass mergers.

Bezanson et al. (2009) argue that models in which the high redshift objects change their sizes but not their masses by the present time (e.g. Fan et al. 2008) lie well below the z= 0 counts. Because this results in an order of magnitude fewer counts than observed at z= 0, such models, while viable, do not represent the primary growth mechanism of massive galaxies. Other models invoke minor (dry) mergers (e.g. Bernardi 2009). If every one of the objects with M≥ 1011 M at z∼ 2 merged with other objects of much smaller mass, then the abundance of these objects would not change, but their masses would: the expected evolution of the population with M≥ 1011 M at z∼ 2 is shown by the horizontal shaded region. The fractional mass increase by a minor merger is expected to lead to a size increase that is larger by a factor of 2 (e.g. Bernardi 2009). The observed size change suggests that the masses have not increased by more than a factor of about 2: this is the vertical dashed line labelled ‘minor mergers’. The horizontal shaded region intersects this vertical line at abundances which are about a factor of 5 smaller than our z= 0 counts, so this model is also viable. On the other hand, if every one of the objects with M≥ 1011 M at z∼ 2 merged with another of the same mass – a major dry merger – then this would shift the counts downwards and to the right, as shown by the shaded curved region. In this case, the fractional mass and size changes are equal, so the observed size increase requires mass growth by a factor of 5. This is the vertical line labelled ‘equal mass mergers’. The intersection of the curved shaded region with this dashed line lies above previous estimates of the z= 0 abundances; this leads Bezanson et al. (2009) to conclude that major mergers could not be the dominant evolution mode at the massive end. While we believe this an overly simplistic model, here we are simply pointing out that our higher abundances suggest that their conclusion should be revisited.

5.4 Morphological dependence of the IMF

Recently, Calura et al. (2009) have argued that a number of observations are better reproduced if one assumes a Salpeter (1955) IMF for ellipticals and a Scalo (1986) IMF for spirals. (A Salpeter IMF for ellipticals is also preferred by Treu et al. 2010, which appeared while our paper was being referred.) Whereas the M*/L–colour relation for a Scalo IMF is similar to that for the Chabrier IMF which we have been using, the relation for the Salpeter IMF is offset by 0.25 dex (see Table 2). Since we have found a method to separate Es, S0s and Spirals we can incorporate such a dependence easily.

For ellipticals, i.e. objects selected following Hyde & Bernardi (2009), we compute M* by adding 0.25 to the right-hand side of equation (6). For S0s, i.e. objects with Cr > 2.86 that were not identified as ellipticals, we compute M* by adding 0.2 to equation (6), since S0s are closer to ellipticals than to spirals. For all other objects, we use equation (6) as before. The open red triangles in Fig. 25 show the stellar mass function which results. Smooth curves show the observed fits to equation (9); the best-fitting parameters are reported in Table 5.

Figure 25

Top: solid black circles show the stellar mass function φ(M*) estimated from cmodel magnitudes and gr colours (equation 6). Open red triangles show φ(M*) if we use a Salpeter IMF for elliptical galaxies, the Salpeter IMF offset by −0.05 dex in M*/L for a given colour for S0s, and the Salpeter IMF with an offset of −0.25 for the remaining galaxies (see text for details). Open blue squares show φ(Mdyn) where Mdyn= 5Reσ2/G. Solid and dashed lines show our observed fits (equation 9) computed using log10M*/M > 10.5 and log10M*/M > 8.6, respectively. These fits are reported in Table 5. Dash–dotted green line shows the dynamical mass function of Shankar et al. (2006) based on dynamical mass-to-light ratios. Bottom: same as top panel, but now all quantities have been normalized by the observed fit to the black circles (from Table B2 computed for log10M*/M > 8.6).

Figure 25

Top: solid black circles show the stellar mass function φ(M*) estimated from cmodel magnitudes and gr colours (equation 6). Open red triangles show φ(M*) if we use a Salpeter IMF for elliptical galaxies, the Salpeter IMF offset by −0.05 dex in M*/L for a given colour for S0s, and the Salpeter IMF with an offset of −0.25 for the remaining galaxies (see text for details). Open blue squares show φ(Mdyn) where Mdyn= 5Reσ2/G. Solid and dashed lines show our observed fits (equation 9) computed using log10M*/M > 10.5 and log10M*/M > 8.6, respectively. These fits are reported in Table 5. Dash–dotted green line shows the dynamical mass function of Shankar et al. (2006) based on dynamical mass-to-light ratios. Bottom: same as top panel, but now all quantities have been normalized by the observed fit to the black circles (from Table B2 computed for log10M*/M > 8.6).

Table 5

Top two rows: parameters of φ(M*) fit to log10M*/M > 8.5 (top) and > 10.5 (bottom) derived from fitting equations (9) (in brackets) and (10) to the observed counts based on cmodel magnitudes and a Salpeter IMF for elliptical galaxies, an offset of −0.05 from the Salpeter IMF for S0s and an offset of −0.25 for the remaining galaxies. Bottom two rows: similar to top two rows but for φ(Mdyn), where Mdyn= 5Reσ2/G.

Sample φ*/10−2 Mpc−3 M*/109 M α β ρ*/109 M Mpc−3 
ΔIMF (35.196) 130.824 ± 76.761 (69.09) 75.52 ± 7.01 (0.01) 0.01 ± 0.02 (0.657) 0.700 ± 0.024 0.382 
ΔIMF (1.797) 1.958 ± 0.907 (38.99) 49.18 ± 15.89 (0.27) 0.23 ± 0.12 (0.595) 0.647 ± 0.046 0.365 
Mdyn (6.066) 6.194 ± 2.699 (17.85) 21.10 ± 9.71 (0.19) 0.18 ± 0.10 (0.485) 0.512 ± 0.041 0.617 
Mdyn (2.135) 1.757 ± 0.474 (1.28) 0.45 ± 0.28 (0.82) 1.12 ± 0.22 (0.361) 0.337 ± 0.018 0.581 
Sample φ*/10−2 Mpc−3 M*/109 M α β ρ*/109 M Mpc−3 
ΔIMF (35.196) 130.824 ± 76.761 (69.09) 75.52 ± 7.01 (0.01) 0.01 ± 0.02 (0.657) 0.700 ± 0.024 0.382 
ΔIMF (1.797) 1.958 ± 0.907 (38.99) 49.18 ± 15.89 (0.27) 0.23 ± 0.12 (0.595) 0.647 ± 0.046 0.365 
Mdyn (6.066) 6.194 ± 2.699 (17.85) 21.10 ± 9.71 (0.19) 0.18 ± 0.10 (0.485) 0.512 ± 0.041 0.617 
Mdyn (2.135) 1.757 ± 0.474 (1.28) 0.45 ± 0.28 (0.82) 1.12 ± 0.22 (0.361) 0.337 ± 0.018 0.581 

Summing up the observed counts to estimate the stellar mass density in the range log10M*/M > 8.6 yields 3.92 × 108 M Mpc−3. Integrating the intrinsic fit (see Table 5) over the entire range of masses gives a similar result: 3.82 × 108 M Mpc−3. The intrinsic fit to log10M*/M > 10.5 gives (2.92, 1.81, 0.97, 0.50) × 108 M Mpc−3 for objects with log10M*/M above (10.5, 11.0, 11.3, 11.5). This means that ∼(75, 46, 25, 13 per cent) of the mass is in systems with M* > (0.3, 1, 2, 3) × 1011 M. These estimated values of the stellar mass density (i.e. from the intrinsic fit to log10M*/M > 10.5), for objects with log10M*/M above (11.0, 11.3, 11.5), are ∼(105, 224, 388) per cent larger than those inferred from stellar masses computed using the cmodel magnitudes but using the Chabrier IMF for all types (solid black circles in Fig. 25).

Finally, we compare φ(M*) with φ(Mdyn) (open blue squares in Fig. 25), where Mdyn= 5Reσ2/G is the dynamical mass. Our fits to φ(Mdyn) are reported in Table 5. At low σ and R the velocity dispersions and sizes are noisy. This makes the mass estimate noisy below ∼4 × 109 M, so we only show results above this mass. Using different IMFs for galaxies of different morphological type reduces the difference between the estimated value of the stellar and dynamical mass especially at larger masses.

We also find this estimate of the stellar mass function to be in reasonably good agreement with the one computed by Shankar et al. (2006) based on dynamical mass-to-light ratios calibrated following Salucci & Persic (1999), Cirasuolo et al. (2005) and references therein, lending further support to the possibility of a Hubble-type dependent IMF.

5.5 The match with the integrated star formation rate

It has been argued that a direct integration of the cosmological SFR overpredicts the local stellar mass density (see e.g. Wilkins, Trentham & Hopkins 2008, and references therein). This has led several authors to invoke some corrections, such as a time-variable IMF. We now readdress this interesting issue by comparing our value for ρ* with that from integrating the SFR.

The stellar mass density at redshift z is given by  

16
formula
where forumla is the cosmological SFR in units of M Mpc−3 yr−1, and fr(t) is the fraction of stellar mass that has been returned to the interstellar medium. For our IMF,  
17
formula
where t is in years (Conroy & Wechsler 2009). Note that this results in smaller remaining mass fractions (∼50 per cent at z∼ 0 instead of the usual 63–70 per cent), than assumed in most previous work. This will be important below.

We specify the SFR as follows. Bouwens et al. (2009) have recently calibrated the SFR over the range 2 < z < 6 using deep optical and infrared data from Advanced Camera for Surveys (ACS)/Near Infrared Camera and Multi-Object Spectrometer (NICMOS) in the GOODS fields, UBVi dropout Lyman break galaxies and ultraluminous infrared galaxies (ULIRGs) data from Caputi et al. (2007). The solid squares in Fig. 26 show their measurements, decreased by −0.25 dex to correct from their assumed Salpeter to a Chabrier IMF. These values of the SFR are much lower than simple extrapolations of the SFRs by Hopkins & Beacom (2006) and Fardal et al. (2007), shown by dotted and dashed lines, respectively. We also note that the Fardal et al. fit matches well the updated SFR recent Bouwens et al. estimates in the range 2.5 < z < 4. In detail, the curves show the parametrization of Cole et al. (2001),  

18
formula
Fardal et al. (2007) set a= 0.0103, b= 0.088, c= 2.4, d= 2.8, γ= 1, and we then multiply the total by 0.708 (−0.1 dex) to correct from the assumed diet-Salpeter to our Chabrier IMF. Hopkins & Beacom (2006) set a= 0.014, b= 0.11, c= 1.4, d= 2.2, γ= 0.7 (all parameters defined for h= 0.7). We convert from their IMF (from Baldry & Glazebrook 2003) to the Chabrier IMF by multiplying by 1.135 (i.e. 0.055 dex, see Table 2). Based on detailed spectral modelling, Bouwens et al. (2009) concluded that the discrepancy is due to dust extinction for star-forming galaxies in this redshift range being smaller then previously assumed. (However, we note that gamma-ray burst based estimates from, e.g. Kistler et al. 2009, suggest this is not a closed issue.) To improve the match with Bouwens et al., we use the Fardal et al. values at z < 3.65, but set a= 0.0134, b= 0.0908, c= 3.1, d= 6.5, γ= 0.7 at z > 3.65. This is shown by the solid curve.

Figure 26

Cosmological SFR as given by Fardal et al. (2007; long-dashed line), and by Hopkins & Beacom (2006; dotted line), compared to the dust-corrected UV data by Bouwens et al. (2009; solid squares). The solid line is our ‘corrected’ Fardal et al. SFR fit tuned to match the data at z > 4.

Figure 26

Cosmological SFR as given by Fardal et al. (2007; long-dashed line), and by Hopkins & Beacom (2006; dotted line), compared to the dust-corrected UV data by Bouwens et al. (2009; solid squares). The solid line is our ‘corrected’ Fardal et al. SFR fit tuned to match the data at z > 4.

The dotted, dashed and solid curves in Fig. 27 show the result of inserting these three models for the SFR (Hopkins & Beacom, Fardal et al. and Fardal et al. corrected) into equation (16). The grey band bracketing the solid curve shows the typical ∼15 per cent 1σ uncertainty (estimated by Fardal et al. 2007) associated with the SFR fit. The figure also shows a compilation of estimates of the stellar mass density over a range of redshifts. Our own estimate of the local ρ* value is shown by the filled triangle (filled circle shows the observed rather than intrinsic value); it is in good agreement with the one from Bell et al. (2003) (corrected to a Chabrier IMF), and is slightly larger than those from Panter et al. (2007), and Li & White (2009). Comparison of the measured ρ*(z) values with our new estimate of the integrated SFR shows that the measurements lie only slightly below the integrated SFR at all epochs, with the discrepancy smallest at z > 2 and at z∼ 0. Also note that K- or NIR-selected high-z galaxies might be missing a significant population of highly obscured, dust enshrouded, forming galaxies in the range 0.5 < z < 2.

Figure 27

Comparison of the expected stellar mass density based on the SFR, with the measured values at a range of redshifts. Filled triangle shows our determination of the intrinsic local stellar mass density; filled circle shows the observed value. Recent determinations at higher redshifts by other groups (as labelled) are also shown. Dotted, dashed and solid curves show the result of inserting the SFRs shown in the previous figure in equation (16).

Figure 27

Comparison of the expected stellar mass density based on the SFR, with the measured values at a range of redshifts. Filled triangle shows our determination of the intrinsic local stellar mass density; filled circle shows the observed value. Recent determinations at higher redshifts by other groups (as labelled) are also shown. Dotted, dashed and solid curves show the result of inserting the SFRs shown in the previous figure in equation (16).

The improvement with respect to previous works is due to the combined effects of a larger recycling factor and smaller high-z SFR, both of which act to reduce the value of the integral (also see discussion in Shankar et al. 2006). Despite the good agreement with the Fardal et al. estimate, we note that other SFR fits (e.g. Hopkins & Beacom) yield substantially higher values for the local stellar mass density. Clearly, systematic differences such as this one must be resolved before this issue is completely settled.

6 THE AGE–SIZE RELATION

The previous sections studied how the distribution of L, M*, σ and R depends on morphology or concentration. The present section shows one example of a correlation between observables which is particularly sensitive to morphology. As a result, how one chooses to select ‘red’ sequence galaxies matters greatly.

The age–size relation has been the subject of recent interest, in particular because, for early-type galaxies, the correlation between galaxy luminosity and size does not depend on age (Shankar & Bernardi 2009). This is somewhat surprising, because it has been known for some time that early-type galaxies with large velocity dispersions tend to be older (e.g. Trager et al. 2000; Cattaneo & Bernardi 2003; Bernardi et al. 2005; Thomas et al. 2005; Jimenez et al. 2007), and the virial theorem implies that velocity dispersion and size are correlated.

To remove the effects of this correlation, Shankar et al. (2009d) and van der Wel et al. (2009) studied the age–size correlation at fixed velocity dispersion. They find that this relation is almost flat (with a zero-point that depends on the velocity dispersion, of course). At fixed dynamical mass, however, Shankar et al. still find no relation, whereas van der Wel et al. find a significant anticorrelation: smaller galaxies are older. What should be made of this discrepancy?

Although both groups claim to be studying early-type galaxies, the details of how they selected their samples are different: Shankar et al. follow Hyde & Bernardi (2009); the results from the previous sections suggest that this sample should be dominated by ellipticals. van der Wel et al. use the sample of Graves, Faber & Schiavon (2009): this has 0.04 ≤z≤ 0.08, 15 ≤mr,Pet≤ 18, i-band concentration >2.5; and likelihood of deV profile >1.03 that of the exponential (this likelihood is output by the SDSS photometric pipeline), no detected emission lines (EW Hα≤ 0.3Å and O ii ≤ 1.7); and spectra of sufficient S/N that velocity dispersions were measured (following Bernardi et al. 2003). Because the requirements on the profile shape are significantly less stringent than those of Hyde & Bernardi, one might expect this sample to include more S0s and Sas. Moreover, recall that age is a strong function of morphological type (e.g. Fig. 10), so, to see if this matters, we have studied how the age–size relation depends on morphological type.

Fig. 28 shows this relation, at fixed σ, for the full Fukugita sample (bottom right) and for the different morphological types (other panels). The relation for the full sample is approximately flat, except at σ < 100 km s−1. However, when divided by morphological type, age (at fixed σ) is slightly correlated with size for ellipticals, but the relation is more flat for S0s and Sas. (The mean age is a slightly increasing function of σ for the later types Sb-d.) Thus, the flatness of the age–size relation at fixed σ in the full sample hides the fact that the relation actually depends on morphology.

Figure 28

Correlation between age and size at fixed velocity dispersion (as labelled) for types E, S0, Sa, Sb, Sc-Sd and All in the Fukugita et al. (2007) sample.

Figure 28

Correlation between age and size at fixed velocity dispersion (as labelled) for types E, S0, Sa, Sb, Sc-Sd and All in the Fukugita et al. (2007) sample.

Although this is a subtle effect for the relation at fixed σ, the dependence on morphology is much more pronounced when studying galaxies at fixed Mdyn≡ 5Reσ2/G. Fig. 29 shows that, in the full sample, this relation is flat for Mdyn > 1011.5 M, but decreases strongly for lower masses, except at Mdyn < 1010.5 M, where it appears to be curved. The stronger dependence here is easily understood from the previous figure: since dynamical mass is ∝Rσ2, as one moves along lines of constant mass in the direction of increasing R, one is moving in the direction of decreasing σ. In Fig. 28, this means that one must step downwards by one bin in log σ for every 0.4 dex to the right in log R. For ellipticals, age is an increasing function of size at fixed σ; hence, the net effect of moving up and to the right (at fixed σ), and then stepping down to lower σ, produces an approximately flat age–size relation for fixed Mdyn. For Sas, on the other hand, keeping Mdyn fixed corresponds to shifting down and to the right (at fixed σ), and then stepping downwards to the lower σ bin; the net result is that age decreases strongly as size increases. What is remarkable is that the ellipticals show precisely the scaling with Mdyn reported by Shankar et al. (2009d), whereas the S0s and Sa's show that reported by van der Wel et al. (2009).

Figure 29

Correlation between age and size at fixed dynamical mass (as labelled) for types E, S0, Sa, Sb, Sc-Sd and All in the Fukugita et al. (2007) sample.

Figure 29

Correlation between age and size at fixed dynamical mass (as labelled) for types E, S0, Sa, Sb, Sc-Sd and All in the Fukugita et al. (2007) sample.

7 DISCUSSION

We compared samples selected using simple selection algorithms based on available photometric and spectroscopic information with those based on morphological information. Requiring concentration indices Cr≥ 2.6 selects a mix in which E+S0+Sa's account for about two-thirds of the objects; if Cr≥ 2.86 instead, then two-thirds of the sample comes from E+S0s; whereas Es alone account for more than two-thirds of a sample selected following Hyde & Bernardi (2009) (Figs 12 and 13, and Table 3). E's alone account for about 40, 50 and 75 per cent of the total stellar mass in samples selected in these three ways.

The reddest objects at intermediate luminosities or stellar masses are edge-on discs (Fig. 14). As a result, samples selected on the basis of colour alone, or cuts which run parallel to the red sequence are badly contaminated by such objects. However, simply adding the additional requirement that the axis ratio b/a≥ 0.6 is an easy way to remove such red edge-on discs from the ‘red’ sequence; the resulting sample is similar to requiring Cr≥ 2.86. This may provide a simple way to select relatively clean early-type samples in higher redshift data sets (e.g. DEEP2, zCosmos). Our measurements provide the low-redshift benchmarks against which such future higher redshift measurements can be compared.

We showed how the distribution of luminosity, stellar mass, size and velocity dispersion in the local universe is partitioned up amongst different morphological types, and we compared these distributions with those based on simple selection algorithms based on available photometric and spectroscopic information (Figs 15, 1719). We described our measurements by assuming that the intrinsic distributions have the form given by equation (9). We showed how measurement errors bias the fitted parameters (equation 10), and used this to devise a simple method which removes this bias. The results, which are reported in tabular form in Appendix B, show that ellipticals contain ∼20 per cent of the luminosity density and 25 per cent of the stellar mass density in the local universe, and have mean sizes of the order of 3.2 kpc. Including S0s increases these numbers to 33 per cent and 40 per cent; adding Sas results in further increases to 50 and 60 per cent, respectively. These numbers are in broad agreement with those from the Millennium Galaxy Survey of about 104 objects in 37.5 deg2. Driver et al. (2007) report that 15 ± 5 per cent of the stellar mass density is in ellipticals, and adding bulges increases this to 44 ± 9 per cent.

Our stellar mass function has more massive objects than other recent determinations (e.g. Cole et al. 2001; Bell et al. 2003; Panter et al. 2007; Li & White 2009), similarly shifted to a Chabrier (2003) IMF (Figs 3 and 22). The mass scale on which the discrepancy arises is of order where some previous work had only a handful of objects – our substantially larger volume is necessary to provide a more reliable estimate of these abundances. Using stellar masses estimated from cmodel luminosities, which are more reliable than Petrosian luminosities at the large masses where the discrepancy in φ(M*) is largest, gives stellar mass densities in objects more massive than (1, 2, 3) × 1011 M that are larger by more than ∼(20, 50, 100) per cent compared to Bell et al. (2003) (Fig. 23).

This analysis required that we study the systematic differences between the stellar mass estimates based on gr colour (our equation 6, following Bell et al. 2003), colours in multiple bands (Blanton & Roweis 2007), and on spectral features (Gallazzi et al. 2005). (See Gallazzi & Bell 2009, which appeared while our work was being referred, for a discussion of the pros and cons of these various approaches, and of the accuracy to which stellar masses can currently be derived.) The gr and Gallazzi et al. estimates are generally in good agreement (Fig. A1), although the spectral-based estimates suffer slightly from aperture effects which are complicated by the magnitude limit of the survey (Figs A2, A5 and 3). The Blanton et al. estimates are in good agreement with the other two provided one uses LRG-based templates to estimate masses at the most massive end (Figs A3, A4 and 22). At lower masses, some combination of the LRG and other templates is required. Ignoring the LRG templates altogether (e.g. Li & White 2009) results in systematic underestimates of as much as 0.1 dex or more (Fig. A3), severely compromising estimates of the number of stars currently locked up in massive galaxies (Fig. 3). If we compare our estimate of the stellar mass density in objects more massive than (1, 2, 3) × 1011 M with those from the Li & White (2009) fit, then our values are ∼(140, 230, 400) per cent larger.

Figure A1

Comparison of stellar masses computed following Bell et al. (2003)[our equations 6 (solid circles) and 7 (open squares) with Petrosian r-band luminosity], and Gallazzi et al. (2005), for objects with Cr > 2.86.

Figure A1

Comparison of stellar masses computed following Bell et al. (2003)[our equations 6 (solid circles) and 7 (open squares) with Petrosian r-band luminosity], and Gallazzi et al. (2005), for objects with Cr > 2.86.

Figure A2

Aperture effects on the Gallazzi et al. (2005) stellar mass estimate. Top: redshift dependence of M*Gallazzi/M*Bell (left) and the angular half-light radius (right) for objects with Cr > 2.86 in three different bins of M*Bell as indicated. Bottom: similar, but now for a few narrow bins in luminosity.

Figure A2

Aperture effects on the Gallazzi et al. (2005) stellar mass estimate. Top: redshift dependence of M*Gallazzi/M*Bell (left) and the angular half-light radius (right) for objects with Cr > 2.86 in three different bins of M*Bell as indicated. Bottom: similar, but now for a few narrow bins in luminosity.

Figure A5

Same as Fig. A2, but now for the full sample (i.e. no cut on concentration index).

Figure A5

Same as Fig. A2, but now for the full sample (i.e. no cut on concentration index).

Figure A4

Comparison of our various stellar masses in the full sample (i.e., no cut on concentration index).

Figure A4

Comparison of our various stellar masses in the full sample (i.e., no cut on concentration index).

Allowing more high-mass objects means that major dry mergers may remain a viable formation mechanism at the high-mass end (Fig. 24). It also relieves the tension between estimates of the evolution of the most massive galaxies which are based on clustering (which predict some merging, and so some increase in stellar mass; Wake et al. 2008; Brown et al. 2008) and those based on abundances (for which comparison of high redshift measurements with the previous z∼ 0 measurements indicated little evolution; Wake et al. 2006; Brown et al. 2007; Cool et al. 2008). This discrepancy may be related to the origin of intercluster light (e.g. Skibba, Sheth & Martino 2007; Bernardi 2009); our measurement of a larger local abundance in galaxies reduces the amount of stellar mass that must be stored in the ICL.

It has been argued that a number of observations are better reproduced if one assumes a different IMF for elliptical and spiral galaxies (e.g. Calura et al. 2009). We showed that this acts to further increase the abundance of massive galaxies (Fig. 25), and reduces the difference between stellar and dynamical mass, especially at larger masses. At M*≥ 1011 M, the increase due to the change in IMF is a factor of 2 with respect to models which assume a fixed IMF.

If we sum up the observed counts to estimate the stellar mass density in the range 8.6 < log10M* < 12.2 M (M* from equation 6 using cmodel magnitudes), then the result is 3.05 × 108 M Mpc−3. Using our fit to the observed distribution (values between round brackets in Table B2) gives a similar value (3.06 × 108 M Mpc−3) and a slightly smaller value if one uses the intrinsic fit (2.89 × 108 M Mpc−3, see Table B2). Our values are ∼15 and 30 per cent larger than those reported by Panter et al. (2007) and Li & White (2009), respectively. If we allow a type-dependent IMF, the total stellar mass density increases by a further 30 per cent.

However, although our stellar mass function has more M* > 1011 M objects than other recent determinations, our estimate of the total stellar mass density is similar to that measured by Bell et al. (2003). It is about 20 per cent smaller than the value reported by Driver et al. (2007) (once shifted to the same IMF, for which we have chosen Chabrier 2003; see Table 2). This is because differences at the mid/faint end contribute more to the total stellar density than the difference we measured at the massive end.

It has been suggested that direct integration of the cosmological SFR overpredicts the total local estimate of the stellar mass density (see e.g. Wilkins et al. 2008, and references therein). However, we showed that recent determinations of the recycling factor (equation 16) and the high-z star formation rate (Fig. 26) result in better agreement (Fig. 27). This is because the former yields smaller remaining masses, and the latter produces fewer stars formed in the first place.

Our measurements also show that the most luminous or most massive galaxies, which one might identify with BCGs, are less concentrated and have smaller b/a ratios than slightly less luminous or massive objects (Figs 4, 6 and 12). Their light profile is also not well represented by a pure deVaucoleur law. This is consistent with results in Bernardi et al. (2008) and Bernardi (2009) who suggest that these are signatures of formation histories with recent radial mergers. In this context, note that we showed how to define seeing-corrected sizes, using quantities output by the SDSS pipeline that closely approximate deVaucouleur bulge + Exponential disc decompositions (equations 2 and 3). Our cmodel sizes represent a substantial improvement over Petrosian sizes (which are not seeing corrected) and pure deV or Exp sizes (Figs 1 and 2).

And finally, our study of the age–size correlation resolves a discrepancy in the literature: whereas Shankar et al. (2009d) report no correlation at fixed Mdyn, van der Wel et al. (2009) report that larger galaxies tend to be younger. We showed that ellipticals follow the scaling reported by Shankar et al. scaling, whereas S0s and Sas follow that of van der Wel et al. (Fig. 29), suggesting that Shankar et al. select a sample dominated by galaxies with elliptical morphologies, whereas van der Wel et al. include more S0s and Sas. These conclusions about the differences between the samples are consistent with how the samples were actually selected.

Since van der Wel et al. use their measurements to constrain a model for early-type galaxy formation, this is an instance in which having morphological information matters greatly for the physical interpretation of the data. Our results indicate that models of early-type galaxy formation should distinguish between ellipticals and S0s because, in the projection of the age–size–mass correlation shown in Fig. 29, the S0s and Sas are very different from the other morphological types. Whether the smaller sizes for older S0s are due to the gradual stripping away of a younger disc is an open question.

van der Wel et al. (2009) use their observation that the age–size relation at fixed σ is flat to motivate a model in which early-type galaxy formation requires a critical velocity dispersion (which they allow to be redshift dependent). The same logic applied here suggests that while this may be reasonable for S0s or Sas, it is not well motivated for ellipticals (Fig. 28). However, it might be interesting to explore a model in which elliptical formation requires a critical (possibly redshift dependent) dynamical mass rather than velocity dispersion (which may be redshift dependent). This is interesting because, in hierarchical models, the phenomenon known as down-sizing (e.g. Cowie et al. 1996; Heavens et al. 2004; Sheth et al. 2006; Jimenez et al. 2007) is then easily understood (Sheth 2003).

We thank Anna Gallazzi, Eric Bell, Cheng Li, Danilo Marchesini, Eyal Neistein and Simon White for interesting discussions. We would also like to thank the members of the APC in Paris 7 Diderot and the Max-Planck Institut für Astronomie, Heidelberg, for their hospitality while this work was being completed. MB is grateful for support provided by NASA grant LTSA-NNG06GC19G and NASA ADP/NNX09AD02G. FS acknowledges support from the Alexander von Humboldt Foundation. RKS is supported in part by NSF-AST 0908241.

Funding for the Sloan Digital Sky Survey (SDSS) and SDSS-II Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society and the Higher Education Funding Council for England. The SDSS website is http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, The University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory and the University of Washington.

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Appendices

APPENDIX A: COMPARISON OF STELLAR MASS ESTIMATES

This section shows results from a study of how the different M* estimates compare with one another. We first show results for galaxies that have Cr≥ 2.86 for two reasons. First, because this is expected to provide a sample dominated by massive galaxies, and these are expected to be more homogeneous. And secondly, we are particularly interested in studying the high-mass end of the stellar mass function, because this is the end which is first observed in higher redshift data sets.

Fig. A1 shows that the M*Bell and M*Gallazzi estimates are in good agreement, although M*Bell tends to be larger at M*Bell > 1011M. The sharp downturn at small masses is due to aperture effects. This is because M*Bell is based on the model colour; for otherwise similar galaxies, this samples the same fraction of a galaxy's light (that within Re) whatever its distance. On the other hand, M*Gallazzi is based on the light which enters the SDSS fibre (radius 1.5 arcsec); this samples a distance-dependent fraction of the light of a galaxy. For small low-mass galaxies which have significant discs, the fibre samples the bulge of the nearby objects, and the discs of the more distant ones. Since M*/L is larger for bulges than discs, the more distant objects of a given M*Bell will have smaller M*Gallazzi. Here, of course, we have explicitly selected against discy galaxies, but colour gradients will produce the same effect.

Fig. A2 shows this effect explicitly: at fixed M*Bell, the ratio M*Gallazzi/M*Bell decreases with increasing distance (this is a more dramatic effect for the lower mass systems in samples which include discs – see Fig. A5). The panel on the right, which shows the angular size for objects of a fixed M*Bell increasing at high z, illustrates a curious selection effect (the upturn at higher redshift for the middle stellar mass bin, which is not present if when we study bins in luminosity). For a given M*Bell, the highest redshift objects in a magnitude-limited survey will be bluer; since colour is primarily determined by velocity dispersion, these objects will have smaller than average velocity dispersions, and larger sizes. Although this is a small effect for the massive objects we have selected here, it is more dramatic in samples which include discs (as we show shortly; see Fig. A5).

Fig. A3 shows how these estimates compare with those from Blanton & Roweis (2007). The top-left panel shows that M*Model is ∼0.04 dex larger than M*Petro (top-left panel). Because the two estimates are based on the same mass-to-light ratio, this offset is entirely due to the fact that Petrosian magnitudes are 0.1 mag fainter than (the more realistic) model magnitudes. The middle-left panel shows that M*Bell/M*Model is approximately independent of mass for gr, and strongly mass dependent for ri. But perhaps most importantly, note that M*Model is typically 0.1 dex smaller than the gr based M*Bell. This offset would have been even larger if we had compared to M*Petro– the fairer comparison, because our M*Bell here is based on the Petrosian magnitude. We argue shortly that this offset reflects the fact that the templates used to estimate M*Model are not appropriate for large masses. In contrast, M*Gallazzi/M*Model is strongly mass dependent at small masses – this is because aperture effects matter for M*Gallazzi but not for M*Model. The behaviour at larger masses is a combination of aperture and template effects, but note again that M*Model is about 0.1 dex low.

The panels on the right show that M*LRG fares much better. The top-right panel shows that it is always substantially larger than M*Model, and the two panels below it show that M*LRG is much more like M*Bell and M*Gallazzi than is M*Model. Comparison with the corresponding panels on the left shows that much of the offset is removed, strongly suggesting that neither M*Petro nor M*Model are reliable at the high-mass end. This is not surprising – massive galaxies are expected to be old, so a 10-Gyr template should provide a more accurate mass estimate than ones that are restricted to ages of ∼7 Gyr or less. This will be important in Section 5. At the low-mass end, M*LRG is larger than either of the other estimates, suggesting that it overestimates the true mass. However, at these smaller masses, using M*Model exclusively produces masses which lie below M*Gallazzi and M*Bell. This suggests that the LRG template remains the better description for a non-negligible fraction of low-mass objects. Therefore, in practice, to use the Blanton & Roweis estimates, one must devise a method for choosing between M*Model and M*LRG.

Fig. A4 compares the different estimates of the stellar mass for the full sample. This gives similar results to when we restricted to Cr≥ 2.86, although the offsets are more dramatic. In this case, the selection effect associated with studying fixed M*Bell makes the aperture effect associated with M*Gallazzi appear more complex than it really is: the large bump at the highest redshift associated with a given M*Bell is not present when one studies objects at fixed luminosity (compare top and bottom panels of Fig. A5).

APPENDIX B: TABLES

This appendix provides tables (Tables B1–B4) which summarize the results of fitting equation (10) to the measured luminosity, stellar mass, size and velocity dispersion distributions shown in Figs 15, 1719. These measurements were obtained using cmodel magnitudes and sizes. We described our measurements by assuming that the intrinsic distributions have the form given by equation (9). The values between round brackets show the parameters obtained fitting equation (9) to the data ignoring measurement errors.