aa r X i v : . [ a s t r o - ph . C O ] J un Photometric properties and luminosity function of nearby massiveearly-type galaxies
Y. Q. He , , , X. Y. Xia , C. N. Hao , Y. P. Jing , S. Mao , , Cheng Li Received ; accepted University of Chinese Academy of Sciences, Beijing 100049, China National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road,Chaoyang District, Beijing 100012, China Tianjin Astrophysics Center, Tianjin Normal University, Tianjin 300387, China; E-mail:[email protected] Center for Astronomy and Astrophysics, Department of Physics and Astronomy, Shang-hai Jiao Tong University, Shanghai 200240, China Jodrell Bank Centre for Astrophysics, University of Manchester, Alan Turing Building,Manchester M13 9PL, UK Partner Group of the Max Planck Institute for Astrophysics at the Shanghai Astronom-ical Observatory and Key Laboratory for Research in Galaxies and Cosmology of ChineseAcademy of Sciences, Nandan Road 80, Shanghai 200030, China 2 –
ABSTRACT
We perform photometric analyses for a bright early-type galaxy (ETG) sam-ple with 2949 galaxies ( M r < − . M r < −
23 mag), our Petrosian magnitudes, and isophotal magnitudes to 25mag / arcsec and 1% of the sky brightness are on average 0.16 mag, 0.20 mag,and 0.26 mag brighter than the SDSS Petrosian values, respectively. In the firstcase the underestimations are caused by overestimations in the sky backgroundby the SDSS PHOTO algorithm, while the latter two are also due to deeperphotometry. Similarly, the typical half-light radii ( r ) measured by the SDSSalgorithm are smaller than our measurements. As a result, the bright-end of the r -band luminosity function is found to decline more slowly than previous works.Our measured luminosity densities at the bright end are more than one orderof magnitude higher than those of Blanton et al. (2003), and the stellar massdensities at M ∗ ∼ × M ⊙ and M ∗ ∼ M ⊙ are a few tenths and a factor offew higher than those of Bernardi et al. (2010). These results may significantlyalleviate the tension in the assembly of massive galaxies between observationsand predictions of the hierarchical structure formation model. Subject headings: galaxies: elliptical and lenticular, cD - galaxies: luminosity function,mass function - galaxies: photometry
1. INTRODUCTION
The properties of nearby early type galaxies (ETGs), especially the brightest ones,offer important clues to understanding the cosmic assembly history of massive galaxies. Interms of their morphologies, colors, stellar population content and scaling relations, ETGsappear to be relatively simple systems compared with spirals and other galaxies. Thereare, however, a lot of renewed interests in their dynamical properties, particularly from therecent integral field unit surveys (e.g. Emsellem et al. 2007; Brough et al. 2010; Cappellariet al. 2012). There is also a debate on their stellar mass assembly processes (Renzini 2006;Scarlata et al. 2007), and so the understanding of ETGs remains a particularly interestingissue.In the past two decades, the concept of “downsizing” (Cowie et al. 1996; Gavazzi& Scodeggio 1996; Fontanot et al. 2009) for galaxy formation has been widely discussed.According to this scenario, the epoch of star formation in ETGs depends on the galaxymass, namely, the most massive ETGs formed their stars on a shorter time scale and atearlier times. Furthermore, large ground-based and space-based imaging and spectroscopicsurveys at low and high redshifts found that the mass function shows a weak evolution formassive ETGs since redshift ∼ M ∗ ≥ M ⊙ at highredshift are more compact with effective radii a factor of ∼ ∼ × M ⊙ < M ∗ < × M ⊙ , whereas the evolution can still besubstantial for ETGs more massive than 4 × M ⊙ . It should also be noted that suchstudies have not focused on the evolution of ETGs at the high mass end with M ∗ ≥ M ⊙ .As pointed out by Naab (2012), the most massive ETGs or their progenitors start formingtheir stars at redshift ∼ § r / surface brightness profile. To avoid underestimatingthe luminosities of the brightest cluster galaxies (BCGs), von der Linden et al. (2007)used isophotal photometry to 23 mag / arcsec in the r -band, rather than the Petrosianmagnitude. Therefore, aperture photometry offers a viable alternative to describe thephotometric properties of ETGs.Underestimating the luminosities of luminous ETGs will lead to an underestimate ofthe stellar mass density at the bright end. Using their own sky background subtractionalgorithm (cmodel) and stellar mass estimation, Bernardi et al. (2010) found a highernumber of very massive galaxies than previous works. The excess can be up to a factor of ∼
10 when the stellar mass M ∗ is larger than 5 × M ⊙ , which is highly significant.In this work, we perform accurate photometry for a complete sample of nearby brightearly-type galaxies with M r < − . M ∗ > M ⊙ . The morphologies of thesegalaxies are taken from the Galaxy Zoo project which classified nearly 900,000 SDSSgalaxies (Lintott et al. 2011). We perform our own sky background subtraction, whichleads to much more accurate photometry of ETGs. We then compare the luminosities andsizes measured by different methods and investigate the luminosity function as well as thestellar mass density for the bright ETGs.The structure of this paper is as follows. In § § §
4, we compare our measured luminositiesand sizes of bright ETGs with the SDSS Petrosian magnitudes and sizes, and present the 6 –results of the r -band luminosity function and the stellar mass density. We finish the paperwith a summary in §
5. In this paper, we adopt a Hubble constant of H = 70 km s − Mpc − ,matter density parameter Ω m = 0 . Λ = 0 .
2. SAMPLE
Our bright early type galaxy (ETG) sample is drawn from the morphological catalogueof Galaxy Zoo 1 (Lintott et al. 2011). The Galaxy Zoo 1 project performed visualmorphological classifications for nearly 900,000 galaxies from the Sloan Digital Sky Survey(SDSS, York et al. 2000), in which 667,945 galaxies are from the Main Galaxy Sampleof SDSS (MGS, Strauss et al. 2002). The MGS includes galaxies with r -band Petrosianmagnitude r ≤ .
77 and r -band Petrosian half-light surface brightness µ ≤ . . All 667,945 galaxies from the MGS have spectroscopic redshift in the rangeof 0 . < z < .
25 and u, g, r, i, z band photometry based on SDSS DR7 (Lintott et al.2011).Given that the SDSS spectroscopy survey is incomplete for bright galaxies with redshiftless than 0.05 (Stoughton et al. 2002; Strauss et al. 2002; Schawinski et al. 2007; Kavirajet al. 2007) and reliable photometric analyses with high signal to noise (S/N) ratios canonly be performed for galaxies with r ≤
16 mag (Fukugita et al. 2007), we constrain ourbright ETG sample to z > .
05 and r <
16 mag. Since we are only concerned with thephotometric properties of the luminous ETGs, we further restrict our sample to brightETGs with M r < − . M r , are calculated usingthe equation M r = m r − D L /
10 pc) − A − k , where D L is the luminosity distance, A isthe Galactic extinction obtained from the photometric catalogue of SDSS DR7 and k is thek-correction derived using the IDL KCORRECT algorithm of Blanton et al. (2007). Finallythere are 7930 bright ETGs in the redshift range of 0 . < z < .
15 and M r < − . . < z < . M r < − . . < z < . M r < −
23 mag and 108 ETGs with 0 . < z < .
15 for M r < − . < V /V max > test (Schmidt 1968) for these subsamples. Theaverage value < V /V max > are 0.51, 0.50 and 0.51 for the three subsamples, respectively,which indicates that all the three subsamples are spatially homogeneous. The photometricanalysis in this work is based on these three subsamples. In total, there are 2949 ETGsbrighter than − −
23 mag. Fig. 2shows the spatial distribution of our sample galaxies on the sky in Galactic coordinates:the total area is 9055 degree , corresponding to 22% of the whole sky. Fig. 3 presents thehistograms showing the distributions of apparent magnitude and spectroscopic redshiftfor 2949 early-type galaxies. The median Petrosian magnitude is 15.37 mag and medianredshift is 0.087, respectively.In order to verify the early-type morphology of our sample galaxies, we visuallyinspected the SDSS r -band images for all 2949 galaxies. In addition, we also checked othercommonly used classification criteria for early-type galaxies. All our sample galaxies have g − r > . C r = r /r larger than 2.86 (Bernardi et al. 2010), where r and r are the radiicontaining 90% and 50% of Petrosian flux. Therefore, our selected sample galaxies are allETGs. 8 –
3. DATA REDUCTION3.1. Estimation of the Sky Background
The corrected frame fpC-images in the r -band for our sample galaxies are directlyobtained from the SDSS DR7 Data Archive Server. The images have been pre-processedby the SDSS photometric pipeline (PHOTO), which includes bias subtraction, flat-fieldingand bad pixels correction (cosmic rays removal, bad columns and bleed trails). In order toobtain the SDSS photometric flux calibration information during observations, such as thephotometric zeropoint a , the first-order extinction coefficient k and the airmass X , we alsodownloaded the calibrated field statistic file, named as tsField from the archive.As mentioned in §
1, the SDSS photometric reduction systematically underestimatesthe luminosities and half-light radii for bright ETGs, which is mainly caused by inadequatemasking of their neighbors or extended stellar halos (Aihara et al. 2011), leading to anoverestimate of the sky background. In this work we perform the masking more carefullyand estimate the sky background model following Liu et al. (2008), which has beensuccessfully used to measure the luminosities and half-light radii for the brightest clustergalaxies (BCGs) that are located in crowded fields and usually have extended faint stellarhalos.In the following, we outline our sky background subtraction approach. In order toobtain the sky background, we first masked out all objects detected by SExtractor (Bertin& Arounts 1996) in the corrected frames with 2048 × ′ . × ′ . − −
23 mag, which mostly include ETGs residing in 9 –crowded fields, having extended stellar halos or are close to foreground bright stars. Forthe 2949 ETGs brighter than − −
23 mag, the percentages are evenhigher, about 18%, 10%, 5%, respectively. For these ETGs, we modify the mask imagesmanually. For comparison, Fig. 4 shows three cases (top, middle and bottom rows) betweenthe masked images generated by the automatic algorithm of SExtractor (middle panels)and those by hand (right panels). The left panels of Fig. 4 show the original true colorimages for the example galaxies. The top row shows a galaxy in a crowded field. Fromthe top middle panel, it can be seen that SExtractor cannot separate the target galaxyfrom its neighbors well. Therefore we manually flag the nearby objects with circles that arelarge enough to cover the objects completely, as shown in the top right panel. The secondrow shows a bright ETG with an extended stellar halo. From the middle panel, it is clearthat the flagged area generated by SExtractor is not big enough to cover the whole stellarhalo. Hence, the stellar halo would be considered as the background and subtracted fromthe galaxy itself, leading to underestimates of the luminosity and half-light radius of thegalaxy. Following the shape of the target ETG, we use an ellipse to mask the entire stellarenvelope, as shown in the middle right panel. SExtractor also cannot handle well galaxiessurrounded by bright foreground stars, especially those with diffractive spikes (bottompanels). Therefore, we use long rectangles to mask the star spikes (bottom right). Themasked images generated manually can provide not only more accurate sky backgroundimages but also good masks for the surface photometry (see § ×
51 pixels. This filter size is selected tobe larger than the sizes of most objects in the frame but still sufficiently small so that thevariation of the sky background within the region is still reproduced (i.e., not smoothedout). After the median filtering is performed, masked regions smaller than the median 10 –box filter are replaced with the surrounding sky background, whereas part of the maskedregions larger than the median box filter remain flagged. With the small field of view(13 ′ . × ′ . z = a + b ∗ x + c ∗ y ) using the IRAF/IMSURFIT task. The skybackground map is typically tilted with a spatial variation of ∼ − ×
501 pixels centered on the galaxy ofinterests. The trimmed mask image with the target galaxy un-flagged will be used to probethe masked regions in the isophote fitting.
After the sky background subtraction, we perform surface photometry for our brightETGs in order to estimate the luminosities and sizes of sample galaxies. As is well known,the Petrosian magnitude and Petrosian size are most commonly used to describe the galaxyflux and half-light radius for the analyses based on SDSS database, because they do notdepend on the model fitting to galaxies. The Petrosian radius r p is defined as the radius r at which the ratio of the local surface brightness averaged over an annulus between 0 . r and1 . r to the mean surface brightness within r equals to 0.2 and the Petrosian magnitudeis the integrated flux within 2 r p (Petrosian 1976). However, the Petrosian magnitudemisses the light outside 2 r p (Petrosian aperture) because of its dependence on the surfacebrightness profile of galaxies (Graham et al. 2005), which leads to underestimates of thefluxes and sizes for galaxies with extended stellar halos. In this work, we will measure 11 –not only the Petrosian magnitude and Petrosian half-light radius, but also the isophotalmagnitude and half-light radius to deeper isophote limits at 25 mag / arcsec and 1% of skybrightness, corresponding to ∼
26 mag / arcsec in most cases (Bernardi et al. 2007).We perform the surface photometry analysis following Wu et al. (2005) and Liu et al.(2008). The procedures are briefly described below. First, we use ISOPHOTE/ELLIPSE taskin IRAF to fit each of the trimmed sky-subtracted images excluding the masked regions inthe fitting. The surface brightness of the target galaxy is fitted by a series of elliptical annuliin a logarithmic step of 0.1 along the semi-major axis. The annuli chosen in the outer partsof image is larger, which can suppress the shot noise in the outer regions where the signalto noise ratio (S/N) is much lower. The output of
ISOPHOTE/ELLIPSE is the mean intensityin each isophote annulus. Then we integrate the surface brightness profile to isophotallimits of 25 mag / arcsec or 1% of sky brightness to obtain the apparent magnitudes andhalf-light radius r . We also measure the Petrosian magnitudes and r based on our skybackground subtracted images. In our analysis, the equivalent radius √ ab of an ellipse isused for all the radial profiles, where a and b are the semi-major and semi-minor radii ofthe ellipse. The cosmological dimming is also taken into account for the surface brightnessprofiles. The observational errors in the surface brightness profile include random errors(e.g., the shot noise of the object and sky background, readout noise and noise contributedby data reduction) and the error from sky background subtraction.
4. RESULTS4.1. Luminosities of bright ETGs in the local universe
The SDSS database provides the largest galaxy sample with both photometric andspectroscopic information in the local universe. However, the underestimation of the 12 –luminosities and sizes for the bright ETGs prevents us from correctly understanding theirproperties and assembly history.For brevity in later discussions, we define∆ m p = m p , sdss7 − m p , ∆ m = m p , sdss7 − m , ∆ m = m p , sdss7 − m , (1)where m p , m and m are our measured Petrosian magnitude, isophotal magnitudes withsurface brightness measured to 25 mag/arcsec and 1% of sky brightness, and m p , sdss7 is thePetrosian magnitude from the SDSS DR7 pipeline in the r -band.The top and middle panels in Fig. 5 show ∆ m p (defined eq. 1) as a function of theSDSS Petrosian apparent and absolute magnitudes, respectively, while the bottom panelshows the histogram of ∆ m p . Clearly for more luminous ETGs, the luminosity differencebetween the SDSS and our measurements is larger. The mean and median values of theluminosity differences are 0.16 mag and 0.14 mag, respectively. It shows that the algorithmused in the SDSS DR7 has systematically overestimated the sky background for brightETGs, leading to underestimates in the luminosities of bright ETGs, especially for thebrightest ones.Addressing the same issue, in the SDSS DR8, Aihara et. al. (2011) have re-processedall SDSS imaging data using a more sophisticated sky background subtraction algorithmand obtained significant improvement. The left and right panels of Fig. 6 are histogramsof luminosity difference between the SDSS DR8 and SDSS DR7, and between ourmeasurements and SDSS DR8, respectively. We can see that the median values ofluminosity difference for these two cases are − − − − m measured to the surfacebrightness of 25 mag / arcsec are generally larger than the Petrosian values for BCGs. Wethus also compare the Petrosian and isophotal magnitudes for our sample ETGs. Thetop and middle panels of Fig. 7 show ∆ m (defined in eq. 1), as a function of the SDSSPetrosian apparent and absolute magnitudes, respectively. It is obvious that ∆ m increaseswith the apparent and absolute magnitudes of ETGs. For M p , sdss7 < −
23 mag, the meanand median differences are 0.20 mag and 0.17 mag, respectively, which are larger than thosefor ∆ m p (0.16 and 0.14 mag, respectively).Bernardi et al. (2007) investigated the galaxy luminosity by integrating the best-fitmodel of galaxy surface brightness profile to 1% of sky brightness. The top and middlepanels of Fig. 8 show ∆ m (defined in eq. 1) as a function of the SDSS Petrosian apparentand absolute magnitudes, respectively. The same trend is seen: ∆ m increases as ETGsbecome more luminous. For M p , sdss7 < −
23 mag, the mean and median values of ∆ m are0.26 mag and 0.23 mag, respectively. Note that the photometric limit of 1% sky is ∼ / arcsec , almost one magnitude deeper than 25 mag / arcsec .We further examine the percentage of ETGs with large luminosity difference betweenthe SDSS Petrosian and our measured magnitudes. For the 2949 ETGs brighter than − . m p , ∆ m and ∆ m larger than 0.3 mag.For the 1053 ETGs brighter than −
23 mag, the fractions are even higher, 11%, 19% and 14 –31%, respectively. For more luminous ETGs, the underestimation by the SDSS algorithmis more severe. In addition, the deeper the photometric measurements, the larger theunderestimations.We point out that to the photometric limit of ∼
26 mag / arcsec the light still belongsto galaxies as shown by Tal & van Dokkum (2011). They stacked more than 42000 SDSSimages of LRGs (Luminous Red Galaxies), reaching a depth of ∼
30 mag / arcsec , andfound that the stellar light out to 100 kpc is physically associated with galaxies, insteadof inter-cluster or inter-group light. Our photometric measurement to 1% sky brightnessreaches at most 100 kpc (50 kpc on average, see § Galaxy size is one of the most important parameters of galaxy properties. A reliabledetermination of the sizes of bright ETGs in the local universe provides the basic calibrationfor investigating the size evolution, an area of active study in recent years (e.g. Szomoru etal. 2012; McLure et al. 2012; Trujillo 2012). The aforementioned works compare galaxysizes at high redshift to those of the local universe by Shen et al. (2003) based on the SDSSPHOTO algorithm, who found a power-law relation between the galaxy luminosity andsize (Shen et al. 2003). However, if the luminosities of galaxies have been underestimated,galaxy sizes may have been underestimated too. In this section, we will discuss thesize difference between the SDSS and our measurements based on different luminosityestimations. For convenience, we define r , p , r , and r , as our Petrosian half-lightradius, isophotal half-light radii to 25 mag/arcsec and 1% of the sky brightness. Thesevalues will be compared with r , sdss7 , the Petrosian half-light radius from the SDSS DR7pipeline in the r -band. 15 –Fig. 9 presents histograms of the half-light radii measured by different photometricmethods. Following Shen et al. (2003), a log-normal function is fitted to the half-light radiidistribution. The log-normal function is defined as f ( r, ¯ r, σ ln r ) = 1 √ πσ ln r exp (cid:20) − ln ( r/ ¯ r )2 σ r (cid:21) drr , (2)which is characterized by the median ¯ r and the dispersion σ ln r . We find that the best-fitvalues for r , sdss7 , r , p , r , , r , distributions are ¯ r = 8 . , . , . , .
82 kpc and σ ln r = 0 . , . , . , .
32. From Fig. 9, it can be clearly seen that the largest r , sdss7 valueis smaller than 20 kpc, while our measured r , p , r , and r , can be as large as 30 kpcand a large fraction ( ∼ m p , ∆ m and ∆ m , as a function of r , sdss7 respectively. We can see from Fig. 10 that there is a clear trend that ∆ m increases as ETGs become larger.We further examine the images of the brightest ETGs ( M r < −
23 mag) with differencesbetween the r and the SDSS DR7 measurement larger than 10 kpc and ∆ m > . α = 0 . ± .
03) is steeper than that in the top left panel ( α = 0 . ± . / arcsec and to 1% of skybrightness, respectively. We can see from the bottom two panels of Fig. 12 that the slope forthe correlation between log r , and M ( α = 0 . ± .
03) is somewhat steeper than thecorrelation between log r , and M ( α = 0 . ± . r and M r relation for ETGs is much larger than thevalue 0.67 found by Shen at al. (2003) that is widely adopted by recent works on the sizeevolution for bright ETGs. If we use our measured sizes of ETGs, the size evolution sinceredshift 2 will be somewhat larger. However, due to the surface brightness dimming, it maybe difficult to perform photometry down to 26 mag / arcsec in the rest-frame r -band forETGs at redshift 2. As a result, we should be more cautious in discussing the size evolutionby explicitly taking into account the survey surface brightness limit. r -band LF and stellar mass density A basic way to investigate galaxy properties and their evolution is by studying theluminosity function (LF). There are already many LF studies using different samples andapproaches at different redshifts (see the recent review paper by Johnston 2011). Giventhat the luminosities of the brightest galaxies ( − ∼
10% to 40%), it is worth revisiting the LF at the bright end in the local universe. 17 –In this work, we construct the galaxy luminosity function at the bright end, utilizingthe non-parametric 1 /V max method (Schmidt 1968; Felten 1976; Eales 1993). Our brightETG sample includes 2949 galaxies consisting of three subsamples as described in §
2, inwhich we have already discussed the homogeneity for this sample. Briefly, the 1 /V max isthe inverse of the maximum volume, to which the galaxy could have been detected. TheLF is obtained by integrating 1 /V max in different luminosity bins for the whole sample ofgalaxies. Given that our ETGs contain three subsamples, we calculate the LF in threevolumes separately, and then average them to obtain the final LF. In addition, SDSS fibercollisions lead to ∼
7% incompleteness for the spectroscopic sample (Bernardi et al. 2010),we multiply the 1 /V max counts by a factor of 1 / .
93 to obtain the final LF.The top panel of Fig. 13 shows our r -band luminosity function at the bright end. Thegreen triangles, red circles and blue solid squares represent the luminosity function for ourmeasured Petrosian magnitude, isophotal magnitudes measured to the surface brightnessof 25 mag / arcsec and 1% of the sky brightness, respectively. For comparison, we alsoplot the luminosity functions from Blanton et al. (2003) and Bernardi et al. (2010) inthe top panel of Fig. 13. Blanton et al. (2003) used a sample of 147,986 galaxies (fromSDSS EDR) and the maximum likelihood method to calculate the luminosity function at z = 0 .
1. Their sample is much larger, but includes all morphological types. However, at thebright end (e.g. the galaxies brighter than − r -band luminosity of galaxies in Blanton et al. (2003) are directlyobtained from the SDSS catalogue based on the SDSS PHOTO algorithm. Bernardi et al.(2010), on the other hand, used an ETG sample of galaxies selected from ∼ . < m p , sdss < . C r ≥ .
86, which is aconservative way to select ETGs from SDSS. The luminosities of ETGs in Bernardi et al.(2010) are calculated using the cmodel in the SDSS pipeline with their own sky backgroundsubtraction method. 18 –It is clear from the top panel of Fig. 13 that at the bright end ( M r < − . / arcsec and 1% of the sky background(denoted by φ p , φ and φ , respectively) at several r -band luminosities. The bottompanel of Fig. 13 shows the ratios of these galaxy luminosity densities to that measured byBlanton et al. (2003), denoted as φ p , Blanton , as a function of r -band luminosities. It is clearthat the luminosity density ratios increase with the luminosity of ETGs. For ETGs with M r = − . ± .
46, 8 . ± .
40 and 10 . ± .
56, respectively; forETGs with M r = −
24 mag, they go up to 153 ±
14, 197 ±
16 and 259 ±
22, respectively.It demonstrates that the luminosity density of the brightest ETG calculated based on theSDSS catalogue has been seriously underestimated. Note that for the three magnitudes M p , M and M , the number of ETGs brighter than − −
24 mag is 41, 56 and 90, and so the number statisticsare reasonably good. The underestimates in the luminosity density result in a significantunderestimate of the integrated luminosity density. In particular, the integrated luminositydensity down to M r < − . M p , M , M are about 20%,40%, 50% higher than that from the SDSS Petrosian luminosity, respectively.Next we estimate the stellar mass M ∗ of the ETGs following Bernardi et al. (2010),utilizing the equationlog M ∗ /M ⊙ = 1 . g − r ) − . − . M r − . − . z, (3) 19 –where M ∗ is the stellar mass (in solar units), g − r is the rest-frame color, M r is theabsolute magnitude, and z is the redshift. This equation has already taken into account thek-correction and evolution correction, and the initial mass function (IMF) is assumed to beof the Chabrier (2003) form (see § g − r color and r -band Petrosian magnigtude, which have been adapted tothe Chabrier IMF here. A proper model fitting to colors and magnitudes measured usingour photometric methods may give different coefficients in eq. (3). However, to facilitatecomparison with the stellar mass function obtained by Bernardi et al. (2010), we use thesame equation as their study. Under the assumption that the impact of our photometricalgorithm on the r -band photometry is the same as that on the g -band, we use the SDSSmodel g − r color as a surrogate of colors measured by our methods . The stellar masses M ∗ , p , M ∗ , and M ∗ , are obtained using our measured luminosities M p , M , M ,respectively.Table 2 gives the stellar mass densities, φ M ∗ , p , φ M ∗ , and φ M ∗ , at several stellarmasses of ETGs for our measured magnitudes. The top panel of Fig. 14 shows the stellarmass function for massive ETGs. We also plot the stellar mass function from Blanton &Roweis (2007) and Bernardi et al. (2010) in the top panel of Fig. 14 for comparison. Ascan be seen, the slope of our measured stellar mass functions is shallower than those ofBernardi et al. (2010) and Blanton & Roweis (2007). Given that Bernardi et al. (2010)have already compared their stellar mass density with previous observational results, in thebottom panel of Fig. 14 we just compare our result with Bernardi et al. (2010). It showsthat the stellar mass densities ratios of our measurements to that of Bernardi et al. (2010), In the SDSS Data Release 2 paper (Abazajian et al. 2004) and SDSS web page, modelmagnitudes are recommended to be used for the measures of the colors of extended objects. 20 –written as φ M ∗ , Bernardi , as a function of stellar mass. We can see from the bottom panel ofFig. 14 that the ratios are larger for more massive ETGs. The ratios, φ M ∗ , p /φ M ∗ , Bernardi , φ M ∗ , /φ M ∗ , Bernardi and φ M ∗ , /φ M ∗ , Bernardi are 1.2 ± ± ± M ∗ ∼ × M ⊙ ; for M ∗ ∼ M ⊙ , the ratios are even larger, 2.1 ± ± ±
5. SUMMARY
In this work, we performed photometric analyses for a complete and homogeneousbright ETGs sample with 2949 early-type galaxies ( M r < − . M < −
23 mag), our Petrosian magnitudes,and isophotal magnitudes to 25 mag / arcsec and 1% of the sky brightness are on 21 –average 0.16 mag, 0.20 mag, and 0.26 mag brighter than the SDSS Petrosian values,respectively. In the first case the underestimations are due to overestimations in thesky background by the SDSS PHOTO algorithm, while the latter two are also causedby an additional effect of reaching deeper photometry. Such underestimation is moresevere as ETGs become more luminous. Our results also demonstrate that as weintegrate to deeper surface brightness, we recover more luminosity of galaxies.2. We also find that the sizes of ETGs (half-light radius r ) measured by the SDSSalgorithm are smaller than those measured by us. The largest r of bright ETGs inthe SDSS catalogue is ∼
20 kpc, while our measured r can be as large as 30 kpc andfor a large fraction ( ∼ r is larger than 15 kpc. In addition,we find that the slope in the size-luminosity relation at the bright end is much steeperthan that found by Shen et al. (2003).3. Based on our selected complete and homogeneous sample of 2949 bright early-typegalaxies, we construct the luminosity function. We find that the LF at the bright endis much shallower than those of Blanton et al. (2003) and Bernardi et al. (2010). Theluminosity density at − −
24 mag) measured in this work is one order (twoorders) of magnitude higher than that of Blanton et al. (2003). As a result, the ratiosof the integrated luminosity density for bright galaxies ( M r < − . M p , M , M and the SDSS Petrosian luminosity are1.2, 1.4 and 1.5, respectively. Similarly, the stellar mass density of ETGs is a fewtenths to a factor of few higher than that of Bernardi et al. (2010) for stellar massfrom ∼ × M ⊙ to ∼ M ⊙ . Therefore, our method recovers substantiallymore luminosity for bright ETGs, which may alleviate the contradiction betweenhierarchical galaxy formation theories and current observations. 22 –Our results suggest that very careful photometry needs to be performed to obtain the LFat the bright end which has significant impact on the stellar mass function for massivegalaxies. Previous claims that the massive end of the stellar mass function has a weakevolution since redshift ∼ L ∗ REFERENCES
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M., Arag´on-Salamanca, A., et al. 2011, MNRAS, 412, 246 27 –Wu, H., Shao, Z. Y., Mo, H. J., Xia, X. Y., Deng, Z. G. 2005, ApJ, 622, 244York, D. G., Adelman, J., Anderson, J. E., et al. 2000, AJ, 120, 1579This manuscript was prepared with the AAS L A TEX macros v5.2. 28 –Fig. 1.— Three volume-limited subsamples selected from Galaxy Zoo 1 early-type galaxiesin the redshift vs. absolute Petrosian magnitude plane in the r -band of SDSS DR7, shownin blue, cyan and magenta boxes, respectively. The total number of galaxies is 2949 ETGs.The red lines show the observed flux limits at r = 16 mag and 17.77 mag, respectively.The number of early-type galaxies in each subsample within the redshift range and lowerluminosity limit are shown in the bottom right. 29 –Fig. 2.— The sky coverage for our volume-limited bright early-type galaxies in the Aitoffprojection in Galactic coordinates. The total sky coverage is about 9055 degree . 30 –Fig. 3.— Distributions of the SDSS Petrosian apparent magnitude in the r -band (left panel)and redshift (right panel) for 2949 early-type galaxies with M r < − . median=0.09median=0.14 Fig. 5.— The difference ∆ m p between the SDSS Petrosian magnitude and our measuredPetrosian magnitude as a function of the SDSS apparent Petrosian magnitude m p , sdss7 (toppanel) and Petrosian absolute magnitude M p , sdss7 (middle panel). The red data points witherror bars are the median, lower (25 per cent) and upper (75 per cent) quartiles for binnedgalaxies. The bin width is 0.4 mag for the apparent Petrosian magnitude except the lastbin which has a width of 0.6 mag to include all the remaining objects (top panel); the binwidth is 0.2 mag for the absolute magnitude except the last bin which has a width of 0.7mag to include all the remaining objects (middle panel). The medians shown in red pointsare 0.081, 0.081, 0.098, 0.114, 0.145, 0.216 mag in top panel, and 0.065, 0.078, 0.101, 0.138,0.179, 0.188, 0.225 mag in middle panel. The bottom panel shows the histograms of themagnitude difference with the mean and median values indicated in the top right, as blacksolid line for galaxies with M p , sdss7 < − M p , sdss7 < −
23 mag. 33 –Fig. 6.— The distributions of the difference between the SDSS DR8 and SDSS DR7 Petrosianmagnitudes (left panel), and the difference between our measured Petrosian magnitude andthe SDSS DR8 (right panel). The mean and median values for each distribution are shownin the top left of the panel. 34 – median=0.14median=0.17
Fig. 7.— The difference ∆ m between the SDSS Petrosian magnitude and our isophotalmagnitude to 25 mag/arcsec as a function of the SDSS apparent m p , sdss7 (top panel) andabsolute M p , sdss7 (middle panel) isophotal magnitudes. See Fig. 5 for an explanation of thered data points with error bars in the top and middle panels. The medians shown in redpoints are 0.112, 0.122, 0.151, 0.175, 0.220, 0.278 mag in top panel, and 0.111, 0.131, 0.142,0.172, 0.180, 0.213, 0.219 mag in middle panel. The bottom panel shows the histogram ofmagnitude difference distribution with the mean and median values indicated in the topright, as black solid line for galaxies with M p , sdss7 < − M p , sdss7 < −
23 mag. 35 – median=0.20median=0.23
Fig. 8.— The difference ∆ m between the SDSS Petrosian magnitude and our isophotalmagnitude with surface brightness limit at 1% of sky brightness as a function of the SDSSapparent m p , sdss7 (top panel) and absolute M p , sdss7 (middle panel) magnitudes. See Fig. 5for an explanation of the red data points with error bars in the top and middle panels. Themedians shown in red points are 0.170, 0.184, 0.209, 0.241, 0.280, 0.326 mag in top panel,and 0.166, 0.131, 0.188, 0.201, 0.237, 0.242, 0.270, 0.267 mag in middle panel. The bottompanel shows the histogram of magnitude difference distribution with the mean and medianvalues indicated in the top right, as black solid line for galaxies with M p , sdss7 < − M p , sdss7 < −
23 mag. 36 –Fig. 9.— The histograms of half-light radii ( r ) measured by different photometric methods.The black and green dotted lines represent the Petrosian r distributions based on SDSSand our own measurements, respectively. While the red solid and blue dot-dashed linesshow the r distributions based on isophotal measurement to 25 mag / arcsec and 1% of skybrightness, respectively. Log-normal (see eq. 2) fits are shown for each distribution withcorresponding colors, and the medians and dispersions are shown in the top right. 37 –Fig. 10.— The magnitude differences between the SDSS Petrosian and our measured Pet-rosian, isophotal magnitudes with surface brightness limits at 25 mag / arcsec and 1% of skybrightness respectively, as a function of the SDSS Petrosian half-light radius r , sdss7 . Thered data points with error bars are the median, lower (25 per cent) and upper (75 per cent)quartiles for galaxies in bins of width 2 kpc in half-light radius except the first bin which hasa width of 4 kpc and the last bin which has a width of 9 kpc to include all the remainingobjects. The medians shown in red points are 0.056, 0.093, 0.160, 0.234, 0.270 mag in toppanel, and 0.117, 0.125, 0.166, 0.205, 0.269 mag in middle panel, and 0.169, 0.189, 0.232,0.277, 0.314 mag in bottom panel. 38 –Fig. 11.— Examples of bright ETGs with extended halos and corresponding surface bright-ness profiles. The red solid line shows the best fit S´ersic law convolved with the point spreadfunction. The dotted and dashed lines in the middle panels represent the surface brightnessof 25 mag / arcsec and 1% of sky brightness respectively. The corresponding S´ersic law in-dex n is shown in the top right of each panel. The bottom panels show the residuals of theobserved surface brightness from the S´ersic model. 39 –Fig. 12.— The grayscale representation of the size-luminosity relation for 2949 bright ETGs.The four absolute magnitudes used are the SDSS Petrosian magnitude (top left), our ownPetrosian magnitude (top right), and aperture magnitudes to 25 mag / arcsec (bottom left)and 1% of sky brightness (bottom right), respectively. The red line in each panel is thebest power-law fit for the size-luminosity relation. The power-law index α ( r ∝ L α ) isshown in the bottom right of each panel. The rightmost vertical grayscale bar reflects thecorresponding number of galaxies. 40 –Fig. 13.— The top panel shows the r -band luminosity function at the bright end calculatedbased on our measured luminosities for 2949 ETGs brighter than M r < − . and 1% of sky brightness respectively. The magenta dot-dashed and black solidlines give the fits by Blanton et al. (2003) and Bernardi et al. (2010). The points for objectsbrighter than the luminosity shown by the cyan dotted line are not affected by photometry.The bottom panel shows the galaxy luminosity density ratios as a function of the r -bandluminosity. The green triangles, red circles, and blue solid squares show the ratios for ourmeasured Petrosian, isophotal luminosities to 25 mag/arcsec and 1% of sky brightness tothat from Blanton et al. (2003). 41 –Fig. 14.— The top panel shows the stellar mass function for massive ETGs. The stellarmasses are estimated from our measured Petrosian (green triangles), isophotal luminosities to25 mag/arcsec (red circles) and 1% of sky brightness (blue solid squares), respectively. Themagenta dot-dashed and black solid lines represent the best-fit stellar mass functions fromBlanton & Roweis (2007) and Bernardi et al. (2010), respectively. The navy blue dashedline represents the prediction from the semi-analytic models of the Millennium Simulationof Guo et al. (2011). The points for objects more massive than the stellar mass shown bythe cyan dotted line are not affected by the photometry. The bottom panel shows the ratiosbetween our measured stellar mass densities and those of Bernardi et al. (2010) as a functionof stellar mass; the symbols are the same as in the top panel.able 1. Luminosity densities for the Petrosian magnitude ( φ p ) and isophotal magnitudesto 25 mag/arcsec ( φ ) and 1% of the sky background ( φ ). M r φ p φ φ (mag) (10 − Mpc − mag − ) (10 − Mpc − mag − ) (10 − Mpc − mag − )-22.75 a ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± a The luminosity density at M r = − .
75 mag is affected by the photometry. able 2. Stellar mass densities for the Petrosian magnitude ( φ M ∗ , p ) and isophotalmagnitudes to 25 mag/arcsec ( φ M ∗ , ) and 1% of the sky background ( φ M ∗ , ). log M ∗ /M ⊙ φ M ∗ , p φ M ∗ , φ M ∗ , (10 − Mpc − dex − ) (10 − Mpc − dex − ) (10 − Mpc − dex − )11.425 a ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± a The stellar mass density at log M ∗ /M ⊙ = 11 ..