Mon.Not.R.Astron.Soc.000,1–15(0000) Printed8January2009 (MNLATEXstylefilev2.2) Curvature in the scaling relations of early-type galaxies ⋆ Joseph B. Hyde & Mariangela Bernardi Department of Physics & Astronomy, Universityof Pennsylvania, 209 S. 33rd St., Philadelphia, PA 19104, USA 9 0 8January2009 0 2 n ABSTRACT a We select a sample of about 50,000 early-type galaxies from the Sloan Digital Sky J Survey(SDSS),calibratefittingformulaewhichcorrectforknownproblemswithpho- 8 tometric reductions of extended objects, apply these corrections, and then measure a number of pairwise scaling relations in the corrected sample. We show that, because ] they are not seeing corrected, the use of Petrosian-based quantities in magnitude h limited surveys leads to biases, and suggest that this is one reason why Petrosian- p basedanalysesof BCGs havefailed to find significantdifferences fromthe bulk of the - o early-typepopulation.These biasesarenotpresentwhenseeing-correctedparameters r derived from deVaucouleur fits are used. Most of the scaling relations we study show t s evidence for curvature: the most luminous galaxies have smaller velocity dispersions, a larger sizes, and fainter surface brightnesses than expected if there were no curva- [ ture.Thesestatementsremaintrueifwereplaceluminositieswithstellarmasses;they 2 suggest that dissipation is less important at the massive end. There is curvature in v the dynamical to stellar mass relation as well: the ratio of dynamical to stellar mass 2 increases as stellar mass increases, but it curves upwards from this scaling both at 2 small and large stellar masses. In all cases, the curvature at low masses becomes ap- 9 parent when the sample becomes dominated by objects with stellar masses smaller 4 than 3×1010M⊙. We quantify all these trends using second order polynomials; these . 0 generallyprovidesignificantlybetterdescriptionofthedatathanlinearfits,exceptat 1 the least luminous end. 8 Keywords: methods:analytical-galaxies:formation-galaxies:haloes-darkmatter 0 : - large scale structure of the universe v i X r a 1 INTRODUCTION Measurements of brightest cluster galaxy scaling rela- tionshaveshownthemtobeunusual(Malumuth&Kirshner Early-type galaxy observables, luminosities L, half-light 1981,1985;Oegerle&Hoessel1991;Postman&Lauer1995). radii R , mean surface brightnesses I , colors, and veloc- e e Andstatistically significantdetectionsofcurvatureinmany ity dispersions σ, are strongly correlated with one another. scalingrelationsacrosstheentirepopulationhavenowbeen Thesescalingrelationsareusuallydescribedassinglepower- made(Zaritskyetal.2006;Laueretal.2007;Bernardietal. laws (R L3/5, σ L1/4, g r σ0.4, R I−0.8), sug- e ∝ ∝ − ∝ e ∝ e 2007; Desrocheset al.2007; Liu etal. 2008; butseevonder gestingasingleformationmechanismacrossthepopulation. Linden et al. 2007). The main goal of this paper is to ex- However, galaxy formation models suggest that the bright- ploitthelargesamplesizeprovidedbytheSloanDigitalSky est galaxies in clusters had unusual formation histories (De Survey(hereafterSDSS)tomakeprecisionmeasurementsof Lucia et al. 2006; Almeida et al. 2008), so they may follow thecurvaturein these scaling relations. different scaling laws (Boylan-Kolchin et al. 2005; Robert- Section2describeshowweselectasampleofearly-type son et al. 2006; Hopkins et al. 2008a). Formation histories, galaxies from theSDSS.Italsodiscusses thecorrections we and the importance of gaseous dissipation and/or gas rich apply to account for the fact that the SDSS photometric mergers are also expected to havebeen different depending reductions are unreliable for extended objects. These are on the mass range of the galaxy (e.g. Mihos & Hernquist particularly important since the curvaturewe would like to 1993; Naab et al. 2006; Hopkins et al. 2008b, Ciotti et al. measure is small (else it would have been seen in smaller 2007). So, one might reasonably expect to see departures samples), so the photometric and spectroscopic parameters fromthesinglepower-lawscalingrelations,especiallyatthe at theextremes of thesample must be reliable. extremes of thepopulation. Section 3 quantifies the curvature in a number of pairwise scaling relations for this sample. It also shows that, in ⋆ E-mail:jhyde,bernardm@physics.upenn.edu the relations which involve luminosity, the curvature per- 2 J. Hyde & M. Bernardi sists ifonereplaces luminosities with stellar masses. Afinal section summarizes our findings. An Appendix discusses why, because they are not seeing-corrected,Petrosianbasedquantitiesareill-suitedfor precisionmeasurementsindeep,magnitude-limited,ground- baseddatasets.ItalsoshowswhythePetrosian-based anal- ysisofBCGsbyvonderLindenetal.(2007)yieldedanoma- lous results. 2 THE SAMPLE We start from a sample of 376471 galaxies based on the Fourth Data Release (DR4) of the SDSS but with parameters updated to theSDSSDR6values (Adelman-McCarthy et al. 2008). From this sample we extract 46410 early-type galaxies following the procedure described below. We use SDSS deVaucouleur magnitudes and sizes, model colors, and aperture corrected velocity dispersions to r /8 unless e stated otherwise. The SDSS also outputs Petrosian magnitudes and sizes. However, these are not seeing corrected, and Appendix A shows that this introduces systematic biases, so we do not use them in what follows. Throughout, angular diameter and luminosity distances were computed from themeasured redshiftsassuming aHubbleconstant of H0 =70 km s−1 Mpc−1 in a geometrically flat background modeldominated byacosmological constant atthepresent time: (Ω0,Λ0)=(0.3,0.7). Aboutnotation:weuseR tospecifyradiusinkpc,and e r to specify angular size in arcseconds. For surface bright- e nesses, we use thefollowing definition. L µ = 2.5log (I )= 2.5log e − 10 e − 10„2πR2« e = m+5log10(re)+2.5log10(2π)−10log10(1+z)(1) Figure 1. Comparison of the velocity dispersion measurements from the SDSS-DR6 and the IDLspec2d reduction. Vertical wheremistheevolution,reddening,andk-correctedappar- dashedlineshowsourcutatsmallσ 60kms−1.Symbolswith entmagnitude.WeuseM todenotetheabsolutemagnitude ∼ x errorbarsshowthemedianvalueanditsuncertainty,dashedand in band x,and M∗ to denotestellar mass in solar units. dottedcurvesshowtheregionswhichenclose68%and95%ofthe objects. 2.1 Selecting early-types To obtain a sample of early-type galaxies we first select beencorrectedintheDR6release(seeDR6documentation, the subset of galaxies which are very well-fit by a deVau- or discussion in Bernardi 2007). We compared the values couleur profile in the g and r bands (g-band fracDev = 1 given by the official SDSS-DR6 database and those com- and r-band fracDev = 1); this gives about 100603 objects. putedinthePrincetonreduction(IDLspec2d).Thetwosets To avoid contamination by later-type galaxies we also re- ofmeasurementsstillshowweaksystematictrends.Theup- quire the spectrum to be of “early-type”, by setting eClass per panel of Figure 1 shows that at fixed σDR6 the median < 0 (see SDSS documentation for details of this classifica- values of σIDLspec2d agree quite well with the σDR6 values, tion). This slightly reduces the sample, to 94934 galaxies. except at large σDR6 > 320 km s−1, where σDR6 is slightly Since the SDSS spectroscopic survey is magnitude limited, larger. However, the bottom panel shows that, when com- we require that r band deVaucouleur magnitudes satisfy paredatfixedσ ,asystematictrendismoreevident IDLspec2d 14.5 < mr < 17.5−. (Spectra are actually taken for objects – especially at small σIDLspec2d <120 km s−1. To minimize having Petrosian magnitude m <17.77; our more con- systematics,wedecidedtoaveragetheDR6andIDLspec2d r,Pet servative cut is designed to account for the fact that the velocity dispersion measurements. Petrosian quantity systematically underestimates the total At the low end, we select galaxies with σ >60 km s−1 lightinadeVaucouleurprofile.)Thiscutleaves70417galax- (seeSection3.4,Bernardietal.2003candHyde&Bernardi ies.Ofthese,64492havestellarmassestimatesfromGallazzi 2008 for discussion of biases introduced by eliminating ob- et al. (2005). jects based on their σ). At the high end we select galax- We would also like to study the velocity dispersions of ies with σ < 400 km s−1 to avoid contamination from theseobjects. Oneoftheimportant differencesbetween the double/multiple superpositions (see Bernardi et al. 2006b, SDSS-DR6 and previous releases is that the low velocity 2008).Themaximumσ ofasinglegalaxygiveninBernardi dispersions (σ < 150 km s−1) were biased high; this has et al. 2008, and confirmed by Salviander et al. 2008 using Curvature in scaling relations 3 Figure 2. Distribution of b/a in the r-band as a function of angular (left) and physical size (right). Contours show regions of equal probabilitydensity.Startingatmaximumdensity,eachlinerepresentsafactorof√2decreaseindensity.Thechangefromsolidtodotted linesmarksthepointwhichencloses68%ofthesample.Thereappearstobeaseparatepopulationofsmallb/a<0.6objectsatphysical sizessmallerthan a=10h−1 kpc. Figure3.Histogramshowingdistributionofb/aintheg (solid) andr (dashed) bands. − Figure 4. Dependence of axis-ratio b/a on a in the r band for − − galaxies withb/a>0.6inthe g band. Contours show the joint − distribution of b/a and a with the same conventions for spacing high resolution spectra, is about 430 km s−1. In addition, andlinestylesasinFigure2. the SDSS-DR6only reports velocity dispersions if the S/N in the spectrum in the restframe 4000 5700 ˚A is larger − than10orwiththestatusflagequalto4(i.e.thistendsto with b/a<0.6. The objects with small b/a make up about exclude galaxies with emission lines). To avoid introducing twentypercent ofourfullearly-typesample, buttheyarea a bias from these cuts, we have also estimated velocity dis- largerfractionofthesmallerfaintergalaxiesthantheyareof persionsforalltheremainingobjects.Itturnsoutthatthis thebright. There are good physicalreasons to suspect that cutdoesnotchangethecorrelationsstudiedhere(northose theseobjectsareadifferentpopulation (e.g.rotationalsup- intheFundamentalPlanestudyofHyde&Bernardi2008). portisnecessaryifb/a<0.6),so,inwhatfollows,weremove Thenetresult istochangeoursample from 64492 to64343 all objects with b/a < 0.6 in the g band from the sample. − objects. About 25 objects have colors which lie outside the This leaves 51379 objects. Many of thefigures which follow range 0.4 g r 1.3 which we do not believe, so we show quantities in the r band, for which this cut appears ≤ − ≤ − excludethem from thesample. lesssharp.Figure3showsthatthedistributionofb/ainthe Figure 2 suggests that we should make one more cut. twobandsisverysimilar: thesmall differencesbetweenb/a The left hand panel shows the distribution of axis ratios in g and r is due, in part, to measurement error. b/a in the sample, as a function of the angular half light Figure 4 shows b/a vs a in the r band in our sample − radius (left) and physical scale (right). There is clear evi- after applying this cut. Comparison with Figure 2 shows dence for two populations, particularly when plotted as a that the ‘second’ population at low b/a has been removed. function of physicalscale: one with b/a>0.6 and theother In the sample which remains, there is a weak tendency for 4 J. Hyde & M. Bernardi the largest galaxies to have slightly smaller b/a. The impli- cations are discussed further in Bernardi et al. (2008) who show that the mean b/a drops slightly at the highest luminosities and σ. Finally, we must account for thefact that objects were brighterinthepastby0.9zmagsbecauseofstellarevolution (e.g.Bernardietal.2003a).This,withk-corrections,adjusts slightly the actual values of the magnitude limit which we shouldusewhencomputingeffectivevolumes.Infact,these effectsworkinoppositedirections,sotheneteffectissmall. However,sinceourgoalistoquantifysmalleffectsinalarge sample, itisnecessary todothiscarefully.Thenetresult is to reduce the sample size by about ten percent, to 46410. As a check that this has been done correctly, we perform the test suggested by Schmidt (1968). If V is the survey i volumebetweenobjectiandtheobserver,andVmax,i isthe total survey volume over which the object could have been seen, then the average value of (V/Vmax) should be 0.5: we find V/Vmax =0.499.(Theluminosityevolutionisslightly h i largerthan,butstatisticallyconsistentwith,the0.85z mags reported byBernardi et al. 2003b.) These cuts are similar to those described by Bernardi et al. (2003a, 2006a), who providefurtherdetails about the motivation for each cut, except for: i) the cut on b/a; ii) the inclusion of velocity dispersions from low S/N spectra or with the status flag not-equal to 4; and iii) the inclusion of velocity dispersions at 60 < σ < 90 km s−1. In addition, for the present study, we have chosen to be more conservative. Previously, we required Petrosian magnitudes 14.5 < mr,Pet < 17.75. Changing to a brighter limit makes our final sample size considerably smaller than if we had used the Bernardi et al. (2006a) selection. In addition, we previously required fracDev > 0.8 in the r band; we now − requirefracDev =1 in r aswell as in g,becausenon-early- type features are expected to be more obvious in the g − band.ThismoreconservativechoiceforfracDeveliminated about20000objects(doingtheselection basedong butnot r makes little difference, because requiring fracDev = 1 is quite stringent). To quantify the effect of these additional cuts, we have applied them to the Bernardi et al. (2006a) sample,andfoundthatthecomovingnumberdensityofob- jects is reduced to about 0.4 times that in Bernardi et al. (2006a).Noneoftheresultswhichfollowaresensitivetothe valueof thecomoving numberdensity. 2.2 Corrections to photometry Wemustaddressanotherissuebeforeproceeding.Thisisbe- causetheSDSSreductionsareknowntosufferfromskysub- traction problems, particularly for large objects (Adelman- McCarthy et al. 2008). To illustrate, Figure 5 compares Figure 5. Comparison of GALMORPH and SDSS photometric SDSS photometric reductions with those of our own code, reductions.Althoughthetwoareingoodagreementatsmallre, where the bulk of the objects lie, the SDSS underestimates the GALMORPH,forasubsetofthefullsample( 5500galax- ∼ totalflux(top)andthehalf-lightradius(middle)oflargeobjects. ies used by Bernardi et al. 2003a plus 180 Bright Clus- ∼ However, both pipelines return consistent values of the quantity ter Galaxies analyzed by Bernardi et al. 2007). The GAL- MORPH reductions do not suffer from the sky subtraction log10(re)−0.3µewhichSagliaetal.(2001)callFP.Symbolswith error bars show the mean relation and its uncertainty, dashed problem. This figureshows that while thetwo pipelines are and dotted curves show the regions which enclose 68% and 95% inexcellentagreement forsmall objects, theSDSSunderes- of the objects, and smooth solid curves show the fits given by timatesthesizesandbrightnessesoflargeobjects.However, equations (4)and(5). thequantityIP=log10(r ) 0.3µ ,identifiedbySagliaet e e − al. (2001) (they called it FP), is not substantially changed. In all the panels, symbols with error bars show the mean Curvature in scaling relations 5 Figure6.DistributionofsizesobtainedbytheSDSSphotometric reduction(solid)andcorrectedfollowingEq.5(dotted).Although thetwoareingoodagreementatsmallre,wherethebulkofthe objectslie,thecorrectedsizesarelargerforthesmallfractionof largere. relation andtheerroron themean,dashedcurvesshowthe region encompassing 68% of the objects, and dotted curves enclose 95%. Unfortunately, it is computationally expensive to run GALMORPHontheentireDR6datarelease.Therefore,we havefitsmoothcurvestothetrendsshowninFigure5(solid curves)andweusethesefitstocorrecttheSDSSreductions as follows. Given rSDSS we set re = rSDSS+∆rfit(rSDSS) (2) Figure 7. Observed (top) and corrected (bottom) sizes in the m = mSDSS+∆mfit(rSDSS), (3) SDSS derived from deVaucouleur fits to the light profiles shown asafunctionofr-bandluminosityMr.(Inallcases,∆log10Re≡ where rSDSS = rdeVqab is the SDSS-measured deVau- ltoogp10toRbeo+tt0o.m22Minrt+he4.u2p4p.)erg-p,arn-,eli.-,Tahnedbzo-bttaonmdspaarneeslhoomwnitsfrtohme couleur radius expressed as a geometric mean (√ab) of the z-band. In each panel, for each band, different lines show data semimajor (a) and semiminor (b) axes of a half-light con- fromthe followingredshiftbins: 0.072.4) w−hereitcurvesslightlydownwards. become less well-defined at small M∗. This happens to be 10 J. Hyde & M. Bernardi Figure 11. As for previous figure, but for stellar mass surface-brightness vs stellar mass (left) and surface-brightness vs luminosity (right). Figure 12.Asforpreviousfigure,butnowforthesize-velocitydispersionandsize-surface-brightnessrelations. thesamemassscale whichKauffmannetal.(2003) identify damental role in the Fundamental Plane: the R σ and e − as being special. R µ relations.Figure12showsthatboththeserelations e e − are curved. Notice that accounting for selection effects is Itispossiblethatourearly-typesampleiscontaminated important –thefilled and emptysymbols, which includeor by a different population at thelow mass end – despite the ignoretheV−1 weight,traceverydifferentrelations.Tofirst fact that we have already tried to reduce such an effect by max order,thezero-pointsofthetworelationsaremorestrongly removingobjectswithb/a<0.6andselectinggalaxieswith affected than is the slope. Since it is the zero-point of the the g and r-band fracDev = 1. If we include those objects FPwhichisusedtoestimateevolutioninsmallhigh-redshift with g and r-band fracDev > 0.8 and/or do not remove samples,Figure12suggeststhat,withoutduecare,onemay objects with small b/a, then the quadratic remains a good simply bemeasuring selection effects (apoint also madeby fiteven at small M∗ or L.This isalso trueif wesimply use Bernardi et al. 2003c). thecuts given byKauffmann et al. (2003): R90/R50>2.86, R50 > 1.6 and µ50 < 23 mag/arcsec2 in the r-band. We Neitheroftheserelationsisaswell-fitbyaquadraticat will returnto thisin thenext subsection, butnote that the theextremes;e.g.,whenweightedbyV−1,theR µ rela- curvature at the luminous, massive end of the sample, is max e− e tion curves away significantly from the quadratic at both highly significant. large and small µ (see also Nigoche-Netro et al. 2008). e We turn now to the scaling relations which play a fun- Curvature at large µ (or σ) is not surprising: we already e

Curvature in the scaling relations of early-type galaxies

Similar Curvature in the scaling relations of early-type galaxies