Curvature in the scaling relations of early-type galaxies

0.62 MB·English

by  J. B. Hyde

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 pair-
wise scaling relations for this sample. It also shows that, in
⋆ E-mail:jhyde,[email protected] 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 parame-
ters 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 magni-
tudes and sizes. However, these are not seeing corrected,
and Appendix A shows that this introduces systematic bi-
ases, 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 lumi-
nosities 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 inclu-
sion 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: 0.07<z 0.1, 0.1<z 0.13,
taining ellipse. rdeV, the semimajor axis length, and ab, 0.15<z≤0.18,0.22<z≤0.25and0.25<≤z≤0.35. ≤
the axis ratio, are obtained from the SDSS catalogs where
they are referred to as “deVRAD” and “devAB”, respec-
ent magnitudes will affect these luminosities (they increase
tively. ∆rfit(rSDSS) is zero if rSDSS < 2 arcseconds and
slightly on average), we add equation (3) to Gallazzi et al.
∆mfit(rSDSS) is zero if rSDSS<1.5 arcseconds, otherwise
stellar mass estimates (accounting for the conversion from
∆mfit(rSDSS) = 0.024279 rSDSS rSDSS 2 (4) luminosity to magnitude, m=−2.5log10L). The net effect
mags − 71.1734 −“ 26.5 ” istoslightlyincreasesomeofthestellarmasses,butwehave
∆rfit(rSDSS) = 0.181571 rSDSS + rSDSS 2 (5) not included a plot showing this increase.
arcsec − 4.5213 “3.9165” Figure 6 shows the distribution of the effective radii
of the 46410 early-type galaxies (i.e., whatever their b/a
We propagate the errors similarly. Note that
value)beforeandafterapplyingthecorrectiongiveninequa-
0.6 ln(10)rSDSS∆mfit provides a good approxima-
− tion (5). Although the two are in good agreement at small
tionto∆r ,asonemightexpectgiventhebottompanelin
fit
r , where thebulk of the objects lie, the corrected sizes are
Figure5;itslightlyunderestimates∆rfit atlargerSDSS.We e
larger for thesmall fraction of objects with large r .
havetestedthesecorrections, andfoundthemapplicableto e
the SDSS g, r, and i bands. Throughout the paper we will
denote angular size in arcseconds with r and physical size
e 2.3 Corrections to sizes
in h−1 kpc with R . These values refer to geometric-mean
e
deVaucouleur radii, corrected as described in this section We make one final correction to the sizes, which is aimed
and Section 2.3. at accounting for the fact that the early-type galaxies have
Later in this paper, we will use stellar mass estimates color gradients: on average, their optical half-light radii are
from Gallazzi etal.(2005).Thesearederivedbyestimating larger in bluer bands. If not accounted for, a population
a stellar mass-to-light ratio, and then multiplying by the of intrinsically identical objects will appear to be slightly
estimated luminosity. Since our corrections to the appar- butsystematically larger if they are at higherredshift. Our
6 J. Hyde & M. Bernardi
sample is large enough that we must correct for this effect,
themoralequivalentofthek-correction totheluminosities.
Wedo so following Bernardi et al. (2003a).
Weestimatetherest-frameradiusineachbandbyinter-
polatingtheobservedradiiinadjacent bands.Forexample,
toestimatetherest-frameg-bandradius,weusethefollow-
ing expression:
(1+z)λ λ
Re,g,rest = λg gλ−r r (Re,g,obs−Re,r,obs)+Re,r,obs
−
(6)
where λ = 4686˚A,6165˚A,7481˚A,8931˚A are the
g,r,i,z
{ }
average wavelengths of the SDSS filters, and z is the
spectroscopically-determined redshift.
Figure7showswhythiscorrectionisnecessary.Itshows
the sizes of objects in different bands as a function of r-
band(k-andevolution-corrected)luminosity.Toreducethe
range of sizes, we have subtracted-off a luminosity depen-
dentfactor:weactuallyshow∆log R log (R /kpc)+
10 e ≡ 10 e
0.221M +4.239 (thereason for theexact choice ofparam-
r
eters will become clear shortly – see Table 1 – but note
that, for the present purpose, the exact choice is not im-
portant). The upper panel shows the observed sizes: from
top to bottom, thesets of symbols and lines show g-,r-,i-,
and z-bands. For each band, different lines show data from
a number of redshift bins: 0.07 < z 0.1, 0.1 < z 0.13,
≤ ≤
0.15 < z 0.18, 0.22 < z 0.25 and 0.25 < z 0.35. In
≤ ≤ ≤
anyredshiftbin,thesizesareclearlylargering-andsmaller
in i- than they are in the r-band. The bottom panel shows
rest-framesizes,forg-,r-,andi-bands.Weomitthez-band
since longer wavelength observations would be necessary to
reconstruct thez-bandrest-frame size. Therest-frame sizes
areindeedlargerinthebluerbands,withthedifferenceper-
hapsslightlysmalleratlargeluminosity.Theredshiftdepen-
dence of the size-luminosity relation is not apparent when
using thetheobserved radii (upper-panel),butcan beseen
whenusingtherest-frameradii(lowerpanel);thisisstudied
furtherin Bernardi (2009).
TheSDSSalsooutputsnon-parametricPetrosian sizes.
However, in contrast to the deVaucouleur-fitted quantities,
these sizes are not corrected for the effects of seeing. The
Appendixshowsthat,incontrasttothedeVaucouleursizes,
thePetrosiansizesofobjectsathigherredshiftaresystemat-
icallylargerthanthosefromthemodelbasedfits.Thisisnot
surprising if seeing has compromised the Petrosian-based
measurements, suggesting that, if this is not accounted for,
thentheuseof Petrosian sizeslimits orbiases theprecision
measurements which large sample sizes would otherwise al-
low.
3 CURVATURE IN SCALING RELATIONS
We now turn to measurements of a number of scaling rela- Figure 10. As for previous figure, but now for the ratio of dy-
namical mass to light versus luminosity (top) and dynamical to
tions.Itturnsoutthatcurvatureisoftenevenmoreobvious
if we replace luminosity with stellar mass, so we will often stellarmassversusM∗(middle)andMdyn/LorMdyn/M∗versus
σ (bottom).
showsuchrelationssidebyside.Unlesswespecifyotherwise,
the luminosity is always from the r-band. We will some-
times use a shorthand for the r-band quantities: R,V,I,M
for log R , log σ, µ , M .
10 e 10 e r
Curvature in scaling relations 7
Table 1. Coefficients of best-fitting relations of the form Y X =p0+p1X+p2X2 to pairwisescaling relations. Fits were made to
h | i
the binned points, not to the objects themselves. Linear fits (p2 = 0) were made to the galaxies and restricted to the range 10.5 <
log10(M∗/M⊙) < 11.5 and −23 < Mr < −20.5. The errors on the fitted coefficients are random errors: they depend on slope of the
relation, its scatter, and the sample size. These are smaller than those produced by systematic effects (e.g., using σDR6 or σIDLspec2d
ratherthantheiraverage). Typicalsystematicserrorsareafewtimeslargerthantherandomerrors.
Relation p0 p1 p2 ∆χ2ν
RM∗ 4.79 0.02 0.489 0.002
h | i − ± ± − −
RM∗ 7.55 0.44 1.84 0.08 0.110 0.004 44.39
h | i ± − ± ±
V M∗ 0.86 0.02 0.286 0.002
h | i − ± ± − −
V M∗ 5.97 0.27 1.24 0.05 0.044 0.002 19.20
h | i − ± ± − ±
I∗M∗ 21.77 0.09 0.077 0.009
h | i − ± − ± − −
I∗M∗ 42.11 2.23 12.13 0.41 0.57 0.02 48.37
h | i ± − ± ±
RM 4.24 0.02 0.221 0.001
h | i − ± − ± − −
RM 4.72 0.32 0.63 0.03 0.020 0.001 38.70
h | i ± ± ±
V M 0.32 0.01 0.119 0.001
h | i − ± − ± − −
V M 2.97 0.23 0.37 0.02 0.006 0.001 6.21
h | i − ± − ± − ±
I M 17.37 0.08 0.104 0.004
h | i ± − ± − −
I M 61.57 1.61 4.10 0.15 0.099 0.003 37.96
h | i ± ± ±
hMdyn/Lr|Lri −1.50±0.03 0.200±0.003 − −
hMdyn/Lr|Lri 5.34±0.70 −1.10±0.13 0.062±0.001 5.74
hMdyn/M∗|M∗i −0.46±0.02 0.062±0.002 − −
hMdyn/M∗|M∗i 2.25±0.55 −0.48±0.10 0.027±0.005 1.44
RV 1.42 0.02 0.835 0.008
h | i − ± ± − −
RV 2.46 0.23 2.79 0.20 0.84 0.05 12.16
h | i ± − ± ±
I R 4.41 0.03 0.246 0.001
h | i − ± ± − −
I R 24.60 0.57 2.30 0.06 0.052 0.002 62.77
h | i − ± ± − ±
Figure 8. Half-lightradius vs stellar mass andluminosity(left and right). Symbols with errorbars show the medianinsmall massor
luminositybins,anddashedanddottedlinesshowtheregionswhichcontain68%and95%oftheobjectsineachbin.Curvesshowfitsof
theform Y X =p0+p1X+p2X2 totheserelations;best-fitcoefficientsareprovidedinTable1.Curvedfitsweremadetothebinned
h | i
counts(symbolswitherrorbars),ratherthantotheobjectsthemselves.Straightlinesshowlinearfitstotheserelations;theseweremade
ttohethVem−ag1xa-lawxeiiegshtaenddlirneestarric(tceodnntoectthede trhaningelin1e0).5an<dloqgu1a0d(rMat∗ic/Mfit⊙s)(fi<lle1d1.c5iracnleds)−.23<Mr <−20.5. Bottom panels show residuals from
8 J. Hyde & M. Bernardi
Figure 9.Sameaspreviousfigure,butforvelocitydispersion.
3.1 Curvature in pairwise relations ThesymbolsinthepanelsontherightofFigures8and9
show the curvature in the size-luminosity and σ-luminosity
A simple test for curvature is as follows. In a magnitude
relations, respectively (e.g. Lauer et al. 2007; Bernardi et
limited survey, the more luminous objects are seen out to
al. 2007). This curvature is also evident when shown as a
bigger volumes than the fainter objects, meaning that the
function of stellar mass (panelson theleft), socurvaturein
ratioofthenumberofluminoustofaintobjectsseeninsuch
theM∗/L relation is not theonly cause.
asurveyislargerthantheratioofthetruenumberdensities
Theformatofthisfigureissimilartomanythatfollow.
oftheseobjects.Onecanaccountforthisover-abundanceby
The filled circles in the top panel show the median size in
weighting each object by 1/Vmax(L), the inverse of the vol-
a number of narrow bins in luminosity, when the objects
ume over which it could have been seen; this down-weights
are weighted by V−1. Dashed and dotted curves show the
luminous galaxies relative to fainter ones. If a scaling rela- max
regionswhich include68% and95% oftheweighted counts.
tion is a straight line, e.g., if the mean log(size) increases
linearly as log(luminosity) increases, then the slope of the For statistics at fixed luminosity (as in this case), the Vmax
weighting makes no difference because, at fixed luminosity,
regression of logR on logL will not depend on whether or
allgalaxieshavethesameweight.InFigure12,whichshows
notoneincludesthisweightingterm.(Thiswillalsobetrue
theR σ and R µ relations, thedifferenceissignificant.
for logσ on M, etc.) However, if the intrinsic scaling re- e
− −
There, we use open squares to show the median weighted
lation is curved, then the slope of the regression line will
depend on which weighting was used. This happens to be count when no Vmax weighting is used.
trueinourdataset,indicatingthattherelationsarecurved: Straightsolidanddashedlinesshowtheresultoffitting
the slope of the logR M relation is 0.221 0.001 when straightlinestothedata,withandwithoutVmax weighting.
weighting by V−1, bu−t 0.241 when−not. Th±ese numbers For distributions at fixed luminosity (as in this case), the
max −
are 0.119 0.001 and 0.117 for the logσ M relation. two should be the same if theunderlying scaling relation is
− ± − −
The difference is more dramatic for the µ M relation: notcurved.Ifitis,thenthefittotheunweightedpointswill
e
−
0.104 0.004 and 0.207. Here, we reported the uncer- reflecttheslopeoftherelationathigherluminosities.Inthe
− ± −
tainties due to random errors; uncertainties in the param- case of the size-luminosity relation (Figure 8), the dashed
eters due to systematics errors are a few times larger (see lineissteeperthanthesolid,consistentwiththesteepening
Section 3.2). oftherelationatlargeLshownbythesymbols.Inthiscase
only, we also show the linear fits to the R L and R M∗
− −
relations reported by Shen et al. (2003) (see their Table 1,
3.2 The size-luminosity relation
Fig. 6 and Table 1, Fig. 11 respectively). This shows that
Recent work shows that the correlation between size and theirfits are close to ours when we ignore theVmax weight-
luminosity relation has evolved significantly since z = 2 ing,butnotethattheirsampleisselectedratherdifferently,
and their sizes are from Sersic, rather than deVaucouleur
(e.g.Cimattietal.2008;vanDokkumetal.2008).Bernardi
fitsto thelight profile.
(2009) shows that the sizes of luminous early-type galaxies
(M 23 mag) are still evolving at low redshift and that Toquantifythecurvature,wefit2ndorderpolynomials
r
∼−
satellites are on average 8% smaller than central galax- tothese and a numberof otherscaling relations (which fol-
∼
ies.However,thislastresultisnotseenforlowerluminosity low). Inall cases ourfitsareslightly non-standardbecause,
early-typegalaxies(e.g.Weinmannetal.2009). Duetothis toemphasizethecurvatureintheserelations, wewouldlike
recent interest, we study this relation first,before consider- thefitstobesensitivetothetailsofthedistribution.There-
ing others. fore, we have fit to the binned counts (i.e., the symbols)
Curvature in scaling relations 9
shownintheFigures,ratherthantotheobjectsthemselves. Theflatteningoftheσ Lrelation isconsistentwithprevi-
−
ForfitsatfixedL,thisisequivalenttoweightingeachobject, ouswork,butnoticethatthiseffectisevenmorepronounced
notjustbyVm−a1x(L),butby[Vmax(L)φ(L)]−1 =[Nobs(L)]−1. for theσ−M∗ relation.
In effect, this upweights the tails of the distribution. The
fittingminimizes χ2,defined as thesum of thesquared dis- Comparison withFigure8showsthat,atlarge M∗ and
L, the R L and σ L relations curve in opposite senses.
tances to the binned points shown. If, instead, we weighted Sincedyn−amicalmas−sM Rσ2,itisinterestingtocheck
dyn
each of the binned points by (the inverse of its) error bar ∝
if the M L relation is well-fit by a simple power-law.
when defining χ2, then, because this additional weight is We do thdiysnin−Figure 10, where we set M 5R σ2/G=
dyn e
proportional tothenumberofcountsin thebin,thiswould 4.65 1010h−1M⊙(Re/h−1kpc)(σ/200 km s≡−1)2. The top
betheequivalentofweightingeachobject,ratherthaneach ×
panel shows that, at both small and large L, the M L
dyn
binned point, equally. relation curves upwards from the Rσ2 L0.2 scaling (s−olid
Thebottompanelshowsresidualsfromthelinear(V−1 ∝
max line). The middle panel shows that curvature remains if
weighted)andquadraticfits;inallcases, theresidualsfrom
one replaces L with M∗, indicating that more than stellar
thequadraticfitsshowfewer,ifany,trends.Table1summa-
population related effects are responsible. The slight rise at
rizestheresultsofthesefits.Althoughthecoefficientsofthe
small masses is not implausible: star formation is expected
quadratictermappeartobeverydifferentfromthosewhen
to be less efficient at small masses. However, we view this
simply fitting a line, this is because we are reporting fits of
withcaution:thevelocitydispersionsatthesmall-massend
tmheeafnorvmaluheYs|Xfirist=, apn0d+fipt1hXy|x+ip=2Xp02+. Hpa1dxw+epr2e[mx2ov−edhxt2hie] aargeemtroenredufonrceMrtdayinn/(Me.∗g.toBeinrncraeradsie2w00it7h).MT∗h,earelthisouagnhaivteris-
instead (where x = X −hXi, etc.) then p1 would be very weak: Mdyn/M∗ Mdyn M∗0.062±0.006 (the error 0.06 on
similar to linear fit,indicating that thecurvatureis small. h | i ∝
the slope was computed accounting for systematics errors
To quantify if a quadratic is a significantly better fit
– the uncertainty from random errors is smaller 0.002).
than a straight line, we fit both to the binned counts. We ∼
χth2encoamndpaχr2eχ2qdueando/t(Ne bthines−m3in)iwmiitzhedχ2lvinael/u(eNsboinfsχ−22,)a,nwdhtehree ThMhedycno/rMre∗la|Mtiodnynbie∝twMeend0y.1nM7±d0y.n01/M(s∗eewFitihguMredy1n1iisnsHtryodnege&r:
quad line Bernardi 2008). If there were no scatter around this rela-
fivatsluweeorfeχt2ofNorbinthsebiqnunaeddraptoicinitss.mIunchallclcoasseers,tothuisnirteyduthcaedn Mtio0n.2,;thbeencawuesewothueldreexispescctathtMerd,ynt/hMis∗s|Mca∗liin∝g iMs∗0s.h17a/l0lo.8w3e∝r,
for the linear fits. (Note that we do not use the coefficients ∗
M0.06±0.01. Even if this relation were a simple power
of the linear fits to the unbinned counts that are shown in ∝ ∗
law without curvature, it would indicate that stars make
Table1,since,forthistest,wewanttotreatboththelinear
up a smaller fraction of the total mass of a galaxy at large
andquadraticfitsequally.Ofcourse,usingthesecoefficients
masses.Thisprovidesanimportantpieceofinformation for
whencomputingχ2 doesnotchangeourconclusions,since
line adiabaticcontractionbasedmodelsofscalinglaws(e.g.Pad-
theycan, at best, producethesame valueof χ2 which we
line manabhan et al. 2004; Lintott et al. 2005).
havejust described.)
Finally,awordontheerrorsonthefittedcoefficientsis Toconnectwithpreviouswork,thebottompanelshows
inorder.Thenumberswequotearerandomerrors:theyde- how Mdyn/L (triangles) and Mdyn/M∗ (filled circles) scale
pendonslopeoftherelation,itsscatter,andthesamplesize. with σ; both relations are slightly curved. Except at the
These are smaller than systematic effects: e.g., using σDR6 largestσ,ourdataarerelativelywelldescribedbythecurved
or σIDLspec2d rather than their average makes a difference Mdyn/L−σ relation reported by Zaritsky et al. (2006) (we
which is larger than the random error. Typical systematics have shifted our measurements downwards by 0.44 dex be-
errorsareafewtimeslargerthanrandomerrors–therefore, causeourdataareinr whereastheirfitwasinI,i.e.weuse
it is important to separate thetwo typesof error. r I = 1.1. Note that we have also subtracted 0.3 from
− −
Figure 8 shows that, compared to a single power law, their fit since they used the effective light Le = L/2, while
the R L relation curves significantly towards larger sizes weusethetotallight L).ThefactthatMdyn/Lisasteeper
at larg−e L or M∗, consistent with previous work. At small functionofσthanisMdyn/M∗canbeunderstoodasfollows.
aLpporeaMrt∗o(sMcartt<er−sli1g9h.t5lyordolowgn1w0aMrd∗s/Mfro⊙m<th1e0q.2u)a,dtrhaeticdafitta. Fthirastt,Mn∗o/tLe tinhcarteMasedsynw/iLth=in(cMredaysni/nMgg∗)(Mr c∗o/lLo)r.(Te.hge.nB,enlloetet
−
Although thereare few objects in this tail, so the measure- al.2004). However,g r color isstrongly correlated with σ
−
ment is noisier, it is possible that this indicates that the –largeσimpliesreddercolors(Bernardietal.2005),andso
sample is slightly contaminated at small L. We will return M∗/Lincreaseswithσ.Therefore,Mdyn/Lincreaseswithσ
to this shortly. because Mdyn/M∗ does and because M∗/L does so as well.
Before we move on to other scaling relations, we note Asa final studyof curvaturein relations which involve
that results based on Petrosian quantities are shown in the luminosity,wenowturntotheµe Landµ∗ M∗relations.
Athpepmenfudrixth,ewr.here we also discuss why we do not consider (2W.5elogdefi(n2eπ)µ;∗th=is−i2s.5tlhoeg1s0t(eMlla∗r/−Mm⊙a)ss+s5ulrof−ga1c0e(Rber/igkhptcn)es+s
10
within the half light radius.) Figure 11 shows that in this
case too, there is significant curvature. However, the panel
3.3 Other scaling relations
on the left shows that at M∗ < 3 1010M⊙, the µ∗ M∗
× −
Figure 9 shows the σ L relation in this sample. This re- relation becomes rather well fit by the linear relation, al-
−
lation is actually rather well described by a single power- though it also becomes significantly noisier. A look back at
law, except at Mr < 23 (where the mean value of the other relations which involve M∗ shows that they too
log10(σ/km s−1)>2.4) 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