A Revised Model for the Formation of Disk Galaxies: Low Spin and Dark-Halo Expansion

1 MB·

by  Aaron A. Dutton

THEASTROPHYSICALJOURNAL654:27-52,2007JANUARY1
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
AREVISEDMODELFORTHEFORMATIONOFDISKGALAXIES:
LOWSPINANDDARK-HALOEXPANSION
AARONA.DUTTON
DepartmentofPhysics,SwissFederalInstituteofTechnology(ETHZurich),CH-8093Zurich,Switzerland
FRANKC.VANDENBOSCH
Max-Planck-InstitutfürAstronomie,Königstuhl17,69117Heidelberg,Germany
AVISHAIDEKEL
7 RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem,Israel
0 AND
0
STÉPHANECOURTEAU
2
DepartmentofPhysics,Eng.Physics&Astronomy,Queen’sUniversity,Kingston,ONK7L3N6,Canada

n THEASTROPHYSICALJOURNAL654:27-52,2007January1
a
J ABSTRACT

9 Weuseobservedrotationvelocity-luminosity(VL)andsize-luminosity(RL)relationstosingleoutaspecific
scenariofordiskgalaxyformationintheΛCDMcosmology.Ourmodelinvolvesfourindependentlog-normal

2 randomvariables: dark-haloconcentrationc, diskspin λ , diskmassfractionm , andstellar mass-to-light
gal gal
v ratio Υ . A simultaneous match of the VL and RL zero points with adiabatic contraction requires low-c
I
3
halos, butthis modelhasV 1.8V (whereV andV are the circularvelocityat 2.2disk scale lengths
5 2.2∼ vir 2.2 vir
and the virial radius, respectively) which will be unable to match the luminosity function (LF). Similarly
5
models without adiabatic contraction but standard c also predict high values of V /V . Models in which
4 2.2 vir
diskformationinducesanexpansionratherthanthecommonlyassumedcontractionofthedark-matterhalos
0
6 haveV2.2 1.2VvirwhichallowsasimultaneousfitoftheLF.Thismayresultfromnon-spherical,clumpygas
∼
0 accretion,wheredynamicalfrictiontransfersenergyfromthegastothedarkmatter. Thismodelrequireslow
/ λgal and mgal values, contraryto naiveexpectations. However, the low λgal is consistentwith the notionthat
h
diskgalaxiespredominantlysurviveinhaloswithaquietmergerhistory,whilealowm isalsoindicatedby
p gal
galaxy-galaxylensing.ThesmallerthanexpectedscatterintheRLrelation,andthelackofcorrelationbetween
-
o theresidualsoftheVLandRLrelations,respectively,implythatthescatterinλgal andincneedtobesmaller
r thanpredictedforΛCDMhalos,againconsistentwiththeideathatdiskgalaxiespreferentiallyresideinhalos
st withaquietmergerhistory.
a Subject headings: galaxies: formation — galaxies: fundamentalparameters — galaxies: spiral — galaxies:
:
v structure
i
X
r 1. INTRODUCTION above. Inwhatfollowswereferto therelationsbetweenthe
a globaldiskparametersastheVL,RL,andRV relations.
In the standard, cold dark matter (CDM) based model for
Inthepast,theVLrelation,alsoknownastheTully-Fisher
disk formation, set out by Fall & Efstathiou (1980), disks
relation (Tully & Fisher 1977), has received much attention
form out of gas that slowly cools out of a hot gaseoushalo,
asadistanceindicatorowingtotherelativelysmallobserved
associated with the dark matter potential well, while main-
scatter. Althoughnumerousstudieshaveaddressedtheorigin
taining its specific angular momentum. During this process
oftheVLrelation,noconsensushasbeenreached. Inpartic-
thedarkmatterhalocontractstoconserveitsadiabaticinvari-
ular,itiscurrentlystillunderdebatewhethertheoriginofthe
ants(Blumenthaletal.1986).Becauseofthecentrifugalbar-
VL relation is mainly governedby initial cosmologicalcon-
rierthegassettlesinarotationallysupporteddiskwhosesize
ditions(e.g., Eisenstein & Loeb1996; Avila-Reese, Firmani
isproportionaltoboththesizeandangularmomentumofthe
& Hernández 1998), or by the detailed processes governing
darkmatterhalo(Mo,Mao&White,1998;hereafterMMW).
star formation (Silk 1997; Heavens & Jiminez 1999) and/or
Consequently,thestructureanddynamicsofdiskgalaxiesare
feedback(e.g., Kauffmann,White &Guiderdoni1993;Cole
expectedtobestronglycorrelatedwiththepropertiesoftheir
etal.1994;Elizondoetal.1999;Natarajan1999).Inthemost
darkmatterhalos. Inparticular,thecorrelationsbetweenthe
recentmodels(vandenBosch2000;2002;Navarro&Stein-
observable,structuralparametersofdiskgalaxies,rotationve-
metz 2000; Firmani& Avila-Reese 2000, hereafter FA00) it
locity,V, size, R, andluminosity,L, are expectedtobe a re-
is typically understoodthat both initial conditionsand bary-
flection of the virial properties of dark matter halos, which
onicphysicsrelatedtostarformationandfeedbackmustplay
scaleasV R M1/3. Slightdeviationsfromthesescal-
vir∝ vir∝ vir an important role. Reproducing the VL zero point has also
ingsareexpectedfromthefactthatmoremassivehalosare,on
beenalongstandingproblemforCDMbasedgalaxyforma-
average,lessconcentrated.Anyfurtherdeviationsmusteither
tion models. In particular no model has been able to simul-
reflectsomeaspectsofthebaryonicphysicsrelatedtogalaxy
taneously match the luminosity function and VL zero point
formation,orsignalafailureinthestandardpictureoutlined usingstandardΛCDM parameters(Coleetal. 2000;Benson
2 Duttonetal. 2006
et al. 2003; Yang, Mo, & van den Bosch 2003). This prob- of the standardΛCDM cosmology. In particular,their
lem can be traced to the high values ofV/V expected for density profiles have an NFW form with a concentra-
vir
ΛCDM halos once the effects of the baryons, such as adia- tionparametercthatdeclinessystematicallywithmass.
baticcontraction(Blumenthaletal.1986),aretakenintoac-
count.Allsolutionstothisproblemrequireachangetoeither 2. The baryons form an exponential disk in centrifugal
the standard cosmological model or the standard picture of equilibrium,whichisspecifiedbyamassfractionm
gal
galaxyformation. andaspinparameterλ .
gal
Anotherpotentialproblemfordiskformationmodelsisthe
formationoflargeenoughdisks. Therelativefragilityofdisk 3. Thegalaxymassfractionistreatedasafreeparameter
galaxiesandthestronglyorderedmotionoftheirstarsandgas with a mean m Mαm. Positive values of α are
is generally interpreted as evidence for a relatively smooth expectedfromfgeaeld∝backvierffects. m
formationhistorywithoutviolentmergerprocesses.Thesub-
sampleofdarkmatterhaloswithoutrecentmajormergersis
4. A bulge is includedbased on a self-regulatingmecha-
knowntohavesystematicallylowspinparameters(D’Onghia
nismthatensuresdiskstability.
&Burkert2004).However,standardmodelsfortheformation
ofextendedrotatingdisksseemtorequirehighspinparame-
5. Theinteractionbetweenthebaryonsandthedarkmat-
tersinordertoreproducethezeropointoftheRLrelation.
ter halo is modeled with a generalized adiabatic con-
InadditiontotheslopesandzeropointsoftheVLandRL
tractionmodelwhichalsoallowsforhaloexpansion.
relations,additionalconstraintscomefromthescatterinthese
tworelations. Inparticular,Courteau&Rix(1999;hereafter
6. Stars form in the disk once above a threshold surface
CR99) have shown that the residuals of the VL relation, at
density.
fixed L, are virtually uncorrelated with the residuals of the
RLrelation,atfixedL(seealsoMcGaugh2005). Thisisan-
7. TheI-bandstellarmass-to-lightratioΥ increaseswith
otherwayofexpressingthe factthattheVLrelationisinde- I
luminosityasconstrainedbyobservations.
pendent of surface brightness (see Courteau et al. 2006 for
details), andimpliesthatthesize ofa diskatgivenLhasno
relevanceto itsrotationvelocity. Thisis a puzzlingresultto 8. Themodelparametersc,λgal,mgalandΥI areassumed
explain,asonewouldnaivelyexpectthatamoreconcentrated tobeindependentlog-normalrandomvariables.
disk also results in a higher rotation velocity. CR99, there-
The data and models are introducedin §2 and §3, respec-
fore, interpreted the weak residual correlation as indicating
tively.In§3.4weoutlinetheconversionbetweenstellarmass
that,onaverage,highsurfacebrightness(HSB)disksaresub-
and luminosity; In §3.5 we detail the computation of our
maximal, so that the disk only contributes mildly to the ob-
model scaling relations. In §4 we discuss how to construct
servedrotationvelocity. Ifconfirmedthisputsconstraintson
thestellarmass-to-lightratio,Υ,andthusonthestellarinitial modelsthatmatchtheslopes,zeropoints,scatterandresidual
correlation.In§5weexplorethemodelparameterspace,and
massfunction(IMF).FA00,ontheotherhand,claimthatthe
advocatearevisedmodelfordiskformation. We summarize
weakresidualcorrelationowestothe surfacedensitydepen-
ourresultsin§6.
dence of star formation, such that at a given baryonicmass,
lowersurfacedensitygalaxieshavelowerstellarmasses,and
hence lower luminosities, which compensate for the some- 2. THEDATA
whatlowerrotationvelocities. However,intheirmodelboth We compareourmodelswith the largedata setof 1600
the highest and lowest surface brightness galaxies lie above local disk galaxies compiled by Courteau et al. (200∼6). For
themeanVLrelation,contrarytoobservations,andtheirmod- each of these galaxies a rotation velocityV, a luminosity L,
els do notreproducethe zeropointof theVL relation or the a disk scale lengthR, anda centralsurface brightnessofthe
amountofscatterintheVLandRLrelations. disk µ , are available. The data set is compiled from three
0
Althoughvariousstudieshaveattemptedtoexplaintheori- independent samples: Mathewson, Ford, & Buchhorn 1992
ginoftheVLortheRLrelation,thetruechallengeliesinfind- (hereafter,MAT);Courteauetal.2000(hereafter,Shellflow);
inga self-consistentmodelofdisk formationthatcanmatch andDaleetal.1999(hereafter,SCII).
both relations as well as the galaxy luminosity function si- AllgalaxyluminositiesaremeasuredintheI-band,andcor-
multaneously. Findingsuch a modelis a non-trivialtask, as respondto total luminosities(i.e., disk plus bulge). In addi-
allcurrentmodelsfailtodoso. Inthispaperweexaminethe tion,forasignificantsub-samplewealsohaveopticalcolors
parameter space of such models within the standard ΛCDM available,whichweusetoestimatestellarmass-to-lightratios
cosmology. Wesimultaneouslymatchtheslopes,zeropoints (see§3.4below). Diskscalelengthsandsurfacebrightnesses
and residuals of the VL and RL relations. We also investi- aredeterminedfromtheI-bandphotometry. Inwhatfollows,
gate what each of these modelspredict for the meanV/Vvir. wheneverwerefertosurfacebrightnesswemeanthecentral
In orderto be able to reproducethe observedabundancesof surface brightness, µ , of the exponential disk fitted to the
0,I
diskgalaxies,thisrationeedstoberelativelylow 1.2. We data.
∼
showthatthisrestrictionseverelylimitstheallowableparam- DuetothecomplicationsinvolvedwithinterpretingHIline
eterspace,favoringamodelwithlowspinparameterλ ,low widths,weonlyuserotationvelocitiesderivedfromresolved
gal
galaxymassfractionmgal,andwithhaloexpansionratherthan Hα rotation curves. This reduces the full sample by 300
halocontraction. galaxies. For the MAT and Shellflow samples the ro∼tation
Thekeyingredientsofourmodelareasfollows: curvesare fitted with a parametricfunction(Courteau1997)
which is then evaluated at 2.2 disk scale lengths. For the
1. Disk galaxies form in spherical dark-matter halos SCIIsamplethevelocitiesaremeasuredattheopticalradius
whose properties are drawn from N-body simulations (equivalentto3.2scalelengthsforanexponentialdisk).
ARevisedModelforDiskGalaxyFormation 3
FIG. 1.—ObservedI-bandVLRscalingrelationsusingdatafromCourteauetal.2006. Bi-weightedorthogonalleastsquaresfitsaregivenbythesolidblack
lines,with2-sigmadeviationsgivenbythedashedlines. Theopenblackcircleswitherrorbarsshowthemeanand2-sigmascatteroftheVLandRLrelations
binnedat0.3dexintervalsinLI.Thecolorsandpointtypescorrespondtoextrapolateddiskcentralsurfacebrightness,µ0,Iasindicatedinthetoprightpanel.
2.1. Corrections A . Asmallk-correction,A ,isalsoappliedsuchthat
ext k
Inordertohomogenizethesedatasamplesasmuchaspos- m =m - A - A - A . (3)
I I,obs int ext k
sible, we have derivedthe inclinationand cosmologicalcor-
Internalextinctions are computed using the line-width (W =
rectionsto the velocities, luminositiesand scale lengthsin a
2V)dependentrelationfromTullyetal.(1998):
uniformway,asdescribedbelow.
Theobservedrotationvelocities,Vobs,arecorrectedforin- Aint=γI(W)log(a/b)= 0.92+1.63(logW- 2.5) log(a/b)
clinationandcosmologicalbroadeningusing (4)
V witha/bthemajor-to-mi(cid:2)noraxisratio. Theexter(cid:3)nal(Galac-
V = obs . (1) tic) extinction is computed using the dust maps of Schlegel,
(1+z)sini
Finkbeiner,&Davis(1998),whilethek-correctionsarecom-
Theinclination,i,iscomputedusing puted using the line width dependent formalism of Willick
etal.(1997).
1- (b/a)2 Theabsolutemagnitudes,M ,arecomputedusing
sini= , (2) I
s 1- q20 M =m - 5logD - 25; D =VCMB(1+z), (5)
I I L L
where b/a is the minor-to-majoraxis ratio, and q is the in- 100h
0
trinsic thickness of the disk. We assume q = 0.2, and set withV thesystemicvelocityofthegalaxyinthereference
0 CMB
i=90◦ifb/a<q . frameatrestwiththecosmicmicrowavebackground(Kogut
0
Theapparentmagnitudes,m , arecorrectedforbothinter- etal.1993). TheI-bandluminositiesarecomputedfromM
I I
nal extinction, A , and external (i.e., Galactic) extinction, usinganabsolutemagnitudefortheSunofM =4.19.
int I,⊙
4 Duttonetal. 2006
FIG. 2.—ResidualcorrelationsfromtheobservedI-bandVLRrelationsinFig.1. Theblacklinesshowweightedleast-squaresfits. TheRLresidualsshowa
clearcorrelationwithsurfacebrightness,whiletheVLresidualsshownone.TheresidualsoftheVLandRLrelationareonlyweaklycorrelated.Thedottedline
hasaslopeof-0.5expectedforapureexponentialdisk(CR99).
Diskscalelengthsarecorrectedforinclinationusing 2.2. ScalingRelations
R =R /[1+0.4log(a/b)], (6) WenowusethedatadescribedabovetoinvestigatetheVL
I I,obs
and RL relations of disk galaxies. Despite the fact that our
(Giovanellietal.1994),andconvertedintokiloparsecsusing
sampleconsistsofsubsamplesthatuseslightlydifferenttech-
theangulardiameterdistanceD =V /[100h(1+z)].
A CMB niquesordefinitionsforthescalelengthsandrotationveloci-
Finally,centralsurfacebrightnessesarecorrectedforincli-
ties,wefindthat,withtheuniforminclinationandextinction
nation, Galactic extinction, and cosmological dimming (per
corrections described above, each subsample yieldsVL and
unitfrequencyinterval)using
RLrelationsthatareconsistentwitheachother(seeCourteau
µ =µ +0.5log(a/b)- A - 2.5log(1+z)3. (7) etal. 2006fordetails). We thereforecombinethethreesub-
0,I 0,I,obs ext
samplestoasinglesampleof 1300galaxies.
Thefactorof0.5in frontofthe log(a/b)termisempirically ∼
TheresultingVLR relationsareshownin Fig.1. Thesolid
determinedbydemandingthattheresidualsoftherelationbe-
lines show the mean relations, which have been determined
tweencentralsurfacebrightnessandrotationvelocityhasno
using bi-weighted orthogonalleast-squares fits. The dashed
inclination dependence1. Following Giovanelli et al. (1997)
lines show the 2σ scatter. The open circles with error bars
weassumeanuncertaintyof15%inγ ,a/b,A ,andA ,and
I ext k showthedatameanand2σscatterinseparateluminositybins
propagatetheerrors.Notincludingdistanceuncertainties,the
witha widthof0.3dex. TheseshowthattheVL andRLre-
averageerrorsontheobservablesareσ 0.08,σ 0.1,
lnV ≃ lnL≃ lationsarewellfittedwithasinglepower-lawovertherange
andσ 0.14.
lnR≃ consideredhere. TheRV relationhasthelargestscatter, and
isthustheleastwelldetermined. Forconsistencyweusethe
1Foradiskofzerothicknessoneexpectsthefactortobebetween0,for
anopticallythickdisk,and2.5,foranopticallythindisk. meanVLandRLrelationstodeterminethemeanRV relation.
ARevisedModelforDiskGalaxyFormation 5
FIG.3.—ComparisonofourI-bandVLRscalingrelationswiththosefromtheliterature.Thelong-dashedlinesshowtheVLrelationfromGiovanellietal.1997
(G97),andthedottedlinesshowtheRVrelationfromCourteau(1996,1997).TheredlinesarederivedfromPizagnoetal.(2005).
Themeanrelationsthusderivedare: definition2 dominated by scatter in surface brightness. The
residualsoftheVLrelation,however,shownosignificantcor-
V L
log =0.296log I +2.162, (8) relation with surface brightness at all (lower left panel). In
[kms- 1] [1010.3h-702L⊙] addition,theVLandRLresidualsareonlyveryweaklycorre-
lated(lowerrightpanel): theslopeoftheresidualcorrelation
log RI =0.322log LI +0.455. (9) is γ ≡d[∆logV(L)]/d[∆logR(L)]=- 0.08±0.03,in agree-
[h- 1kpc] [1010.3h- 2L ] mentwithCR99.
70 70 ⊙
Fig.3showsacomparisonoftheVLRrelationsderivedhere
withpreviousstudies. Theblacklinesandpointtypesarethe
R V
log I =1.086log - 1.894. (10) sameasinFig.1. Thelong-dashedanddottedlinesshowthe
[h- 1kpc] [kms- 1]
70 VLrelationfromGiovanellietal.(1997),andtheRV relation
that fits the data of Courteau (1996, 1997). Note that these
where R is the de-projected disk scale length in the photo-
I VL and RL relations, which are in excellent agreementwith
metricI-band,andh70=H0/(70kms- 1Mpc- 1). ourmeanscalingrelations,wereusedbyMMWandFA00as
ThecolorcodinginFig.1denotesthecentralsurfacebright- modelconstraints. The redlines in Fig. 3 show theVLR re-
ness of the disk in the I-band, µ0,I. Note that the scatter in lationsofPizagnoetal.(2005). Thelatter studyis basedon
the VL relation is not correlated with µ0,I. This is also ap- a sample of 81 disk dominated (B/D < 0.11) galaxies, with
parentfromFig.2whichshowsthecorrelationsbetweenthe
residuals of the VL and RL relations at constant L (∆logV 2 AfamilyofpureexponentialdiskswithconstantΥI has,ataconstant
and∆logR,respectively)andsurfacebrightness,µ0,I. Again LI, ∆logR∝0.2µ0,I, whichis different fromtheobserved slope of0.13.
thesymbolsarecolorcodedaccordingtoµ . Asis evident Thisowestotherelatively smallrangeofluminosities sampled, relativeto
0,I theamountofscatterintheRLrelation.
fromtheupperleftpanel,the scatterin theRLrelationisby
6 Duttonetal. 2006
luminositiesintheSloani-band,diskscalelengthsmeasured tiondlnρ/dlnr=- 2,andρ =ρ(r ). Theoverallshapeofthe
s s
with bulge-to-disk decompositions, and velocities measured densitydistributioncanbecharacterizedbytheso-calledcon-
at2.2diskscalelengthsfromHαrotationcurves.Weconvert centrationparameterc=R /r . HereR isthevirialradius,
vir s vir
from L to L using logL =logL +0.4(i- I)- 0.4(i- I) whichisdefinedastheradiusinsideofwhichtheaveragehalo
i I I i ⊙
0.036, assuming M = 4.56, and (i- I) 0.46 (Courtea≃u density is ∆ times the critical density for closure. For the
i,⊙ vir
etal.2006).NotethattheslopeoftheVLr≃elationofPizagno ΛCDM cosmology adopted here ∆ 100 (Bryan & Nor-
vir
≃
et al. (2005) is somewhat steeper than ours, while their RL man1998). InadditiontothevirialradiusR wealsodefine
vir
relationisoffsetfromourstowardslargerdiskscalelengths. thevirialvelocityV asthecircularvelocityatthevirialra-
vir
Thesedifferencesmaybeduetoacombinationoftheirrela- dius,i.e.,V2 =GM /R ,withGthegravitationalconstant.
vir vir vir
tivelysmallsamplesize(81galaxies),theirbulge-to-diskratio The total angular momentum of a halo, J , is commonly
vir
selectioncriteria,anddifferentinclinationcorrections. These expressedintermsofthedimensionlessspinparameter:
differencesdonotsignificantimpactonourmainconclusions,
J E 1/2 J /M
aswediscussinAppendixA λ= vir| | = vir vir f1/2 (12)
3. DISKFORMATIONMODELS GMv5i/r2 √2RvirVvir c
Here E is the halo’s energy, and f measures the deviation
Inthestandardpictureofdiskformation(e.g.,Fall&Efs- c
ofE fromthatofasingularisothermalspherewiththesame
tathiou 1980), disks form inside virializeddark matter halos
mass,andisgivenby
throughthecoolingofthebaryonicmaterial. Ourmodelsare
basedonthisstandardpicture,andcloselyfollowMMW,but c1- 1/(1+c)2- 2ln(1+c)/(1+c)
with some additional ingredients. In particular, we include fc= 2 c/(1+c)- ln(1+c) 2 (13)
a model for the formation of bulges, we use a prescription
for star formationwhich separatesthe disk into gaseousand (seeMMW).
(cid:2) (cid:3)
stellar components,we usean improvedandgeneralizedde- We assume that the galaxy that forms consists of a bulge
scriptionforadiabaticcontraction,andweuseempiricalrela- and a disk. Our algorithmto ascribe a bulge-to-diskratio is
tionstoconvertourmodelstoobservablequantitiesfordirect discussed at the end of this section. We define the baryonic
comparison with the data described above. All these model massofthetotalgalaxyas
ingredientsarediscussedinmoredetailbelow.
M M +M =m M (14)
Althoughwe refer to our models as ‘disk formationmod- gal≡ d b gal vir
els’ they are completely ‘static’ (i.e., the model is not actu- withMdandMbthemassesofthediskandbulge,respectively,
allyevolved). Foragivenspecificangularmomentumofthe and0<mgal< fbar. Wedefinethebulge-to-diskmassratioas
baryonic material out of which the disk forms and a given Θ Mb/Md∼sothat
≡
potential due to the dark matter, the structural properties of 1
the resulting disk are computed assuming (i) that disks are Md= 1+ΘmgalMvir (15)
exponential, and (ii) that the specific angular momentum of
Θ
the baryonicmaterial is conserved. Alternatively, one could
M = m M . (16)
inprincipleconsidermore‘dynamic’models,thatfollowthe b 1+Θ gal vir
actualformationofthediskgalaxiesstartingathighredshifts Inaddition,wewritethatthetotalangularmomentumofthe
(e.g., Firmani & Avila-Reese 2000; Avila-Reese & Firmani baryonsoutofwhichthediskplusbulgeformis
2000; van den Bosch 2001, 2002). However, as long as the
J j J . (17)
formationofthediskissufficientlyquiescent,thefinalstruc- gal≡ gal vir
tureofthediskshouldbeindependentofitsactualformation AsshownbyvandenBoschetal.(2002),thespecificangular
history:thestructuralpropertiesofthefinaldiskarebasically momentumdistributionofthetotalbaryonicmass(including
justgovernedbytheprincipleofdynamical,centrifugalequi- thosebaryonsinthehalothatdonotpartakeintheformation
librium.Inamoredynamicapproach,onecanactuallymodel ofthediskplusbulge)isvirtuallyidenticaltothatofthedark
the star formation history of the disk, which is not possible matter. Therefore,onealsoexpectsthat0< jgal< fbar.
with our static model. However, the star formation history Asdiscussedbelow,weassumethatthebulge∼formsoutof
mainlygovernsthefinalmass-to-lightratioofthestars,which diskinstabilities. LetJb= jbJvir indicatetheoriginalangular
we set usingempiricalrelations. Themodelsdescribedhere momentum of the baryonic material out of which the bulge
shouldthusbeapplicableindependentofthedetailedforma- forms.Weassume,however,thatthebulgeformationprocess
tionhistoryaslongasthetwoassumptionsmentionedabove transfersthisangularmomentumtothediskplusthehalo,so
aresatisfied. thatthefinalangularmomentumofthebulgeiszero. Indeed,
UnlessstatedotherwiseweadoptaΛCDMcosmologywith suchanangularmomentumtransferisobservedinnumerical
Ωm=0.3,Ωb=0.044,ΩΛ=0.7,h=H0/(100kms- 1Mpc- 1)= simulations (e.g., Hohl 1971; Debattista et al. 2006). If we
0.7 and with a scale-invariant initial power spectrum with a define ftasthefractionofJbthatistransferredtothedisk,we
normalization σ =0.9. The baryonic mass fraction of this obtainthatthefinalangularmomentumofthediskisequalto
8
cosmology fbar≃0.15. Jd= jgal- (1- ft)jb Jvir (18)
3.1. DiskFormation Thespecificangularmomentumofthefinaldiskistherefore
(cid:2) (cid:3)
WemodeldarkmatterhalosassphereswithaNFWdensity J j J
d =(1+Θ)[1- f ] gal vir (19)
distribution lost
M m M
ρ(r)= 4ρs (11) d (cid:18) gal(cid:19) vir
(r/r )(1+r/r)2 wherewehaveintroducedtheparameter
s s
(Navarro, Frenk & White 1997) where rs is a characteristic f =(1- f) jb (20)
radiusatwhichthelogarithmicslopeofthedensitydistribu- lost t j
(cid:18) gal(cid:19)
ARevisedModelforDiskGalaxyFormation 7
which expressesthe fraction of the total angular momentum MMWfordetails). Whencomputingthetotalcircularveloc-
ofthematerialoutofwhichthebulgeplusdiskformsthathas ityV(R)weusethat
beenlosttothehalo.Formodelingpurposesitismoreuseful
V2(R)=V2(R)+V2(R)+V2 (R) (27)
todefinetheparameter d b DM
1+Θ J /M Here
f f =(1- f) b b (21)
x≡ lost(cid:18) Θ (cid:19) t (cid:18)Jgal/Mgal(cid:19) Vd2(R)= GRMd2y2[I0(y)K0(y)- I1(y)K1(y)] (28)
which expresses the ratio of the specific angular momentum d
that has been lost to the halo due to bulge formation to the withy=R/(2Rd),andInandKnaremodifiedBesselfunctions
totalspecificangularmomentumofthematerialoutofwhich (Freeman1970). Thecircularvelocitiesofthebulge,Vb,and
thediskplusbulgehaveformed. Forexample, fx =1means the darkmatter halo,VDM, are computedassumingspherical
thatbulgeformationdoesnotchangethespecificangularmo- symmetry, whereby the mass distribution of the dark matter
mentumofthedisk. Theextreme f =0occurswhenthedisk halo is adjusted for adiabatic contraction (see §3.2 below).
x
loses mass but not angular momentum during bulge forma- The density distribution of the bulge is assumed to follow a
tion. Note thatunlike f , theparameter f canin principle Hernquist(1990)profile
lost x
belargerthanunity. Inpractice,however,thebulgeislikely
M r
b b
to form out of material with relatively low specific angular ρb= 2π r(r+r )3 (29)
momentum(e.g.,Norman,Sellwood&Hassan1996;vanden b
Boschetal.2002). Furthermore,thefractionofangularmo- Inprojection,thisissimilartoadeVaucouleursprofile(i.e.,a
mentumthatis lostto the halois expectedto be fairlysmall SersicprofilewithSersicindexn=4),withahalflightradius,
(e.g., Weinberg 1985; Debattista & Sellwood 2000; Valen- R =1.8152r (Hernquist 1990). Throughoutwe adopt the
eff b
zuela&Klypin2003;O’Neill&Dubinski2003),sothat f is relationbetweenR andM fromShenetal.(2003):
t eff b
expectedtobeclosetounity.
ForadiskwithasurfacedensityΣ(R)andacircularveloc- logReff= -- 51..5241++00..5164llooggMMb ((llooggMMb>1100..33)) (30)
ityV(R)thetotalangularmomentumis (cid:26) b b≤
Rvir Althoughbulgesoflate-typediskgalaxiesarebetterdescribed
J =2π Σ(R)RV(R)RdR. (22) by exponential profiles (e.g. Courteau, de Jong & Broeils
d
Z0 1996),whatmatters mostfor ourpurposesis the totalbulge
Throughoutweassumethatthediskthatformshasanexpo- mass;itsdistributionisonlyofsecondaryimportance.
nentialsurfacedensitydistributionΣ(R)=Σ exp(- R/R ),so Thecomputationof the actualbulge-to-diskratio for each
0 d
that model galaxy is based on the fact that self-gravitating disks
J =2M R V f (23) are unstable againstglobalinstabilities (e.g., bar formation).
d d d vir V
We follow the approachof van den Bosch (1998, 2000)and
with
Avila-Reese & Firmani (2000) and assume that an unstable
f = 1 Rvir/Rde- uu2V(uRd)du (24) disktransformspartof itsdisk materialinto a bulgecompo-
V
2Z0 Vvir nent in a self-regulating fashion, such that the final disk is
(cf., MMW). If we combine equations (12), (19), and (23) marginallystable. Bars, whichareconsideredthetransitions
weobtainthefollowingexpressionforthescalelengthofthe objects in this scenario, are thus expected to be fairly com-
disk: mon,asobserved.
1 We define β(R)=V (R)/V(R), and considerthe disk to be
R = 1+(1- f )Θ λ R (25) d
d f √2f x gal vir stableaslongas
V c
wherewehavedefinedλga(cid:2)l astheeffect(cid:3)ivespinparameterof βmax= max β(R)<βcrit (31)
thematerialoutofwhichthebulgeplusdiskform: 0≤R≤Rvir
(Christodoulou,Shlosman&Tohline1995).Theactualvalue
j
λgal gal λ (26) ofβcrit dependsonthegasmassfractionofthedisk,butfalls
≡(cid:18)mgal(cid:19) roughly in the range 0.52<βcrit <0.70. For a given value
Fromequation(25)itisevidentthatbulgeformationimpacts ofβcrit,weuseaniterative∼techniq∼uetofindthebulge-to-disk
thefinalscalelengthofthedisk.Typically,for fx>1thedisk ratioforwhichβmax=βcrit.
sizewilldecrease,while f <1causesanincreaseinR . Note
x d 3.2. AdiabaticContraction
thatthetransitiondoesnotoccurexactlyat f =1,becausethe
x
presenceofabulgecomponentmodifies f . Numericalsimu- Whenbaryonscoolandconcentrateinthecenterofadark
V
lationsindicatethatbulgeformationcausesanincreaseindisk matter halo they deepen and modify the shape of the gravi-
scalelengths(Debattistaetal.2006),suggestingthat f <1. tational potential. If this process is slow with respect to the
x
Shenetal.(2003)consideredafairlysimilarmodelbutwith (local) dynamical time of the halo, the halo will contract to
theassumptionthatthematerialthatformsthebulgehasthe conserve its adiabatic invariants. For the idealized case of
same specific angularmomentumas the disk. Their favored a spherical halo in which all dark matter particles move on
model is equivalent to our model with f =0.5. However, circularorbits, the adiabaticinvariantsreduceto the specific
x
given the argumentsabove, we expect f to be smaller than angularmomentum,rV(r). If,inaddition,thedistributionof
x
this. In whatfollowswe adopta fiducialvalueof f =0.25, the baryons has spherical symmetry, this reduces further to
x
althoughnoneofourresultsareverysensitivetothisparticu- rM(r),withM(r)theenclosedmasswithinradiusr(Blumen-
larchoice. thaletal.1986;hereafterBFFP).
Note thatsince the computationof f requiresknowledge In realistic dark matter halos, however, the particles typ-
V
of R , this set of equations has to be solved iteratively (see ically move on highly eccentric orbits (Ghigna et al. 1998;
d
8 Duttonetal. 2006
Birnboim 2006; Kaufmann et al. 2006; Keres et al. 2005).
Wedeferadetailedstudyoftheeffectsofsuchcoldinfallon
the contraction of the halo to a future paper. Here we con-
siderasimplemodificationoftheBFFPadiabaticcontraction
formalismthatallowsus, witha singletunableparameter,to
considerreducedcontraction,nocontraction,orevenexpan-
sion.
OurmethodstartsfromtheBFFP formalism,accordingto
which a dark matter particle initially (i.e., before the forma-
tion of the disk plus bulge) at radius r settles at a radius r,
i f
where
r M(r)=r M(r). (32)
f f f i i i
Here M(r) and M(r) are the initial and final mass distribu-
i f
tions. If we assume that initially the baryonshave the same
(normalized) density distribution as the dark matter, then
M(r) is simply givenby the initial halo profile (e.g. NFW).
i
Forthefinalmassdistributionwehavethat
M(r)=M (r)+M (r)+M (r)
f f d f b f DM,f f
=M (r)+M (r)+(1- m )M(r) (33)
FIG. 4.—Ratio ofobserved to virial circular velocities, V2.2/Vvir, as a d f b f gal i i
functionoftheconcentrationparameter,forvariousformsofadiabaticcon- HereM (r)isthemassoftheexponentialdiskenclosedwithin
traction. Allmodelshaveλgal=0.048andmgal=0.05. Theverticaldotted sphericadl shells of radius r, M (r) is the similarly enclosed
linesshowthemeancaccordingtomodelofBullocketal.(2001a)forha- b
loswithvirialmass,Mvir=1013,1012,&1011h- 1M⊙.Thehorizontaldotted massoftheHernquistbulge,andthesecondequalityfollows
linesshowV2.2/Vvirratiosof1to1.8atintervalsof0.2. Therelationusing from the assumption that adiabatic contraction occurs with-
thestandard(Blumenthaletal.1986)adiabaticinvariant,rM(r),isgivenby outshellcrossing. Equations(32)and(33) canbesolvedit-
thebluesolidline. Themodifiedadiabaticinvariant,rM(¯r),asproposedby erativelyfor the contractionfactor Γ(r) r/r, which then
Gnedinetal.2004resultsinonlya≃0.02dex(≃5%)reductioninV2.2(red i ≡ f i
shortdashedline).Mostofthisreductionistakenbackifweusespecifican- allowsforacomputationofthemassdistributionofthecon-
gularmomentumastheadiabaticinvariant,andtakeintoaccountthediskge- tracteddarkmatterhalo.
ometrywhencomputingVcirc(dot-dashedredline).Foralloftheseadiabatic Ourmodificationconsistsofsimplydefiningtheactualre-
contractionmodels,standardconcentrationparametersyieldV2.2/Vvir≃1.6. lationbetweenr andr as
Theblacklong-dashedlinesshowmodelswherewehaveartificiallymodi- f i
fiedthecontractionfactor(seeequation34).Startingfromthestandardmodel r =Γνr (34)
(ν=1),therM(¯r)relationisapproximatelyreproducedwithν=0.8,ν=0 f i
resultsinnoadiabatic contraction ofthehalo, whileν=- 1giveshaloex- withν afreeparameter: ν=1yieldsthestandardBFFPcon-
pansion. Thegreendot-longdashedlineshowsVmax/Vvir,whereVmaxisthe traction, ν =0 corresponds to no adiabatic contraction, and
maximumhalocircularvelocity(withoutadiabaticcontraction).
ν < 0 models an expansion of the dark matter halo. As a
specificexampleofanexpansionmodel,ν=- 1resultsinan
vandenBoschetal.1999). Takingthisintoaccountreduces expansionfactorthatis equalto the contractionfactorinthe
the effect of adiabatic contraction (Wilson 2003). Gnedin BFFPmodel.
etal.(2004)giveamodifiedadiabaticinvariant,rM(¯r),where Fig. 4 shows the impact of various adiabatic contraction
¯r 0.85(r/Rvir)0.8istheorbitaveragedradius.Usingthisadi- formalisms on the ratio of the total circular velocity at 2.2
≃
abatic invariant results in somewhat less contraction of the disk scale lengths, V , to the virial velocity, V . We con-
2.2 vir
halo than in the standard BFFP formalism. An additional sider models with an effective spin parameter λ = 0.048
gal
problemisthatdisksarenotspherical.Therefore,inprinciple and a baryonic mass fraction m = 0.05 (no bulge forma-
gal
one should use rV(r) as the adiabatic invariant, rather than tion is considered here). We apply the various contrac-
rM(r). It is well known that the circular velocity curve of tion formalisms described above and compute the resulting
athinexponentialdiskriseslessrapidlybutreachesahigher V /V as a function of the halo concentration parameter
2.2 vir
peakvelocitythanaspherewiththesameenclosedmass(e.g., c. The three vertical dashed lines (from left to right) in-
Binney&Tremaine1987). Therefore,usingrV(r)asanadi- dicate the mean halo concentration expected for halos of
abaticinvariant,ratherthanrM(r), resultsin a strongercon- massM =1013,1012,&1011h- 1M inthemodelofBullock
vir ⊙
tractionofthedarkmatterhaloat2.2diskscalelengths. etal.(2001a).
Inadditiontothesesomewhatsubtleproblemsforthestan- Note that with the standard BFFP adiabatic contraction
dard BFFP formalism, one may also question whether adia- (ν =1), and a mean halo concentrationparameter for a halo
baticcontractionreallyoccursduringdiskformation.Ifdisks of mass M = 1012h- 1M , we expect that V /V 1.7,
vir ⊙ 2.2 vir
are not built by smooth, relatively slow, spherical infall, the whileV /V 1.45 without adiabatic contraction (ν≃=0).
2.2 vir
adiabaticcontractioncould,inprinciple,becounter-balanced Intheextremec≃asewherethehaloisadiabaticallyexpanded
byavarietyofprocesses.Thesemayevengoasfarastocause (ν=- 1)theratioisfurtherloweredtoV /V 1.3. Asan
2.2 vir
anactualexpansionofthedarkmatterdistribution. Onesuch illustrationwealsoplotV /V ,whereV ist≃hemaximum
max vir max
processisdynamicalfriction(e.g.El-Zant,Shlosman&Hoff- circularvelocityofaNFWhalo,whichisrelatedtothehalo
man 2001; Ma & Boylan-Kolchin 2004; Mo & Mao 2004; concentrationas
Tonini, Lapi & Salucci 2006), which could be significant if
V c
disks are built by relativelybig clumps. Indeed, the physics max 0.465 . (35)
V ∼ ln(1+c)- c/(1+c)
of gas cooling suggests that disks may have formed out of vir r
clumpy,coldstreamsratherthanfromasmoothcoolingflow Note that for the typicalconcentrationof galaxysized halos
(Birnboim& Dekel 2003; Maller & Bullock 2004; Dekel& V V ifν - 1.
2.2 max
≃ ≃
ARevisedModelforDiskGalaxyFormation 9
Theshortdashed(red)lineinFig.4showstheresultsforthe formation time scale is short compared to the time scale on
adiabatic invariant rM(¯r) proposed by Gnedin et al. (2004). which the disk accretes new gas. Consequently, these mod-
Asmentionedabove,thisadiabaticinvariantresultsinasome- elsindeedpredictthatthegasdiskhasbeendepletedbystar
whatsmalleroverallcontractionthanthestandardBFFPfor- formationdowntoΣ =Σ (vandenBosch2001). Aswe
gas crit
malism. Note that we can accurately model the Gnedin willseelater,theinclusionofastarformationthresholdden-
etal.formalismbysimplysettingν=0.8. Notonlydoesthis sityprovesacrucialingredientinsolvingtwoproblemsofthe
reproduceV butalsotheshapeofthecircularvelocitypro- standardMMWmodel.
2.2
file. Finally,thedot-shortdashed(red)lineshowstheresults
fortheadiabaticinvariantrV(¯r),withr¯theorbitaveragedra- 3.4. ConversionfromMasstoLight
diusofGnedinetal.(2004).Thisadiabaticinvariantaccounts
In order to compare our models with observations, we
forboththe eccentricityof the orbitsofthe darkmatter par- need to convert both the stellar masses and the stellar disk
ticlesandforthe(non-spherical)geometryofthedisk. Note
scalelengthsintotheobservedI-bandluminositiesandscale
thatthe resultingV2.2/Vvir is virtually indistinguishablefrom lengths,respectively.
whatone obtainswith the standard BFFP formalism: taking
The conversion from mass to light is conventionallydone
accountof the non-sphericityof the disk completelycancels viathemass-to-lightratio,Υ M /L. Bell&deJong(2001)
∗
theimpactofnon-circularorbits. showedthatΥcanbeestimate≡dfromopticalcolors.Thesere-
lationshavebeenupdatedbyBelletal.(2003a)andPortinari,
3.3. StarFormation
Sommer-Larsen & Tantalo (2004). The main uncertainty in
Disks are made up of stars and cold gas. In the high- thismethodisthenormalization,reflectingtheunknownstel-
est surface brightness galaxies the gas fraction (defined as larIMF.Additionaluncertaintiesarisefromdetailsrelatedto
the ratio of cold gas mass to total disk mass) is small, typ- the amount of dust extinction and the star formation histo-
ically 10%, so assuming a pure stellar disk is reasonable. ries. Upperlimitsonthenormalizationcanbeobtainedfrom
∼
However, the gas fraction increases with decreasing surface maximal disk fits to observed rotation curves, as shown in
brightnessandluminosity(McGaugh& deBlok1997;Kan- Bell & de Jong (2001), although these may still suffer from
nappan2004)suchthatlowsurfacebrightnessgalaxieshave distanceuncertainties. Moreaccurateestimatesfromrotation
comparableamountsofmassincoldgasandstars. Thedata curvemass modelingare hinderedby wellknowndegenera-
usedhere, however,onlyprovidesmeasurementsofthestel- cies(e.g.,vanAlbadaetal.1985;vandenBoschetal.2000;
lar disks, while the models describe the distribution of total Duttonetal.2005).
baryonic matter (cold gas plus stars). To allow for a proper Forthesub-sampleofourgalaxieswithopticalcolors,we
comparisonbetweenmodelsanddataweneedaprescription applythe relationsin Bell et al. (2003a),to obtain estimates
forcomputingtheratiobetweenstarsandcoldgasasfunction of Υ . These are based on a ‘diet’-SalpeterIMF, introduced
I
ofradiusinthegalaxy. by (Bell & de Jong 2001). We have verified that, assuming
Kennicutt(1989)hasshownthatstarformationisstrongly thesameIMF,therelationsofPortinarietal.(2004)givevery
suppressed below a critical surface density, which can be similarresultsthatagreetowithin0.05dex. InFig.5weplot
modeledbyasimpleToomrestabilitycriterion: the resulting Υ as a functionof the I-band luminosity. The
I
σ κ(R) solidredlineshowsthebi-weightedorthogonalleastsquares
Σ (R)= gas (36)
crit fittothesepoints,andisgivenby
3.36QG
(cTleoofrmeqreu1en9c6y4,).σHgaesriesκth(Re)v=el√oc2iVtyR(Rd)(i1sp+edrsldniloVnnR(R)o)f21tihsethgeaes,piacnyd- log[M⊙Υ/IL⊙] =0.172+0.144log[1010.3LhI-702L⊙]+∆IMF,
Q is the Toomre stability parameter. Throughout we adopt (38)
σ =6kms- 1andQ=1.5;withthischoiceofparametersthe The intrinsic scatter in this relation is estimated to be 0.1
gas
≃
radius at which star formation is observed to truncate coin- dex (Bell et al. 2003a, Kauffmann et al. 2003a). We model
cideswiththeradiuswhereΣ =Σ (Kennicutt1989). this scatter by drawing Υ for each model disk galaxy from
gas crit I
Wecomputethetotalstellarmassas a log-normal distribution with a mean give by eq. (38) and
RSF with a scatter σlnΥ, which we treat as a free parameter. The
M∗=Mb+2π [Σ(R)- Σcrit(R)]RdR (37) other free parameter, ∆IMF, absorbs our lack of knowledge
Z0 abouttheIMF.Itisequaltozeroforthe‘diet’-SalpeterIMF
whereR isthestar formationtruncationradius,definedby assumedhere,while∆ +0.15foraSalpeter(1955)IMF,
SF IMF
Σ(R )=Σ (R ). Notethatwethusassumethatthebulge and∆ - 0.15foratop≃-heavyIMFlikethatofKennicutt
SF crit SF IMF
≃
ismadeofstarsentirely,andthatallthediskmaterialwitha (1983).SeePortinarietal.(2004)fordetails.
surfacedensityabovethe criticaldensityhasbeenconverted As a consistency check, we apply the Bell et al. (2003a)
into stars. Although this is clearly an over-simplification of relationstothemean(g- r)color-luminosityrelationofPiza-
thecomplicatedphysicsassociatedwithstarformation,there gno et al. (2005). The resulting Υ (L ) is indicated as the
I I
areseveralreasonswhythisisprobablyreasonable.Firstly,it long dashed line in Fig. 5 and agrees with our mean rela-
isaclearimprovementovertheassumptionthatdisksconsist tion to within 0.05 dex. For comparison, we also show the
ofstarsonly,suchasinthestandardMMWmodel.Secondly, Υ adoptedbypreviousstudies. Theshort-dashedlineshows
I
asshownbyvandenBosch(2000,2001),thissimplemodel the relation adopted by FA00; the relatively weak slope of
(i)matchesthegasmassfractionsofdiskgalaxiesasfunction their Υ (L ) relation was derived from a comparison of the
I I
of their surface brightness and luminosity, and (ii) naturally slopes of I-band and H-band Tully-Fisher relations, while
leads to gas disks that are more extended than stellar disks. theirnormalizationisbasedonthesomewhatadhocassump-
Finally,moredetailed,‘dynamic’modelsfordiskformation, tionthatthediskcontributes70percenttothecircularveloc-
whichmodeltheactualstarformationrateusingrealistic,em- ity at 2.2disk scale lengths. Finally, McGaughet al. (2000)
pirically motivated, prescriptions, show that the typical star and MMW both adopted a constant mass-to-light ratio (in-
10 Duttonetal. 2006
bandlight, R , andthusis probablyan underestimateof the
K
truecorrection.
3.5. SamplingStrategy
In order to pursue a meaningfulcomparisonof our model
VLandRLrelationswiththeobservations,weconstructsam-
ples of random realizations of model galaxies. Each model
galaxyisspecifiedbyfourparameters;thevirialmassofthe
halo, M , the halo concentration parameter, c, the effective
vir
spinparameterofthegalaxy,λ ,andthebaryonicmassfrac-
gal
tionofthegalaxy,m .
gal
Previousstudies(e.g.,MMW;Somerville&Primack1999;
Coleetal.2000;vandenBosch2000;Firmani&Avila-Reese
2000; Crotonet al. 2006)haveassumed thatλ =λ. How-
gal
ever, there are numerousreasons why the spin parameter of
thediskmaybedifferentfromthatofitsdarkmatterhalo(see
§5). In this paper we assume that λ follows a log-normal
gal
distribution:
1 ln2(λ /λ¯ ) dλ
ThFeIGp.oi5n.t—sshIo-bwanΥdIsetesltliamramteadssf-rtoom-liogphtticraalticoo,lΥorIs,uvseirnsgusthIe-braenladtilounmsiinnoBsietyll. p(λgal)dλgal= σlnλ√2πexp"- 2σgal2lnλ gal # λggaall, (39)
etal.(2003a). Thesedataformasub-sampleoftheCourteauetal.(2006)
¯
dataset: MAT(green),Shell(blue),andSCII(red). Atypical1σerrorbar andwetreatλgal andσlnλ asfreeparameters. Ifλgal λwe
isgiveninthelowerrightcorner. Thesolidredlinegivesthebi-weighted ¯ ≃
fittothesedata. Asaconsistencycheckthelongdashedlinegivestherela- expectthatλgal 0.042andσlnλ 0.5,whicharethevalues
tioncomputedusingthemean(g- r)color-luminosityrelationfromPizagno obtainedfordar≃kmatterhaloesfro≃mcosmologicalnumerical
etal.(2005)andtheΥi-(g- r)colorrelation fromBelletal.(2003a). We simulations (Bullock et al. 2001b). In what follows we will
transform i-bandluminosities intoI-band luminosities withLI =Li -0.036 considertheseasourfiducialvalues.
(Courteauetal.2006).Thisrelationisinexcellentagreementwithours.For
comparison wegive the ΥI =1.7 estimate ofMcGaugh et al. (2000; dot- Theconcentrationparameter,c,isstronglycorrelatedwith
dashedline),ΥI(LI)relationfromFA00(short-dashedline),andtheadopted thehalosmassaccretionhistory(Wechsler etal. 2002;Zhao
ΥI =1.7h(dottedline)andtheusedΥI =1.19h(dot-longdashedline)by etal.2003;Li,Mo&vandenBosch2005),andthusdepends
MMW.
onbothhalomassandcosmology.Bullocketal.(2001a)and
Eke,Navarro,&Steinmetz(2001)presentanalyticalmodels,
dependent of luminosity). The value adopted by McGaugh calibratedagainstnumericalsimulations,forthecomputation
etal. (2000),Υ =1.7(M/L) , isbasedona stellar popula- ofameanconcentrationgivenahalomassandcosmology.In
I I ⊙
tionsynthesismodelwithaSalpeterIMF,whileMMWbased whatfollowsweusethemodelbyBullocketal.(2001a).Ata
their value, Υ =1.7h(M/L) on the sub-maximal disk ar- givenhalomass,thehaloconcentrationsfollowalog-normal
I I ⊙
guments of Bottema (1993). Note, however, that there is distributionwitha scatterσlnc =0.32(Wechsleretal.2002).
an error in MMW, and that the actual value they used is However,asforthespinparameter,themeanandscatterare
Υ =1.19h(M/L) (H.J.Mo2004,privatecommunication). differentforthesubsetofhaloswithoutarecentmajormerger.
I I ⊙
Notethatthisvalueissmallcomparedtothetypicalmass-to- As shown by Wechsler et al. (2002), the scatter becomes
light ratios shown here. As we will demonstratebelow, this smaller while the mean increases. To be able to accountfor
hasimportantimplicationsfortheMMWresults. this, we consider σlnc a free parameter and compute c(Mvir)
usingtherelationbyBullocketal.(2001a)butmultipliedby
3.4.1. DiskScaleLengths afreeparameterηc.
Unlikeforcandλ,verylittleisknownregardingthebary-
Inorderto computethe I-bandscale lengthsofourmodel
onic mass fractions, m , of galaxies. We account for this
disks,fordirectcomparisonwiththedatadiscussedin§2,we gal
limitationbymodelingthemeanas
proceedasfollows.
Foreachgalaxywefirstcomputethestellarsurfacedensity M αm
profileofthedisk,i.e.,Σ∗=Σ(R)- Σcrit(R)with0 R RSF, mgal(Mvir)=mgal,0 1011.5hv-ir1M (40)
which we fit with an exponential disk over the ra≤dial≤range (cid:18) ⊙(cid:19)
0.15R R 0.80R . withm andα twofreeparameters.Thevalueofα isre-
SF SF gal,0 m m
≤ ≤
Intheoreticalmodelsofdiskgalaxyformation,disksform latedtotherelativeefficienciesofcoolingandfeedbackpro-
insideout. Thisresultsincolorgradients,withprogressively cesses. Typically, cooling results in α <0 while feedback
m
largerscale lengthswhengoingfromstellar massto K-band results in α >0 (e.g., Dekel & Silk 1986; van den Bosch
m
light(assumedtocloselytracetheunderlyingstellarmass)to 2002). Finally, to allow for scatter in the aboverelation, we
B-bandlight.ObservationsshowasimilartrendwithR /R = assume that, at fixed halo mass, m follows a log-normal
R H gal
1.17(MacArthur,Courteau,&Holtzman2003),andR /R = distribution with scatter σ , which we also consider a free
I K lnm
1.13(deJong1996),wheresubscriptsdenotethephotometric parameter.
band. To account for these color gradients we convert our ToconstructVLandRLrelationsforourmodelsweproceed
stellardiskscalelengthstoI-bandscalelengthsusinglogR = asfollows.WeuniformlysamplearangeinlogM .Foreach
I vir
logR +0.05, with R the scale length of the exponentialfit halo,wethendrawvaluesforc,λ ,andm usingthelog-
∗ ∗ gal gal
to the model stellar disk. Note that this assumes the scale normaldistributionsdescribedabove. We theniterateuntila
length in stellar mass, R , is equal to the scale length in K- solutionforthedisk-to-bulgeratioisfound,takingadiabatic
∗