Table Of ContentMon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev2.2)
Towards a Resolution of the Galactic Spin Crisis:
Mergers, Feedback, and Spin Segregation
Ariyeh H. Maller and Avishai Dekel
Racah Institute for Physics, The Hebrew University,Jerusalem 91904, Israel
2
0
0
1February2008
2
n
a ABSTRACT
J
We model in simple terms the angular-momentum problems of galaxy formation in
1 CDM cosmologies, and identify the key elements of a scenario that may solve them.
1
The buildup of angular momentum is modeled via dynamical friction and tidal strip-
ping in a sequence of mergers. We demonstrate how over-cooling in incoming halos
1
leadstoatransferofangularmomentumfromthebaryonstothedarkmatter,incon-
v
flictwithobservations.Byincorporatingasimplerecipeofsupernovafeedback,weare
7
8 abletosolvethe problemsofangularmomentumindisk formation.Gasremovalfrom
1 the numerous small incoming halos, which merge to become the low specific angular
1 momentum (j) component of the product, eliminates the low-j baryons. Heating and
0 puffing-up of the gas in largerincoming halos,combined with efficient tidal stripping,
2 reducestheangularmomentumlossofbaryonsduetodynamicalfriction.Dependence
0 of the feedback effects on the progenitor halo mass implies that the spin of baryons
h/ is typically higher for lower mass halos. The observed low baryonic fraction in dwarf
p galaxiesisusedtocalibratethecharacteristicvelocityassociatedwithsupernovafeed-
- back,yieldingVfb ∼100kms−1,withintherangeoftheoreticalexpectations.Wethen
o
findthatthemodelnaturallyproducestheobserveddistributionofthespinparameter
r
t among dwarf and bright disk galaxies, as well as the j profile inside these galaxies.
s
This suggests that the model indeed captures the main features of a full scenario for
a
: resolving the spin crisis.
v
i Key words: cosmology – dark matter – galaxies:formation– galaxies:spiral
X
r
a
1 INTRODUCTION thenreproducestheobservedsizesandstructureofgalactic
disks.
The sizes of galactic disks are commonly linked to the
The ‘standard’ model of cosmology, which assumes hierar-
angularmomentumoftheirparentdark-matterhalos(Fall&
chical buildup of structure in a universe where the mass is
Efstathiou1980).Thismodelingisbasedonthedistribution
dominated by cold dark matter (CDM), seems to be fac-
ofhalospinparametersasfoundinN-bodysimulations(see
ing intriguing difficulties in explaining some of the robust
Bullock et al. 2001, and references therein), combined with
observed properties of galaxies. Standing out among these
theassumptions that thebaryons and dark matter initially
problems is the inability of galaxy formation models to re-
share the same distribution of specific angular momentum,
produceboththesizesandstructureofdiskgalaxies.Inhy-
j,withinthehalos(asseeninsimulationsbyvandenBosch
drodynamicalsimulationsbaryonsloseasignificantfraction
et al. 2002) and that j is conservedas thebaryonscontract
of their angular momentum leading to overly small disks.
toform thedisk(assuggestedbyMestel1963).Thesizesof
Toavoid this,analytic andsemi-analytic models commonly
disksobtainedundertheseassumptionsareroughlycompa-
assumethereisnoangular momentumloss. Recentstudies,
rable to those observed.
however, show that this leads to a variety of discrepancies
with observations. Taken together this suggests a crisis in However, high-resolution simulations that incorporate
ourunderstandingoftheroleofangularmomentuminform- gas processes find this scenario to be invalid. In particular,
ing disk galaxies. Our aim here is to make progress in the they find that a significant fraction of the angular momen-
efforttoresolvethiscrisisbyfirstreproducingtheproblems tumof thebaryonsis transfered tothedark matter,result-
using a simple model in which the important physical in- ingindisksizesroughlyanorderofmagnitudesmallerthen
gredientsarespelledoutandwellunderstood.Subsequently thoseobserved(Navarro&Steinmetz2000;Sommer-Larsen
we incorporate an additional process into this model which et al. 1999; Navarro& Steinmetz1997; Navarroet al. 1995;
2 Ariyeh H. Maller and Avishai Dekel
Navarro & Benz 1991). This has been refered to as the an- higher spin than average (Gardner 2001; Wechsler 2001).
gular momentum catastrophe. This demonstrates that the exact connection between the
Theangularmomentumcatastropheiscommonlyasso- angular-momentum properties observed in galaxies and in
ciatedwiththeproblemof“over-cooling”inCDM-typesce- N-bodysimulations is unclear.
narios(White&Frenk1991; White&Rees1978). Without We propose that the solution to the crisis is spin seg-
sufficient feedback much of the gas cools quickly, contracts regation — that the angular momentum distribution of
into small halos and then spirals deep into the centers of baryonsdiffersfromthatofthedarkmatterduetogaspro-
bigger halos, efficiently transferring its orbital angular mo- cesses. These processes can either decrease or increase the
mentum to the dark matter (Navarro & Steinmetz 2000). specific angular momentum of the baryons relative to the
It has therefore been speculated that enough energy feed- dark matter. Gas cooling generally results in lower spin for
backfrom supernovamay preventthisover-cooling (Larson the baryons compared to the dark matter, but heating due
1974; White & Rees 1978) and thus reduce the angular- tofeedbackreducesthiseffect,andgasremovalinsmallha-
momentumloss.Indeed,simulationswheregascoolingisar- losactuallyresultsinahigherspinforthebaryonscompared
tificiallysuppressedtillz=1donotsufferfromtheangular to thedark matter.
momentum catastrophe (Eke et al. 2000; Weil et al. 1998). In this paper we work out a simple model to explore
Furthermore, simulations including some forms of feedback these effects. Knowing that in a hierarchical scenario the
show a reduction in the amount of angular momentum lost halo buildup can be largely interpreted as a sequence of
(Sommer-Larsen et al. 1999). However, while feedback has mergersbetweensmallerhalos,ourmodelisbasedonasim-
beenstudiedusingsimplifiedapproximations(e.g.,Dekel& ple algorithm for thebuildup of halo spin by summation of
Silk 1986, hereafter DS),a realistic implementation offeed- the orbital angular momenta of merging satellites (Maller
backhasprovedchallenging(seeThacker&Couchman2000, et al. 2002; Vitvitska et al. 2001). This algorithm has been
and references therein), though some partial progress may found to match well the spin distribution among halos in
havebeenmaderecently(Ferrara&Tolstoy2000;Thacker& N-bodysimulations.Inthispaperweextendthemodeland
Couchman2001;Springel&Hernquist2002). Atthispoint, find that it also reproduces the angular-momentum profile
thefeedbackscenariohasnotyetbeenstudiedinsatisfactory within halos. It therefore provides a useful insight into the
detail, norhas it been confirmed to solve thespin problem, originofthespinproblems,andacluefortheirpossiblesolu-
orproperlyunderstoodinbasicterms.Thismotivatesanat- tion.Wethereforeusethismodelasatoolforincorporating
tempttounderstandthescenarioandhowitmayworkusing therelevant baryonic processes, and especially feedback.
a simple semi-analytic model, in which the basic elements In 2 we describe the model for angular-momentum
§
are easily understood. buildup in halos by mergers. In 3 we model the angular
§
Even if the angular momentum catastrophe can be momentum catastrophe as resulting from over-cooling and
avoided such that the total baryonic spin agrees with the dynamical friction. In 4 we introduce a simple model of
§
totaldark-halospin,thereremaindiscrepanciesbetweenthe feedback. In 5 we study the resultant angular-momentum
§
angular-momentumpropertiesofdarkhalosinN-bodysimu- properties of the baryonic component both in bright and
lationsandthoseofobserveddiskgalaxies.Foremostamong dwarf disk galaxies and compare them to observations. In
theseistheapparentmismatchofthedistributionofspecific 6 we test the robustness of our results to the details of
§
angular momentum within galaxies (loosely termed “the j the feedback model assumed. We conclude and discuss our
profile”)betweenobservationsandN-bodysimulations.Bul- results in 7.
§
lock et al. (2001, hereafter BD) revealed a universal pro-
file with an excess of both low-j and high-j material com-
pared to the observed j distribution in exponential galac-
tic disks. Subsequently, van den Bosch, Burkert & Swaters 2 BUILDUP OF HALO SPIN BY MERGERS
(2001, hereafter BBS) have measured the j distribution in
2.1 Modeling the Spin Parameter
a sample of dwarf disk galaxies, confirming in detail, case
by case, the differences between the j profile predicted by The angular momentum of a galaxy, J, is commonly ex-
the simulated halos and those found in disk galaxies. We pressed in terms of the dimensionless spin parameter λ =
refertothisseconddiscrepancyasthemismatchofangular- J E /GM5/2, where E is the internal energy (Peebles
momentum profiles. 19p69|).|Inpractice,thecomputationandmeasurementofthis
Another spin problem is indicated by the observations quantity,especiallytheenergy,maybeambiguous,e.g.,itis
of de Jong & Lacey (2000), who found that the scatter in not obvious how to define separate spin parameters for the
thelogofthespinparameterinferredfromfittingthespread darkmatterandthebaryons.Furthermore,itintroducesan
of observed disks sizes in a large sample of late-type disk undesired dependence on the specifics of the halo density
galaxies is only 0.36 0.03, while the scatter seen in sim- profile. Instead,following BD, we use themodified spin pa-
±
ulated halos is significantly larger, 0.5 0.05 (BD and ref- rameter
±
erences therein). This could have been explained by a sce-
′ j
nario in which the disks formed in low-spin halos are un- λ = , (1)
stable and thus become early-type galaxies, which implies √2VvirRvir
that disk galaxies occupy only the higher spin halos (Mo, where j = J/M is the specific angular momentum. This
Mao&White1998;vandenBosch1998).However,thissce- quantity is more straightforward to compute for each com-
′ ′
narioisinapparentconflictwiththefindinginN-bodysim- ponent, λdm and λb, and it does not explicitly depend on
ulations that recent major mergers, usually identified with the density profile of the halo. For a density profile of the
′
largespheroidalcomponents,actuallygiverisetohaloswith typesuggested by Navarro et al. (1996, NFW) λ is related
Towards a Resolution of the Galactic Spin Crisis 3
to λ by
λ′=λf−1/2, (2) 1.0
c
wherethefactorf reflectsthedifferencebetweentheenergy
c
ofanNFWhaloandatruncatedsingularisothermalsphere. 0.8
Anexpressionforf asafunctionofthehaloconcentration,
c
cvir, is given in Mo et al. (1998, equation 22). For a NFW v
halo with cvir = 4.5 the values of the two spin parameters M 0.6
/
become equal. j)
<
BD found for simulated halos that, similarly to λ, the (
′ M 0.4
distribution of λ is well fit by a log-normal function,
′ ′ 1 ln2(λ′/λ′0) dλ′
P(λ)dλ = exp , (3)
2πσλ2 (cid:18)− 2σλ2 (cid:19) λ′ 0.2
′ p
with λ0 ≃0.035 (compared to λ0 ≃0.042) and σλ ≃0.5. 0.0
InMalleretal.(2002),weproposedaverysimplemodel
for the buildup of spin in dark halos, termed “the orbital- 0.0 0.2 0.4 0.6 0.8 1.0
mergermodel”.Inthismodel,thespinofafinalhaloissim- j/j
max
plythevectorsumoftheorbitalangularmomentaofallthe
halos merging into its main progenitor throughout its his- Figure1.Dark-haloangular-momentumprofiles:Themassfrac-
tory.Inthispaper,asinMalleretal.(2002),wegenerate500 tionwithspecificangularmomentumlessthenj asafunctionof
randomrealizationsofmergerhistoriesforeachdesiredfinal j/jmax.Thelinesconnectingsymbolsrepresentarandomsample
of profiles produced by our orbital-merger model. The smooth
halomassatz=0,usingthemethodofSomerville&Kolatt
curves are fits to these profiles, where µ is chosen to cross at
(1999) with the slight adjustment of Bullock, Kravtsov &
halfthe total mass.The shadedregionmarks the90% spreadof
Weinberg(2000)toimplementtheExtendedPressSchechter
profilesfoundbyBDinsimulations.
formalism (Lacey & Cole 1993; Bond et al. 1991). The spe-
cific cosmology assumed, without significant loss of gener-
ality, is the “standard” ΛCDM cosmology, with Ωm = 0.3, in which one of the parameters, say jmax, can be replaced
′
ΩΛ =0.7,h=0.7andσ8 =1.0.Foreachofthemergertrees, by λ. This profile is a simple power law, M(< j) ∝ j, for
weperform10realizationsoftheorbitalangularmomentum at least half the mass. When µ is close to unity, it bends
that comes in with each merger. The encounterparameters over towards the high-j end. The distribution of the shape
are taken to be those of typical encounters in simulations, parameter µ was found to be Gaussian in log(µ 1), with
−
andthedirections oftheaddedspinsare drawnat random, a mean of 0.6 and standard deviation of 0.4.
−
allowingforaslightcorrelationbetweentheplanesofsucces- Onecanalsoquantifytheshapeoftheangularmomen-
sive mergers (as seen in simulations, e.g. Dekel et al. 2001; tumprofilebytheratioofthetotalspecificangularmomen-
Porcianietal.2001).Weendupwithadistributionof5000 tum,jtot,tojmax.Therelation betweenthisparameterand
spin values for each halo mass. This distribution is shown µ is
in Fig. 4, in comparison with the log-normal distribution jtot µ
ζ = =1 µ(1 (1 (µ 1)log )) (5)
obtainedbyBDfrom acosmological simulation. Themodel jmax − − − − µ 1
predictions of λ′0 = 0.036 and σλ = 0.53 match rather well −
(BBS eqn.12).
thesimulatedresults.OnecanseeinFigure2ofMalleretal.
We create M(< j) profiles for each of the realizations
(2002) that the match is reasonable even without introduc-
of the orbital-merger model as follows. We divide the mass
ing any correlation between theencounterplanes.
growth of the halo into 20 equal-mass bins and assign to
SimilarresultswereobtainedbyVitvitskaetal.(2001),
each bin the corresponding angular momentum that comes
who draw their encounter parameters statistically from the
inwiththatmass.Whenasatellite’smassisdividedamong
distribution found in N-body simulations, and by Monaco,
severalbins,i=1,n,withafraction f inbin1,weassign a
Theuns & Taffoni (2002), where encounter parameters for
fractionf2ofitsangularmomentumtobin1.Theremaining
each object are provided by Lagrangian perturbative the-
satellite mass and angular momentum are then distributed
ory. We conclude that the simple orbital-merger model is
in an analogous way among bins i = 2,n. This mimics an
successful in reproducing the simulated halo spin distribu-
M(< j) j1/2 profile for the contribution of each satel-
tion, and therefore serves as a useful tool for studying the ∝
lite,inroughagreementwiththepredictionsofatoymodel
angular-momentum buildupin galaxies.
basedondynamicalfrictionandtidalstrippingforanNFW
halo (BD; Dekel et al. 2001). The role of this procedure
2.2 Modeling the Angular-Momentum Profile is to smooth the profile; in most cases it does not change
theglobalshape,whichispredominantlydeterminedbythe
The cumulative mass distribution of specific angular mo- cosmologicalsequenceofmergers.Theresultisthereforenot
mentum within simulated dark halos, M(< j), was found too sensitive to the exact assumed distribution of j due to
byBDtobewellfitbythe2-parameteruniversalfunctional each single merger.
form Fig.1showsasampleofjprofilesproducedbythispro-
M(<j) µj/jmax cedure.Ourprofilesaresomewhatnoisierthenthoseseenin
= , µ>1, (4)
Mvir µ 1+j/jmax simulations, partly because we ignore any exchange of an-
−
4 Ariyeh H. Maller and Avishai Dekel
3.1 Tidal Stripping and Dynamical Friction
1.5
Wecanunderstandthetransferofangularmomentumfrom
thebaryonstothedarkmatterincosmologicalhydrodynam-
ical simulations with a simple model including gas cooling,
dynamicalfrictionandtidalstripping.Inourmodelforhalo
1.0 spin buildupdiscussed abovewehavenot specified how the
)
1) orbital angular momentum of the satellite is converted to
-
µ thespinofthehalo.Clearly thetransferofangularmomen-
(
10 tumoccurs bytheprocesses of dynamicalfriction and tidal
g
o stripping. The dynamical friction acting by the halo parti-
l
P( 0.5 clesontheboundpartofthesatelliteexertsatorquewhich
transfers angular-momentum from the satellite to the halo
and eventually brings the satellite towards the halo center.
Satelliteparticlesthataretidallystrippedandbecomepart
of thehalo before the satellite sinks to thehalo’s centerre-
0.0
tain what is left of their angular momentum at that point
0.01 0.10 1.00 10.00 and add it directly to thehalo.
µ-1 A hint for the typical M(< j) distribution due to a
single merger can be obtained as follows (BD; Dekel et al.
Figure2.Thedistributionofshapeparameterµforhaloangular- 2001). If the satellite moves with a circular velocity Vc(r),
momentum profiles. The histogram describes the distribution in and we assume that the lost mass and j are deposited, on
thesamplereproducedbyourorbital-mergermodel.Thesmooth
average, locally at r, then the resultant spatial profile j(r)
curve is the log-normal fit to the N-body simulation results of
can beobtained byaveraging over shells,
BD.Thereisaslightoffsetinthemeanofthedistributionofour
model(-0.5insteadof-0.6)butthisisnotstatisticallysignificant. 2 d[rVc(r)] dm(r)
4πr ρ(r)j(r)=m(r) + rVc(r) . (6)
dr dr
ThefirsttermreferstotheJtransferasareactiontodynam-
gular momentum between particles. The overall agreement icalfriction,andthesecondistheJ depositassociatedwith
betweenourmodelprofilesandthesimulationresultsofBD the tidal mass transfer. We need to estimate the momen-
suggeststhattherearrangementofangularmomentumonly tarysurvivingsatellitemass m(r)inordertocomputej(r),
smoothes the profile without changing its global shape. We which can then be invertedto obtain the desired M(<j).
fiteachofourprofilestoeq.(4)bythesimplemethodofre- The tidal mass loss at halo radius r can be crudely
quiringanexact matchat half thevirial mass(Fig. 1).The estimatedbyevaluatingthetidalradiusℓt ofthesatelliteat
same results follow if we use ζ to compute µ by inverting r using thecrude resonance condition,
eqn.5.Weobtain adistribution ofµvaluesthatareconsis-
3 3
tent with the results of BD (Fig. 2). As in BD, they show m(ℓt)/ℓt =M(r)/r , (7)
no dependence on halo mass or redshift. Thus, the orbital-
where m(ℓ) and M(r) are the mass profiles of the satellite
merger model, which reproduces the correct distribution of
and halo respectively (e. g. Weinberg 1994). If, for simplic-
spin values among halos, also recovers reasonably well the
ity, these two profiles are self-similar, then the resonance
angular-momentum profileinside halos.
condition implies
Ourorbital-merger model reveals an important feature
in the buildup of halo angular momentum. We see in the ℓt/ℓvir =r/Rvir, (8)
typical realization that the final halo spin and its direction
are predominantly determined by one last major merger. where ℓvir and Rvir are the virial radii of the satellite and
thehalorespectively.Thismeansthattheremainingbound
Most of the high-j mass in the final distribution comes in
mass of thesatellite when it is at r obeys
withthismerger.Themanysmallersatellites,eachpossibly
carrying ahigh j, come in at differentdirections and there-
m[ℓt(r)] M(r). (9)
fore tend to sum up to a low spin. Thus, most of the low-j ∝
fraction of the halo mass originates in minor mergers. This Amoreaccuraterecipe fortidalstripping, basedon studies
feature provides an important clue for a possible solution with N-body simulations of mergers, reveal that eq. (9) is
to the spin crisis, as follows. If small satellites lose a large valid as a crude approximation for a wide range of merger
fraction of their gas before they merge into the halo then parameters(Dekeletal.,inpreparation).Wethereforeadopt
the final gas distribution in the galaxy lacks the low j ma- it in our model.
terial. Furthermore,if moreof thegalactic gas originates in Ifthehalodensityprofileisisothermal, M(r) r,then
∝
big satellites then the spin of the baryonic component may substituting eq. (9) in eq. (6) yields j(r) r, which corre-
∝
behigher then that of thedark matter. spondstoM(<j) j.AmorerealisticNFWdensityprofile
leadsintheouterh∝alotoajprofileclosertoM(<j) j1/2,
∝
(BD; Dekeletal.2001)whichiswhatweusetosmooththe
profileintheorbital-mergeralgorithm.Thequalitativesim-
ilarity between the profile predicted by this toy model and
3 REPRODUCING THE ANGULAR
the typical j profile seen in simulations (BD) provides fur-
MOMENTUM CATASTROPHE
thersupporttothegeneralpictureofJ buildupbymergers.
Towards a Resolution of the Galactic Spin Crisis 5
3.2 Spin Segregation
Over−Cooling Spin Crisis
We can now explore the effect of cooling in the satellites
on thefinal angular momentum of the baryons. We assume
that at an early time the baryons follow the dark-matter Gas Cooling
distribution. As the gas within a halo cools radiatively, it
contractstoamorecompactconfiguration characterized by
a smaller radius, Rb, compared to the extent of the dark
matter,Rdm.Suchaspatialsegregation inasatellitebefore DM
it merges with a bigger halo naturally leads to a difference
inthespinsofbaryonsanddarkmatterintheproducthalo. (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
The bound part of the satellite is continuously transferring (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
orbitalangularmomentumtothedarkmatterbydynamical (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
friction.Intheextremecase,theinnersatellitethatremains
bound all the way to the halo center eventually loses all
its orbital angular momentum by this mechanism. At the
moment of stripping, the escaping satellite material stops
being affected by dynamical friction, and it joins the halo Tidal Stripping
withtheangularmomentumthatitisleftwithatthatpoint.
Thespatialsegregationinthesatellitethusimpliesthat
thej-richmassstrippedattheearlystagesofthemergerin DM
theouterhaloisdominatedbydarkmatter,whilethemore
compact baryonic component survives longer as a bound
satellite and loses more of its orbital angular momentum
by dynamical friction. The result is a net transfer from the Dynamical
baryons to the dark-matter component. This process is il- Friction
lustrated in a schematic diagram, Fig. 3.
Using eqn. 8 above, we obtain that the orbital angular
momentum added to the merger product with the baryons
is related to the amount gained by thedark matter via
Figure 3.Aschematic illustrationof howover-cooling inmerg-
Rb ingsatellitesleadstotheangularmomentumcatastrophe.Thegas
∆Jb = Rdm∆Jdm. (10) contraction within the incoming satellite makes the gas immune
againsttidalstripping:itspiralsallthewayintothehalo center
Weignoreanytransferofangularmomentumfromthesatel-
whilelosingallofitsorbitalangularmomentumtothedarkhalo
litetothebaryonsalreadyinthehaloasbaryonsinthemore
duetodynamicalfriction.Thedarkmatter,whichdominatesthe
massive halo are already more centrally concentrated than
outer regions of the satellite, is gradually stripped in the outer
those in the satellite. In the case of maximum cooling the parts of the halo, thus retaining part of its orbital angular mo-
baryons dominate the center of the halo which we approx- mentum.
imate as Rb = fbRdm, where fb is the universal baryon
fraction. Adoptingfb =0.13 (Tytleretal. 1999), weobtain
that the spin of the baryons is reduced by almost an order
4 FEEDBACK
of magnitude, and thus reproduces the angular momentum
catastropheasseeninhydrodynamicalcosmological simula- Itisgenerally assumed thatsomeform ofheatingmayplay
tions. In Fig. 4 we show the resultant probability distribu- an important role in preventing over-cooling and therefore
tionofthespinparameterasobtainedinourorbital-merger angular-momentumloss.Themostimportantsourceofheat-
model with maximum cooling; it has λ′0 =0.005. ingislikelytobesupernovafeedbackthoughreionizationby
Thissimplemodelingofthespinproblemmakesiteasy theUVbackground,tidalheating,andrampressuremayall
to understand why it is usually assumed that some form of contribute. Semi-analytic models assume different ad hoc
energyinputintothegasmayremedytheproblem;itwould recipes for the amount of heating as a function of star-
delaythecooling,increaseRb,andthusreducethebaryonic formation rate, though hydrodynamical simulations have
spin loss. been largely unsuccessful so far in implanting feedback in
Note that in common semi-analytic models of galaxy awaythatpreventsover-cooling(seeThacker&Couchman
formation the gas is assumed to be either in a hot phase or 2000, and references therein). Our approach in the present
in a cold phase, with the hot phase extending out to Rdm work is to avoid the details of star formation and feedback
and the cold phase condensed. This implies that the cold recipes and ratheruse avery simple prescription for theef-
gas in the satellite looses most of its angular momentum fect of feedback in a satellite halo as a function of its virial
to dynamical friction, while the hot gas retains most of its velocity,Vsat.
angular momentum. In this case the fraction of baryons in Followingtheanalysisin IVofDS,weassumethatthe
§
thehotphase, f ,determinestheorbital angular momen- totalsupernovaenergypumpedintothegasperunitmassis
hot
tum added as baryonic angular momentum to the merger arobustquantity.Thisisbasedontheassumptionsthatthe
product, ∆Jb = fhot∆Jdm. Thus, one can relate our dis- supernovarateisroughlyproportionaltothesamegasmass
cussion here to other semi-analytic modeling by identifying thatislaterbeingaffectedbyit,thatthestar-formationrate
fhot =Rb/Rdm. is inversely proportional to the dynamical time, and that
6 Ariyeh H. Maller and Avishai Dekel
0.15
40
30 λ′ = 0.036
0 0.10
σ = 0.53
λ′ λ
λ′)d 20 λ′0 = 0.005 /Jbdm
( σ = 0.51 J
P λ
0.05
10
0 0.00
0.00 0.02 0.04 0.06 0.08 0 50 100 150 200
λ′ v (km/s)
satt
Figure 4. The effect of over-cooling on the spin distribution Figure5. Theeffectoffeedbackontheratioofbaryonicangular
of baryons compared to the dark matter. Since the baryons are momentum tothatofthedarkmatter asafunctionofthevirial
tightly bound in the satellite centers, they spiral into the inner velocity of the merging satellite. In the model described by the
partofthebighalowithoutbeingstrippedandthusloosemostof thick curve, the feedback is characterized by Vfb = 95kms−1,
their angular momentum to the haloby dynamical friction.The with the exponents γ1 = γ2 = 1. If the baryons in the satel-
modelreproducesthedeficiencyofangularmomentuminbaryons lite are distributed like the dark matter (middle range), then
ofalmostanorderofmagnitude asseeninhydrodynamical sim- Jb/Jdm = fb, the cosmological baryon fraction. In larger satel-
ulations. liteswithVsat>Vfb,thebaryonsarelessextendedthenthedark
matter and thus loose more angular momentum by dynamical
friction before they are stripped into the halo. In smaller satel-
the supernova remnant transfers most of its energy to the lites with Vsat < Vfb/√2, gas is blown out and therefore Jb is
interstellarmediumbytheendofits“adiabatic” phase.DS reduced, though the specific angular momentum of the baryons
estimatedasacrudeupperlimitthatthesupernovainputis remainsthe sameasthat ofthedarkmatter. Theshadedregion
shows the range of models we consider in 6, all constrained to
comparabletothekineticenergyperunitmassofanisother- §
mal halo with virial velocity 100kms−1. The actual free matchthebaryonicfractionindwarfgalaxies.
∼
parametercharacterizingourfeedbackmodel,Vfb,isdefined
as thevirial velocity of ahalo for which theenergy inputis
with γ2 another arbitrary exponent.
sufficienttoheatallthegastothevirialtemperatureofthe
halo. Based on DS we expect Vfb to be of order 100kms−1 Westartwiththesimplestarbitrarychoiceγ1 =γ2=1.
Later, in section 6, we will explore the robustness of the
or less, butwe keep thisestimate in mind only as ageneral
results to different assumptions about choices of values for
constraintwhichdoesnotenterourcurrentanalysis.Theac-
γ1 and γ2.
tual relation between Vfb and the supernova input depends
The only remaining free parameter of the feedback
on theinitial state of the gas before thesupernovaburst.
model is Vfb. We set it by requiring that halos with a virial
Wethusparameterizetheeffectoffeedbackonthespa- velocity of 60kms−1 have an average fd = 0.04, as mea-
tial extent of the baryons in a halo by the ratio of Vsat to
sured in the data of BBS. Notethat we are concerned with
Vfb. The limit Vsat >>Vfb, massive halos with deep poten-
thestateofthebaryonsonlywhenthesatellitemergerswith
tialwells,correspondstomaximumcooling,Rb Rdm.For
≪ thehalo;it isthereforesufficientthatthefeedbackbeeffec-
smaller halos with Vsat Vfb we expect the heating to bal-
≃ tive for a short duration, perhaps triggered by the merging
ancethecoolingandyieldRb Rdm.Ourmodelistherefore
≃ process itself.
a simple interpolation between theselimits,
The effects of heating and blowout on the angular mo-
Vfb γ1 mentum of baryons versus dark matter, according to our
Rb = Rdm (Vsat >Vfb), (11)
(cid:16)Vsat(cid:17) model, in shown in Figure 5. For massive satellite with
with γ1 an arbitrary exponent. Vsat >Vfb,thebaryonsretainlessangularmomentumasthe
mass increases, because of dynamical friction. Small satel-
If Vfb is larger than Vsat, the feedback can cause gas
ejection from the satellite. For Vsat <<Vfb we expect total lites with Vsat <√2Vfb allow a blowout of a larger fraction
oftheirgasasthemassdecreases,thusaddingtothemerger
blowout, and we assume that partial blowout starts occur-
productlessbaryonsandthereforelessbaryonicangularmo-
ringforhaloswherethepotentialenergyofthegasiscompa-
rable to the energy input from supernova, Vs2at = (1/2)Vf2b. mentum, while the specific angular momentum of the two
components remains thesame.
We therefore parameterize the amount of gas that remains
In Fig. 6 wedemonstrate theeffectsof thedifferent in-
in thehalo by another simple interpolation,
gredients of this feedback scheme, with Vfb =95kms−1, on
γ2
fd = Vsat (Vsat <Vfb/√2), (12) the baryonic spin-parameter distribution. We do it for two
(cid:18)√2Vfb(cid:19) kindsof final halo masses, corresponding tovirial velocities
Towards a Resolution of the Galactic Spin Crisis 7
40 Heating Blowout Both
30 λ′ = 0.028 λ′ = 0.051 λ′ = 0.040
0 0 0
σ = 0.46 σ = 0.47 σ = 0.43
λ′ λ λ λ
d λ′ = 0.036 λ′ = 0.067 λ′ = 0.068
λ′) 20 σ0 = 0.54 σ0 = 0.60 σ0 = 0.61
P( λ λ λ
10
0
0. 00 0. 05 0.1 0 0.0 0 0. 05 0. 10 0. 00 0. 05 0. 10 0. 15
λ′
Figure6.Theeffectsofheatingandblowoutinourmodelonthedistributionofbaryonicspinparameterλ′.Log-normalfitsareshown
foreachhistogram,withthecorrespondingmeanandscatterquoted.Ineachpanelthehigherpeakedcurveandhighertwonumbersare
the bright galaxies (Vvir =220kms−1) and the lower peaked curve and lower two numbers are the dwarf galaxies (Vvir =60kms−1).
Left: Only heating is included in our feedback recipe (blowout ignored). Center: Only blowout is included (cooling ignored). Right:
Thewholefeedbackrecipeisincluded,withcooling,heatingandblowout.
of60and220kms−1.Werefertotheseasrepresentingdwarf 5 MODEL VERSUS OBSERVATIONS
and bright galaxies respectively.
We now confront the model predictions with observations,
The left panel of Fig. 6 shows the effects of adding
inparticularthoseofBBS.Wefirstcalibratethevalueofthe
heating only, ignoring blowout. The dwarf galaxies are all
one free parameter Vfb by the observed gas deficit in dwarf
built up by satellites of Vsat < Vfb, so there is full heat-
galaxies. We then address the distribution of the baryonic
ing, Rb Rdm The result is that the spin distribution of
≃ spinparameterin thesegalaxies, aswellastheangularmo-
the baryons resembles that of the dark-matter. In bright
mentum profiles within dwarf and bright disks.
galaxies, the partial heating raises the baryonic spin to
′
λ0 =0.028, which is still lower than the typical spin of the
dark matter dueto thepartial cooling.
5.1 Observations
Thecenterpanelshowstheeffectsoffullheating(Rb
≃
Rdm)andblowout,asifthereisnocooling.Forthetwokinds Generally, galaxy formation modelers have compared their
′
of galaxies, the baryonic λ distribution becomes signifi- predicted disk sizes to the distribution of disk scale lengths
cantlyhigherthenthatofthedarkmatter,withλ′0=0.051 foundinlargesamplesoflatetypegalaxies(Courteau1996;
′
for the bright galaxies and a very high λ0 = 0.067 for the Mathewson et al. 1992). A more detailed comparison has
dwarf galaxies. beenperformedbydeJong&Lacey(2000)wheretheycon-
Finally, the right panel shows the result of applying structedthebivariatespacedensityofgalaxiesasafunction
the full feedback scheme, considering cooling, heating and of luminosity and size. This allows the spread in observed
blowout together. The baryons in the bright galaxies now disk sizes to be compared to models and they found that
have spins comparable and even slightly higher then their thespreadcorresponded toaσ ofonly0.36 0.03. Ideally
λ
′ ±
dark-matter halos, with λ0 = 0.040 (and σλ = 0.43), while one would like to compare the actual angular momentum
the baryons in dwarf galaxies now have spins higher by in models with observed galaxies instead of just properties
50%, with λ′0=0.068 (and σλ =0.61). ∼ derived from the angular momentum.
Thus, the baryonic spin in dwarf galaxies, which are The only observational attempt so far to measure spin
made by mergers of small satellites, is dominated by the parameters for galactic disks is by BBS, which therefore
blowoutofthegasfromthesesatellites, anditendsupwith serves as the main target for our modeling effort in this
′ ′
λb > λdm. For bigger galaxies, which are largely made of paper. BBS used fits to the rotation curves and assumed
bigger satellites, the dominant effect preventing the over- an NFW profile for the halos (Navarro et al. 1995) in or-
coolingspincatastropheistheheating,withsomecontribu- der to determine the halo virial quantities Rvir and Vvir for
tion from blowout. This yields a baryonic spin distribution 14 dwarf galaxies, with an average Vvir 60kms−1. They
≃
similar to that of the dark matter halos, in general agree- also assumed a mass-to-light ratio of unity in the R band,
ment with observations. but their results are not very sensitive to this assumption
because dwarf galaxies are dominated by their dark matter
halos. BBS then determined for each galaxy the baryonic
8 Ariyeh H. Maller and Avishai Dekel
50
25
V = 60
vir Dark Matter
40 V = 220
vir 20
fd 30 λ’ 15
d d
(f)d λ’)
P 20 (
P 10
10 5 Baryons
0 0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.05 0.10 0.15 0.20
f λ’
d
Figure7.Theprobabilitydistributionofthefractionofmassin Figure 8.Thedistributionofλ′ inthesampleofdwarfgalaxies
baryonsfdforthedataofBBS(shaded),incomparisonwithour byBBS(shadedhistogram)comparedwithourmodelpredictions
model predictions with Vfb =95kms−1 fordwarf galaxies (left) withVfb=95kms−1 forthebaryonsindwarfgalaxies(curveon
andforbrightgalaxies(right).Themodelpredictionforthedwarf theright),andthesimulationresultfordarkhalosbyBD(curve
galaxies,withsignificantblowout, isingoodagreement withthe onthe left). Theinclusionofblowout produces a λ′ distribution
BBSdata.Thebrightgalaxiesretainmostoftheirbaryons. ingoodagreementwiththedata.
′
spin parameter λb which we have converted to λb using encouraging that the obtained value for Vfb is in the range
their values for cvir and equation 2. This leads to an av- of expected values for supernova-drivenwinds (DS).
′
erage value of λb 0.063, significantly larger than that of
≃
simulated dark-matter halos. They also estimated the ra-
tio fd of disk mass (stars + gas) to dark-matter mass, and
5.3 Spin Parameter of Baryons
found an average of fd 0.04, about a factor of 3 smaller
≃
thantheuniversalbaryonicfraction(Tytleretal.1999).Fi- Next, we compare predicted and observed distributions of
nally, they measured the j distribution in each galaxy, and spin parameters for dwarf galaxies. The distribution of λ′b
confirmed themismatch of angular-momentum profiles. We for the BBS disks of dwarf galaxies is plotted in Figure 8
discard one galaxy (UGC4499) from thesamplebecause its as a shaded histogram. Shown first in comparison is the
rotation-curvefit yields strikingly anomalous results. λ′ distribution for dark halos in cosmological ΛCDM sim-
ulations from BD. Contrary to comments made by BBS,
these two distributions are not in agreement — the ob-
5.2 Baryon Fraction servedspinsaresignificantlyhigher,withanaverageof0.063
compared to 0.035 in the dark-matter simulations. Finally
The distribution of fd values for the dwarf galaxies of BBS
shown in Figure 8 is our model prediction for the baryonic
is displayed in Fig. 7 as a shaded histogram. These values
spin distribution in dwarf galaxies (Vvir =60kms−1), with
are significantly lower than the generally adopted univer-
sal value of fb 0.13, which suggests significant baryonic Vfb = 95kms−1. The effect of blowout brings the baryonic
≃ spin distribution into good agreement with the observed
mass loss from theseobjects. Shown forcomparison are the
sample. Note that this distribution is quite different then
correspondingmodelpredictionsfordwarfandbrightgalax-
′
ies. We have chosen a value of Vfb = 95kms−1 in order to that of the dark matter, with λ0 = 0.068 and σλ = 0.61.
Wefinditremarkablethattheagreementwiththeobserved
match themean oftheBBS measurementsfor dwarf galax-
spindistributionfollows automaticallyfrom theadjustment
ies, fd 0.04, but one can see in Fig. 7 that the resulting
spread≃infd valuesisalsoingoodagreementwiththedwarf ofVfb tomatchtheobservedbaryonfractionindwarfgalax-
ies.Inthenextsectionwedemonstratethattheresultisalso
galaxies data.Forbrightgalaxies, fd istypicallylowerthan
independentof the details of thefeedback model used.
theuniversalvaluebyonly 30 40%,reflectingthelimited
fraction of small merging sa∼telli−tes wholost theirgas.1 It is In the context of the scatter issue mentioned above we
′
note that the predicted spread in λ values for baryons in
brightgalaxiesissmallerthanthespreadforthedark-matter
halos(σ =0.43versus0.5),andismorelikethespreadob-
1 Thereisonedwarfgalaxywithanfd valuemuchhigherthen λ
served by de Jong & Lacey (2000). We note also that the
therestofthe distribution.Thisgalaxy hasthe lowestvirialve-
locity in the sample, Vvir = 43kms−1, and the second highest high baryonic spin values obtained in our model for both
concentrationparameter,cvir=31,suggestingthattheremaybe dwarf and bright disks suggests that one does not need to
something unusual about this galaxy, either in its history or in eliminatelow-spinhalosinordertomatchtheobserveddis-
theBBSfittoitrotationcurve. tribution of disk sizes [e.g., as donein Mo et al. (1998)].
Towards a Resolution of the Galactic Spin Crisis 9
0.8
BBS data dwarf galaxies bright galaxies
0.6
0.4
0.2
0.0
0.6
0.4
MMMM
0.2
dddd
)))) 0.0
MMMM
0.6
((((
FFFF
0.4
0.2
0.0
0.6
0.4
0.2
0.0
0 1 2 3 0 1 2 3 0 1 2 3 4
jjjj////jjjj
ttttooootttt
Figure9.Examplesofbaryonicspecificangular-momentumprofilesfortheBBSdata(leftcolumn)andforourmodeldwarfandbright
galaxies (center and right columns respectively). Also drawn is the average (µ=1.25) dark-matter j-profileseen in the simulations of
BD (dashed curve). The BBS profiles clearly lack the high and low j tails seen inthe simulations. Likewise, many of our model dwarf
galaxiesalsolackthismaterialbecauseofgasloss.Thebrightgalaxies,whichretainmostoftheirbaryons,have j profilesmoresimilar
to the dark matter. However, insome cases they are consistent withthe profileof an exponential diskwith aflat rotation curve (solid
curveinrightcolumn).
5.4 Angular-Momentum Profile of Baryons theblowout preferentially removes gas from small satellites
— those satellites that merge to form the low-j halo ma-
BBS also measured thespecific angular momentum profiles terial. The high-j tail tends to be reduced in the baryons,
oftheirdwarfgalaxies, whichtheyfindtobequitedifferent because this tail is often the result of a small satellite that
from the profiles of dark halos found by BD. The dwarf comes in with its orbital angular momentum aligned with
galaxies in their sample show a systematic behavior, with the halo spin, and now has lost its gas. While there is gen-
a low baryonic fraction, and a significant deficit of angular eral agreement between the model dwarf galaxies and the
momentumatthetwoendsofthedistributioncomparedto BBS data, sometimes the model galaxies show a spike of
thehaloj profilesofBD.Examplesofsomeofthej profiles low j material that is not seen in the data. We can think
from their data are shown in theleft column of Fig. 9. of several alternative reasons for this apparent discrepancy.
Either the predicted spike corresponds to a baryonic com-
We construct a baryonic j profile in each of our model
ponentthatBBSfailtoobserve(suchasfainthalostars),or
realizations, following the same method used to produce
thesegalaxieswhichshowaspikedonotbecomediskdwarf
dark-matter j profiles in 2.2, but now including feedback
§ galaxies and aretherefore missing from theBBS sample, or
and gas loss. Examples of such model profiles for dwarf
galaxies of Vvir = 60kms−1 are shown in the central col- ourmodelismissingacertainelementthatshouldeliminate
thevery low-j spikemore efficiently.
umn of Fig. 9. The low-j tail of the halo distribution is
missing in the baryons in many cases, as expected, because TherightcolumnofFig.9showsthemodelpredictions
10 Ariyeh H. Maller and Avishai Dekel
Table1.Thedependenceofourresultsonthemodelparameters.
12 For a given γ2 the scale Vfb is set by requiring the average fd
in dwarf galaxies (Vvir = 60kms−1) to be 0.4. Then we find
10 the lowest value of γ1 that creates a spin distribution in bright
galaxies(Vvir=220kms−1)withamedian 0.035.
≥
8 model dwarfgalaxies brightgalaxies
γ2 Vfb γ1 fd λ′0 σλ fd λ′0 σλ
ζ
)d 6 0.5 185 3.0 0.4 0.061 0.56 0.8 0.053 0.50
ζ( 0.8 130 3.0 0.4 0.063 0.56 0.8 0.035 0.53
P
1.0 95 1.5 0.4 0.067 0.61 0.9 0.035 0.44
4 3.0 60 0.5 0.4 0.082 0.64 0.9 0.036 0.42
2
6 ROBUSTNESS OF THE FEEDBACK MODEL
0 Inthissectionweexaminethesensitivityofourresultsinthe
previoussectiontoourextremelysimplemodeloffeedback.
0.00 0.10 0.20 0.30 0.40 0.50
ζ = j /j We explore a range of values for the exponents γ1 and γ2.
tot max Foreachchoiceofexponents,wedetermineVfbsuchthatthe
Figure10.Probabilitydistributionofthespin-profileshapepa- mean value of fd for the dwarf galaxies is the same as the
BBS value of 0.04 We find that the results of 5 for dwarf
rameter ζ =jtot/jmax (the BBS alternative to µ of BD) forthe §
dwarf-galaxydataofBBS(shadedhistogram)incomparisonwith galaxiesremainpracticallyunchanged.Theresultsforbright
ourmodelpredictionsforthebaryonsindwarf(thickhistogram) galaxies do depend on the value of γ1, though a value can
andbright(thinhistogram)galaxies.Shownalsoistheresultfor always be chosen such that the median spin for the bright
simulatedhalosbyBD(smoothcurve).Themodelpredictionfor galaxies is at least as high as that of thedark halos.
baryonsisinreasonableagreementwiththedata,thoughthepre- InTable1weshowarangeofvaluesofγ2andthecorre-
dictedspreadissomewhatlarger.Brightgalaxiesshowaslightly spondingvaluesofVfb necessarytogiveourdwarfhalosthe
shifteddistributiontohighervaluesofζ.
rightbaryonfractionfd.Thenwelistthelargestvalueofγ1
′
forwhich brightgalaxies haveamedianλb 0.035, assug-
≥
for a sample of bright galaxies, Vvir =220kms−1. They re- gestedobservationally.Amaximumvalueof3.0isconsidered
for both exponents as higher values produce indistinguish-
tain most of their baryons,so theirprofiles are less affected
′
by blowout, and they are more similar to the dark-matter ableresults.Theresultingvaluesoffd,λ0 andσλ arelisted
forthetwotypesofgalaxies.Weindeedseeinthetablethat
j-profiles seen in the simulations of BD. However, in some
casestheyshowareductionofbothlowandhighj material varying the exponents γ1 and γ2 has a small effect on the
resulting baryonic spin distribution for dwarf galaxies. We
for the same reasons that it is seen in the dwarf galaxies.
conclude that any feedback recipe in which an appropriate
For comparison, we also show in the right column of Fig. 9
amount of gas is being ejected from dwarf galaxies should
thejprofileofanexponentialdiskwithaflatrotationcurve
resolve the angular-momentum crisis for dwarf galaxies in
truncated at 4.5 disk scale lengths. The j-profile mismatch
a way similar to our results in 5. The angular momentum
problem in bright galaxies is partly remedied by the com- §
mon presence of low-j bulge components in such galaxies. catastrophe for bright galaxies is prevented if γ1 is suffi-
ciently small.
Thefactthatsometimesthelow-j materialislostevenfrom
bright galaxies allows some of them to become pure expo- If γ2 is as small as 0.5, as in the first row of Table 1,
nentialdisks,asobserved,withouttheneedtohideanylow-j thentherequiredvalueofVfbneededtofitthemeanvalueof
material in a bulge. fd indwarfgalaxiesisveryhigh(185kms−1).Thisvelocity
issignificantlyhigherthanthecrudeupperlimitallowedby
Fig.10showsthedistributionsofthequantityζ,which
characterizes theshapeofthej-profile.It comparestheob- the energetics (DS). Also, for such a large Vfb value bright
′
galaxieshaveamedianλ muchhigherthenthedarkmatter
servedvaluesintheBBSdatasetwithourmodelpredictions
for the baryons in dwarf and in bright galaxies. Given the even for the maximum value of γ1. Thus, the model yields
crudeness of our model, the agreement between our model sensible results only if γ2 is limited to therange 0.8−3.0.
The shaded region in Fig. 5 shows the range of curves
for dwarf galaxies and the data is quite reasonable. The
correspondingtotherangespannedinTable1.Wenotethat
somewhatsmallerspreadinthedatacomparedtothemodel
a tight constraint is enforced by the data on this range of
predictions may suggest that either selection effects or gas
feedbackmodels,thathalosof 35kms−1mustlooseabout
processes tend to favor a particular value of ζ. Also shown ∼
half of theirbaryons.
forcomparisonisthedistributionofζ fordarkhalosasmea-
sured by BD in simulations. One sees that the values of ζ
for bright galaxies are slightly shifted to higher values then
the dark halos. For dwarf galaxies the shift is a significant 7 CONCLUSION
effect,reflectingthemismatchbetweenthej-profilesofdark
Weused a simple model to address thetwo issues that cre-
halos and observed galaxies.
atetheangular-momentumcrisisofgalaxyformationwithin
the CDM scenario. One, the angular momentum catastro-
phe is that the spin of the baryonic component in galax-
