How supernova feedback turns dark matter cusps into cores

1.3 MB·English

by  Andrew Pontzen

Mon.Not.R.Astron.Soc.000,000–000(0000) PrintedJanuary27,2013 (MNLATEXstylefilev2.2)
How supernova feedback turns dark matter cusps into cores
Andrew Pontzen1,2,3(cid:63), Fabio Governato4
1KavliInstituteforCosmologyandInstituteofAstronomy,MadingleyRoad,CambridgeCB30HA,UK
2EmmanuelCollege,StAndrew’sStreet,Cambridge,CB23APUK
3OxfordAstrophysics,DenysWilkinsonBuilding,KebleRoad,OxfordOX13RH,UK
4AstronomyDepartment,UniversityofWashington,Seattle,WA98195,USA
2
Accepted—.Received—;inoriginalform—
1
0
2
ABSTRACT
n
We propose and successfully test against new cosmological simulations a novel analytical
a
descriptionofthephysicalprocessesassociatedwiththeoriginofcoreddarkmatterdensity
J
profiles.Inthesimulations,thepotentialinthecentralkiloparsecchangesonsub-dynamical
0
timescalesovertheredshiftinterval4>z>2asrepeated,energeticfeedbackgenerateslarge
2
underdense bubbles of expanding gas from centrally-concentrated bursts of star formation.

] Themodeldemonstrateshowfluctuationsinthecentralpotentialirreversiblytransferenergy
O into collisionless particles, thus generating a dark matter core. A supply of gas undergoing
C collapseandrapidexpansionisthereforetheessentialingredient.Theframework,basedona
novelimpulsiveapproximation,breakswiththerelianceonadiabaticapproximationswhich
.
h areinappropriateintherapidly-changinglimit.Itshowsthatbothoutflowsandgalacticfoun-
p tainscangiverisetocusp-flattening,evenwhenonlyafewpercentofthebaryonsformstars.
-
Dwarfgalaxiesmaintaintheircoretothepresenttime.Themodelsuggeststhatconstantden-
o
r sitydarkmattercoreswillbegeneratedinsystemsofawidemassrangeifcentralstarbursts
t orAGNphasesaresufficientlyfrequentandenergetic.
s
a
[ 1 INTRODUCTION 2008)stronglysupportedthenotionofstellarfeedbackandenergy
transferfrombaryonstodarkmatterasthegeneratorofcores,but
2 Over the last two decades, galaxies formed in numerical simula-
didnotfullyexplainthephysicalmechanismbehindthistransfer
v tionsbasedontheinflationaryΛCDMparadigmhavesufferedfrom orfollowtheevolutionofdwarfgalaxiestoz=0toensurethatthe
9 anumberofwell-documentedmismatcheswithobservedsystems.
coreswerelong-lived.
9 One of the most prominent of these has been the rotation curves
4
ofdisk-dominateddwarfgalaxies(e.g.Moore1994;Flores&Pri- Recently,simulationswereabletoproducerealistic,present-
0
. mack1994;formorerecentupdatesseeSimonetal.2005,Ohetal. daycoreddwarfgalaxieswithinafullycosmologicalcontext(Gov-
6 2011b and references therein). The observed kinematics imply a ernatoetal.2010,henceforthG10).Thesesimulationsresolvein-
0 constantdensitycoreofdarkmatterinteriorto1kpc,whereassim- dividualstarformation‘clumps’atthedensityofmolecularclouds
1 plephysicalargumentsandsimulationssuggestthatthecolddark leading to galaxies that are additionally realistic because, like
1 matter density should be increasing roughly as ρ ∝r−1 to vastly manyobserveddwarfs(Dutton2009),theyhavenobulge–acon-
:
v smallerradii(e.g.Dubinski&Carlberg1991;Navarroetal.1996b). sequence of preferentially expelling low-angular-momentum gas
i Sincesomeoftheearliestworkondarkmatterprofilesithas fromtheprogenitorsvianaturallyoccurringgalacticwinds(Brook
X
been suggested that sufficiently violent baryonic processes might etal.2011;seealsoBullocketal.2001andvandenBoschetal.
r
a be responsible for heating dark matter cusps into cores (Flores 2001).Ohetal.(2011a)confirmedthattheseeffectsbringthesim-
& Primack 1994). The proposed mechanisms identified in these ulationsintoexcellentagreementwithobservationalconstraintson
papers fall in two broad categories: supernova-driven flattening stellar and HI content as well as those on overall mass distribu-
(Navarroetal.1996a;Gelato&Sommer-Larsen1999;Binneyetal. tion.Bytestingagainstdark-matter-onlyruns,G10providedstrong
2001; Gnedin & Zhao 2002; Mo & Mao 2004; Read & Gilmore supporttoamodelwherethecoreflatteningisgeneratedbybary-
2005;Mashchenkoetal.2006,2008),anddynamicalfrictionfrom oniceffects,inparticularbyrapidgasmotions;moreover,asuite
infallingbaryonicclumpsordiskinstablities(El-Zantetal.2001; ofcomparisonsimulationsrevealedthattheseeffectsonlybecome
Weinberg & Katz 2002; Tonini et al. 2006; Romano-D´ıaz et al. significantifstarsformindenseclumps(∼100amucm−3),sug-
2008, 2009; Pasetto et al. 2010; Goerdt et al. 2010; Cole et al. gestingthatenergyinjectionhastobeconcentratedinlocalpatches
2011). Within the former category, most early works focused on (seealsoCeverino&Klypin2009).
asingle,explosivemass-lossevent.Itthenbecameclearthateven
withextremeparameters,suchaneventtransferredinsufficienten- Thecomparisonofsimulationswithrecentobservations(Oh
ergytodarkmatterparticles(Gnedin&Zhao2002).Ontheother etal.2011a)highlightedthatfeedbackmustoccurinnumerousrel-
hand, Read & Gilmore (2005) showed that several more moder- ativelymildeventstoallowthindiskstoform.Howevergiventhe
atelyviolent burstscouldbeeffective increatingacore. Increas- lackofanalyticframeworkforunderstandingthemicrophysicsof
ingly sophisticated numerical work by Mashchenko et al. (2006, thisprocess,theprecisemechanismofsupernova-drivencuspflat-
(cid:13)c 0000RAS
2 A.PontzenandF.Governato
teningwasnotfurtherelucidatedinG10.Providingsuchaframe- Saitohetal.2008).Inparticular,onlyathighformationthresholds
workistheaimofthepresentwork. exceeding∼10cm−3 willsupernovafeedbacknaturallygiverise
The remainder of this paper describes how to model the ef- tobulkgasmotionsandoutflows(Ceverino&Klypin2009).This
fects of small, central starbursts which create pockets of rapidly followsbecauseindividualhigh-densityclumpsareefficientincon-
expandinggasandstrongfluctuationsinthelocalpotential.Over vertinggastostars,ultimatelyleadingtovastoverpressurizationof
time these repeated processes gradually transfer energy from the theclumpfromthehighlocaldensityofsupernovae.Theparticular
gastothedarkmattercomponent.Thishasmuchincommonwith thresholdvalueof100cm−3forHTischosenforconsistencywith
theviewofMashchenkoetal.(2006,2008)butplacesmoreem- G10,inwhichitwasarguedthatthisisthehighestdensitywhich
phasis on disrupting clumps as opposed to pushing them around, canbeconsideredphysicalatourpresentresolution.Wehavealso
and clarifies that resonance1 is not required. Because our picture verified that when H physics is consistently included in simula-
2
canbemodelledmathematically,weareabletovalidateitagainst tions(Christensenetal,submitted;Governatoetal,submitted)star
thesimulations,showingthattheenvisionedprocessindeedcreates formation indeed proceeds only at high densities even in the ab-
coreswithintheG10simulations. senceofanexplicitthreshold.
Thispaperisorganizedasfollows.Section2introducesim- By contrast, LT’s threshold of 0.1cm−3 represents an ap-
provedsimulationsbasedonthoseofG10,anddiscussesthechar- proximate historical norm for galaxy formation simulations (e.g.
acteristicsofthesesimulationswhichpredictcusp-flattening,thus Navarro&White1993;Katzetal.1996)broadlycompatiblewith
motivating a study of orbits in rapidly changing potentials (Sec- theobservedcut-offinstarformationatlowcolumndensitiesaver-
tion3).Theinitialdiscussionis,forsimplicity,limitedtopower- agedover∼kpcscales(e.g.Kennicuttetal.2007).Untilrecently
law potentials but Section 4 removes this restriction, presenting LTwouldhavebeenthemostmotivatedchoice;however,withthe
moregeneralequationstoexplainthedetailedsimulationresults. additionofmetal-linecoolingatincreasingresolutionwenowpre-
We relate our work to the wider literature and conclude in Sec- fertheHTsimulations,usingLTasareferencetounderstandwhy
tion5,alsodiscussingtherealismoftheunderlyinghydrodynami- oldersimulationsdidnotproducetheeffectsofinteresthere.
calevolutionwithinthesimulatedgalaxies.Inacompanionpaper As expected following G10, LT remains cusped, unlike HT
(Governato et al, in prep) we will discuss the scaling of the dark whichdevelopsa1kpcdarkmattercoreatz=0inbothofitstwo
mattercoreswithgalaxymasses. mostmassivehalos.Inallcasesthecodeconsistentlyfollowsthe
feedbackeffectsofthestellarpopulations(Stinsonetal.2006)so
that,afteradelayof∼10Myr,significantamountsofthermalen-
2 THESIMULATIONS ergy are deposited into the surrounding gas. By z=2, when HT
has developed a stable core, LT and HT runs have formed an al-
Thesmoothedparticlehydrodynamics(SPH)simulations,runus- mostidenticalmassofstars(7×107M(cid:12))andthereforethesame
ing the Gasoline code (Wadsley et al. 2004), are closely related quantity of supernova energy has been released (7×1056ergs).2
toandimproveuponthosedescribedinmoredetailbyG10.The The failure of LT to lose its cusp thus reflects a difference in the
emphasisofthepresentworkisoninterpretingdynamicaleffects, couplingmechanism,notintheabsoluteenergydeposition.
ratherthandiscussingthenumericalmethodsindepth;howeversee Figure 1 gives immediate insight into the difference be-
Section5forbriefcommentsoncomputationalaccuracy.Ournew tween HT (cusp-flattening) and LT (cusp-preserving) simulations
runsoutputmoreregulartimestepsandincludetheeffectsofmetal- byshowingtheirspherically-averagedhalodensityprofilesshortly
linecoolingaccordingtotheprescriptionofShenetal.(2010).The beforethecuspbeginstoflatteninHT,atz=4.Solid,dashedand
simulations in this paper focus on the region hosting the galaxy dottedlinesindicaterespectivelydarkmatter,gasandstellarden-
denoted‘DG1’inG10.The‘zoom’technique(e.g.Katz&White sity;thickredandthinbluelinesrepresenttheHTandLTrunsin
1993)allowsforahighmassresolutionofMp=3×103M(cid:12) (for turn.IntheLTrun,thegasdensitycannotexceedthethresholdof
gasparticles)andMp=1.6×104M(cid:12)(darkmatter)withasoftening 0.1cm−3:starsareabletoformanddepositsupernovaenergy,pre-
of86pcinafullΛCDMcosmologicalcontext.Weconductedanal- ventingfurthercooling.IntheHTrun,bycontrast,thegasdensity
ysisonthetwomostmassivesystemswithinthisregion:DG1itself risesmonotonicallytowardsthecentrebecausemostofthegasis
(Mvir =3.7×1010M(cid:12) at z=0) and a somewhat smaller galaxy noteligibletoformstars.Theresultisthatthecentralgasdensity
(Mvir =1.3×1010M(cid:12) at z=0). Most results will be presented slightlyexceedsthatofthedarkmatter.Itisnaturaltosupposefrom
forthelattercase,becausetheformerundergoesamajormergerat thisthatthehalowillhaveaqualitativelydifferentreactiontothe
z=3. Although our model does predict the correct flattening for presenceofbaryonsinthetworuns.(Thisworkfocusesonexpan-
DG1,itsvolatilemergerhistorywouldintroduceunduecomplexi- sionofdarkmatterorbits,butweverifiedthatthesameprocesses
tiesintoourdiscussion. operate on the similarly collisionless star particles in the simula-
In the first run, denoted HT (“high threshold”), stars are al- tion,theinterestingimplicationsofwhichareleftforfuturestudy.)
lowedtoformonlyathydrogendensitiesexceeding100cm−3.The FocusingontheHTsimulation,Figure2(upperpanel)shows
secondrun,LT(“lowthreshold”),isidenticaltothefirstexceptthat thebaryonicmassenclosedwithin0.2,0.5and1.0kpcasafunction
itallowsstarstoformatdensitiesexceeding0.1cm−3. oftime.Fromaround1.7Gyrafterthebigbang,thedensitynearthe
Adopting the higher threshold for star formation is strongly centreundergoesorder-of-magnitudefluctuations.First,gasflows
motivatedbyobservationalevidencethatmolecularcloudsformat in,coolsandcondensesnearthecentreofthepotentialwell.Then,
such densities (Bigiel et al. 2008; Tacconi et al. 2010). HT thus asthedensityofclumpsrisesabovethe100cm−3 threshold,star
exhibits more realistic behaviour of the interstellar medium (e.g. formationisallowedtoproceed.
1 AlthoughMashchenkoetal.(2008)describedtheirmodelas‘resonant’,
theyhavesincestatedthattheydidnotmeantoinvokeaformalresonance, 2 Note,however,thatbyz=0theLTsimulationforms4×109M(cid:12)instars
butratherthenotionofchangesinthepotentialoccurringonroughlythe comparedagainsttheHTsimulation’s5×108M(cid:12).OnlytheHTsimulation
dynamicaltimescale(Wadsley,pri.comm.). formsarealisticdwarfgalaxy(Ohetal.2011a).
(cid:13)c 0000RAS,MNRAS000,000–000
Howsupernovafeedbackturnsdarkmattercuspsintocores 3
Figure1.Sphericallyaveragedhalodensityprofilesforhighstar-formation
threshold(HT,thickredlines)andlowthreshold(LT,thinbluelines)simu-
lationsatz=4,shortlybeforethedarkmatterprofilestartstoflatteninthe
high-thresholdmodel.Solid,dashedanddottedlinesshowrespectivelydark
matter,gasandstellarcontent.Inthelowthresholdcase,darkmatterdom-
inatesbyordersofmagnitudeateveryradius.Inthehighthresholdcase,
thegasreachesacomparabledensitytothecollisionlessmatterinthecen-
tralregions.Gaseousprocessescanthereforecauseheatingofcollisionless
components(darkmatterandstars)inHTbutnotLTruns.
Figure2.(Upperpanel)Thebaryonicmassinteriorto,fromtoplinetobot-
tom,1kpc,500pcand200pc(HTsimulation).Burstycentralstarformation
coupledtostrongsupernovafeedbackcausescoherent,rapidoscillationsin
Oncesupernovaenergyisdumpedintothegas,thermalexpan-
thepotentialinteriorto1kpc.Theorbitaltimeoftypicaldarkmatterparti-
sionformsanunderdensebubbleofuptoseveralkiloparsecsdiam- clesinteriorto1kpcis∼>25Myr.Bycontrastthesimulatedsupernovabub-
eter(lowerpanel,Figure2).Theenergyisinitiallydepositedwithin blescanencompasstheinnerkiloparsecinaround3Myr,fartoorapidlyfor
asmallvolume(aconsequenceofthehighstarformationthreshold) theadiabaticapproximationtobevalid.Thelowerpanelshowsthedisk-
whichthenreachestemperaturesof∼108K.Thegasisvastlyover- planedensityduringthestarbursteventatt=2.56Gyr,z=2.67.Alarge
pressurizedrelativetoitssurroundings,soexpandsatclosetothe underdensebubblehasformedatthecentreofthediskthroughthermalex-
thermal velocity (typically reaching ∼300km/s(cid:39)0.3kpc/Myr). pansionofgasheatedbymultiplesupernovaexplosions.Thecrossmarks
Comparedtotheorbitaltimescales,whichare∼>25Myr,thebub- thehalocentre.
bleformationiseffectivelyinstantaneous.
Theadiabaticcoolingfromtheexpansionisincluded,butthe
3 ANALYTICALMODEL
bubblesnonethelessremainhot(∼106K)bythetimetheyreach
rough pressure equilibrium with the remainder of the disk. They ThisSectiondiscusseshowtheenergyofasingledarkmatterpar-
are then sufficiently underdense (∼10−2cm−3) that the radiative ticle(orstar)changesinresponsetoafluctuatingpotentialsourced
coolingtimescaleisupto100Myr;thebubblecanthereforepersist bygassubjecttoprocessesdescribedabove.Tworestrictionsonthe
forthislengthoftime,afterwhichitcoolsbackintothediskifithas calculationareimposedthroughoutthepaper:
notactuallyescapedinagalacticwind.Therapid,repeatedfluctua-
(i) thepotentialisassumedtobesphericallysymmetric;
tionsinthecentralmasscontentofthesimulatedgalaxyaresimilar
(ii) thetracerparticlesareassumedtobemassless,i.e.thepo-
tothoseshowninFigure3ofMashchenkoetal.(2008).However
tentialisalwaysexternal.
wehaveverifiedthat,inourcase,thisisduetothegasbeingheated
and expanding outward rather than remaining in rapidly-moving The latter condition represents a decision to focus on the micro-
coherentclumpsassuggestedbyMashchenkoetal.(2008). physical mechanism via which particles gain energy, rather than
The overall picture for the dark matter is insensitive to the the subsequent evolution of the self-gravitating system. The first
ultimatedestinationofthegas,requiringonlytheintermittentvari- conditioncouldinfutureberelaxed,butmakescalculationsmuch
ations explained above; the present work makes no assumptions simplerbecauseparticlesorbitinthe1Deffectivepotential
aboutmassloss.Note,however,thatsignificantwindsdoexistin
j2
the simulations; the final baryon fraction in the galaxies is only V (r;j,t)=V(r;t)+ , (1)
eff 2r2
25%ofthecosmicvalue(G10)andonly3%ofthebaryonshave
beenturnedintostars.Thewindshavebeenshowntobeimportant whereV(r;t) is the time-dependent physical potential and j is a
inmatchingthestarformationrates,distributionofstarsandfinal conservedangularmomentum.
baryonfractionsofthedwarfgalaxies(seeMcGaughetal.2010; Forsimplicitywewilltemporarilyimposetwofurtherrestric-
Brooketal.2011;Ohetal.2011a). tionswhichwilllaterberemovedinSection4:
(cid:13)c 0000RAS,MNRAS000,000–000
4 A.PontzenandF.Governato
(iii) onlythenormalizationofthepotentialchanges,i.e.itsfunc- initialblowoutandeventualrecollapsepredictable(thisaperiodic-
tionalformisfixed; ityisillustratedbyFigure2).ByTaylorexpanding,wefindthatthe
(iv) thefunctionalformofthepotentialisapowerlaw. expectedfinalenergyoftheorbitisgivenby
Togethertheseimplythattheunderlyingpotentialin(1)isspecified (cid:18)∆V (cid:19)2 2n
byV(r;t)=V0(t)rn.Someusefultestcasesfallintothisexactform: (cid:104)Ef(cid:105)=E0+(cid:104)∆E1+∆E2(cid:105)(cid:39)E0+ V00 (2+n)2E0, (6)
aKeplerianorbithasn=−1whileaharmonicoscillatorimplies
whichisalwaysanenergygainforboundorbitssinceE <0for
n=2. The final case will be of particular interest chiefly for its 0
n<0.Theenergygainissecondorderinthepotentialchange∆V ,
analyticsimplicity,butwealsonotethatitcorrespondstoassuming 0
butlinearintheenergy.Onemayverifythat,ifthepotentiallyfirst
aspatiallyconstantdensityofmatter.
changessuddenlybutthengradually(i.e.adiabatically)relaxesto
The rate of change of the total energy of a particle orbit-
itsoriginalstate,wewillalsoseeanincreaseinexpectedfinalen-
ing within the potential, dE/dt, is given by the partial derivative
ergy of the same magnitude. The essential point is for the initial
∂V/∂t| , where r(t) denotes the solution to the equations of
r(t)
changetoberapid;theenergyshiftwillthenfollow.
motion. In the limit of an instantaneous change in the potential
Thespecialcaseoftheharmonicoscillator(n=2)ishelpful
V →V+∆V occurring at time t, the total energy of the particle
thereforechangesby∆E=∆V(r(t))=∆V r(t)n. indemonstratingtheoriginofthisenergyshift,becauseitsdynam-
0
ics are especially simple. The potential is separable in Cartesian
Assuming we have no prior knowledge of the phase of the
coordinateswhichmeansthatwecanassumetheone-dimensional
particle, the virial theorem (e.g. Goldstein et al. 2002) states that
subcase without loss of generality. Then, for the case of sudden
theexpectedpotentialenergyis
discretejumps,theanalyticformofthesolutioniswritten
2E
(cid:104)V(cid:105)= 0 , (2)
2+n x(t)=Acos(ωt+ψ), (7)
whereE0 isthetotalenergyoftheparticleandtheresultisinde- where ω2(t)=2V0(t) while A(t) and ψ(t) specify the amplitude
pendentof j.Thisandfollowingequationsarealsothereforevalid and phase of the oscillation, which change discontinuously with
intheone-dimensional(j=0)subcase. the potential. The new values of A and ψ after any jump can be
If suddenly V0 →V0+∆V0, the energy after the potential determinedeitherthroughenergyargumentsasaboveorbyrequir-
changeisE0+∆E1,where ingcontinuityofbothxandx˙.Withoutlossofgenerality,weletω
(cid:104)∆E1(cid:105)=∆V0(cid:104)rn(cid:105)= 22+E0n∆VV0. (3) cthheanjugmepfr,oAm1,ωis0gtoivωen1abtytt=he0e.xTphreesasmiopnlitudeofthetrajectoryafter
0
(cid:34) (cid:35)
Thefiducialadiabaticlimitcanbeattainedfromherebyassuming ω2−ω2
∆V0and∆E1tobeinfinitesimalandintegratingoveraseriesofsuch A21=A20 1+ 0ω2 1 sin2ψ0 , (8)
1
changestakingV smoothlytoV .Thisyieldsafinal,finitechange
0 f
inenergy: whichisexplicitlydependentontheorbitalphaseψ0atwhichthe
discontinuityoccurs.Thecorrespondingchangeofenergyis
(cid:18)V (cid:19)2/(2+n)
f
Ef,adiabatic=E0 V0 . (4) ∆E1=−E0ω02ω−2ω12sin2ψ0. (9)
Asexpectedintheadiabaticlimit,equation(4)impliesnoenergy 0
shift,regardlessoftheintermediatestates,ifthefinalpotentialis Wecannowanalyzethechangesinasingletrajectorywhich
thesameastheinitial.Thisisthecentralproblemoftheadiabatic leadustorecovertheexpectedgaininorbitalenergyunderasingle
approximation.Figure2showsthatthefinaldistributionofgaswill blowout-recollapsecycle,equation(6).Figure3(thickline)shows
indeedbeverysimilartotheinitial,andthereforethattheadiabatic an example orbit for which ω2 =10ω2. The initial amplitude is
0 1
prediction will be for no change in the final distribution of dark unityuntil,atacertaintime,thepotentialflattensoutasmassislost
matter. frominsidetheorbit.Intuitively,orfromequation(9),thismustal-
HoweverSection2showedthatpotentialchangesinthesim- waysinvolvealossofenergyfortheparticle(sinceω2>ω2);see
0 1
ulationstakeplaceontimescalesmuchshorterthanthedynamical also panel 1 at the top of Figure 3. However, according to equa-
time,becausetheexpansionspeedsofthesupernova-inducedbub- tion (8), the new potential is sufficiently flattened that the orbital
bles are much larger than the local circular velocity. As the gas amplitudeisnowlargerthanunity(panel2).Thisiscrucialbecause
expandsandleavesthegalaxycentre,thepotentialundergoesase- theenergygainmadewhenthepotentialchangeislaterreversed(ω
riesoflarge,instantaneousjumps,invalidatingtheadiabaticresult returnsfromω1toω0)willbeaccordinglylarger,scalingwith(cid:104)x2(cid:105)
givenbyequation(4). (panel3).Applyingequations(8)and(9)forthereversejump(i.e.
Insteadofintegrating,oneshouldthereforerecursivelyapply withω1 andω0 exchanged)allowthispicturetobedirectlyveri-
equation (3) to each finite change. If, for instance, the potential fied.
switchesimmediatelybacktoitsoriginaldepth,thesecondshiftin This makes more concrete the assertion of equation (6) that
energyisgivenby one always expects to gain energy during a series of potential
changes.Yetseenfromanotherperspective,suchaclaimstillneeds
2(E +(cid:104)∆E (cid:105)) −∆V
(cid:104)∆E (cid:105)= 0 1 0 , (5) tobereconciledwiththeunderlyingdynamicswhicharefullytime-
2 2+n V +∆V
0 0 reversible.Forinstance,thetime-reverseofFigure3(viewingthe
wheretheangularbracketsindicateaveragingovertheorbitalphase figure from right to left) represents an equally valid trajectory of
ofourchosentrajectoryduringboththeinitialandfinalinstanta- theforwardsdynamicsandyetlosesenergy.
neous potential jumps. These conditions are justified because the Under time-reversal, the blowout phase maps onto the rec-
initialblowoutisnotcausallyconnectedtothelocationofasingle ollapse phase and vice-versa. The irreversibility thus arises not
tracerparticle,noristheexactfractionalnumberoforbitsbetween fromdynamicaldifferencesbutstatisticaldifferencesbetweenthe
(cid:13)c 0000RAS,MNRAS000,000–000
Howsupernovafeedbackturnsdarkmattercuspsintocores 5
qualitativelydifferentresultstobeexpectedfromgradualvariation
asopposedtosuddenjumps.
V(x ) V(x ) V(x )
ΔE> |ΔE|
2 1
ΔE 4 VALIDATINGTHEANALYTICMODELAGAINST
1
x x x SIMULATIONS
1. Small energy loss 2. Orbit expands in 3. Large energy gain
shallow potential Totestthepictureexpoundedabovewestartbygeneratingatime-
dependenteffectivetoypotentialfromthesimulations(Section2).
Thisisgivenbyequation(1),withV(r;t)calculatedfromthespher-
icallyaverageddensityprofile.ThestartingenergyE andthevalue
0
nt x of jcanbedeterminedbyspecifyinginitialorbitalparameters.The
e angularmomentumisnecessarilyconservedbecauseofthespher-
m
e icalsymmetryofthemodelling(restriction1ofSection3).Inthe
c
a
pl simulationsthechangesinpotentialarenotexactlysymmetric(e.g.
s
Di lowerpanelofFigure2);howeverwewillseebelowthat,forthe
e
blowout recollaps pawpueprlprl.ooxseimsaotfiocnalwcuhliacthinegnfroearcl-esspcaocnesdtaenntsijtyacpturoafilllyesw,othrkesseyxmtrmemeterliyc
Asbefore,theenergyshiftforonejumpisgivenbyaveraging
Figure3.Themechanismforinjectingenergyintothedarkmatterorbits, over possible orbital phases. However the potential Vsphere is no
illustratedbytheexactsolutionforatime-varyingharmonicoscillatorpo- longeranexactpowerlaw,sothecalculationrequiredis
tential.Thelowerpanelshows(solidline)asolutiontotheequationsof
motionwhereω2=1(blue)atearlyandlatetimes,whileatintermediate (cid:104)∆E(cid:105) = (cid:82)∆Veff(r(t))dt
timesω2=0.1(red)mimickingbaryonicblowoutandrecondensation.The (cid:82)dt
changesinpotentialoccurinstantaneously;inthiscasethefinalamplitude (cid:44)
(cid:90) ∆V (r)dr (cid:90) dr
oftheoscillationisapproximatelytwicethatoftheinitialorbit.Thedashed = eff , (12)
(cid:112) (cid:112)
lineshowsthesolutionwhenthepotentialchangessmoothlyoverseveral E−Veff(r) E−Veff(r)
orbitalperiods;thisgivesadiabaticbehaviour,sothatthefinalorbitregains
where the time integrals are evaluated over an orbital period; af-
theinitialamplitude,demonstratingthenecessityforrelativelysuddenpo-
terchangingvariablestorthiscorrespondstointegratingoverthe
tentialjumps.Theinsetfigures(top)illustratehowthepost-blowoutorbit
regionwheretheintegrandisreal.Equation(12)agreeswithequa-
expansionimpliesthatthelate-timeenergygaindominatesovertheinitial
energyloss. tion(3)forthespecialcaseofpower-lawpotentials.
TheremainderofthisSectionappliesexpression(12)recur-
sivelytoatime-seriesofpotentialsfromtheHT(cusp-flattening)
forwards-blowout and reverse-recollapse pictures. In particular a simulation, at each step updating ∆V, E andVeff appropriately3.
uniformpriorontheorbitalphasebeforeatransitionisalwaysas- Theenergygainisevaluatedateverystoredsimulationtimestep;
sumed, 2πp(ψ0)=1. On the other hand the phase ψ0(cid:48) after the therelevantoutputsarewritteneveryδt(cid:39)27Myr.Thuschanges
transitionisdeterminedby occurringontimescales(cid:54)δtwillimplicitlybeclassifiedas“rapid”
(composed of one jump) whereas those occurring on timescales
tanψ0(cid:48) = ωω10 tanψ0, (10) m(cid:29)anδyt swmilalllasutteopms)a.tiWcahlliylebtehetrbeoautenddaarsy“baedtiwabeeantict”he(sceomlimpoitssedcaonf-
and,accordingly,theprobabilitydistributionfunctionofψ(cid:48) is not be uniquely defined, the change in behaviour must occur at
0
aroundtheorbitalperiodforaparticle,whichisindeed∼25Myr.
2πp(ψ(cid:48))=(cid:18)ω0cos2ψ(cid:48)+ω1sin2ψ(cid:48)(cid:19)−1. (11) We verified by running checks with only every second timestep
0 ω1 0 ω0 0 (δt(cid:39)54Myr)thattheresultspresentedareinsensitivetothepre-
cisetime-slicing.
The precise functional form (11) is not crucial, only that p(ψ(cid:48))
0 ThesolidlinesinFigure4showtheresultingmeanradius(cid:104)r(cid:105)
cannotbetakentobeuniform.Afterthesuddenbaryonicblowout,
oforbitsasafunctionoftime,where
collisionlessparticlesentertheirneworbitinaspecialphase–pref-
(cid:44)
erentiallynearpericentre–sothattheysubsequentlymigrateout- (cid:90) rdr (cid:90) dr
(cid:104)r(cid:105)= . (13)
wardsinunison. (cid:112)E(t)−V (r;j,t) (cid:112)E(t)−V (r;j,t)
eff eff
It is this difference in knowledge of phases before and after
suddenchangesthatallowsirreversibilityintherealuniversetoap- Thevaluesof jandE0 foreachorbitarechosenbyrequiringthe
pearinthemodel.Onlyifallcollisionlessparticleswereneartheir initial motion to be circular at a range of different radii. As time
pericentre just before the baryons returned would the statistical progresses, the orbits starting interior to 1kpc migrate outwards,
propertiesofthereversedpicturematchthoseoftheactualmodel. reflectinganetgaininenergy.Orbitsoutsidethisradiusarelargely
Whilethisisdynamicallypossible,itisstatisticallyunlikely.
Finally note that if the changes in potential are introduced
3 BecauseE takesarandomwalk,amoreaccurateresultisinprinciple
gradually, the process should become adiabatic and hence re-
attainablebykeepingtrackofitsevolvingdistributionfunctionratherthan
versible. The dashed line in Figure 3 shows a numerical solution
justitsexpectedvalue.Ourapproachhereisakintotakingthefirsttermin
forwhichω changessmoothlyoverseveralorbitaltimesfromω0 aFokker-Planckanalysisandwillbeanexcellentapproximationbecause
to ω1, then back to ω0. As expected from equation (4), the final theenergyshiftsareapproximatelylinear,ascanbeshownbygeneralizing
orbital amplitude is the same as its initial value, confirming the equation(3).
(cid:13)c 0000RAS,MNRAS000,000–000
6 A.PontzenandF.Governato
Redshift
4.23.9 3.6 3.3 3.0 2.7 2.4 2.1
100
c
p
k
/
fi
bit
or
r
›
10-1
1.5 2.0 2.5 3.0 3.5
t/Gyr
Figure4.Usingthesphericallyaveragedpotentialfromthesimulations, Figure5.Thesphericallyaverageddarkmatterdensityasafunctionofra-
wemodeltheexpansionoforbitsoftestparticlesatdifferentinitialradii dius,measuredatz=2whenthecorehasformedintheHTsimulations
(solidlines).Orbitsstartingsignificantlywithintheinnerkiloparsecmigrate (thick dotted line). The solid line shows the density profile at this time
outwardsoverseveralgigayears,whereasthosestartingoutsideakiloparsec accordingtoourmodel(seetextfordetails);thisisseentobeinexcel-
do not feel the rapid potential variations and so remain near their initial lentagreementwiththeHTsimulation.Theadiabaticmodel(dashedline)
radius.Ourmodelthusexplainstheflatteningofcentraldensitycuspsinto failstocorrectlymodelthecuspflattening,demonstratingtheneedforthe
kiloparsec-scalecoresinsmallgalaxiesthroughradialoutwardsmigration. improvedmodellingpresentedhere.TheLTcomparisonsimulation(dash-
Asexpectedthereversible,adiabaticmodel(illustratedfortheinnermost dottedline)alsoremainscuspedasexplainedinSection2.
orbitbythedashedline)doesnotcorrectlymodeltheheatingeffectofvery
rapidpotentialvariationsintheinnerpartsofthehalo.
distributionforeachparticle,
1
p(r;E,j)∝ , (15)
unaffected.IntheLTrun,bycontrast,notracerparticlesgainen- (cid:112)
E−V (r;j)
eff
ergy;thosethatstartoncircularorbits,forinstance,arepredicted
toremainatthesameradiusfortheentirerun. wascalculatednumerically.Thesumofthenormalizedprobability
Figure2impliesthatthecentralbaryonicpotentialreturnsto distributionsforallparticlesthenimpliesadensityprofileaccord-
its approximate original shape at the end of each starburst cycle, ingto
because the gas affected by the supernovae has cooled back into 1
thedisk(orhasflowedout,replacedbyfreshgas).Intheadiabatic ρmodel(r)∝ r2∑p(r;Ei,ji), (16)
limit, where all potential changes occur slowly, the final orbital i
parametersshouldreturntotheirinitialvalues.Indeedbymaking where the sum is over all tracer particles. Time evolution of
∆Veffinfinitesimalandintegrating(12)oneobtains ρmodel(r)arisesfromupdatingVsphereandeachEiateverytimestep
according to equation (12); or, for comparison, by solving equa-
(cid:90) (cid:112)
E(t)−V (r;j,t)dr=constant, (14) tion(14)toderivethebehaviourintheadiabaticlimit.
eff
Starting at z=4, the distribution function is evolved in this
where again the integral is taken over the real region of the inte- way to z=2; the resulting density profiles are illustrated in Fig-
grand.Thisisthegeneralizationofequation(4),andisexactlythe ure5.Thethicksolidlineshowsourmainmodel[i.e.itisderived
adiabaticinvariantderivedthroughtheaction-angleapproach(e.g. fromequation(12)],andisseentobeinexcellentagreementwith
Binney&Tremaine1987).ItimpliesthatE =E ifthepo- the output of the simulation (dotted line). The dotted line shows
final initial
tentialreturnstoitsinitialformviaaseriesofslowchanges. theresultsofmodellingthebaryoniceffectsusingtheadiabaticap-
Demandingtheadiabaticinvariant(14)isconstantyieldsthe proximation[i.e.equation(14)];thecuspremains,contrarytothe
orbital migration in the ‘gradual outflows’ scenario. The dashed resultsofthesimulation.Thisreaffirmsthattheadiabaticapproxi-
lineinFigure4showsthattheresultderivedinthislimitisasex- mationdoesnotcaptureimportantaspectsoftheimpactofbaryons
pected:althoughtemporarychangesintheorbitalradiusdooccur, onthedarkmatter.Finally,thedash-dottedlineshowstheprofile
they do not persist over time. This underlines the difference be- fromtheLT(lowstarformationthreshold)simulationwhich,asex-
tweenournewmodel(whereatracerparticlepicksupenergyfrom plainedinSection2,retainsitscuspandisthereforeinapproximate
baryons)andtheolderadiabaticcalculations(wheretheenergyof agreementwiththeadiabaticallyevolvedcase.
atracerparticleisconserved). Thecalculationdescribedbyequation(12)involvescalculat-
AlthoughFigure4showsthatorbitsgainenergy,itcannotbe ingtheparticledistributionfunctionforeveryintermediatestep.It
useddirectlytoinferthefinalinnerprofileofthedarkmatter.To is possible, therefore, to monitor the rate at which the cusp flat-
drawconclusionsabouttheevolutionoftheslope,weevolvedthe tens and compare it against the simulations. Figure 6 shows the
energyof∼90000orbitscorrespondingtoalldarkmatterparticles timeevolutionofthemeasuredlogarithmicslopeat500pcforboth
in the halo at z=4. At each timestep, the full radial probability themodeldensityprofile,equation(16),andthesimulateddensity
(cid:13)c 0000RAS,MNRAS000,000–000
Howsupernovafeedbackturnsdarkmattercuspsintocores 7
Redshift ergy transfer is inherently irreversible. The G10 simulations do
4.23.9 3.6 3.3 3.0 2.7 2.4 2.1 exhibit galactic-scale outflows which remove 75% of baryons by
z=0.Theseoutflowshaveotherimportanteffects(e.g.Brooketal.
0.2 2011), yet the mass involved is only around a third of the mass
involvedinthecentralblowout-recollapsecycle.Therelationbe-
c) 0.4 tweenthegalacticoutflowsandthelocalfeedbackwillbethefocus
p
0 offuturework.
0 The picture is related to the more established view that re-
5 0.6
(r movalofbaryonsthroughgalacticsuperwindscouldcausethecen-
g traldarkmatterprofiletoflatten(e.g.Navarroetal.1996a;Gnedin
o 0.8
l & Zhao 2002; Read & Gilmore 2005). However, our model con-
d
/ firms that extreme, violent mass-loss events are not necessary, as
ρ
1.0
g suggestedby(Mashchenkoetal.2006,2008).Thisisimportantbe-
o
l causethemoremoderateheatingeventsallowretentionofbaryons
d
1.2 andtheformationofathinstellardisk.
Model Dynamical friction from infalling baryonic clumps (El-Zant
1.4 Simulation et al. 2001; Mo & Mao 2004; Romano-D´ıaz et al. 2009; Goerdt
1.5 2.0 2.5 3.0 3.5 etal.2010)doesnotappeartoplayadominantroleinoursimu-
t/Gyr lations.IntheLTsimulations,nocoresform;thereforeeffectsof
dynamicalfrictionareruledoutexceptfromthedensestclumpsin
Figure6.Theevolutionofthelogarithmicslopeofthehalointherawsim- HT.IntheHTsimulations,anydenseinfallingclumpsaretypically
ulationdata(dottedline)andaccordingtoourmodel(solidline),measured
disrupted long before they reach the inner kiloparsec. To test the
at500pcinbothcases.Values∼<−1indicateaconventionalNFW-style effectofdynamicalfriction,onewouldfirstneedtoremovetheex-
cusp;shallowerprofilesareindicatedbyvaluesclosertozero.Theflatten-
plosiveeventsassociatedwithfeedback.Howeverasthefeedback
ingoftheprofileinthesimulationshaspreviouslybeendemonstratedto
energyisdecreased,denseclumpsstarttopileupingalaxiesuntil
agreewithobservationaldata(Ohetal.2011a).Ourphysicalmodelforthe
originoftheflatteningisinexcellentagreementwiththedetailedsimulation timeintegrationbecomescomputationallyunfeasible,atthesame
results. time creating an unrealistic dense central bulge. Additionally we
do not have the resolution to produce star clusters (Goerdt et al.
2010)whichcouldbemorerobusttodisruptionthangasclumps.
profile.Astimeprogresses,boththesimulatedandmodeldensity Asaresultwearenotrulingoutdynamicalfrictionasanagentfor
profiles gradually flatten out, with the value rising from <−1.0 weakeningcuspsexceptfortheparticularsimulationsinusehere.
(cusped)to∼−0.4,consistentwithobservations(Ohetal.2011a). TheDG1case(whichundergoesmajormergersatz=3and
Toconclude,theanalyticmodelpresentedherepredictsflat- z=1;seeSection2)hasaverysimilarcusp-flatteninghistoryto
teningoftheslopeatthesamerateasseenintheHTsimulation the galaxy described in this work. The mergers themselves have
(Figures 5 and 6); whereas the adiabatic approximation does not no measurable effect on the halo slope and, as expected through
predictanysignificantflattening(Figure5).Wefurtherverfiedthat argumentsbasedonLiouville’stheorem,coresremainpresentafter
the new model did not predict a change in slope for the LT run themerger(Kazantzidisetal.2006).InGovernatoetal(inprep.)
inwhichthegasdensityremainstoosmalltogeneratesignificant we show that cores form in the large majority of dwarf galaxies,
fluctuationsinthepotential.Overall,thenewmodelaloneprovides irrespectiveoftheirassemblyhistory.
aconvincingexplanationfortheflatteningprocessesseenbyG10. Themodelisreminiscentoftheviolentrelaxationenvisioned
byLynden-Bell(1967),althoughouranalysisdoesnotattemptto
generatethegravitationalpotentialself-consistently.Itwouldthus
bedesirableinthefuturetoworkthemicrophysicsintoabroader,
5 CONCLUSIONS
analyticaldescriptionoftheevolutionofaself-consistentdarkmat-
Wehaveproposedanewanalyticmodelthataccountsfortheflat- ter distribution function. We would further like to investigate the
tening of dark matter cusps into cores. Energy is transferred into importanceofdeparturesfromexactsphericalsymmetry.Thesim-
dark matter particle orbits through repeated, rapid oscillations of ulated supernova explosions are rarely exactly on the axis of the
thecentralgravitationalpotential.Theseoscillationsarecausedby disk, so even axisymmetry is violated. This will lead to the non-
recurrent,concentratedburstsofstarformationwhichinducerapid conservationofangularmomentum,whichmustbeunderstoodbe-
expansion of gas through supernova feedback heating. We veri- forewecaninvestigatetheimpactoftheprocessontheanisotropy
fiedthatthisprocessquantitativelyaccountsforcusp-flatteningin oftheorbits(Toninietal.2006).
anovelsetofsimulationssimilartothoseinG10.Thesimulations Stars,likedarkmatterparticles,arecollisionlessandtherefore
includetheeffectsofmetal-linecooling,butlikethoseinG10form should be subject to the same migratory processes outlined here.
thin stellar disks and have a galactic star formation efficiency of Stars forming near the centre of the HT galaxies indeed migrate
onlyafewpercent.Acomparisonsimulation(LT)withlowerstar outwards;itcouldbenaturalinthiscontextthatthescale-lengthof
formationdensitythresholddoesnotformacore,despiteforming thedarkmattercoresandthestellardisksareapproximatelyequal,
tentimesasmanystarsbyz=0.Themodelcorrectlypredictsno anobservationalrelationnotedbyGentileetal.(2009).Indeedthe
cusp-flatteninginthiscase(Figure6)confirmingourinterpretation scale-lengthsofbothstellardiskanddarkmattercoreofthedwarf
thatforcorestoformthesupernovaenergymustbeinjectedina galaxiesdiscussedhereareapproximately1kpc(G10;Brooksetal.
concentratedregion(Figure1). 2011).Tomakethislinkconvincingwillrequireamoresystematic
The baryons do not have to escape the system completely, studyofscalingwithmass(Brooksetalinprep).
but only temporarily vacate the central regions, because the en- Whiletheanalyticmodelwehavedescribedisindependentof
(cid:13)c 0000RAS,MNRAS000,000–000
8 A.PontzenandF.Governato
the detailed gas dynamics, to obtain results the simulated hydro- Bullock,J.S.,Dekel,A.,Kolatt,T.S.,Kravtsov,A.V.,Klypin,
dynamicsareusedasaninput.Thecusp-flatteningeffectisthere- A.A.,Porciani,C.,&Primack,J.R.2001,ApJ,555,240
foreonlyachievedinrealityiftherapidgasmotionspredictedby Ceverino,D.,&Klypin,A.2009,ApJ,695,292
the SPH code are reproduced. Mesh-based codes (Teyssier 2002; Cole,D.R.,Dehnen,W.,&Wilkinson,M.I.2011,MNRAS,416,
O’Shea et al. 2004; Springel 2010) highlight potential shortcom- 1118
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related to sharp density contrasts (Agertz et al. 2007; Bauer & Dutton,A.A.2009,MNRAS,396,121
Springel 2011). In future we intend to address the sensitivity of El-Zant,A.,Shlosman,I.,&Hoffman,Y.2001,ApJ,560,636
ourresultstotheseinaccuraciesthroughcomparisonwithalterna- Flores,R.A.,&Primack,J.R.1994,ApJ,427,L1
tive codes and use of forthcoming improvements to the Gasoline Gelato,S.,&Sommer-Larsen,J.1999,MNRAS,303,321
SPHenginethatreduceartificialsurfacetension(seealsoRead& Gentile, G., Famaey, B., Zhao, H., & Salucci, P. 2009, Nature,
Hayfield2011).ThroughdirectcomparisonofvariouscodesScan- 461,627
napieco et al. (2011) conclude that, for most practical purposes, Gnedin,O.Y.,&Zhao,H.2002,MNRAS,333,299
thechoiceofsub-gridmodelapproximation(viawhichsupernova Goerdt, T., Moore, B., Read, J. I., & Stadel, J. 2010, ApJ, 725,
energy is coupled to the gas) is more critical than the numerical 1707
technique. It will be of interest to determine whether other feed- Goldstein,H.,Poole,C.,&Safko,J.2002,ClassicalMechanics,
back mechanisms and numerical methods can reproduce our re- 3rdedition(AddisonWesley)
sults.IndeedrecentlyMartizzietal.(2011)reportedtheformation Governato,F.,etal.2010,Nature,463,203
ofdarkmattercoresinAMRsimulationsthroughgasfluctuations Katz,N.,Weinberg,D.H.,&Hernquist,L.1996,ApJS,105,19
verysimilartoours,butdrivenbyAGNactivityinclusters. Katz,N.,&White,S.D.M.1993,ApJ,412,455
We are currently investigating how the cores scale for Kazantzidis,S.,Zentner,A.R.,&Kravtsov,A.V.2006,ApJ,641,
109M(cid:12) <Mvir <1012M(cid:12) (Governato et al, in prep). At higher 647
massestheresultswilldependonadetailedinterplaybetweenthe Kennicutt,Jr.,R.C.,etal.2007,ApJ,671,333
deepeningdarkmatterpotential,increasedstarformationrates,and Lynden-Bell,D.1967,MNRAS,136,101
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ofrunningsuitablesimulationstotacklethemostmassivesystems submitted,arXiv:1112.2752
atsufficientresolutionisformidable,butonethatwehopetotackle Mashchenko, S., Couchman, H. M. P., & Wadsley, J. 2006, Na-
induecourse. ture,442,539
Mashchenko,S.,Wadsley,J.,&Couchman,H.M.P.2008,Sci-
ence,319,174
McGaugh,S.S.,Schombert,J.M.,deBlok,W.J.G.,&Zagursky,
M.J.2010,ApJ,708,L14
Mo,H.J.,&Mao,S.2004,MNRAS,353,829
ACKNOWLEDGEMENTS Moore,B.1994,Nature,370,629
Navarro,J.F.,Eke,V.R.,&Frenk,C.S.1996a,MNRAS,283,
Wethanktheanonymousrefereeforhelpfulcomments.APthanks L72
Steven Gratton for many helpful discussions and James Wadsley, Navarro,J.F.,Frenk,C.S.,&White,S.D.M.1996b,ApJ,462,
Justin Read, Jorge Pen˜arrubia, Alyson Brooks, Fergus Simpson, 563
HiranyaPeiris,GaryMamonandMaxPettiniforcommentsona Navarro,J.F.,&White,S.D.M.1993,MNRAS,265,271
draftversionofthepaper.FGacknowledgessupportfromaNSF Oh,S.,Brook,C.,Governato,F.,Brinks,E.,Mayer,L.,deBlok,
grantAST-0607819andNASAATPNNX08AG84G. W.J.G.,Brooks,A.,&Walter,F.2011a,AJ,142,24
The analysis was performed using the pynbody pack- Oh,S.-H.,deBlok,W.J.G.,Brinks,E.,Walter,F.,&Kennicutt,
age(http://code.google.com/p/pynbody).Simulations Jr.,R.C.2011b,AJ,141,193
andanalysiswereperformedusingtheDarwinSupercomputerof O’Shea,B.W.,Bryan,G.,Bordner,J.,Norman,M.L.,Abel,T.,
theUniversityofCambridgeHighPerformanceComputingService Harkness,R.,&Kritsuk,A.2004,inAdaptiveMeshRefinement
(http://www.hpc.cam.ac.uk),providedbyDellInc.using –TheoryandApplications,ed.T.Plewa,T.Linde,&V.Weirs
StrategicResearchInfrastructureFundingfromtheHigherEduca- (Springer Lecture Notes in Computational Science and Engi-
tionFundingCouncilforEngland. neering)
Pasetto,S.,Grebel,E.K.,Berczik,P.,Spurzem,R.,&Dehnen,W.
2010,A&A,514,A47+
Read,J.I.,&Gilmore,G.2005,MNRAS,356,107
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