Thermal Conduction in Simulated Galaxy Clusters

0.15 MB·

by  K. Dolag

TOBESUBMITTEDTOASTROPHYSICALJOURNALLETTERS
PreprinttypesetusingLATEXstyleemulateapjv.4/9/03
THERMALCONDUCTIONINSIMULATEDGALAXYCLUSTERS
K.DOLAG1,M.JUBELGAS2,V.SPRINGEL2,S.BORGANI3,E.RASIA1
TobesubmittedtoAstrophysicalJournalLetters
ABSTRACT
Westudytheformationofclustersofgalaxiesusinghigh-resolutionhydrodynamiccosmologicalsimulations
thatincludetheeffectofthermalconductionwithaneffectiveisotropicconductivityof1/3theclassicalSpitzer
value.Wefindthat,bothforahot(T ≃12keV)andseveralcold(T ≃2keV)galaxyclusters,thebaryonic
ew ew
fractionconvertedintostarsdoesnotchangesignificantlywhenthermalconductionisincluded. However,the
temperatureprofilesaremodified,particularlyinoursimulatedhotsystem,whereanextendedisothermalcore
4
isreadilyformed.Asaconsequenceofheatflowingfromtheinnerregionsoftheclusterbothtoitsouterparts
0
0 andintoitsinnermostresolvedregions,theentropyprofileisalteredaswell. Thiseffectisalmostnegligible
2 forthecoldcluster,asexpectedbasedonthestrongtemperaturedependenceoftheconductivity. Ourresults
demonstratethatwhilethermalconductioncanhaveasignificantinfluenceonthepropertiesoftheintra–cluster
n
mediumofrichgalaxyclusters,itappearsunlikelytoprovidebyitselfasolutionfortheovercoolingproblem
a
inclusters,ortoexplainthecurrentdiscrepanciesbetweentheobservedandsimulatedpropertiesoftheintra–
J
clustermedium.
2
Subjectheadings:conduction—cosmology:theory—galaxies:clusters—methods:numerical
2

1
1. INTRODUCTION fromthe outercluster regionsto the (slightly cooler)central
v
gas,therebylargelyoffsettingitscoolinglossesandstabiliz-
0 Over the last few years, spatially resolved spectroscopic
ingtheICM(cf. alsoSoker2003). However,conductioncan
7 observationswith the XMM–Newton and Chandra satellites
onlyhaveasignificanteffectiftheconductivityκoftheICM
4 have provided invaluable information about the structure of
1 cooling gas in central cluster regions. Contrary to expec- is a sizable fraction of the Spitzer (1962) value, κsp, appro-
0 tations based on the standard cooling–flow model (Fabian priateforanunmagnetizedplasma.Inthepresenceofamag-
4 1994),theseobservationshaveruledoutthepresenceofsig- neticfield,conductionisheavilysuppressedorthogonaltothe
0 fieldlines,sothatforatangledmagneticfield,oneusuallyex-
nificant amounts of star formation and cold gas at tempera-
/ pectsarelativelylow,effectivelyisotropicconductivity,with
h turesbelow1/3rd–1/4thoftheclustervirialtemperature(e.g.,
the amount of suppression depending on the field topology.
p Petersonetal. 2001;Molendi&Pizzolato2001;Böhringeret
However,Narayan&Medvedev(2001)haveshownthatfora
- al. 2002). The spectroscopicallymeasured mass–deposition
o rates are ∼ 10 times smaller than those inferred from the chaoticallytangledmagneticfield,conductivitiesintherange
r κ∼(0.2–0.5)κ canberecovered. Suchfieldconfigurations
t spikesofX-rayemissivityseeninrelaxedclusters(e.g.,Mc- sp
s mayquiteplausiblyariseinclustersofgalaxiesasaresultof
Namara et al. 2001; David et al. 2001). Furthermore,
a turbulence, so that high conductivities in some parts of the
: measurementsoftemperatureprofilesforrelaxedhotclusters
v (T∼>3keV)showthattheyfollowanapproximatelyuniversal ICMmaybeviabledespitethepresenceofmagneticfields.
i Using simple analytic models based on the assumption of
X profile: gasis almostisothermalonscales belowone–fourth
a local balance between radiative cooling and thermal con-
ofthevirialradius(DeGrandi&Molendi2002;Pratt&Ar-
r duction, Zakamska & Narayan (2003) and Voigt & Fabian
a naud2002),withasmoothdeclineoftemperaturetowardsthe
(2003)wereabletoreproducetheobservationaldataforsev-
innermostregions(e.g.,Allen,Schmidt&Fabian2001;John-
eral clusters, including their detailed temperature profiles.
stone et al. 2002; Ettori et al. 2002). These results consis-
They treated the effective isotropic conductivity κ as a fit
tentlyindicatethatsomeheatingmechanismoperatesinclus-
parameterand foundgoodfits for severalclusters with sub–
ter cores, supplyingsufficientenergyto the gas to preventit
fromcoolingtolow(∼<1keV)temperatures. Spitzervalues,whilesomeimpliedunphysicallylargesuper–
Spitzerconductivities.Theseresultsinterestinglysuggestthat
Direct hydrodynamical simulations of cluster formation
conduction may play an important role, while also hinting
have so far failed to reproduce these features. In particular,
thatyetanotherheatingmechanismmaybepresent(cf. also
simulationsthatincludecoolingandstarformationfindanin-
Medvedev et al. 2003). For example, the energy feedback
creaseofthegastemperatureinthecentralregions(e.g.,Katz
fromacentralAGNmaysupplementconductioninadouble
& White 1993); here central gas does cool out of the ICM
heating model(Ruszkowski& Begelman 2002, Brighenti&
and loses its pressure support, so that gas flows toward the
Mathews2003).
center, undergoingcompressionalheating. This leads to the
In this Letter, we present the first cosmological hydrody-
counterintuitiveresult thatcoolinggeneratesa steepeningof
namical simulations of cluster formation that account self-
thecentraltemperatureprofiles,unlikeobserved.
consistentlyforthermalconduction,aswellasradiativecool-
Narayan&Medvedev(2001)havesuggestedthermalcon-
ing and supernova feedback. Such simulations are essential
duction as a possible heating mechanism for the cores of
to understand the highly non-linear interplay between con-
galaxyclusters. Thisprocesscouldtransportthermalenergy
ductionandcoolingduringtheformationofclusters. Inthis
1DipartimentodiAstronomia,UniversitàdiPadova,Padova,Italy study, we focus on the effect of conduction on the temper-
2Max-Planck-InstitutfürAstrophysik,Garching,Germany ature and entropy structures of clusters with rather different
3DipartimentodiAstronomiadell’UniversitàdiTrieste,Italy temperaturesofTLx≃2and12keV.
2 Dolagetal.
TABLE1. CHARACTISTICSOFSIMULATEDCLUSTERS
Simulation (cid:10)MvCilr1(cid:11) MvCilr2 (cid:10)fcColl1d(cid:11) fcColl2d (cid:10)TMCl1(cid:11) TMCl2 (cid:10)TLCxl1(cid:11) TLCxl2 (cid:10)LCxl1(cid:11) LCxl2
csf 1.13±0.05 22.6 0.269±1.3 0.226 1.32±0.04 9.3 2.28±0.07 11.9 0.47±0.04 38.0
csf+cond. 1.08±0.06 22.6 0.261±1.0 0.226 1.30±0.05 9.8 2.15±0.04 12.3 0.43±0.05 54.6
NOTE. —Propertiesofclusterswhencooling, starformationandfeedback(‘csf’)areincluded, andwhenconductionisconsideredaswell(‘csf+cond’).
Columns2–3:virialmass(1014h- 1M⊙);columns4–5:thefractionofcoldgaswithintheclustervirialregions(stars+gasbelow3×104K);colums6–7:mass
weightedtemperature(keV);columns8–9:emissionweightedtemperature(keV);columns10–11:bolometricX–rayluminosity(1044ergs- 1).
2 4 6 8 10 12 14 2 4 6 8 10 12 14
(counts) (counts)
T [keV] T [keV]
FIG. 1.—Projectedmapsofmass–weightedgastemperature forourhotcluster(Cl2),simulatedbothwithoutandwiththermalconduction(leftandright
panels,respectively).Eachpanelshowsthegaswithinaboxofphysicalside-lengthof8Mpconaside(Rvir≈3.9Mpc),centeredontheclustercenter.
2. NUMERICALSIMULATIONS particle. The finalmass–resolutionof these simulations was
Our simulations were carried out with GADGET-2, a new mDM=1.13×109h- 1M⊙andmgas=1.7×108h- 1M⊙fordark
matterandgaswithinthehigh–resolutionregion,respectively.
version of the parallel TreeSPH simulation code GADGET
The clusters were hence resolved with about 4×106 and
(Springeletal. 2001). Itusesanentropy-conservingformu-
2×105 particles, respectively. Thesubstantiallylowercom-
lationofSPH (Springel&Hernquist2002),andincludesra-
putationalcostof‘Cl1’systemsallowedustosimulate5clus-
diativecooling,heatingbyaUVbackground,andatreatment
terswithina verynarrowmassrange, allyieldingconsistent
ofstarformationandfeedbackprocesses. Thelatterisbased
results. Thegravitationalsofteninglengthwasǫ=5.0h- 1kpc
onasub-resolutionmodelforthemultiphasestructureofthe
(Plummer–equivalent),keptfixedincomovingunits.
interstellar medium (Springel & Hernquist 2003). We have
Foreachcluster,werunsimulationsbothwithandwithout
augmentedthe codewitha new methodfortreatingconduc-
thermal conduction, but we always included radiative cool-
tion in SPH, which is both stable and manifestly conserves
ingwithaprimordialmetallicity,andstarformation. Forthe
thermalenergyevenwhenindividualandadaptivetimesteps
conduction runs, we assume a conductivity of κ = 1/3κ ,
areused. Inourcosmologicalsimulations,weassumeanef- sp
fectiveisotropicconductivityparameterizedasafixedfraction where κsp ∝T5/2 is the temperature-dependentSpitzer rate
oftheSpitzerrate. Wealsoaccountforsaturation,whichcan forafullyionized,unmagnetizedplasma. Ourchoiceforκis
becomerelevantinlow-densitygas. Afulldiscussionofour appropriateinthepresenceofmagnetizeddomainswithran-
numericalimplementationofconductionisgiveninJubelgas, domlyorientedB–fields(e.g.,Sarazin1988),orforachaoti-
Springel&Dolag(2004). callytangledmagneticfield(Narayan&Medvedev2001).
We have performed simulations of galaxy clusters of two
3. RESULTS
widely differing virial mass. We refer to them as ‘Cl1’
(1.1×1014h- 1M⊙) and ‘Cl2’ (2.3×1015h- 1M⊙) systems. Anexpectedgeneraleffectofthermalconductionistomake
the gas more isothermal by smoothing out temperature sub-
The clusters have been extracted from a DM–only simula-
tion with box-size 479h- 1Mpc of a flat ΛCDM model with structureintheICM.ThiseffectisclearlyvisibleinFigure1,
Ω =0.3,h=0.7,σ =0.9andΩ =0.04.Usingthe‘Zoomed where we compare projected temperature maps of our mas-
0 8 b sive cluster, with and withoutconduction. The cluster with-
Initial Conditions’ technique (Tormen et al. 1997), we re-
outconduction(leftpanel)showsarichpatternofsmall-scale
simulated the clusters with higher mass and force resolution
temperaturefluctuations,stemmingfromthecontinuousstir-
by populating their Lagrangian regions in the initial condi-
ringoftheICMbyinfallinggalaxies. Thesefluctuationsare
tionswithmoreparticles,addingadditionalsmall-scalepower
largelywipedoutwhenconductionisincluded(rightpanel).
appropriately. Gaswas introducedin thehigh–resolutionre-
Gas in large infalling galaxiescan stay cooler than the local
gion by splitting each parent particle into a gas and a DM
ICM(priortoram-pressurestripping)inthesimulationwith-
ThermalConductioninSimulatedGalaxyClusters 3
on our simulations, it is not clear whether conductive heat-
ingoftheinnermostpartsofclustersoccursinanysignificant
way. Instead, the dominanteffectseemsto beheattransport
frominnertoouterparts,whichcanbeunderstoodasacon-
sequenceofthefallingtemperaturegradientobtainedinsim-
ulations that only include radiative cooling and star forma-
tion.Whileconductionappearstobeeffectiveinestablishing
anisothermaltemperaturein the core, the innermostregions
subsequentlydonotbecomestill cooler,whichwouldbe re-
quired to turn around the direction of conductive heat flow
and tap the thermal reservoir at larger radii in the way pro-
posedby Zakamska& Narayan(2003). We dothusnotfind
that conduction can really prevent the central cooling flow;
it apparently only transports the energy gained by compres-
sionalheatingofinflowinggastoouterregionsofthecluster.
Infact,theinclusionofconductionmayevenmakethecen-
tral cooling flow stronger, depending on the resulting den-
sity and temperature structure of the inner parts in dynami-
FIG. 2.—Comparisonofprojectedtemperatureprofilesforourhot(Cl2, cal equilibrium. For our hot cluster, this actually seems to
upperpanel)andcoldclusters(Cl1,lowerpanel)whenconductionis,oris
be the case, judging from the bolometric X–ray luminosity,
not, included. Foreachrun, thick lines givetheaverage profiles forthree
orthogonalprojectiondirections, whicharealsoshownindividually asthin which is increased by about 40% at z=0 when conduction
lines.Thebundleoflinesinthelowerpanelillustratesthedispersionamong is included. The colder clusters on the other hand show an
our5simulatedclustersofthismass.Forreference,symbolswitherror-bars essentially unchanged X-ray luminosity, consistent with our
giveobservationaldatabyDeGrandi&Molendi(2002).
previouslyfoundtrends.
However, we note that the fraction of collapsed baryons
(cold gas and stars) in the clusters is essentially indepen-
out conduction, but this same gas is conductively heated as dent of conduction, as seen in Table 1, where we summa-
soonasitentersthehotclusteratmosphereintheothersimu- rize some of the main characteristics of the simulated clus-
lation, leading to much more rapid thermalization. We also ters. This resultis notreally surprisingbecause at high red-
note that the cluster with conduction shows a larger, near- shift, whenmostofthestar formationin theclustergalaxies
isothermalregionnearthecenter,andappearssomewhathot- takesplace, the gastemperaturein the progenitorsystemsis
terinitsouterparts. muchlowerthanthevirialtemperaturereachedeventuallyat
InFigure2,weshowtheprojectedtemperatureprofilesfor z=0, and, therefore, the effect of thermalconductionis ex-
the simulated clusters, compared with observational results pectedtobeweak. Theamountofcollapsedgasthusremains
byDeGrandi&Molendi(2002)forasetof22clusterswith at f ≃0.20–0.25,aboutafactortwolargerthanindicated
cold
T >3keV observed with the Beppo–SAX satellite. For the byobservations(e.g.,Baloghetal. 2001;Lin,Mohr&Stan-
runsthatincludeonlycoolingandstarformation,wefindris- ford2003),suggestingthatstrongerfeedbackprocessesthan
ingtemperatureprofilestowardstheclustercenter,consistent includedinoursimulationsareatworkintherealuniverse.In
with recentsimulationwork(e.g., Lewiset al. 2000,Kay et anycase,conductionappearsunabletoresolvethisovercool-
al. 2002,Tornatoreetal. 2003;Borganietal. 2003).Thisbe- ingproblemonitsown.
haviordisagreeswithobservationalevidenceforthepresence Another piece of information about the thermodynamical
of an isothermal regime at R∼<0.2R180 (here R180 is the ra- propertiesoftheICMisprovidedbyitsentropy(S=T n-e2/3)
diusencompassinganaveragedensity180ρcrit)andasmooth profile,whichweshowinFigure3forourclustersimulations,
decline in the innermost regions (Allen, Schmidt & Fabian also comparedto results for pure gravitationalheating. The
2001;DeGrandi&Molendi2002). effectof coolingis that of selectively removinglow entropy
However,thermalconductionsignificantlyflattensthetem- gasfromthehotdiffusephaseincentralclusterregions,such
perature profiles. In fact, for the hot cluster, an isothermal thatanetentropyincreaseofX–rayemittinggascomparedto
core is created, making it more similar to what is observed. puregravitationalheatingsimulationsisseen. Thishasbeen
Forthecoldersystemontheotherhand,thetemperaturepro- predictedbyanalyticmodelsoftheICM(e.g.Voitetal.2003)
file is almost unaffected, consistent with expectations based andhasalsobeenconfirmedindirecthydrodynamicalsimula-
onthestrongdependenceoftheconductivityonelectrontem- tions. Interestingly,theinclusionofheatconductionappears
perature,whichfavorsthermalconductioninhottergas. This toreversepartofthischangeintheentropyprofile,butonly
also implies that thermal conduction cannot easily account in the hot cluster, where conduction is efficient. Here, the
fortheobservedself–similarityofthetemperatureprofilesof entropydecreasesonscales∼(0.01–0.1)R becauseofcon-
vir
fairlyrelaxedclusters. ductivelossesbothto theouterandinnermostparts. Forthe
Interestingly, we find that the mass–weighted as well as colder systems, only a very weak modificationin the region
emission–weighted temperatures of the two simulated clus- of centralmass drop-outis seen. Hotter systems hence tend
ters change by less than 10% when conduction is included. tobecomemoreisentropicincentralregions,whichisjustthe
Conduction thus mainly appears to re–distribute the overall oppositeofwhatonewouldobserveif,instead,pre-heatingis
thermalenergycontentwithinthecluster,whilenotcausinga responsible for breaking the ICM self-similarity (Borganiet
significantheatlosstotheouterintergalacticmediumaspro- al. 2001),butthetrendisconsistentwitharecentanalysisof
posedbyLoeb(2002). We dohoweverfind thatthe temper- entropyprofilesobtained fromASCA data of galaxygroups
ature of the outer parts of clusters is raised by conduction, andclusters(Ponmanetal. 2003).
as seen in the temperature profiles of Fig. 2. In fact, based
4 Dolagetal.
(c) Conduction does not avoid the ‘overcoolingproblem’.
Evenfor ourhotcluster, whereconductionis quiteefficient,
wefindanessentiallyunchangedbaryonfractionof f ∼>0.2
cold
in cold gas and stars, which is largerthan what is observed.
Thisis becausemostofthe coolingandstar formationtakes
placeathighredshiftwhenthetemperatureofthediffusegas
inhalosislowenoughthatconductionisinefficient.Stronger
feedbackprocessesthanconsideredhere,e.g.energeticgalac-
tic winds, are required to solve this problem. We note that
even for the hot cluster at z=0, we do not find a tempera-
ture structure that would allow central cooling losses to be
offset by heat conduction, making it questionablewhether a
detailedlocalbalancebetweenradiativelossesandheatcon-
ductioncanarisenaturallyinhierarchicalclusterformation.
FIG. 3.—EntropyprofilesfortheCl2cluster(uppercurves)andforthe
average oftheCl1 clusters (lower curves). Thedifferent lines distinguish Whilelargersamplesofsimulatedclusterswillberequired
runswithandwithoutthermalconduction,andforpuregravitationalheating. for a more detailed assessment of the role of thermal con-
duction,ourresults alreadydemonstratethatconductioncan
have a sizeable effect on the observationalcharacteristics of
richgalaxyclusters. However,itsinclusionappearsunlikely
4. CONCLUSIONS to to overcome the current discrepancies between simulated
We have presented self-consistent cosmological hydrody- and observed properties of the ICM. For instance, conduc-
namicalsimulationsoftheformationofgalaxyclusterswhich tion tends to produce different temperature profiles for cold
for the first time included the effect of thermal conduction. and hot clusters, invoking a conflict with the observed self–
similarity. Furthermore, since conduction does not prevent
In particular, we carried out simulations of moderately poor
clusterswithT ≃2.0keV,andofarichsystemwithT ≃ overcooling,the presenceofsomeotherheatingsource,per-
ew ew
12keV. Forbothclustersofbothmasses, wecomparedsim- hapsAGN,appearsstillrequired.
ulations that followed radiative cooling, star formation and Admittedly, our present simulations still lack a realistic
feedback with corresponding ones that also accounted for self-consistent description of the magnetic field structure,
whichcanmakeconductionlessimportant.Spatialvariations
thermalconductionwithaneffectiveisotropicconductivityof
κ=κ /3. Ourmainresultscanbesummarizedasfollows: inthe conductivity,itsinterplaywithgasturbulence,aswell
sp
as potentialeffects of anisotropicconductiondue to ordered
(a) Thermal conduction creates an isothermal core in the
centralregionsof our hotcluster, thusproducinga tempera- fieldcomponents,canthusnotbeproperlytakenintoaccount.
tureprofilesimilartothoseobserved. However,thiseffectis Thisrepresentsamajoruncertaintyinassessingtherelevance
much less pronouncedfor our poorer systems, owing to the ofconductionforrealgalaxyclusters. Itisahighlyinterest-
sensitive temperature dependence of the conductivity. As a ingtaskforfutureworktoreducethisuncertaintybyabetter
theoretical and observational understanding of the magnetic
result,thepresenceofconductiontogetherwithcoolingdoes
notleadtoself–similartemperatureprofiles,unlikeobserved propertiesoftheICM.
forrealclusters. The simulations were carried out on the IBM-SP4 at
(b)Comparedtosimulationswithcoolingonly,conduction CINECA,Bologna,withCPUtimeassignedunderanINAF-
leadsto a smalldecrease of the entropyin most ofthe inner CINECAgrant,andontheIBM-SP3atPadova.Weacknowl-
regionsofthehotgalaxycluster,exceptperhapsfortheinner- edge useful discussions with S. Molendi, and thank G. Tor-
mostpartatR∼<0.01R .Thiscanbeunderstoodasaresultof menforprovidingtheZICcode.K.Dolagacknowledgessup-
vir
heatflowingfromtheseregionsbothtoouterpartsoftheclus- portbyaMarieCurieFellowshipoftheEuropeanCommunity
ter, and at some level also to the innermost regions. Again, program “Human Potential” under contract number MCFI-
thiseffectislargelyabsentinthecoldersimulatedclusters. 2001-01227.
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