Thermodynamical properties of the ICM from hydrodynamical simulations

2.1 MB·English

by  S. Borgani

SSRvmanuscriptNo.
(willbeinsertedbytheeditor)
Thermodynamical properties of the ICM from
hydrodynamical simulations
S.Borgani · A.Diaferio · K.Dolag · S.Schindler
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n Received:1October2007;Accepted:8November2007
a
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7 Abstract Modern hydrodynamical simulations offer nowadays apowerful means to trace
the evolution of the X–ray properties of the intra–cluster medium (ICM) during the cos-

] mological history of the hierarchical build up of galaxy clusters. In this paper we review
h
the current status of these simulations and how their predictions fare in reproducing the
p
- mostrecentX–rayobservationsofclusters.Afterbrieflydiscussingtheshortcomingsofthe
o self–similar model, based on assuming that gravity only drives the evolution of the ICM,
r
t we discuss how the processes of gas cooling and non–gravitational heating are expected
s
tobring model predictions into betteragreement withobservational data.Wethenpresent
a
[ resultsfromthehydrodynamical simulations,performedbydifferentgroups,andhowthey

compare withobservational data.Astermsofcomparison, weuseX–ray scalingrelations
1
betweenmass,luminosity,temperatureandpressure,aswellastheprofilesoftemperature
v
andentropy.Theresultsofthiscomparisoncanbesummarisedasfollows:(a)simulations,
2
3 whichincludegascooling,starformationandsupernovafeedback,aregenerallysuccessful
0 in reproducing the X–ray properties of the ICM outside the core regions; (b) simulations
1 generally fail in reproducing the observed “cool core” structure, in that they have serious
. difficultiesinregulatingovercooling,therebyproducingsteepnegativecentraltemperature
1
0 profiles.Thisdiscrepancycallsfortheneedofintroducingotherphysicalprocesses,suchas
8 energyfeedbackfromactivegalacticnuclei,whichshouldcompensatetheradiativelosses
0
:
v S.Borgani
i DepartmentofAstronomy,UniversityofTrieste,viaTiepolo11,I-34143Trieste,Italy
X E-mail:[email protected]
INAF–NationalInstituteforAstrophysics,Trieste,Italy
r
a INFN–NationalInstituteforNuclearPhysics,SezionediTrieste,Italy
A.Diaferio
DipartimentodiFisicaGenerale“AmedeoAvogadro”,Universita`degliStudidiTorino,Torino,Italy
E-mail:[email protected]
INFN–NationalInstituteforNuclearPhysics,SezionediTorino,Italy
K.Dolag
Max-Planck-Institutfu¨rAstrophysik,Karl-SchwarzschildStrasse1,GarchingbeiMu¨nchen,Germany
E-mail:[email protected]
S.Schindler
Institutfu¨rAstro-undTeilchenphysik,Universita¨tInnsbruck,Technikerstr.25,6020Innsbruck,Austria
E-mail:[email protected]
2
ofthegaswithhighdensity,lowentropyandshortcoolingtime,whichisobservedtoreside
intheinnermostregionsofgalaxyclusters.
Keywords Cosmology: numerical simulations · galaxies: clusters · hydrodynamics ·
X–ray:galaxies
1 Introduction
Clustersofgalaxiesformfromthecollapseofexceptionallyhighdensityperturbationshav-
ing typical size of ∼10 Mpc in a comoving frame. As such, they mark the transition be-
tweentwodistinctregimesinthestudyoftheformationofcosmicstructures.Theevolution
of structures involving larger scales is mainly driven by the action of gravitational insta-
bilityofthedarkmatter(DM)densityperturbations and,assuch,itretainsthememoryof
the initial conditions. Onthe other hand, galaxy–sized structures, which form from initial
fluctuationsonscalesof∼1Mpc,evolveunderthecombinedactionofgravityandofcom-
plexgas–dynamicalandastrophysicalprocesses.Onsuchscales,gascooling,starformation
andthesubsequentreleaseofenergyandmetalfeedbackfromsupernovae(SN)andactive
galacticnuclei(AGN)haveadeepimpactontheobservationalpropertiesofthediffusegas
andofthegalaxypopulation.
Inthissense,clustersofgalaxiescanbeusedasinvaluablecosmologicaltoolsandastro-
physicallaboratories(seeRosatietal.2002andVoit2005forreviews).Thesetwoaspects
are clearly interconnected with each other. From the one hand, the evolution of the pop-
ulation ofgalaxy clusters and theiroverall baryonic content provide inprinciple powerful
constraintsoncosmologicalparameters.Ontheotherhand,forsuchconstraintstoberobust,
one has to understand in detail the physical properties of theintra–cluster medium (ICM)
anditsinteractionwiththegalaxypopulation.
The simplest model to predict the properties of the ICM and their evolution has been
proposedbyKaiser(1986).Thismodelisbasedontheassumptionthattheevolutionofthe
thermodynamicalpropertiesoftheICMisdeterminedonlybygravity,withgasheatedtothe
virialtemperatureofthehostingDMhalosbyaccretionshocks(Bykovetal.2008-Chapter
7,thisvolume). Sincegravitational interactiondoesnotintroduce anypreferredscale,this
modelhasbeencalled“self–similar”1.Asweshalldiscuss,thismodelprovidesprecisepre-
dictionsontheshapeandevolutionofscalingrelationsbetweenX–rayluminosity,entropy,
totalandgasmass,whichhavebeentestedagainstnumericalhydrodynamical simulations
(e.g.,Ekeetal.1998;Bryan&Norman1998).Thesepredictionshavebeenrecognisedfor
severalyearstobeatvariancewithanumberofobservations.Inparticular,theobservedre-
lationbetweenX–rayluminosityandtemperature(e.g.Markevitch1998;Arnaud&Evrard
1999; Osmond&Ponman 2004) is steeper and the measured level of gas entropy higher
thanexpected(e.g.,Ponmanetal.2003;Pratt&Arnaud2005),especiallyforpoorclusters
and groups. This led to the concept that more complex physical processes, related to the
heatingfromastrophysicalsources ofenergyfeedback, andradiativecooling ofthegasin
thecentral clusterregions, play akeyroleindetermining theproperties ofthediffuse hot
baryons.
1 Strictlyspeaking,self–similarityalsorequiresthatnocharacteristicscalesarepresentintheunderlying
cosmologicalmodel.ThismeansthattheUniversemustobeytheEinstein–de-Sitterexpansionlawandthat
theshapeofthepowerspectrumofdensityperturbationsisafeaturelesspowerlaw.Inanycase,theviolation
ofself–similarityintroducedbythestandardcosmologicalmodelisnegligiblewithrespecttothatrelatedto
thenon–gravitationaleffectsactingonthegas.
3
Althoughsemi–analyticalapproaches(e.g.,Tozzi&Norman2001;Voit2005,andref-
erences therein) offer invaluable guidelines to this study, it is only with hydrodynamical
simulationsthatonecancapturethefullcomplexityoftheproblem,soastostudyindetail
theexistinginterplaybetweencosmologicalevolutionandtheastrophysicalprocesses.
Inthelastyears,everimproving codeefficiencyandsupercomputing capabilitieshave
opened the possibility to perform simulations over fairly large dynamical ranges, thus al-
lowing to resolve scales of a few kiloparsecs (kpc), which are relevant for the formation
of single galaxies, while capturing theglobal cosmological environment on scales of tens
or hundreds of Megaparsecs (Mpc), which are relevant for the evolution of galaxy clus-
ters. Starting from first attempts, in which only simplified heating schemes were studied
(e.g.,Navarroetal.1995;Bialeketal.2001;Borganietal.2002),anumberofgroupshave
studied the effect of introducing alsocooling (e.g. Katz&White 1993; Lewisetal. 2000;
Muanwongetal.2001;Dave´ etal.2002;Tornatoreetal.2003),ofmorerealisticsourcesof
energyfeedback(e.g.,Borganietal.2004;Kayetal.2007;Nagaietal.2007a;Sijackietal.
2007),ofthermalconduction(e.g.,Dolagetal.2004),andofnon–thermalpressuresupport
frommagneticfields(e.g.,Dolagetal.2001)andcosmicrays(e.g.,Pfrommeretal.2007).
Inthispaper,wewillreviewtherecentadvancementperformedinthisfieldofcomputa-
tionalcosmologyandcriticallydiscussthecomparisonbetweensimulationpredictionsand
observations, byrestrictingthediscussiontothethermal effects.Assuch,thispapercom-
plements thereviews by Borganietal. 2008 -Chapter 18, this volume, whichreviews the
studyoftheICMchemicalenrichmentandbyDolagetal.2008b-Chapter15,thisvolume,
which reviews the study of the non–thermal properties of the ICM from simulations. We
refertothereviewsbyDolagetal.2008a-Chapter12,thisvolumeforadescriptionofthe
techniquesofnumericalsimulationsandbyKaastraetal.2008-Chapter9,thisvolumefor
anoverviewoftheobservedthermalpropertiesoftheICM.
The scheme of the presentation is as follows. In Sect. 2 we briefly discuss the self–
similarmodeloftheICMandhowtheactionofnon–gravitational heatingandcoolingare
expectedtoalterthepredictions ofthismodel.Sect.3and4overviewtheresultsobtained
onthecomparisonbetweenobservedandsimulatedscalingrelationsandprofilesofX–ray
observablequantities,respectively.InSect.5wesummariseandcriticallydiscusstheresults
presented.
2 ModellingtheICM
2.1 Theself–similarscaling
The simplest model to predict the observable properties of the ICM is based on the as-
sumptionthatgravityonlydeterminesthethermodynamicalpropertiesofthehotdiffusegas
(Kaiser1986).Sincegravitydoesnothavepreferredscales,weexpectclustersofdifferent
sizestobethescaledversionofeachother.Thisisthereasonwhythismodelhasbeencalled
self-similar.
If,atredshiftz,wedefineMD c tobethemasscontainedwithintheradiusrD c,encom-
passing a mean density D c times the critical density r c(z), then MD c (cid:181) r c(z)D crD3c. The
criticaldensityoftheuniversescaleswithredshiftasr (z)=r E2(z),whereE(z)isgiven
c c0
by
E(z) = (1+z)3W m+(1+z)2W k+W L 1/2, (1)
(cid:2) (cid:3)
4
whereW mandW L arethedensityparametersassociatedtothenon–relativisticmatterandto
thecosmologicalconstant,respectively,W k=1−W m−W L andweneglectanycontribution
fromrelativisticspecies.
Therefore, theclustersizerD c scaleswithzand MD c as rD c (cid:181) MD1/c3E−2/3(z),sothat,
assuminghydrostaticequilibrium,clustermassscaleswithtemperatureT as
MD c (cid:181) T3/2E−1(z). (2)
Ifr isthegasdensity,thecorrespondingX–rayluminosityis
gas
r 2
L = gas L (T)dV, (3)
X ZV(cid:18)m mp(cid:19)
whereL (T)(cid:181) T1/2 for pure thermal Bremsstrahlung emission. Ifgasaccretes along with
DMbygravitationalinstabilityduringtheformationoftheclusterhalo,thenweexpectthat
r (r)(cid:181) r (r),sothat
gas DM
LX (cid:181) MD cr cT1/2 (cid:181) T2E(z). (4)
Another useful quantity characterising the thermodynamical properties of the ICM is
theentropy(Voit2005)which,inX–raystudiesoftheICM,isusuallydefinedas
k T
B
K = , (5)
m m r 2/3
p gas
wherek istheBoltzmannconstant,m themeanmolecularweight(≃0.58foraplasmaof
B
primordialcomposition)andm theprotonmass.Withtheabovedefinition,thequantityK
p
istheconstantofproportionality intheequationofstateofanadiabaticmono-atomicgas,
P=Kr 5/3.Usingthethermodynamicdefinitionofspecificentropy,s=c ln(P/r 5/3)(c :
gas V gas V
heat capacity at constant volume), one obtains s=k lnK3/2+s , where s is aconstant.
B 0 0
Anotherquantity,oftencalled“entropy”intheclusterliterature,whichwewillalsousein
thefollowing,is
S = k Tn−2/3, (6)
B e
wheren istheelectronnumberdensity.Accordingtotheself–similarmodel,thisquantity,
e
computedatafixedoverdensityD ,scaleswithtemperatureandredshiftaccordingto
c
SD c (cid:181) T(1+z)−2. (7)
Asalreadymentioned intheintroduction, anumberofobservational factsfromX–ray
data point against the simple self–similar picture. The steeper slope of the L –T relation
X
(Markevitch1998;Arnaud&Evrard1999;Osmond&Ponman2004),L (cid:181) Ta witha ≃3
X
for clustersand possiblylarger forgroups, theexcess entropy inpoor clusters andgroups
(Ponmanetal. 2003; Pratt&Arnaud 2005; Piffarettietal. 2005) and the decreasing trend
ofthegasmassfractioninpoorersystems(Linetal.2003;Sandersonetal.2003)allpoint
towardthepresenceofsomemechanismwhichsignificantlyaffectstheICMthermodynam-
ics.
5
Fig.1 Profilesofreducedentropy,S/T,fornon–radiative simulations(fromBorganietal.2001).Theleft
panelisforsimulationsincludingonlygravitationalheating,thecentralpanelisforrunsincludingSNfeed-
backaspredictedbyasemi–analyticalmodelofgalaxyformationandtherightpaneliswithpre–heatingwith
anentropythresholdatredshiftz=3.Solid,short–dashedandlong–dashedcurvesarefora3keVcluster,
fora1keVgroupandfora0.5keVgroup,respectively.Thedottedstraightlineintheleftpanelshowsthe
analyticalpredictionbyTozzi&Norman(2001)fortheentropyprofileassociatedtogravitationalheating.
2.2 HeatingandcoolingtheICM
The first mechanism, that has been introduced to break the ICM self–similarity, is non–
gravitationalheating(e.g.Evrard&Henry1991;Kaiser1991;Tozzi&Norman2001).The
ideaisthatbyincreasingthegasentropywithagivenextraheatingenergypergasparticle
E prevents gasfromsinkingtothecentreofDMhalos,therebyreducing gasdensityand
h
X–ray emissivity. This effect will be large for small systems, whose virial temperature is
k T .E , while leaving rich clusters with k T ≫E almost unaffected. Therefore, we
B h B h
expectthat theX–rayluminosityandgascontent arerelativelymoresuppressedinpoorer
systems,thusleadingtoasteepeningoftheL –T relation.
X
Thenotionofnon–gravitationalheatinghasbeenfirstimplementedinnon–radiative(i.e.
neglectingtheeffectofcooling)hydrodynamicalsimulationsbyeitherinjectingentropyin
animpulsivewayatagivenredshift(Navarroetal.1995;Bialeketal.2001),orbyadding
energy in a redshift–modulated way, so as to mimic the rate of SN explosions from an
external model of galaxy formation (Borganietal. 2002). In Fig. 1 we show the different
efficiencythatdifferentheatingmechanismshaveinbreakingtheself–similarbehaviourof
theentropyprofilesinobjectsofdifferentmass,rangingfromaVirgo–likeclustertoapoor
galaxy group. According to the self–similar model, the profiles of reduced entropy, S/T,
shouldbeindependent oftheclustermass.Thisisconfirmed bytheleftpanel,whichalso
shows that these profiles have a slope consistent with that predicted by a model in which
gas is shock heated by spherical accretion in a DM halo, under the effect of gravity only
(Tozzi&Norman2001).Thecentralpanelshowsinsteadtheeffectofaddingenergyfrom
SN, whose rate is that predicted by a semi–analytical model of galaxy formation. In this
case, which corresponds to a total heating energy of about 0.3 keV/particle, the effect of
extra heating starts being visible, but only for thesmaller system. It is only withthe pre–
heatingscheme,basedonimposinganentropy floorof50keVcm2,thatself–similarityis
clearly broken. While this heating scheme is effective in reproducing the observed L –T
X
relation,itproduceslargeisentropiccores,apredictionwhichisatvariancewithrespectto
observations(e.g.,Donahueetal.2006).
6
Fig.2 Leftpanel:therelationbetweenentropyandtemperatureforgashavingafixedvalueofthecooling
time(fromVoit&Bryan2001).Thecrosseswitherrorbarsareobservationaldataontheentropymeasured
atone–tenth ofthevirial radiusforclusters andgroupsbyPonmanetal.(1999).Rightpanel:acompari-
sonbetweenobservations(errorbarswithcrosses;Ponmanetal.(1999)andsimulations,includingradiative
coolingandstarformation(numbers),fortheentropyinthecentralregionsofgalaxygroupsandclusters.
Thesolidlineshowsthepredictionoftheself–similarmodel(fromDave´etal.2002).
Although it may look like a paradox, radiative cooling has been also suggested as a
possible alternative to non–gravitational heating to increase the entropy level of the ICM
andsuppressingthegascontentinpoorsystems.AsoriginallysuggestedbyVoit&Bryan
(2001),coolingprovidesaselectiveremovaloflow–entropygasfromthehotX–rayemitting
phase(seealsoWu&Xue2002).Asaconsequence,whiletheglobalentropyofthebaryons
decreases,theentropyoftheX–rayemittinggasincreases.Thisisillustratedintheleftpanel
ofFig.2(fromVoit&Bryan2001).Inthisplot,eachofthetwocurvesseparatestheupper
portion of the entropy–temperature plane, where the gas has cooling time larger than the
ageofthesystem,fromthelowerportion, wheregaswithshortcoolingtimeresides.This
impliesthat onlygashaving arelativelyhigh entropy willbeobservedasX–rayemitting,
whilethelow–entropygaswillbeselectivelyremovedbyradiativecooling.Thecomparison
withobservationaldataofclustersandgroups,alsoreportedinthisplot,suggeststhattheir
entropylevelmaywellbetheresultofthisremovaloflow–entropygasoperatedbyradiative
cooling.Thisanalyticalpredictionhasbeenindeedconfirmedbyradiativehydrodynamical
simulations. The right panel of Fig. 2 shows the results of the simulations by Dave´ etal.
(2002)onthetemperaturedependence ofthecentralentropy ofclustersandgroups. Quite
apparently,theentropylevelinsimulationsiswellabovethepredictionoftheself–similar
model,byarelativeamountwhichincreaseswithdecreasingtemperature,andinreasonable
agreementwiththeobservedentropylevelofpoorclustersandgroups.
Althoughcoolingmaylooklikeanattractivesolution,itsuffersfromthedrawbackthat
atoolargefractionofgasisconvertedintostarsintheabsenceofasourceofheatingenergy
whichregulatesthecoolingrunaway.Indeed,whileobservationsindicatethatonlyabout10
per cent of the baryon content of aclusteris inthe stellarphase(e.g., Baloghetal. 2001;
Linetal. 2003), radiative simulations, like those shown in Fig. 2, convert into stars up to
∼50percentofthegas.
7
Anotherparadoxical consequenceofcoolingisthatitincreasesthetemperature ofthe
hotX–rayemittinggasatthecentreofclusters.ThisisshownintheleftpanelofFig.3(from
Tornatoreetal.2003),whichcomparesthetemperatureprofilesforthenon–radiativerunof
aVirgo–like clusterwithavariety ofradiative runs, based on different ways of supplying
non–gravitational heating. The effect of introducing cooling is clearly that of steepening
thetemperatureprofilesinthecoreregions,whileleavingitunchangedatlargerradii.The
reasonforthisisthatcoolingcausesalackofcentralpressuresupport. Asaconsequence,
gas starts flowing in sub-sonically from more external regions, thereby being heated by
adiabaticcompression.AsweshalldiscussinSect.4,thisfeatureofcoolingmakesitquite
difficulttoreproducethestructureofthecoolcoresobservedingalaxyclusters.
Steepening of the central temperature profiles and overcooling are two aspects of the
same problem. In principle, the solution to this problem should be provided by a suitable
scheme of gas heating which regulates star formation, while maintaining pressurised gas
in the hot phase. The right panel of Fig. 3 compares the temperature–density phase dia-
gramsforgasparticleslyinginthecentralregionofanSPH–simulatedcluster,whenusing
twodifferentfeedbackefficiencies.Thetwosimulationsincludecooling,starformationand
feedbackintheformofgalacticwindspoweredbySNexplosions,followingtheschemein-
troducedbySpringel&Hernquist(2003a).Theupper(green)andthelower(red)cloudsof
hightemperatureparticlescorrespondtoawindvelocityof500kms−1 andof1000kms−1,
respectively. This plot illustrates another paradoxical effect: in the same way that cooling
causes anincreaseofthetemperature ofthehot phase, supplying energy withanefficient
feedbackcausesadecreaseofthetemperature.Thereasonforthisisthatextraenergycom-
pensatesradiativelosses,therebymaintainingthepressuresupportforgaswhichwouldoth-
erwisehaveaveryshortcoolingtime,therebyallowingittosurviveonaloweradiabat.Itis
alsoworth reminding that cooling efficiencyincreaseswiththenumerical resolution (e.g.,
Baloghetal.2001;Borganietal.2006).Therefore,forafeedbackmechanismtoworkprop-
erly, it should be abletostabilisethecooling efficiency inawaywhich is independent of
resolution.
In the light of these results, it is clear that the observed lack of self–similarity in the
X–rayproperties ofclusters cannot besimplyexplained onthegrounds ofasingleeffect.
The emerging picture is that the actionof cooling and offeedback energy, e.g. associated
to SN explosions and AGN, should combine in a self–regulated way. As we shall discuss
inthefollowingsections,hydrodynamicalsimulationsofgalaxyclustersinacosmological
context demonstrate that achieving this heating/cooling balanceisnot easyandrepresents
nowadaysoneofthemostchallengingtasksinthenumericalstudyofclusters.
Asanexample,weshowinFig.4howthegasdensityofasimulatedclusterchanges,
both at z=2 (left panels) and at z=0 (right panels), when cooling and star formation
arecombinedwithdifferentformsofnon–gravitational heating(fromBorganietal.2005).
Thecomparisonofthetopandcentralpanelsshowstheeffectofincreasingthekineticen-
ergy carried by galacticoutflows by afactor of six. Thestronger winds are quiteefficient
in stopping star formation in the small halos, which are washed out, and make the larger
onesslightlypuffier,whilepreservingthegeneralstructureofthecosmicwebsurrounding
theLagrangianclusterregion.Comparingthetopandthebottompanelsshowsinsteadthe
effect ofadding togalacticwinds alsotheeffect ofan entropy floor. Although the energy
budgetofthefeedbackschemesofthecentralandbottompanelsarequitecomparable,the
effectofthegasdistributionisradicallydifferent.Imposing anentropyflooratz=3with
animpulsiveheatinggeneratesamuchsmoothergasdensitydistribution,bothatz=2and
atz=0.Inthiscase,thefilamentarystructureofthegasdistributioniscompletelyerased,
while only the largest halos areable toretain part of their gas content. This demonstrates
8
Fig.3 Leftpanel:temperatureprofilesfromhydrodynamicalsimulationsofa∼3keVgalaxycluster.Inall
panelsthedottedandthesolidcurvescorrespondtoanonradiativerunandtoarunincludingcoolingand
starformation.Theothercurvesarefordifferentrecipesofgasheating(fromTornatoreetal.2003).Right
panel:therelationbetweentemperatureandoverdensityforgasparticleswithin0.1r200forSPHsimulations
ofacluster ofmass≃1014h−1M⊙.Upper(green) points are forarunwhich includes feedback through
galactic windswithavelocity of500 kms−1,whilethelower(red)pointsareforarunbasedonassum-
ingstrongerwinds(sw),withatwiceaslargevelocity.Bothrunsincludeamodelofchemicalenrichment
(Borganietal.2008-Chapter18,thisvolume)whichassumesanInitialMassFunctionforstarformationby
Arimoto&Yoshii1987(AY).Thepointsinthebottomrightcornerarestar–forminggasparticles.
thatafixedamountofenergyfeedbackcanprovidelargelydifferentresultsontheICMther-
modynamicalproperties,dependingontheepochandonthedensityatwhichitisreleased.
3 Scalingrelations
So far, we have qualitatively discussed how simple models of pre–heating and radiative
cooling canreproduce theobserved violation ofself–similarity in theX–ray properties of
galaxyclusters.Inthisandinthefollowingsectionswewillfocusthediscussiononamore
detailedcomparisonbetweensimulationresultsandobservationaldata,andontheimplica-
tions ofthis comparison onour current understanding ofthefeedback mechanisms which
regulatestarformationandtheevolutionofthegalaxypopulation.Asastartingpointforthe
comparisonbetweenobservedandsimulatedX–rayclusterproperties,wedescribehowob-
servablequantitiesarecomputedfromhydrodynamical simulationsandhowtheycompare
totheanalogousquantitiesderivedfromobservationaldata.
AsfortheX–rayluminosity,itiscomputedbysummingthecontributions totheemis-
sivity, e , carried by all the gas elements (particles in a SPH run and cells in an Eulerian
i
grid–based run), L =(cid:229) e , where the sum extends over all the gas elements within the
X i i
regionwhereL iscomputed.Thecontributionfromthei-thgaselementisusuallywritten
X
as
e = n n L (T,Z)dV , (8)
i e,i H,i i i i
wheren andn arethenumberdensitiesofelectronsandofhydrogenatoms,respectively,
e,i H,i
associated to the i-th gas element of given density r , temperature T and metallicity Z.
i i i
9
Fig. 4 The maps of the gas density for simulations of a Virgo–like cluster at z=2 and z=0 (left and
rightpanels,respectively),includingcooling,starformationanddifferentformsofnon–gravitationalheating.
Upperpanelsareforarunwhichincludes galactic windswithavelocity ofabout340kms−1,thecentral
panelsisforgalacticwindswithvelocityofabout830kms−1 andthebottompanelsisforgalacticwinds
asinthetoppanels,butalsoaddingapre–heatingwithanentropythresholdof100keVcm2 atz=3.At
z=0thesizeoftheboxisof11.7h−1Mpc,whileatz=2iscorrespondsto17.5h−1Mpccomoving(from
Borganietal.2005).
10
Furthermore,L (T,Z)isthetemperature–andmetallicity–dependentcoolingfunction(e.g.,
Sutherland&Dopita1993)computedwithinagivenenergyband,whiledV =m/r isthe
i i i
volumeofthei-thgaselement,havingmassm.
i
Asforthetemperature,differentproxiestoitsX–rayobservationaldefinitionhavebeen
proposed in the literature, which differ from each other in the expression for the weight
assignedtoeachgaselement.Ingeneral,theICMtemperaturecanbewrittenas
(cid:229) wT
i i
T = i , (9)
(cid:229) w
i
i
whereT isthetemperatureofthei-gaselement,whichcontributeswiththeweightw.The
i i
mass–weighted definition of temperature, T , is recovered for w =m (m: mass of the
mw i i i
i-thgaselement),whichalsocoincideswiththeelectrontemperatureT forafullyionised
e
plasma. A moreobservation–oriented estimateoftheICMtemperature isprovided bythe
emission–weighteddefinition, T ,whichisobtainedforw =e (e.g.,Evrardetal.1996).
ew i i
Theideaunderlyingthisdefinitionisthateachgaselementshouldcontributetotheoverall
spectrumaccordingtoitsemissivity.
Mazzottaetal.(2004)pointed outthatthethermalcomplexity oftheICMissuchthat
the overall spectrum is given by the superposition of several single–temperature spectra,
eachoneassociatedtoonethermalphase.Inprinciple,thesuperpositionofseveralsingle–
temperaturespectracannotbedescribedbyasingle–temperaturespectrum.However,when
fitting it to a single–temperature model in a typical finite energy band, where X–ray tele-
scopes are sensitive, the cooler gas phases are relatively more important in providing the
high–energy cut–offofthespectrumand,therefore, indetermining thetemperatureresult-
ing from the spectral fit. In order to account for this effect, Mazzottaetal. (2004) intro-
duced a spectroscopic–like temperature, T , which is recovered from Eq. 9 by using the
sl
weightw =r mTa −3/2.Byusinga =0.75,thisexpressionforT wasshowntoreproduce
i i i sl
withinfewpercentthetemperatureobtainedfromthespectroscopicfit,atleastforclusters
withtemperatureabove2–3keV.Morecomplexfittingexpressionshavebeenprovidedby
Vikhlinin(2006),whogeneralisedthespectroscopic–liketemperaturetothecasesoflower
temperatureandarbitrarymetallicity.
3.1 Theluminosity–temperaturerelation
TheL –T relationrepresentedthefirstobservationalevidenceagainsttheself–similarmodel.
X
Thisrelationhasbeenshownbyseveralindependentanalysestohaveaslope,L (cid:181) Ta ,with
X
a ≃3forT &2keV(e.g.,Whiteetal.1997),withindicationsforaflatteningtoa .2.5for
themostmassivesystems(Allen&Fabian1998).Thescatterinthisrelationislargelycon-
tributedbythecool–coreemission,sothatitsignificantlydecreaseswhenexcisingthecores
(Markevitch 1998) or removing cool–core systems (Arnaud&Evrard 1999). A change of
behaviour isalsoobserved at thescalesof groups, T .2keV, whichgenerally displays a
verylargescatter(Osmond&Ponman2004).
HydrodynamicalsimulationsbyBialeketal.(2001)andbyBorganietal.(2002)demon-
stratedthatsimplepre–heatingmodels,basedontheinjectionofentropyatrelativelyhigh
redshift,canreproducetheobservedslopeoftheL –T relation.Dave´ etal.(2002)showed
X
thatasimilarresultcanalsobeachievedwithsimulationsincludingcoolingonly,theprice
to be paid being a large overcooling. Muanwongetal. (2002) and Tornatoreetal. (2003)