The Effects of Radiative Cooling on the Entropy Distribution of Intracluster Gas

0.27 MB·English

by  Ian G. McCarthy

Draftversion February2,2008
PreprinttypesetusingLATEXstyleemulateapjv.4/12/04
THE EFFECTS OF RADIATIVE COOLING ON THE ENTROPY DISTRIBUTION OF INTRACLUSTER GAS
Ian G. McCarthy, Mark A. Fardal, and Arif Babul
DepartmentofPhysics&Astronomy,UniversityofVictoria,Victoria,BC,V8P1A1,Canada;
[email protected],[email protected],[email protected]
Draft versionFebruary 2, 2008
ABSTRACT
High resolution X-ray observations indicate that the entropy profiles in the central regions of some
massive cooling flow clusters are well approximatedby powerlaws. McCarthy and coworkersrecently
accountedfor this trendwith ananalytic modelthat includes the detailedeffects ofradiativecooling.
Interestingly,theseauthorsfoundthatcooling(andsubsequentinflowofthegas)naturallyestablishes
5
approximate steady-state powerlaw entropy profiles in the cores of clusters. However, the origin of
0
0 this behavior and its dependence on initial conditions have yet to be elucidated. In the present
2 study,weexplainthistrendinthecontextoftheself-similarcoolingwavemodeldevelopedpreviously
by Bertschinger (1989). It is shown that the logarithmic slope of the entropy profile in the cores
n
of relaxed cooling flow clusters is given by a simple analytic function that depends only on the
a
logarithmic slopes of the local gravitationalpotential and the cooling function. We discuss a number
J
of potentially interesting uses of the above result, including: (1) a way of measuring the shapes of
8
gravitational potentials of cooling flow clusters (which may, for example, be compared against the

standard hydrostatic equilibrium method); (2) a simple method for constructing realistic analytic
1
cluster models that include the effects of radiative cooling; and (3) a test of the reliability of cooling
v
7 routines implemented in analytic models and hydrodynamic simulations.
3 Subject headings: cosmology: theory — galaxies: clusters: general — X-rays: galaxies: clusters
1
1
0 1. INTRODUCTION establishes approximate powerlaw entropy profiles near
5 the cluster core2. It would be interesting to understand
The presence of strong positive temperature gradients
0 from a physicalperspective what is driving this trend in
(e.g., Allen et al. 2001; De Grandi & Molendi 2002; Pif-
/ the observed and model clusters.
h faretti et al. 2004; Vikhlinin et al. 2004) and the lack
Unfortunately, a rigorous calculation of the effects of
p of an obvious entropy floor (e.g., Pratt & Arnaud 2002;
radiativecooling andthe subsequent (quasi-hydrostatic)
- Mushotzkyetal. 2003;Piffarettietal.2004)inthecores
o inflow of intracluster gas can only formally be obtained
of some clusters have stimulated interest in theoretical
r by numerically solving the time-dependent hydrody-
t models of the intracluster medium (ICM) that include
s namic equations (as approximately done, e.g., in M04).
the effects of radiative cooling. Such models have been
a This makes it a challenge to understand physically why
: showntobeinbroadagreementwiththecurrentsuiteof
v theabovetrendsareestablishedbycooling. Despite this
X-ray observations of clusters, particularly if some form
i inconvenience, there are, however, analytic tools at our
X of feedback/heating is also incorporated into the mod-
disposal that can help to establish a physical picture.
els (e.g., Tozzi & Norman 2001; Voit et al. 2002; Oh &
r For example, Bertschinger (1989) (hereafter, B89) used
a Benson 2003;Borganiet al. 2004;McCarthy et al. 2004,
aself-similarityanalysistoderivethebehaviorofcooling
hereafter M04).
flows in clusters. One of the interesting results derived
Interestingly, by assembling published Chandra and
by Bertschinger is that the logarithmic slopes of the gas
XMM-Newton X-ray data from the literature, M04 un-
density and temperature profiles in the limit r ≪ r
covered a population of relaxed clusters (e.g., A2029, cool
(where r is the radius at which the cooling time of
PKS0745,HydraA)withcentralentropyprofiles(where cool
the gasequalsthe ageofthe cluster)depend onlyonthe
we define the “entropy”, S, as kTn−2/3) that are well
e shapes of the cluster gravitational potential and of the
approximated by powerlaws (see also the recent study
cooling function, Λ(T). The behavior of the gas density
of Piffaretti et al. 2004). M04 were able to account
and temperature profiles can be used to infer how ra-
for these clusters (as well as other clusters that exhibit
diative cooling influences the distribution of intracluster
large entropy cores) with a simple analytic model that
entropy, which is perhaps a more fundamental quantity
includes the detailed effects of cooling and that assumes
since convection will strive to prevent the establishment
the ICM was initially heated prior to cooling. In par-
ofarisingentropyprofiletowardstheclustercenter(see,
ticular, M04 found that if the ICM in a given cluster is
e.g., Voit et al. 2002). Thus, the study of B89 is a good
initiallyheated1byonlyamildamount,coolingnaturally
starting point for our investigationof the effects of cool-
ing and inflow on intracluster entropy.
1 Wenotethatsuch“preheating”doesnotprecludesubsequent
In the present study, we briefly review the self-similar
heating events followingcluster formation, suchas AGNfeedback
initiatedbytheaccretionofcoldgasattheclustercenter. analysisofB89,includingthebasicassumptionsmadein
2 However, the mild amount of heating is sufficient to delay that study and their validity, and use his results to de-
the onset of catastrophic cooling for up to several Gyr and, as a rive how radiative cooling modifies the entropy profiles
result, the predicted global cold gas fractions remain consistent
of clusters. We compare the self-similar solution to the
withobservationallyestablishedlimits.
2 McCarthy et al.
resultsofthe 1-DcoolingmodelofM04forclusters with Other assumptions made in the analysis of B89 in-
powerlawdark matter profiles and that cool via thermal clude spherical symmetry, subsonic inflow, and single-
bremsstrahlung and, indeed, find extremely good agree- phase cooling. Spherical symmetry is expected to be
mentinthelimitr ≪r . Wefurtherdemonstratethat approximately valid, at least in an average sense for a
cool
the self-similar solution should provide an accurate de- reasonably large relaxed cluster sample. Likewise, sub-
scription for cooling (via both bremsstrahlung and line sonic flow should hold for the majority of the gas within
emission) in more realistic dark matter halos. Finally, r , except possibly near the very center where the gas
cool
we discuss anumber of interestinguses ofthis result,in- maybecome transsonic. However,it hasyetto be deter-
cludingmeasuringtheshapesofclustergravitationalpo- minedwhetherornotmulti-phasecoolingisimportantin
tentials (that, e.g., could be compared against the usual clusters. In the absence of significant non-gravitational
hydrostatic equilibrium method), testing the reliability heating,theintraclustergasisthermallyunstable. How-
of numericalcooling routines in analytic models and hy- ever, because the gas flows into the cluster center essen-
drosimulations,andasasimple wayofsettingupinitial tiallyasfastasitcools,weexpectmulti-phasecoolingto
conditions for cluster models with radiative cooling. beimportantonlyneartheverycenter(seeB89). Recent
2. COOLINGFLOWSANDSELF-SIMILARITY:THE spatially-resolvedChandra and XMM-Newton spectra of
the central regions cooling flow clusters have confirmed
BERTSCHINGERSOLUTION
that, probably with the exceptionof the very centralra-
Asnotedabove,ingeneral,thetime-dependenthydro-
dial bin, single-phase models provide at least as good a
dynamicequationsmustbesolvednumericallyinorderto
fit as multi-phase models (e.g., David et al. 2001; Mat-
obtainadetailedpictureoftheeffectsofradiativecooling
sushita et al. 2002). For the present study, we assume
on cluster gas. However, when the time dependence is
single-phase cooling.
duetoasinglephysicalprocessthatcanbecharacterized
Finally, it is implicitly assumed that there are no sig-
byauniquescalelength[inthiscase,thecoolingradius3,
nificant sources of non-gravitational heating (such as
r (t)], similarity solutions can provide useful physical
cool AGN feedback, thermal conduction, and turbulent mix-
insight. Adoptingthisapproach,B89derivedthegeneral
ing)presentinthe ICM.In§4,we givea briefdiscussion
behavior of cooling flows in clusters of varying gravita-
of the potential impact of such heating.
tional potentials (and with varying cooling functions as
Implementing the above assumptions, B89 renormal-
well). We are especially interested in the properties of
ized the hydrodynamic equations by removing any time
his solution in the limit r ≪ r , since they are inde-
cool dependence arising through r (t). A self-similar so-
pendentofthe initialconditionsofthe gas. (Outside the cool
lution is obtained if one neglects the acceleration terms
coolingradiusthe initialconditionsare ofcrucialimpor-
(which is valid since the inflow of gas is highly subsonic)
tance since radiative cooling hasn’t had enough time to
in the renormalized hydro equations. It is straightfor-
significantlymodifythegasthere.) Beforeexaminingthe
ward to derive the limiting behavior of the gas density
solution itself, let us first review the basic assumptions
and temperature profiles in the limit r ≪ r under
made in B89 and their validity. cool
these conditions (see eqns. 2.30 of B89):
Sinceself-similarsolutionscanbecharacterizedbyonly
a single scale length, some simplifying assumptions are
dlogρ −3+(2−α)(1−β) dlogT
required in order to obtain a solution for the properties = , =2−α (1)
of cooling flows. In particular, Bertschinger assumed dlogr 2 dlogr
that the gravitational potential and the cooling func- where we have assumed that dark matter dominates the
tion could be characterized by simple powerlaws that gravitationalpotential and that ρdm ∝r−α and Λ(T)∝
remained fixed as a function of time. Of course, in real- Tβ. Note that the definitions of α and β differ from the
ity, neither of these assumptions are strictly valid. High definitions of these symbols in B89.
resolution numerical simulations indicate that the dark In§1,wedefinedthe“entropy”,S,intermsofgasden-
matter density profiles of clusters (and halos of other sity and temperature. The above equations can, there-
massesaswell)haveacharacteristicscalelength(theso- fore,beusedtoyieldthelogarithmicslopeoftheentropy
called scale radius, r ), where the index of the powerlaw profile within r :
s cool
profile changes from relatively shallow (between ∼ −1
and −1.5; e.g., Navarro, Frenk, & White (NFW) 1997; dlogS 1
γ ≡ = 1− (1−β) (2−α)+1 (2)
Moore et al. 1999) to relatively steep (∼−3). The cool- dlogr (cid:20) 3 (cid:21)
ing function is not scale-free either, owing primarily to
Thus, for β =1/2 (i.e., cooling dominated by thermal
line emission. Thus, one is justified in questioning the
bremsstrahlung),
physicalrelevanceofamodelthatinvokestheseassump-
tions. (However,physicalrelevancemay notbe ofmajor 5
concernifoneissimplytestingthereliabilityofnumerical γ = (2−α)+1 (3)
6
cooling methods.) We examine in §3.2 whether relaxing
which yields γ =1 for a singular isothermalsphere (α=
these assumptions significantly affects the shapes of the
2), γ ≈ 1.4 for a Moore et al. profile in the limit r ≪
resultingentropyprofilesinthecoresofmassiveclusters.
r (α = 1.5), and γ ≈ 1.8 for a NFW profile in the
s
3 The cooling radius grows larger with time. Unfortunately, limit r ≪ rs (α = 1). Note, however, that the value of
the self-similarsolution inB89 isexpressedinterms ofthe initial γ in equation (2) depends only weakly on the shape of
cooling radius. Thus, it might be expected that comparison to the cooling function (that is, for reasonable values of β
observations,whichareusedtoinferthe presentcoolingradius,is ranging from −1/2 to 1/2) for α&1.
somewhat ambiguous. However, as highlighted by B89, this leads
Below,wecomparethissimpleanalyticresultwiththe
to only a small error since, over the course of a cluster’s life, the
coolingradiusgrowsonlybyasmallamount. 1-D cooling model of M04.
Radiative Cooling and Intracluster Entropy 3
3. COMPARISONWITHM04
3.1. Powerlaw clusters
M04 developed a simple radiative cooling code which,
1000
when applied to the entropy injection model clusters
of Babul et al. (2002), successfully and simultaneously
reproduces the luminosity-temperature and luminosity- 100
mass relations (including their associated intrinsic scat-
ter)andyieldsdetailedfitstotheentropy,surfacebright- 10
ness,andtemperatureprofilesofclustersasinferredfrom
recent high resolutionX-ray observations. As alluded to
in §1, these authors found that radiative cooling estab-
lished a powerlaw entropy profile in cores of their model
clusters that experienced only mild preheating (that is, 1000
for those clusters that were initially injected with .300
keV cm2, i.e., the entropy cooling threshold for massive 100
clusters). Itisinterestingtoseewhetherornotthistrend
can be explained by the self-similar solution of B89.
10
In order to test this, we use the model of M04 (see
§2.2 of M04 for a detailed discussion of the model) to
1
track the effects of cooling for a set of clusters with ar- 0.01 0.1 1 0.1 1
bitrary initial conditions (recallthat the solution of B89
within the cooling radius does not depend on initial gas
conditions). For specificity, however,we show results for Fig. 1.— Theeffectsofcoolingandinflowontheentropydistri-
butionofclusters. Solidlinesrepresenttheinitialentropyprofiles.
clustersthathaveatotaldarkmattermassof1015M⊙,a Thedotted, short dashed, and longdashed lines represent the re-
totalgasmassof≈1.5×1014M⊙,andamaximumradius, sultingsteady-state entropyprofileswhentheclustersareevolved
r ,of2.06Mpc. Theclustergasisinitiallyassumedto in dark matter halos that have density profiles characterized by
halo
powerlaw indices of α = 2.0, 1.5, and 1.0, respectively. The var-
be in hydrostatic equilibrium within a dark matter halo
ious panels show the resultingprofiles for different initial entropy
thatischaracterizedby adensityprofileρdm ∝r−α (the distributions. Figure 2 presents a comparison of the central loga-
normalization being set by the mass and radius given rithmicentropyslopestotheBertschingersolution.
above). The cluster gas is then allowed to evolve via ra-
diative cooling and inflow until a stable entropy profile
is achieved. To calculate the cooling rate, we assume 2
a cooling function that scales as Λ(T) ∝ T1/2 with a
normalization that is set by matching a zero metallicity
Raymond-Smith plasma at a temperature of T ≈108 K.
Finally, as in M04, we remove any gas that is able to 1.75
cool below a threshold temperature of 105 K from the
calculationand assume that its dynamical effects on the
cluster gravitational potential are negligible. Clearly, if
1.5
a significant amount of gas is able to completely cool
this assumption will be violated. However, in this case,
we expect these effects will be relevant only for the cen-
tral tens of kpc and will have only a minor effect on the
1.25
overallentropy distribution within the cooling radius.
To show how the effects of cooling are linked to the
underlying gravitational potential, we present Figure 1.
Focusing first on the top left-hand panel, we start with
1
acluster thatis initially characterizedby entropyprofile
that contains a core and a logarithmic slope of γ = 1.1
outside the core. This is the initial slope adopted by
Babul et al. (2002) and M04 and is what one expects 1 1.25 1.5 1.75 2
if the gas is in hydrostatic equilibrium and if its den-
sity profile traces that of the dark matter (e.g., Voit et
Fig. 2.— Comparison of the self-similar solution of B89 with
al. 2002; Williams et al. 2004), an expectation that is
theresultsofM04’scoolingmodel. Theshadedregionreflectsthe
supported by high resolution hydrodynamic simulations uncertaintyinthebestfitpowerlawindicesoftheentropyprofiles
(e.g., Lewis et al. 2000;Voit et al. 2003;Ascasibar et al. (withinrcool)showninFig. 1.
2003;Voit2004). Ascanbeclearlyseen,theslopeofthe
dark matter profile, α, is important in determining the
resulting slope of the entropy profile, γ, within 0.1rhalo The remaining panels of Figure 1, which show the re-
(which corresponds roughly to rcool for these clusters). sults for clusters that initially have steeper entropy pro-
In particular,as the dark matter profile steepens the re- files outside the entropy core, illustrate that the initial
sulting entropy profile becomes more shallow, agreeing gasconditionsoftheclustershavealmostnoeffectonthe
qualitatively and quantitatively with equation (3). resulting steady-state entropy profile within r . Like-
cool
4 McCarthy et al.
1
0.1
0.01
0.01 0.1 1 0.01 0.1 1
Fig. 3.— Theeffectsoflineemissionandrealisticdarkmatterprofilesonthesteady-stateentropyprofile. Left: AclusterwithaMoore
et al. dark matter halo. Right: A cluster with a NFW dark matter halo. In both panels the dotted line represents the initial entropy
distribution, the thick solid lines represent the final entropy distribution assuming cooling with a pure thermal bremsstrahlung cooling
function, andthe thick dashed linesrepresent the final entropy distributionassumingcooling withaRaymond-Smith plasmamodel with
Z=0.3Z⊙. ThethindashedlinesindicatetheslopepredictedbyB89’sself-similarmodelassumingβ=1/2andα=1.5(leftpanel)and
α=1.0(rightpanel). Theyhavebeenarbitrarilynormalizedtocrossthethicklinesatthecoolingradius.
wise,whathappensintheinterioroftheclusterdoesnot tent to which the shape of the resulting entropy profile
significantly influence gas outside of r , as expected. is affected by these assumptions.
cool
We have fitted the entropy profiles within r with Since we are introducing additional scales into the
cool
powerlaws. However, we find that the entropy profiles problem, it is important that we constructrealistic clus-
within r are not exact powerlaws and there is some ter models. We consider two different systems: one with
cool
“wiggle” room in the best fit logarithmic slope, depend- a NFW dark matter profile and one with a Moore et
ingontherangeofradiioverwhichtheprofilesarefitted. al. dark matter profile. Both systems have been cho-
Forexample,thebestfitlogarithmicslopeovertherange sen to have the same total gas and dark matter masses;
0≤r ≤rcool/2 differs slightly from the best fit over the specifically,Mgas(r200)≈1.5×1014M⊙andMdm(r200)=
range rcool/2≤r ≤rcool. We use these two radial inter- 1015M⊙,wherer200 ≈2.06Mpc. Weuseatypicalcluster
vals to roughly quantify the scatter in the best fit slope. dark matter concentration of c ≡ r /r ≈ 3.4 for
NFW 200 s
Figure2presentsacomparisonbetweentheanalyticself- the NFW halo(e.g., Ekeetal.1998;Bullocket al.2001)
similarsolutionofB89andthefitstotheentropyprofiles and c ≡ r /r = c /0.630 ≈ 5.4 for the
Moore 200 (−2) NFW
shown in Fig. 1. The shaded region roughly reflects the Moore et al. halo (see Keeton 2001). The above implies
uncertainty in the best fit powerlaw indices for the pro- a scale radius, r , of ≈ 600 kpc. As for the intracluster
s
files predicted by M04’s cooling model. gas,we turn to the studies of Voit et al. (2003)and Voit
Reassuringly, excellent agreement between the self- (2004). These authors found that the entropy profiles of
similar solution and the 1-D cooling code is obtained. alargesampleofclustersgeneratedwithanumericalsim-
Thus, the self-similar cooling wave model of B89 pro- ulation of a ΛCDM cosmology including hydrodynamics
vides a physical basis for the powerlaw trends found by (e.g., shock heating) but not radiative cooling are ap-
M04. In addition, the agreement in Figure 2 gives us proximatelyself-similarovera wide range ofmasses (see
confidence in the reliability of the 1-Dcooling modelde- Fig. 11ofVoit2004). Theinitialentropydistributionsof
veloped in M04. our model clusters are assumed to be identical to Voit’s
bestfittohissimulatedclusters4. Theinitialgasdensity
3.2. Realistic clusters and temperature distributions are determined through
the equation of hydrostatic equilibrium by applying the
Observed clusters and clusters formed in cosmological
boundary condition that the total amount of gas within
numerical simulations do not have pure powerlaw gravi-
r is equal to that specified above.
tationalpotentials. Furthermore,theICMcontainsasig- 200
nificant quantity of metals and, consequently, cools not 4Atlargeradii,i.e.,forr>0.1r200,Voit(2004)reportsabestfit
only through thermal bremsstrahlung but also through entropyprofileofS(r)∝r1.1. Atsmallradii,however,thereisan
line emission. Line emission has the effect of distorting apparent entropy core whose origin remains uncertain. We have
the cooling function away from the powerlaw form that shrunk this core for computational convenience, since the model
clusters reach steady state more quickly if they have small ini-
is characteristic of bremsstrahlung. For these two rea-
tial entropy cores. However, this modification does not affect our
sons, the physical relevance of the results presented in results or conclusions since the entropy core is contained entirely
§3.1 may be questioned. Below, we investigate the ex-
Radiative Cooling and Intracluster Entropy 5
10
8
6
4
2
0
0.01 0.1 1 0.01 0.1 1
Fig. 4.— Thefinal(steady-state) temperaturedistributionsofthegasintheMooreetal. andNFWhalos.
To compute the effects of cooling, we again make use Mooreetal.haloisroughly3timeslargerthanthatofthe
of the model developed by M04. In order to gauge the NFWhalo. InFigure5,weshowthecoolingfunctionfor
effects of line emission, we explore two different cooling a Raymond-Smith plasma with Z =0.3Z⊙. For temper-
functions: thepurethermalbremsstrahlungfunctionim- atures of kT &2 keV, the cooling function is dominated
plemented in §3.1 and a Raymond-Smith plasma with a by thermal bremsstrahlung and is well approximatedby
metallicity set to 0.3Z⊙. As in the case of the powerlaw a powerlaw; Λ(T) ∝ T1/2. This then explains why the
models, we neglect the dynamical effects of mass drop entropy profile of the Moore et al. halo is unaffected by
out and we run the cooling model until steady-state en- line emission: at any particular time there is virtually
tropyprofilesareachieved. Atsteadystate,wefindthat no gas below 2 keV (except at the exact center where
bothclusters havesimilarglobalemission-weightedtem- gas rapidly cools below the X-ray emitting threshold of
peratures with kTew ≈5 keV. ≈ 105 K). The shallower NFW potential, however, per-
In Figure 3, we plot the initial and final entropy pro- mits some gas to cool below 2 keV before reaching the
files of our model clusters. In both panels, the dotted centerofthecluster. ThethindottedlineinFig. 5shows
linesrepresenttheinitialentropydistributions,thethick thebestfitpowerlawtothecoolingfunctionbetween0.1
solid lines represent the final distributions when cooled keV ≤ kT ≤ 2 keV, which has an index of β ≈ −0.35.
using a pure bremsstrahlung cooling function, and the Using this value for β in equation (2), we are able to
thick dashed lines represent the final distributions when account for the slight (∼10%) deviation in the shape of
cooled using the 0.3Z⊙ Raymond-Smith plasma cooling theentropyprofilewithinthecentral0.01r200 (≈20kpc)
function. The thin dashed lines indicate the slope pre- of the NFW halo.
dictedbyB89’sself-similarmodelassumingβ =1/2and The shape ofthe final entropyprofile is more sensitive
α = 1.5 (left panel) and α = 1.0 (right panel). They to the shape of the gravitational potential than it is to
have been arbitrarily normalized to cross the thick lines theshapeofthecoolingfunction. Thus,wemightexpect
at the cooling radius. self-similar solution to be a poor description of the final
First, consider the role of the cooling function. For shape of entropy profiles in realistic dark matter halos.
the Moore et al. halo, there is virtually no dependence However,thelefthandpanelofFig. 3demonstratesthat
on which cooling function we use. The resulting entropy theshapeoffinalentropyprofileofthe Mooreetal.halo
distribution for the NFW halo, however,is slightly shal- (thick lines) is virtually identical to that of a halo with
lower if we include line emission. This difference can a pure powerlaw profile of α = 1.5 (thin dashed line).
be understood as follows. The gravitational potential of Likewise, with the exception of the small deviation at
the Moore et al. halo is steeper than that of the NFW the center due to line emission, the shape of the entropy
halo. Consequently, gas flowing into the center of the profile in the NFW halo is essentially identical to that
Moore et al. halo requires more thermal support to re- of a halo with a pure powerlaw profile of α = 1.0 (see
maininhydrostaticequilibrium. InFigure4,weplotthe right hand panel of Fig. 3). Recall that in the limit of
final (steady-state) temperature distributions of the two r ≪ r , the logarithmic slopes of the Moore et al. and
s
modelclusters. Note thatthe centraltemperatureofthe NFW halos asymptote to α = 1.5 and 1.0, respectively.
Fig. 3 illustrates that what is relevant is the shape of
withinthecoolingradius. Asdiscussedabove,theresultingsteady- the local gravitational potential (i.e., at r . r ), not
stateentropyprofilewithinrcooldependsonlyontheshapesofthe cool
the shape of the overall potential. Thus, the self-similar
gravitational potential and the cooling function (and not on the
initialpropertiesofthegaswithinrcool).
6 McCarthy et al.
lution is also valid for more realistic dark matter poten-
tials and cooling functions that include line emission, at
least for massive clusters where bremsstrahlung domi-
nates line emission and the typical dark matter scale ra-
dius is much larger than the cooling radius. This result
explains the numerically-derived trends found in M04,
whichprovideagoodfittoanumberofobservedcooling
10 flow clusters. Kaiser & Binney (2003)have also recently
reported that cooling establishes a powerlaw trend be-
tween ICM gas mass and entropy. This trend is also
likely to be explained by the self-similar solution.
The self-similar model of B89 implicitly assumes that
therearenosignificantsourcesofnon-gravitationalheat-
ingpresentintheICM.Heatingintroducesanadditional
scale into the problem and, potentially, violates self-
similarity. Itisclear,however,thatsomeformofheating
has (or is) occurred in real clusters in order to prevent
theso-calledcoolingcrisis(seeBaloghetal.2001). Addi-
tional evidence for heating comes from recent high reso-
lutionChandraimagesofmanycoolingflowclustersthat
1
0.01 0.1 1 10 revealbuoyantly-risingbubblesofhotplasma(e.g.,Heinz
etal.2002;Blantonetal.2003). Thesebubbleswerepre-
sumably blown by a central AGN and must be heating
Fig. 5.— The cooling function for a Raymond-Smith plasma
withZ=0.3Z⊙. Thedotted linerepresents thebestfitpowerlaw the ICM at some level. Other sources of heating, such
(withβ≈−0.35)tothefunctionovertherange0.1keV≤kT ≤2.0 asthermalconduction(e.g.,Medvedev&Narayan2001)
keV. andstirringduetotheorbitalmotionsofclustergalaxies
(e.g.,El-Zantet al.2004),arealsoa possibility. The rel-
evant questions therefore are: (i) Is the level of heating
solutionprovidesanexcellentdescriptionofsteady-state sufficiently high to severely violate self-similarity? (ii)
coolingentropyprofilesinrealisticclustersso longas the If so, is the heating distributed or restricted to only the
cooling radius is smaller than the cluster’s scale radius. very center of the cluster? (iii) In the event the heat-
Thisconditionshouldbemetformosthighmassclusters ing is episodic (such asAGN heating), when didthe last
asthetypicalcoolingradiusofclustersisoforder∼100− heating episode occur (i.e., has enough time passed in
200 kpc (e.g., Peres et al. 1998), while the typical dark order to approximately re-establish a self-similar cool-
matterscaleradiusofmassiveclusters inhighresolution ing flow)? A number of relaxed clusters with published
cosmologicalsimulationsisoforder∼400−700kpc(e.g., entropy profiles show evidence for large entropy cores
Eke et al. 1998;Bullock et al. 2001). (M04). Clearly, such systems cannot be explained by
We conclude that the self-similar solution of B89 the present self-similar cooling model, as these clusters
should provide a good description of the shapes of en- were severely heated and the heating was distributed
tropy profiles (for r . r ) of massive clusters that to large radii. However, M04 (see also Piffaretti et al.
cool
cool via thermal bremsstrahlung and line emission and 2004) also found that several massive cooling flow clus-
that have realistic dark matter profiles. The reason for ters (e.g., A2029, PKS0745, Hydra A) have nearly pure
this isthatself-similarityis onlymildly violatedforhigh powerlaw entropy profiles (except perhaps at very small
mass clusters. The deep gravitational potential wells radii, r . 30 kpc). This likely indicates that the self-
of massive clusters ensure that most of the intracluster similarmodelprovidesareasonablyaccuratedescription
mediumisquite hot(kT &2 keV),evenwithinthe cool- of the cooling gas in this particular subset of observed
ing radius, and thus thermal bremsstrahlung (which is clusters.
scale-free) dominates the X-ray emissivity. The poten- A test of the above hypothesis is to infer the slopes
tial wells themselves have a characteristic scale radius of the gravitational potentials of these clusters by us-
but, for massive clusters, this radius is typically much ing the slopes of their observedentropy profiles together
larger than the cooling radius. Therefore, the central with equation (3). This may then be comparedwith the
cooling flow essentially “feels” only a pure powerlawpo- results obtained using the standard hydrostatic equilib-
tential (as assumed by B89). rium method for inferring the gravitational potential of
clusters. Making use of the clusters with powerlaw en-
4. DISCUSSION tropy profiles from M04 (A2029, PKS0745, Hydra A),
Using the self-similar solution of B89, we have shown we find that the logarithmic slopes of their total matter
that radiative cooling and inflow lead to a characteristic density (dark matter and baryons)profiles are relatively
entropy profile within the cooling radius that depends steep, with 1.3.α.2. This agrees quite well with the
only on the shapes of the cooling function and the grav- recenthydrostaticanalysisofChandradataof10relaxed
itational potential. We have compared the self-similar coolingflowclustersbyArabadjis&Bautz(2004). Thus,
solution to the cooling model of M04 and, reassuringly, for this small sample of clusters, the self-similar model
find excellent agreement for clusters with powerlaw po- appears to provide an apt description of the cooling gas
tentials and that cool via thermal bremsstrahlung. Fur- in the centers of these clusters. A more detailed com-
thermore,wehavedemonstratedthatthe self-similarso- parisonwill soonbe possible as the number ofpublished
Radiative Cooling and Intracluster Entropy 7
cluster entropy profiles is rapidly increasing. expectedforaclusterthatiscoolingradiatively)andthis
Quite independent of how well it describes observed may have some effect on the estimates of the energetic
clusters,theself-similarmodelalsohasanumberofinter- requirements for the prevention of catastrophic cooling.
esting theoretical uses. We briefly discuss but two here. It would be interesting to see whether the estimates of
1.) A simple method for calculating initial conditions theamountofrequiredheatingchangesignificantlywhen
of analytic model clusters with radiative cooling. This more realistic initial conditions (such as those proposed
could serve as a “poor man’s alternative” to a model above) are implemented.
that explicitly takes into account the effects of radiative 2.) A test of the reliability of cooling routines im-
coolingandinflowonintraclustergas. Forexample,are- plementedinanalyticmodelsandhydrodynamicsimula-
alisticsetofinitialconditionscouldbegeneratedbyusing tions. Because the self-similar solution is a simple func-
theresultsofnon-radiativesimulations(e.g.,Lewisetal. tion, it can easily be used to test, for example, how well
2000; Loken et al. 2002; Voit 2004) to describe the gas various formulations of smoothed particle hydrodynam-
at large radii (r > r ) while using the self-similar so- ics (SPH) or mesh-based techniques [such as adaptive
cool
lutionofB89todescribethepropertiesofthe gaswithin meshrefinement(AMR)]treattheeffectsofcooling(see,
r . The normalization of the entropy profile within e.g., Abadi, Bower, & Navarro 2001). We are currently
cool
r (which is not specified by the self-similar solution) undertaking such a study using a variety of popular an-
cool
could be set by matching the non-radiative simulation alytic and hydrodynamic codes (Dalla Vecchia et al. in
results near r . One example of where such initial preparation).
cool
conditionsmightbe usefulis for models thatexplorethe
I.G.M.issupportedbyapostgraduatescholarshipfrom
ability of various heating mechanisms to offset radiative
the Natural Sciences and Engineering Research Coun-
lossesoftheICM.Forexample,anumberofrecentAGN
cil of Canada (NSERC) and A. B. is supported by an
heating simulations (e.g., Quilis, Bower, & Balogh 2001;
NSERC Discovery Grant. A. B. would also like to ac-
Ruszkowski & Begelman 2002) have invoked initial con-
knowledge support from the Leverhulme Trust (UK) in
ditions such as isothermality or temperature profiles de-
the form of the Leverhulme Visiting Professorship.
rived from non-radiative simulation (as opposed to that
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