Table Of ContentAstronomy & Astrophysics manuscript no. AA˙2007˙8428 c ESO 2008
(cid:13)
February 2, 2008
Warming rays in cluster cool cores
8 S. Colafrancesco1,2 and P. Marchegiani2,3
0
0
2 1 ASI-ASDCc/o ESRIN,Via G. Galilei snc, I-00040 Frascati, Italy Email: [email protected]
n 2 INAF- Osservatorio Astronomico di Roma via Frascati 33, I-00040 Monteporzio, Italy. Email: [email protected]
a 3 Dipartimento diFisica, Universit`a diRoma La Sapienza, P.le A.Moro 2, Roma, Italy Email: [email protected]
J
7 Received August 7, 2007 / Accepted January 11, 2008
1
ABSTRACT
]
h
p Context.Cosmicraysareconfinedintheatmospheresofgalaxyclustersand,therefore,theycanplayacrucialroleintheheating
- of their cool cores.
o Aims.Wediscussherethethermalandnon-thermalfeaturesofamodelofcosmicrayheatingofclusters’coresthatcanprovidea
r
t solutiontothecooling-flowproblems.Tothisaim,wegeneralizeamodeloriginallyproposedbyColafrancesco,Dar&DeRujula
s
(2004) and we show that our model predicts specific correlations between the thermal and non-thermal properties of galaxy
a
[ clusters and enables various observational tests.
Methods.Themodelreproducestheobservedtemperaturedistributioninclustersbyusinganenergybalanceconditioninwhich
1 the X-ray energy emitted by clusters is supplied, in a quasi-steady state, by the hadronic cosmic rays, which act as “warming
v
rays” (WRs). The temperature profile of the intracluster (IC) gas is strictly correlated with the pressure distribution of the
7
WRs and, consequently, with the non-thermal emission (radio, hard X-ray and gamma-ray) induced by the interaction of the
3
WRswith theIC gas and the ICmagnetic field.
7
Results. The temperature distribution of the IC gas in both cool-core and non cool-core clusters is successfully predicted from
2
. the measured IC plasma density distribution. Under this contraint, the WR model is also able to reproduce the thermal and
1
non-thermal pressure distribution in clusters, as well as their radial entropy distribution, as shown by the analysis of three
0
clusters studied in details: Perseus, A2199 and Hydra. The WR model provides other observable features of galaxy clusters: a
8 correlation of the pressure ratio (WRs to thermal IC gas) with the inner cluster temperature (P /P ) ∼ (kT )−2/3, a
0 WR th inner
correlationofthegamma-rayluminositywiththeinnerclustertemperatureL ∼(kT )4/3,asubstantialnumberofcool-core
: γ inner
v clustersobservablewiththeGLAST-LATexperiment,asurfacebrightnessofradiohalosin cool-core clustersthatrecoversthe
Xi observed one, a hard X-ray ICS emission from cool-core clusters that is systematically lower than the observed limits and yet
observable with thenext generation high-sensitivity and spatial resolution HXRexperiments likeSimbol-X.
ar Conclusions. The specific theoretical properties and the multi-frequency distribution of the e.m. signals predicted in the WR
model render it quite different from the other models so far proposed for the heating of clusters’ cool-cores. Such differences
makeit possible toproveor disproveour model as an explanation of the cooling-flow problems on thebasis of multi-frequency
observations of galaxy clusters.
Key words.Cosmology; Galaxies: clusters: theory; Cosmic Rays
1. Introduction et al. 2003, Piffaretti et al. 2005, Donahue et al. 2005,
see also Bregman 2004 for a review). The lack of cool-
Theoretical description of the intra-cluster (IC) plasma
ing gas with T <T in cluster cores is also supported
inner
in the central regions of many clusters predicts that it is
by the absence of line emission corresponding to gas be-
coolerthanintheiroutskirts,becauseitradiatesX-raysat
low T , as shown by high spatial resolution imaging
inner
such a rate that the plasma cooling time is much shorter
with Chandra (e.g. McNamara et al. 2000; Fabian 2000;
than the age of the cluster. However, such rapid cooling
Blanton et al. 2001; Allen et al. 2001; Lewis et al. 2002;
is notobservedandthe central-temperaturedepressionin
Blantonetal.2003)andhighspectralresolutionmeasure-
cluster’scoresisnotasdeepasexpectedonthebasisofthe
ments with the XMM-Newton RGS instrument(Peterson
plasma’s cooling rate: the observed central temperature
etal.2001,Tamuraet al.2001,Kaastraetal.2001;Kahn
T settles,infact,atafraction 1/3 1/2oftheouter
inner ∼ − et al. 2002; Peterson et al. 2002) and EPIC instrument
temperature T (see, e.g. McNamara 1997, Peterson
outer (Bo¨hringer et al. 2001, 2002; Molendi & Pizzolato 2001;
Send offprint requests to: S. Colafrancesco
2 S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores
Matsushita et al. 2002). These are referred here as the that the non-thermal pressure in the Perseus cluster (one
cooling–flow (CF) problems. ofthebeststudiedclusterswithcoolcoresandminiradio-
It has been widely recognizedthat some form of heat- halos)isquitesimilar(slightlysteeper)tothethermalone.
ing is necessary to quench the cooling flow and form a In this context, it has been shown that a CR population
warm–core in the IC gas structure. Available sources of in clusters that is radially distributed as the radial pro-
heating in clusters cores are provided by AGN jets and file of the IC gas, is able to recover the observed radial
lobes,pressurewaves,buoyantbubblesandcavities,intra- structure of the IC gas temperature in several clusters
cluster shock waves, leptonic and hadronic cosmic-rays. (Colafrancesco, Dar & DeRujula 2004). The CRs in this
The presence of a central radio source with jet and lobes, lasttheoryarenotconfinedtothe diskofthe Galaxy,but
the mechanical heating provided by pressure waves pro- permeateamuchlargerhalo,beingproducedby”cannon-
ducedbythe jettedAGNs,theformationofbuoyantbub- balls”travellingforkiloparsecsinthe interstellarmedium
bles filled with energetic plasma, the thermal conduc- of galaxies,decelerating by knocking out its constituents,
tion from the hot outer layers of the clusters, may cer- which are thereby accelerated to become CRs2 (see, e.g.,
tainlyalleviatetheCFproblems(seee.g.,B¨ohringeretal. DeRu´jula 2004 for a review, and references therein).
2002;Churazovetal.2002;Ruszkowski&Begelman2002, Beyond the debate on the cannonball picture (see, e.g.,
Fabian 2004, Voit & Donahue 2005, Bru¨ggen & Kaiser Hillas 2006; Dar & DeRu´jula 2006) and its relevance for
2002, Ruszkowski et al. 2004, Vernaleo & Reynolds 2006, the CR origin, we stress here that any mechanism which
Reynolds et al. 2005). producesradialdistributionsofWRssimilartotheprofile
Somedifficultieswithsolutionsalongtheselines,however, of the IC gas can provide quenching of cooling flows by
might stand out: a first difficulty is that not all cool- the action of WRs heating.
ing flow clusters contain a powerful radio source at their Inthis paper we considerandjustify, onthe basis ofa
center (e.g., the cluster RXJ0820.9+0752; Bayer-Kim et consistent picture of CR diffusion in cluster atmospheres
al.2002,andtheclustersA1650andA2244,Donahueetal. (see Appendix A), that the WR number density n (r)
WR
2005),noralargeamountofsphericallydistributedbuoy- can be, in fact, distributed with a radial profile that is
ant bubbles out to the cluster’s core boundary. Another similarto thatofthe IC gas,andfollowsthe generalform
difficulty is that the bremsstrahlung cooling rate is pro- n nα, i.e. it is proportional to a power α of the IC
portional to n2T1/2, with n the plasma’s electron num- gaWsRde∼nsiteyn (r). The totalamountofWRs inourmodel
e e e
ber density1 and T its local temperature: a suitable con- and the slope α of the n (r) profile are left as free pa-
WR
duction and/or heating and pressure-building mechanism rameters that will be subsequently constrained by fitting
must somehow adapt itself to this behaviour,particularly the temperature profileofthe IC gasineachspecificclus-
asafunctionofn (r),whichvariesbyordersofmagnitude ter core. The resulting CR density profile will be further
e
along the clusters’ radii. Other difficulties stand out with testedagainsttheentropyandpressureprofilesintheclus-
the spatial distribution of the energy deposited by AGN ter cores, the spectrum and surface brightness profile of
jets and cavities: on one hand, the energy released from the cluster radio halos and the future observations in the
AGN jets and cavities seems to be sufficient to quench hard X-ray and gamma-raybands.
cooling flow and increase the central gas entropy (e.g., WewillfirstshowthatthepresenceofWRsinclustercores
Sijacki & Springel 2006); on the other hand, it is not yet are able, in fact, to heat the IC gas and reproduce their
clear if such energy can be properly distributed to repro- radial temperature profiles. We will then start from this
ducetheobservedcool-coretemperatureprofiles(see,e.g., constraint to discuss the consequences that such a popu-
discussion by Vernaleo & Reynolds 2005, 2006, Heinz et lation of WR hadrons (protons), produced and stored in
al. 2006, Vernaleo & Reynolds 2007). Small duty cicles the atmospheres of galaxy clusters, has for the gamma-
and a higher fraction of AGNs in cluster cores (see Bird ray emission, for the hard X-ray (HXR) emission and for
et al. 2007) might however weaken the difficulties related the presence of diffuse radio emission of both cool-core
tothe AGNheatingscenarios.Inaddition,there seemsto and non cool-core clusters. We will finally show that it
be a smallAGN activity in groups(Dwarakanath&Nath is possible to use these predictions, in the framework of a
2006). multi-wavelengthobservationalstrategy,tosetconstraints
Sound waves and shocks do not seem to be efficient in on the amount and spatial distribution of WRs in clus-
quenching cooling flows (see, e.g., Fujita & Suzuki 2006). ters, and therefore disentangle between the WR model
Irrespective of the nature of the heating source, it discussed here and other models for the onset of warm-
seemsthatageneralpropertyoftheheatingagent–which cores in galaxy clusters.
derivesfromthesmoothnessandfromthesimilarityofthe The structure of the paper is the following: in Sect.2 we
heating distribution required to quench cooling in several delineate the theoretical framework that is able to model
clusters–isthattheymustbespatiallydistributedinthe the temperature structure of the cluster cool cores and
whole cluster core. In fact, it has been shown (Fabian & determine the main parameters (i.e., the radial distribu-
Sanders2006,Fabianetal.2006,Sanders& Fabian2007) tionandthe densityofWRs)thatfittheclustertempera-
1 Hydrogen in the plasma is fully ionized, and we are using 2 When referring to CRs in CF clusters, we use the expres-
here np ≈ne. sions “CRs” and “WRs” as synonyms.
S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores 3
tureprofiles.InSect.3wewillalsodiscussthedensityand The effect of the “warming rays” on a cluster’s tem-
pressuredistribution of WRs in the clusters’atmospheres peraturedistributionwillbecomputedunderthefollowing
that is consistent with the temperature profile. In Sect.4 simplifying assumptions:
we will apply our procedure to a sample of ten well stud- i) spherical symmetry;
iedclusters.These clustershavebeen chosenaccordingto ii) the radial distribution of the WR density is approxi-
the following criteria: i) 4 clusters which have cool cores mately proportional to a power α of the thermal plasma
and diffuse radio emission (these clusters certainly have density,i.e.n nα,withαandthenormalizationcon-
WR ∝ e
a population of CRs in their centers); ii) 2 clusters with stant being free parameters. In the Appendix A we show
radiohalos(i.e.withCRsintheiratmospheres)but with- thattheeffectofasimplemodelofcosmicraydiffusionin
out cool cores, in order to compare their properties with clusteratmospherescanprovidesuchkindofradialdistri-
theradio-active,cool-coreclusters;iii)4clusterswithcool bution, for quite general CR source terms;
cores but with no evident central diffuse radio emission iii) the temperature of the plasma in its “initial state”
in order to compare these non radio–active clusters with (as the cluster is born) is approximated by a radius-
thepreviousones.InSect.4wealsostudythethermaland independent constant value;
non-thermalpressurestructureoftheselectedclustersand iv) cooling by X-ray emission and heating and pressure-
wewillshowthatthepressureratioP /P foundatthe building by WRs are the dominant evolutionary agents.
WR th
cluster center correlates with the central gas temperature Flows and cluster ageing are relatively unimportant.
T providing evidence for the basic regulation mecha- Given this framework, the rate per unit volume
inner
nism induced by WRs. We will discuss the predictions of at which the IC plasma loses energy by thermal
our model for some specific clusters (like Perseus, A2199 bremsstrahlung emission of X-rays writes as:
and Hydra) for which detailed thermal and non-thermal
dǫ
pressure information is available as well as information =a[n (r)]2 T(r,t);
e
on their radialentropy distribution. In Sect.5 we will dis- dt(cid:12)
(cid:12)X p
cuss the expected gamma-ray luminosity and flux of the (cid:12)
cluster sample here considered that is correlatedwith the a=(cid:12)(cid:12) 211π3 e6√me G¯z¯
33 hm2c3
cool-coretemperaturestructurepredictedinourmodel.In r e
Sects.6 and7 we willpresentanalogousconsiderationsfor 4.8 10−24z¯ 1 ergcm3 , (2)
theexpectedradioandHXRemissionthatareconstrained ∼ × √keV s
by the cluster temperature structure. We will discuss the where z¯ is an average charge of the IC plasma (we have
differences betweenour model and other scenariosfor the approximatedheretheGauntfactorG¯byunity).Wewrite
cool-core heating in Sect.8 and we will resume our main theICgasnumberdensityasn (r)=n g(r), wherethe
e e0
conclusions in Sect.9. function g(r) contains all the radialdepen·dence of the IC
Throughout the paper, we use a flat, vacuum–dominated gas density and is obtained from X-ray observations of
cosmologicalmodelwithΩm =0.3,ΩΛ =0.7andh=0.7. each specific cluster. The rate per unit volume at which
the WRs deposit energy in the IC plasma is proportional
2. Modelling cluster cool cores to n (r) n (r) and it is therefore:
WR e
·
The virial expectation for the temperature of an un- dǫ
=b [n (r)] [n (r)]=b n n [g(r)]1+α ,(3)
magnetized plasma in a spherical cluster with mass M(r) dt(cid:12) · WR · e · WR,0· e0·
within a radius r, is: (cid:12)WR
(cid:12)
sin(cid:12)cewehaveassumedthattheWRnumberdensitywrites
Gm M(r) (cid:12)
kT(r) p as nWR =nWR,0 [g(r)]α, i.e. proportional to [ne(r)]α.
≃ 3r ·
The evolution of the temperature anywhere in the IC
−1
M(r) r plasma satisfies the energy-conservationrelation:
2keV , (1)
≃ 1013M⊙ 100kpc
(cid:20) (cid:21) (cid:20) (cid:21) dT(r,t) dǫ dǫ
3kn (r) = , (4)
(see, e.g. Colafrancesco,Dar & De Rujula 2004). The ob- e
dt dt(cid:12) − dt(cid:12)
servedoutertemperatureTouteroftheICplasma–atlarge (cid:12)(cid:12)WR (cid:12)(cid:12)X
radii where the cooling time should be longer than the with the initial condit(cid:12)ion T(r,(cid:12)0) = Ti. We can rewrite
(cid:12) (cid:12)
cluster’sage–isroughlycompatiblewiththevirialexpec- Eq. (4) as:
tation.In the inner regionsofmany clustersthe tempera-
dT(r,t) an (r)
e
tureissomewhatsmaller( 1/2 1/3ofT )thanthat = dt (5)
∼ − outer − √T (b/a)[n (r)/n (r)] 3k
obtained form Eq. (1), but significantly larger than the − WR e
cooling rate would imply (see, e.g., Peterson et al. 2001). and integrate, to obtain:
To reproduce the observed temperature profile from the
T/T 1 T(r,t) T
inner cool regionto the outer hot region,we will consider ln i f − + i =
a population of WRs spatially distributed in the cluster " pT(r,t)/Tf 1#−s Tf rTf
−
atmosphere (as obtained by solving a transportequation, p bnWR(r)ne(r)
(t t), (6)
see Appendix A) that efficiently heats the IC gas. 6n (r)kT − i
e f
4 S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores
where
kT (b/a)(n /n )g(r)α−1 2 (7)
f WR,0 e0
≡
is the s(cid:2)teady temperature first re(cid:3)ached in the central re-
gionoftheclusteratsmallradiir,wherethedensityn (r)
e
is high, and at asymptotically late time in the periphery,
where the density n (r) is vanishingly small. The r.h.s.
e
of Eq. (6) has been written in a way that shows how the
evolutionofT(r,t)isdeterminedbytheratiobetweenthe
WR deposition rate, bn (r)n (r), and the thermal en-
WR e
ergyper unitvolume,n (r)kT ,inthe inner regionwhere
e f
the cluster’s temperature is T .
f
The temperature evolution for a cluster similar to
A2199 (we assume here indeed the same IC gas density
profile of A2199 and its outer temperature) as described
by Eq.(6), is shownin Fig. 1 at epoch t=0,t /3,t and
U U
3t , wheret is the currentageofthe Universe.This fig-
U U
ureshowsthat,inourmodel,aclusterstartsasanisother-
malsystemathighredshifts(i.e.,t 0)andthenawarm-
≈
coredevelopswithtimeundertheinfluenceofcoolingand
heating effects balancing each other in order to provide a
stationary configuration at each epoch. We note that the
specificchoiceofthe initialepochti isnotveryimportant Fig.1. The evolution of the IC gas temperature profile pre-
inthesolutionofeq.(6),becausetheepocht(redshiftz)at dicted for a cluster similar to A2199 is shown at times t = 0
whichtheclusterisobservedismuchlarger(smaller)than (solid),t /3(dashed),t (dotted)and3t (dot-dashed),with
U U U
theinitialepocht (redshiftz z).Wehavenonetheless t beingtheageoftheUniverse.Weassumeaninitialtemper-
i i ≫ U
calculated the temperature evolution by assuming differ- ature of 4.2 keV and a density profile equal to that observed
ent values of the initial redshift, 50 z(ti) 200,and we in A2199. Values α = 0.83 and nWR,0 = 2.3·10−3 cm−3 are
found that no appreciable differenc≤e appea≤rs in the final assumed for thisspecific calculation.
state of the temperature radial distribution.
Incool-coreclustersthecentraldensityishighenough is therefore irrelevant in our calculations for spectral in-
fortheequilibriumbetweenX-raycoolingandWRheating dices s>2.
to have been reached at the cluster’s center in a time of The quantity N can be derived by assuming that
WR,0
theorderofitscurrentage.Insuchacase,Tf ≃Tinner,i.e. the WR heating rate b·nWR·ne provided by the energy
the presently observed inner-cluster temperature. For all lost by the WRs when they interactwith the IC medium,
the clusters we study, the cooling time in the outer parts recovers the IC gas temperature profile through eq.(6).
of the cluster is much longer than tU, so that Ti ≃Touter. The fundamental processes by which the warming
According to the value of Ti, of the IC gas density ne rays can heat the IC gas are mainly Coulomb and
and of the time t, Eq.(6) predicts that some clusters have hadronic interactions with the plasma nuclei and mag-
reached this stage, while others have not. netohydro-dynamical(MHD)wavesexcitedbyCRproton
The procedure here described is used to fit the tem- anisotropies.
perature profile of clusters with and without cool cores. For moderately relativistic CR nuclei with a chargez,
Given the bremsstrahlung energy loss rate, the tempera- velocityv=βcandLorentzfactorγ,theCoulombenergy-
ture radial profile of each cluster depends on the density loss rate is:
and on the spectrum n (E,r) of WRs in clusters.
WR dE 1 2m c2β2γ2
Kz2Z2 ln e β2 , (9)
− dt ≈ β I −
(cid:20) p (cid:21)
3. Warming Rays in cool core clusters
where Z2 is the (suitably averaged)squaredchargeof the
We consider a spectrum of relativistic WRs (protons) as plasma’s nuclei, K =4πn r2m c3 and r =e2/m c2
e e e e e ≃
given by: 2.82 fm. Here I =h¯ω , with ω =[4πn e2/m ]1/2 the
p p p e e
plasma frequency.3 The heating rate given in Eq. (9), for
N (E,r)=N E−s[g(r)]α , (8)
WR WR,0 typical plasma densities near the center of clusters (n
e
∼
with s typically in the range ≈ 2.3 − 3.3 (see, e.g., 3 It has been noticed by Colafrancesco, Dar & DeRujula
Marchegiani et al. 2007), and we use s = 2.7 as a ref-
(2004)thatahighlyionizedplasmaismuchmoreefficientthan
erence value, hereafter, for energies larger than Emin = non-ionized gas in slowing down the CR nuclei. Energy losses
mpc2 [1 + 3.4 10−5(kT/keV)] (see, e.g. Furlanetto & via Coulomb collisions with neutral atoms become significant
· ·
Loeb 2002). The value of Emax is usually very large and only for energy transfers larger than theirionization potential
S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores 5
10−2 cm−3),is 4timeslargerthanforneutralhydrogen. The total WR loss rate is givenby the sum of eqs. (9)
∼
For protons (z=1) with a typical WR energy (γ=2) and and (10)
for a plasma consisting of 0.93% H, 0.07% He (by
∼ ∼ dE
number) and traces of heavier elements, one finds Z2 n (A +A ) (11)
1.14 and dE/dt 10−17 GeV/s, for the quoted electro∼n (cid:18) dt (cid:19)tot ≃ e coul hadr
≈
number density.
with
Hadronic collisions between WRs and plasma nuclei
1 2m c2β2Γ2
constitute an energy-loss mechanism which is approxi- A = 4πr2m c3z2Z2 ln e β2
mately as efficient as that given in Eq. (9). Here we con- coul e e β (cid:20) Ip − (cid:21)
sider the collisions of WR protons with ambient protons, A 0.7σ cE , (12)
hadr pp
≃
thediscussionofproton–nucleonandnucleon–nucleoncol-
andisapproximatelyproportionalton (byneglectingthe
lisions being far too cumbersome to be justified by the e
logarithmic dependence on n contained in ω in eq.9).
small admixtures of nuclei heavier than protons in the e p
The value of b (see eq.3) is found by convolving the
WR and IC constituencies. For values γ > 2 of the inci-
dent proton, the pp cross section is σ 4 10−26 cm2, energylossratewiththeWRspectrumandequatingwith
pp
∼ · the warming rate
and it is dominantly inelastic. In practically all of these
collisions, the incident proton survives unscathed, as the
dǫ Emax dE
“leading”(mostenergetic)final-stateparticle,carryingon N (E,r) dE , (13)
WR
average ≈ 70% of the incident proton’s energy. Thus, a dt(cid:12)(cid:12)WR ≡ZEmin (cid:18) dt (cid:19)tot
proton’s energy-loss rate by hadronic collisions in a hy- (cid:12)
wh(cid:12)ich writes as
drogenic plasma is (cid:12)
b n (r) n (r) N gα(r)n (r)I , (14)
dE · WR · e ≡ WR,0 e A
0.7n σ cE (10)
e pp
− dt ∼ where
(Mannheim&Schlickeiser1994),thatreadsdE/dt 1.6 Emax
10−17 GeV/s, for our reference value of ne and∼γ = 2×. IA ≡ZEmin E−s(Acoul+Ahadr)dE . (15)
This is very close to the Coulomb energy loss previously
Since the WR density at the cluster’s center is given by
computed for the same reference parameters.
An additionalprocesswhichhas been discussedinthe Emax
literatureistheeffectoftheMHDwavesexcitedbyproton nWR,0 = dENWR,0E−s , (16)
anisotropy,whichmightoccurasrelativistic,chargedpar- ZEmin
ticle stream faster than the local Alfven speed (see, e.g. we obtain
Skilling 1971); these waves can be dissipated efficiently
I
A
in the background ICM via Landau damping and other b= . (17)
EmaxdEE−s
dissipative mechanisms(see, e.g.,Foote & Kulsrud1979). Emin
The effect of this mechanism in cluster atmospheres has
InserRting the previous expression of b in eq.(6) we obtain
beenstudiedindetailsbyLoewensteinetal.(1991):under
the value ofn requiredto fit the temperature profile,
the assumptions that the wavevelocity is large enoughto WR,0
and in turn the pressure of the WRs as
heat the IC gas significantly, and that the Coulomb and
hadronic heating can be neglected, these authors found 1 γmax
P (r)= m c N (γ,r)βγv(γ)dγ , (18)
that the thermal pressure becomes overwhelmed by the WR 3 p WR
Zγmin
CR proton pressure. Therefore, they concluded that such
CR heating mechanism alone is not able to stabilize the where γ =Ep/(mpc2).
cooling flows.
Following the previous considerations, we assume 4. The effect of WRs in cluster cores
therefore that the Coulomb and hadronic heating are the
dominant sources and we search for the amount and spa- 4.1. Temperature profiles
tial distribution of WRs required to produce the temper-
The solution of eq.(6) provides a fit to the cluster’s tem-
ature structure observedin a sample of clusters with cool
perature profile by adjusting the free parameters α and
cores(i.e. where heating is required)and radiohalos(i.e.,
n . We have applied this analysis to ten well stud-
where CRs are required). In Sect. 4.3 we will check our WR,0
ied clusters, eight of which are cool-core (CC) clusters
results by comparing the non-thermal and thermal pres-
and two are non cool-core (NCC) clusters (see Table 1).
sures predicted in our model with those observed in the
Specifically,weconsidered4clusterswithcool-coreswhich
same clusters.
have central radio emission (usually of the class of mini
I.Thecorrespondingsmallimpactparametersimplyrelatively radio-halos): these are A2199, Perseus, A2390 and RX
small cross sections. FortheICplasma, therole of I isplayed J1347.5-1145.Forcomparison,wealsoconsideredtwowell
by themuch smaller quantityIp. knownnon cool-coreclusters with radiohalos (Coma and
6 S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores
Cluster z Tinner Touter Notes Ref.
keV keV
A262 0.0162 0.95 2.4 CC [1]
A2199 0.030 1.6 4.2 CC; RH [2]
A133 0.057 2.05 4.61 CC; Relic [1]
Perseus 0.0179 3.04 6.7 CC; RH [3]
Hydra 0.0539 3.1 4.0 CC [4]
A1795 0.0625 3.5 6.5 CC [1]
A2390 0.2304 4.89 9.81 CC; RH [1]
RX J1347.5-1145 0.451 7.04 19.4 CC; RH [5]
Coma 0.023 8.2 8.2 NCC; RH [6]
A2163 0.203 14.6 14.6 NCC; RH [7]
Table 1. The list of clusters analyzed. Redshift z, inner observed temperature Tinner and outer observed temperature Touter
are given. The last two columns show which cluster has a radio halo (CC= cool-core; NCC= non cool-core; RH= radio halo)
and the references of the data we used for the fit. [1] Vikhlinin et al. (2005); [2] Johnstone et al. (2002); [3] Churazov et al.
(2003); [4] David et al. (2001); [5] Allen et al. (2002); [6] Briel et al. (1992); [7] Elbaz et al. (1995).
A2163) and some of the best studied clusters with cool- This correlationcanbe wellrepresentedwithapower-law
cores but no evident diffuse radio emission, i.e. A262, function P /P (kT )δ with δ 2/3.
WR th inner
∝ ≈−
A133 (which has a relic radio emission feature), Hydra Such correlation is due to the fact that in clusters with
and A1795. cooler inner temperature, a high IC gas density triggers
ForeachclusterinTab.1,wefoundthevaluesofαand fast gas cooling; therefore, a stronger warming action
n whichbest fit the observedtemperature profile,by (compared to the thermal energy) is required to quench
WR,0
assuming that initially (i.e. at high z when the cluster the cooling and to adjust the temperature profile to the
formed) each cluster was isothermal with temperature T observedonebytheenergy-conservationrelationineq.(6).
i
equal to the outer observed temperature, and that the ThecentralpressureratioP /P thatisrequiredinor-
WR th
evolutionofthe temperature proceedsaccordingto eq.(6) dertoequilibrategascoolingandthensettletheICgasat
and finishes at the redshift z at which each cluster is ob- theinnertemperaturevalueT isobtainedfromeq.(7)
inner
served. and reads
The WR density n (r) found through this procedure is
WR
the one required to heat the IC gas at the level shown by 1 a PWR
P /P =
WR th
the temperature profile by using eq.(6). 2√kTinner(cid:18)b(cid:19)nWR,0
We notice that the WR density in cool-core clusters 1 a
is required to be flatter (i.e. with values α < 1) than the = εWR (19)
2√kT b h i
IC gas density to provide the heating rate necessary to inner(cid:18) (cid:19)
quench the gas cooling and reproduce the cluster’s tem-
The correlation P /P (kT )−2/3 is explained
peratureprofile.Ontheotherhand,clusterswithoutcool- WR th ∝ inner
by the combined temperature dependence of the term
coresrequireα 1becausetheydidnotyetdevelopedgas
cooling conditio≈ns (see discussion at the end of Sect.2). ∝ (kTinner)−1/2 and of the quantity (ab)hεWRi ∼
(kT )−1/6,whichdependsonT throughthelower
inner inner
limit of integration E (T ) in the expressions of b
min inner
4.2. Pressure and density of WRs and ε (see Fig.3). Note that the T dependence
WR inner
h i
of b and ε arises only from the spectral properties of
The ratio of the central WR pressure PWR to the IC gas WRs in houWrRmiodel, while the scaling with (kT )−1/2
thermal pressure P is found by using eq.(18) and the inner
th arises from the condition set in eq.(6).
expressionP =2n (r)kT(r) (by including the contribu-
th e
Theothercorrelation,whichisnaturallyrelatedtothe
tion of both thermal protons and electrons). The value of
previousone,isfound–asexpected–betweenthecentral
thebest-fitcentralpressureratioP /P foundforeach
WR th
WR density n and the value of kT (see Fig.4).
cluster considered in this study is reported in Table 2. WR,0 inner
In fact, from eq.(7) we obtain that
Theresultsofourfittingproceduretotheclustertem-
perature profiles indicate that there is a strong correla-
tionbetweentheinnerclustertemperatureTinner andthe b nWR,0 2
kT = . (20)
pressure ratio PWR/Pth at the cluster center (see Fig. 2). f (cid:18)a · ne0 (cid:19)
S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores 7
Cluster α nWR,0 PWR/Pth Fγ Lγ FHXR
cm−3 cm−2 s−1 erg s−1 erg cm−2 s−1
A262 0.83 2.20×10−3 1.23 3.89×10−9 1.43×1042 3.87×10−14
A2199 0.83 2.31×10−3 0.92 8.43×10−9 1.08×1043 3.06×10−13
A133 0.84 4.56×10−4 0.77 7.30×10−10 3.53×1042 6.10×10−15
Perseus 0.91 4.98×10−4 0.54 2.20×10−8 9.91×1042 1.59×10−13
Hydra 0.97 6.24×10−4 0.57 3.46×10−9 1.49×1043 2.57×10−14
A1795 0.96 5.55×10−4 0.50 3.17×10−9 1.86×1043 2.41×10−14
A2390 0.94 2.21×10−4 0.41 1.41×10−10 1.39×1043 6.17×10−16
RX J1347.5-1145 0.89 9.20×10−4 0.33 5.07×10−10 2.37×1044 8.66×10−16
Coma 1.00 1.07×10−5 0.29 1.65×10−8 1.23×1043 1.67×10−13
A2163 1.00 1.42×10−5 0.21 1.31×10−9 9.71×1043 4.85×10−15
Table 2. Col.1: cluster name; Col.2: α; Col.3: nWR,0 (in units of cm−3); Col.4: the pressure ratio PWR/Pth at the cluster
center; Col.5: the gamma-ray flux F in units of cm−2 s−1; Col.6: the gamma-ray luminosity L in units of erg s−1. (The
γ γ
gamma-ray luminosity and flux are integrated in the 0.1–10 GeV band). Col.7: the HXR flux F in units of erg cm−2 s−1
HXR
integrated in the10–50 keV band.
Fig.2. Thecorrelation ofthecentralpressureratioP /P
WR th
with the inner temperature of the cool core T in keV
inner
is given for each cluster in Tab.1. The fit P /P ∝
(kTinner)−2/3 is shown as a solid line. Blue dots reWfeRr tothcool- Fig.3.Thecontributionoftheterm(ab)hεWRi(dashedcurve)
tothecorrelationofthepressureratioP /P withtheinner
core clusters and green dots to non cool-core clusters. WR th
temperatureofthecoolcoreT isshown.Thesolidcurveis
inner
theconvolutionof theterms(a)hε i and(kT )−1/2 (see
b WR inner
for r 0. Since the WRs are normalized to obtain T text for details).
f
→ ∼
T , we can write
inner
I peratures this dependence is I /I (kT )−1.7. The
nWR,0 ∼a kTinner ·ne0· IE (21) conclusion is that E A ∝ inner
A
p n
where IE = EEmmianxdEE−s and IA is defined in eq.(15). WneR0,0 ∝(kTinner)−1.2, (22)
Then the ratio I /I depends on E and, as a conse-
R E A min
quence, on kT ; we found that for our range of tem- which is confirmed by our results (see Fig. 5).
inner
8 S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores
For Perseus we find that a central ratio P /P
WR th
≈
0.54 is required in order to reproduce the inner tempera-
ture profile; such a value is very well consistent with that
foundbetweenthenon-thermalandthethermalpressures
in the analysis of Sanders & Fabian (2007). It is also re-
markable that our model is able to reproduce the steeper
spatialprofile of the non-thermalpressure w.r.t.the ther-
mal pressure profile found by these authors in their anal-
ysis of the central regions of Perseus (see Fig.6). We note
also that the non-thermal pressure we find in this cluster
islowerthanthethermalone:inthis respect,wecancon-
clude that our model has not the problems shown by sce-
narioofCRheatingprovidedbywavesexcitedbyprotons
anisotropy (see Loewenstein et al. 1991). In our frame-
work, the non-thermal pressure component found in the
core of Perseus can be entirely attributed to WRs. These
results bring our WR heating model to a strong consis-
tency level with the results of the available X-ray obser-
vations (e.g. Sanders & Fabian 2007). As a consequence
Fig.4.Thecorrelation ofthecentralWRdensitynWR,0 with
the inner temperature of the cool core T . Blue dots refer
inner
to cool-core clusters and green dots tonon cool-core clusters.
Fig.6. The radial profile of the WR pressure (dashed curve)
of Perseus is compared with that of the thermal gas (solid
curve). The radial profile of the WR pressure resembles the
Fig.5. Thecorrelation of theratio nWR,0/ne0 with theinner radial profile of the non-thermalpressure found by Sanders &
temperatureofthecoolcoreTinner.Thedashedlineshowsthe Fabian (2007).
trend nWR,0/ne0 ∝(kTinner)−1.2.
4.3. Pressure profiles: the cases of Perseus and A2199 of the presence of WRs in the core of Perseus cluster, we
expect a non-thermal HXR emission due to ICS of the
We compare the pressure profiles of the (non-thermal) secondaryelectronsfromthecentralregionofPerseus:we
WRs and of the thermal components obtained in our will discuss in Sect.7 the predictions of our model for the
model by fitting the IC gas temperature profile to the HXRemissioninthe2-10keVbandwheretheinstruments
data available for Perseus and A2199. on board Chandra and XMM-Netwon are sensitive.
S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores 9
We have also studied the energy density profile of the
WRsandofthethermalICgasinA2199.TheWRenergy
density has been evaluated as
γmax
E (r)=m c2 N (γ,r)(γ 1)dγ (23)
WR p WR
−
Zγmin
andagaindependsontheWRspatialdistributionthatfits
theclustertemperatureprofile.ThethermalICgasenergy
density is evaluated as E = 3n (r)kT(r) (by including
th e
the contribution of both thermal protons and electrons).
The comparison of the two energy densities as evaluated
in our model is shown in the upper panel of Fig.7. Our
model predicts that the WR energy density at the center
of A2199 is approximately equal to the IC gas thermal
energy in order to fit the cluster’s temperature profile.
Similarly to the case of Perseus, we find a radial profile
of the WR energy density that decays faster with radius
with respect to the thermal energy density.
It is possible to recoverthe energy density ratio of A2199
as presented in the model of Guo & Oh (2007) by notic-
ing that the minimum energy cutoff chosen by these last
authors is 706 MeV, i.e. much higher than the value
≈
of E assumed in this paper (see Sect.3). In fact, when
min
we cut the WR spectrum at this higher value of E ,
min
we recover an energy density ratio which is quite close to
the value found in Guo & Oh (2007). We verified that
the residual difference in this ratio is due to the marginal
effect of the thermal conduction in their model.
4.4. Entropy profiles
The WR heating in the cluster core can be also respon-
sible for the flattening of the IC gas entropy profile as
observed in cool-core clusters. We have compared the en-
tropy profiles predicted in our model for Perseus, A2199
and Hydra to the available data for these clusters (see
e.g., Churazov et al. 2003, Caffi et al. 2004, David et al.
2001, respectively) and we found a very good agreement
(see Fig.8). This is expected, indeed, because our model
is able to reproduce the cluster temperature profile given
the observed density profile of the cluster and therefore
their combination S = kT /n2/3 which is referred to
inner e
as the IC gas entropy. This analysis bring further robust-
nesstotheabilityofourmodeltoexplainthestructureof
clusters’ cores in terms of a balance between WR heating Fig.7. The radial profile of theWR energy density of A2199
and IC gas cooling. iscomparedwiththatofthethermalgas.Thetwopanelsrefer
to i) a proton spectrum with γ =1+3.4×10−5(kT/keV)
min
(see,e.g.,Furlanetto&Loeb2002);ii)aprotonspectrumwith
5. Gamma-ray emission of cool core clusters
E =706 MeV.
kin,min
The distribution of WRs that heat the IC gas to recover
the observed temperature profiles in their inner regions
inevitably produces a gamma-ray emission due to vari- by the decay of the π0 γγ produced in the interac-
→
ous emission mechanisms: i) pp π0 γ+γ; ii) inverse tions between relativistic protons and the thermal gas
→ →
Comptonscattering(ICS)duetosecondaryelectronspro- target protons (see, e.g., Blasi & Colafrancesco 1999;
ducedbypp π± e±;iii)non-thermalbremsstrahlung Marchegianietal.2007forthedetailsofthecalculations).
→ →
emission of the same secondary electrons. The contribution of secondary electrons to the gamma-
From the WR spectra discussed in this paper (see ray emission via ICS of CMB photons and non-thermal
Sect.4), we calculated the gamma-ray emission produced bremsstrahlung is negligible w.r.t. the π0 γγ flux (see
→
10 S. Colafrancesco and P. Marchegiani: Cosmic rays in cluster cool cores
Sect.2 used to fit the temperature profile of the cluster.
Therefore,the cluster’s gamma-rayemission is univocally
determined by this condition and it is strongly related to
thedensityandtemperaturestructureofthecluster’scool
core.
The total gamma-ray flux and luminosity consistent
with the cool-core temperature profile of the considered
clusters are reported in Table 2. The cluster gamma-
ray luminosity predicted in our model shows an over-
all increase with the value of T even though with
inner
some scatter. The total cluster gamma-ray luminosity
due to pp π0 γγ decay scales with the den-
→ →
sity of the WRs and of the IC gas according to L
γ
∝
n (r)n (r)R3, and it is integrated out to the radius at
WR p
whichwe see targetthermalprotonsfromthe clusters’X-
ray images. The condition imposed by the WR heating
model to fit the inner temperature profile yields a scal-
ing L √kT (a/b)n2 R3 which can be rewritten as
γ ∝ inner e0
L √kT (a/b)L /(kT)1/2andyieldsascalingL
γ inner X γ
∝ ∝
(kT )β(a/b) by using the observed L Tβ relation
inner X
∼
(see Arnaud 2005 for a review) and a value kT
inner
≈
(1/3 1/2)kT. The scaling of L with (kT )1.3 that
γ inner
−
derives form the previous considerationsreproduces quite
well the overall increase of the gamma-ray luminosity of
the ten clusters in Tab.1 as a function of their inner core
temperature kT .
inner
However, a scatter in the L T relation shown in
γ inner
−
Fig.9 is indeed expected because the cluster gamma-ray
luminositydependsonthedifferentvaluesofαandonthe
differentvaluesofthe integrationvolume R3 chosenfor
∝
each specific cluster. The choice of different integration
volumes for the considered clusters are motivated by the
different extensions of the target protons in the thermal
gas to produce effective pp π0 γγ collisions, as in-
→ →
dicated by the available X-ray data for each cluster (see
Tab.1).AsystematicincreaseofL withT ishowever
γ inner
shownbythedataandcanbeunderstoodbytheprevious
scaling-law considerations.
We have also calculated the gamma-ray flux expected
from each cluster in Table 2 and we comparedthe results
withtheGLAST-LATinstrumentsensitivity(seeFig.10).
Fig.8. The radial profile of the IC gas entropy S of Perseus
While all the gamma-ray fluxes are lower than the avail-
(upperpanel), A2199 (mid panel) and Hydra(lower panel) as
ableEGRETlimits(Reimeretal.2003),itisevidentthat
predictedintheWRmodel.Thecurvesreproducetheobserved
6 clusters with F 3 10−9 photons cm−2 s−1 (A262,
entropyprofilesofthethreeclustersasderivedbyChurazovet γ ≥ ×
A2199,Perseus,Hydra,A1795andComa),and 7 clusters
al. (2003) for Perseus, by Caffi et al. (2004) for A2199 and by
withF 1.2 10−9 photonscm−2 s−1 (the previousclus-
David et al. (2001) for Hydra. γ
≥ ·
ters plus A2163) could be detected by the GLAST-LAT
experiment in 1 year and 5 years at the 5σ confidence
also Marchegiani et al. 2007) and, therefore, we do not level, respectively.
consider them in the following discussion. The possible gamma-ray detection of these clusters will
Notethat,atvariancewithpreviousworksthatconsid- allowustoprobeindetailsourmodelfortheWRheating
eredthe non-thermal(radio,HXR and gamma-ray)emis- of the cluster cores and consequently set constraints on
sion of CRs in clusters’ atmospheres (e.g., Colafrancesco the parameters n and α of the WR heating model.
WR,0
& Blasi 1998, Blasi & Colafrancesco 1999, Miniati et al. Westressthatthegamma-rayemissionevaluatedhere
2001a, 2001b, Pfrommer & Ensslin 2004, Marchegiani, for cool-core clusters refer to the truly diffuse component
Perola & Colafrancesco 2007, Pfrommer et al. 2007), arisingfrompp π0 γγdecay.Wedonotconsiderhere
→ →
the density of the WRs is not a free parameter in our the possible contribution due to the gamma-ray emission
model, because it is fixed by the procedure presented in fromthe centralengineofthe radiogalaxiesand/orfrom
