Heating Rate Profiles in Galaxy Clusters

0.34 MB·English

by  Edward C. D. Pope

Mon.Not.R.Astron.Soc.000,1–13 (2005) Printed5February2008 (MNLATEXstylefilev2.2)
Heating Rate Profiles in Galaxy Clusters
Edward C.D. Pope1,2⋆, Georgi Pavlovski1, Christian R. Kaiser1, Hans Fangohr2
1School of Physics & Astronomy, University of Southampton, UK, SO17 1BJ
2School of Engineering Sciences, Universityof Southampton, UK, SO17 1BJ
6
0 5February2008
0
2
n ABSTRACT
a
J Inrecentyearsevidencehasaccumulatedsuggestingthatthegasingalaxyclusters

5 is heated by non-gravitationalprocesses.Here we calculate the heating rates required
to maintaina physicallymotived mass flow rate, in a sample of sevengalaxyclusters.

1 We employ the spectroscopic mass deposition rates as an observational input along
v withtemperatureanddensitydataforeachcluster.Onenergeticgroundswefindthat
6 thermal conduction could provide the necessary heating for A2199, Perseus, A1795
9
and A478. However,the suppressionfactor, of the clasicalSpitzer value, is a different
0
functionofradiusforeachcluster.Basedontheobservationsofplasmabubbleswealso
1
calculate the duty cycles for each AGN, in the absence of thermal conduction, which
0
can provide the required energy input. With the exception of Hydra-A it appears
6
that each of the other AGNs in our sample require duty cycles of roughly 106 −
0
/ 107 yrs to provide their steady-state heating requirements. If these duty cycles are
h
unrealistic,thismayimplythatmanygalaxyclustersmustbeheatedbyverypowerful
p
Hydra-Atypeeventsinterspersedbetweenmorefrequentsmaller-scaleoutbursts.The
-
o suppression factors for the thermal conductivity required for combined heating by
r AGN and thermal conduction are generally acceptable. However, these suppression
t
s factors still require ‘fine-tuning‘ of the thermal conductivity as a function of radius.
a As a consequence of this work we present the AGN duty cycle as a cooling flow
:
v diagnostic.
i
X Key words: hydrodynamics, cooling flows, galaxies: active, galax-
ies:clusters:individual:(Virgo, A2199, Perseus, Hydra-A, A2597, A1795, A478)
r
a
1 INTRODUCTION There is observational evidence that the temperature
profiles of galaxy cluster atmospheres are well described by
The radiative cooling times of the hot, X-ray emitting gas
the same mathematical function across a range in redshift
in the centres of many galaxy clusters may be as short as
(e.g. Allen et al. 2001). Given such similarity it is possible
106 years.Intheclassical model(seee.g.Fabian1994,fora
thattheatmospheresofgalaxyclustersareinaquasi-steady
review), where radiative losses occur unopposed, a cooling
stateandthatobservableparameterssuchastheradialtem-
flow develops in which the gas cools below X-ray temper-
perature profile do not vary signicantly over the lifetime of
atures and accretes onto the central cluster galaxy where
a cluster. This universal temperature profile would be diffi-
it accumulates in molecular clouds and subsequently forms
culttosustainifthedensityprofilesvariedsignificantlywith
stars. However, studies have demonstrated both a lack of
time.Inaddition,iflargequantitiesofgasweredepositedin
the expected cold gas at temperatures well below 1-2keV
the central regions one might expect that the density pro-
(e.g.Edge2001)andthatspectoscopically determinedmass
fileswouldbemuchmorecentrallypeakedthanobservations
deposition rates are a factor of 5-10 less than the classi-
suggest. The culmination of this argument is that, at least
cally determined values (Voigt & Fabian 2004). These find-
inthecentralregions, theinward flowrateofmassmustbe
ings suggest that the cluster gas is being reheated in order
roughlyindependentofradiustoensurethatmassisnotde-
toproducetheobservedminimumtemperaturesandthere-
posited at anyparticular location which would significantly
duction in excess star formation compared to expectations.
alter thedensity profile.
Asyetitisunclearwhetherthisheatingoccurscontinuously
Oneparticularproblemwiththisisthatallofthemass
or periodically.
flowinginthisregionmustbedepositedattheclustercentre.
Therefore,unlessthemassflowrateissmallthiswouldresult
⋆ E-mail:[email protected] in large quantities of gas at thecluster centre.
2 E.C.D. Pope, G. Pavlovski, C.R. Kaiser, H. Fangohr
In contrast, spectroscopically determined mass depo- effectsofmagneticfieldsaretakenintoaccountistoinclude
sition rates suggests that mass is deposited at a roughly a suppression factor, f, in the Spitzer formula, which indi-
constant rate throughout the cluster, at resolved radii catestheactual valueasafraction ofthefull Spitzervalue.
(Voigt & Fabian 2004). However, a roughly constant mass Severalauthorshavefoundtemperatureprofilesinsome
deposition rate is indicative of a mass flow rate that is galaxy clusters that are compatible with thermal conduc-
roughly proportional to radius, rather than constant as the tion, and values of the suppression factor that are physi-
density profiles suggest. Yet, if such a flow were persistent cally meaningful (e.g. Zakamska & Narayan 2003). In con-
this proportionality would result in density profiles which trast, more detailed work in which the Virgo cluster was
contradict the observational evidence by being more cen- simulated using complete hydrodynamics, including radia-
trally peaked. tivecooling andheatingbythermalconductionhasdemon-
Ifwearetoreconciletheimplicationsofthedensitypro- strated that thermal conduction cannot prevent the oc-
filesandmassdepositionratesthenonepossibleexplanation curence of a cooling catastrophe, in this particular exam-
is that the mass flow rate exhibits two different asymptotic ple(Pope et al.2005).Furthermore,themassflowratesun-
properties.Thatis,neartheclustercentreorwithinthecen- dersuchcircumstancesarenotconstantwithradiussothat
tralgalaxythemassflowrateshouldberoughlyconstantso massbuildsupinparticularareasandthereforechangesthe
thatthedensityprofilesareessentiallyleftunchanged,while densitydistribution,makingit muchmore centrally peaked
themassdeposited attheclustercentreissufficientlysmall than observed in real clusters.
toagreewithobservations.Atlargerradiithemassflowrate To answer the questions regarding the necessary heat-
should still be proportional to radius and satisfy the obser- ing rates and mechanisms involved in galaxy clusters we
vationsonresolvablescaleswhichimplyaconstantmassde- construct a simple model based on two main assumptions:
position rate. The relationship between the mass flow rates galaxy clusters are in approximate steady state, and the
and the mass deposition rates are defined section 3. mass flow rates fulfill the constraints provided by observa-
The two main candidates for heating clus- tionsatbothlargeandsmallradii.Toensurethis,theradial
ter atmospheres are: Active Galactic Nuclei (AGN) behaviour of the mass flow rates is modelled using an ob-
(e.g. Tabor & Binney 1993; Churazov et al. 2001; servationally motivated, but empirical function consistent
Bru¨ggen & Kaiser 2002; Bru¨ggen 2003) and thermal with the observations. Using mathematical functions fitted
conduction (e.g. Gaetz 1989; Zakamska & Narayan 2003; tothetemperatureanddensitydataofsevengalaxyclusters
Voigt et al. 2002; Voigt & Fabian 2004). Heating by AGN wederivetherequiredheatingrateswithinregionswhichare
is thought to occur through the dissipation of the internal consistentwiththoseforwhichthemassflowrateswerede-
energyofplasmabubblesinflatedbytheAGNatthecentre termined. We also compare the observed energy currently
of the cooling flow. Since these bubbles are less dense than available in the form of plasma bubbles with the heating
the ambient gas, they are buoyant and rise through the requirements for each cluster. From the heating rate pro-
intracluster medium (ICM) stirring and exciting sound fileswecalculatethethermalconductionsuppression factor
waves in the surounding gas (e.g. Ruszkowski & Begelman as a radial function for each cluster in order that we may
2002;Fabian et al.2002).Thisenergymaybedissipatedby determinewhich clusters could be heated by this process.
means of a turbulent cascade, or viscous processes if they The plan for this paper is as follows. In Section 2 we
are significant. Deep in the central galaxy other processes give the details of the parameters used to fit observational
such as supernovae and stellar winds will also have some temperature and density data for seven galaxy clusters. In
impact on theambient gas. Section 3and 4wedescribe themodel usedto estimate the
However,AGNareonlyperiodicallyactivewhichcould required heating, and thermal conduction suppression fac-
result in similarly periodic heating rates, thus making the tors. The heating rates we have calculated are compared
possibility of a totally steady state unlikely. In this case with observational estimates of both AGN heating and a
one could imagine a scenario in which a quasi steady-state combinationofAGNheatingandthermalconductioninSec-
is possible where temperature and density profiles oscillate tions5and6.InSection7wesummarizeourmainfindings.
around their average values. It is also possible that dissipa- The results are given for a cosmology with H0 =
tionoftheplasmabubblesmayoccurovertimescaleslonger 70kms−1Mpc−1, ΩM =0.3 and ΩΛ =0.7.
than the AGN duty cycle thus providing almost continous
heating (e.g. Reynoldset al. 2005).
Thermal conduction may also play a significant role in
transferringenergytowardscentralregionsofgalaxyclusters
giventhelargetemperaturegradientswhichareobservedin
2 FUNCTIONS FITTED TO OBSERVATIONAL
manyclusters.Infact,severalauthors(e.g.Voigt et al.2002;
DATA
Voigt & Fabian 2004) have shown that on energy grounds
alone it may be possible for thermal conduction to provide Inthissectionwegivedetailsofthefunctionsusedtoderive
thenecessary heating in some clusters. themassdepositionandheatingrateprofiles,thereferences
However,theexactvalueofthethermalconductivityre- to the observational data, and show a comparison with the
mains uncertain.Thetheoretical valuefor thethermal con- data in figure 1.
ductivity of a fully ionised, unmagnetised plasma is calcu- Wehaveonlyfittedourownfunctionstothedatawhen
lated by Spitzer (1962). Magnetic fields, which are thought theoriginalauthorshadnotincludedconfidenceintervalsfor
to exist in the cluster gas, based on Faraday rotation mea- their model parameters, but we give the functions used to
suresofclusters(e.g.Carilli & Taylor2002),maygreatlyal- fit all of the cluster atmospheres. The values of the fitted
terthisvalue.Thestandardmethodbywhichtheunknown parameters are given in tables 1 and 2.
Heating Rate Profiles in Galaxy Clusters 3
Name n0(cm−3) n1(cm−3) β0 β1 r0(kpc) r1(kpc)
Virgo 0.089±0.011 0.019±0.002 1.52±0.32 0.705 5±1 23.3±4.3
Perseus 0.071±0.003 0.81±0.04 28.5±2.7
Hydra 0.07±0.02 0.72±0.004 18.6±0.5
A2597 0.13±0.07 0.15±0.10 0.79±0.06 1.0±2.6 43±14
A2199 0.19 0.75 1
A1795 0.066±0.067 0.24±0.11 0.41±0.13 57±37 12±4.9
A478 0.153±0.019 0.55±0.05 0.41±0.34 145±32 6.6±1.9
Table 1. Summary of best fit parameters for density profiles with 1-sigmaerrors obtained from the least-
squares fitting procedure. We do not give errors the model function parameters for A2199 since we were
unabletoobtaintheobservational data.Seetable2forreferences.
The Virgo cluster‘s electron number density data was We fitted the temperature data for A2597, A1795
fitted byGhizzardi et al. (2004) using a double β-profile, and A478 using the same function as Dennis& Chandran
(2005),
n= [1+(rn/0r0)2]β0 + [1+(rn/1r1)2]β1. (1) T =T0− 1+(rT/1rct)2 δ. (7)
Forsimplicity,thePerseus(seeSanders et al.2004)and (cid:2) (cid:3)
Hydra(David et al.2001)densitydatawerefittedwithsin- The best-fit temperature profile for the Hydra cluster
gle β-profiles, is given by David et al. (2001) as a power-law,
r δ
n= [1+(rn/0r0)2]β0. (2) T =T0(cid:18)rct(cid:19) . (8)
TheA2199temperaturedatawasfoundbytheoriginal
We were unable to obtain the observational data for
authors(Johnstone et al.2002)tobedescribedbyapower-
A2199 and so relied on the fits presented by the original
lawwithoutacharacteristiclength-scale.Forconsistencywe
authors (Johnstone et al. 2002) who found that a simple
use equation (8) to represent the temperature distribution
power-lawwassufficienttodescribethedensitydistribution,
in this cluster.
Although we have provided 1-sigma errors for the fit-
r −β0 ted parameters that describe the temperature and density
n=n0(cid:18)r0(cid:19) . (3) distribution of the gas in galaxy clusters, we will not give
errorsforanyquantitiesderivedfromtheseparameters.The
ForA2597,A1795andA478(seeMcNamara et al.2001;
reason for this is that the errors and gradients of the tem-
Ettori et al. 2002; Sunet al. 2003, respectively) we employ
perature and density are all model dependent. In addition,
the function used by Dennis & Chandran (2005) to fit the
the calculations are complex, making the standard propa-
density distributions,
gation of errors impractical and error estimates themselves
n= n0 1 . (4) veryuncertain.Inaddition,althoughwecanmakeverycon-
[1+(r/r0)2]β0 [1+(r/r1)2]β1 servative estimates of the confidence limits, of a particular
derived quantity,they are too vague to beof importance.
As above, for the cluster gas temperature profiles we
haveonly fittedfunctions tothedata whentheoriginal au-
thors had not included confidence intervals, but give the
details for all clusters in oursample.
3 THE MODEL
Ghizzardi et al. (2004) found that the Virgo tempera-
turedata is best described by a Gaussian curve, 3.1 General heating rates
1 r2 Starting from the assumption that the atmospheres of
T =T0−T1exp(cid:18)− 2rct2(cid:19). (5) galaxy clusters are spherically symmetric, and in a quasi
steady-state,it ispossible toderivewhatthetimeaveraged
Churazov et al. (2003) fitted the Perseus temperature
heating rate, as a function of radius, must be in order to
data with thefunction,
maintain theobserved temperature and density profiles,
3
T =T0(cid:20)1δ++(cid:0)rr//rrcctt(cid:1)3(cid:21). (6) h=n2Λrad+ r12ddr(cid:20)4Mπ˙ (cid:18)25µkmbTp +φ(cid:19)(cid:21), (9)
(cid:0) (cid:1)
4 E.C.D. Pope, G. Pavlovski, C.R. Kaiser, H. Fangohr
Name T0(keV) T1(keV) rct(kpc) δ Reference
Virgo 2.4±0.1 0.77±0.10 23±4 1
Perseus 8.7±1.1 63±5 2.7±0.3 2
Hydra 2.73±0.07 10 0.12±0.01 3
A2597 4.1±0.1 2.0±0.2 16±10 0.7±0.4 4,5
A2199 1.0 1 0.29 6
A1795 7.2±1.9 5.1±2.2 40±34 0.5±0.7 7,5
A478 9.7±1.5 7.6±1.7 16±5 0.3±0.2 8,5
Table 2. Summary of temperature parameters fitted to the data with 1-sigma errors obtained from the
least-squares fitting procedure. References:-(1) Ghizzardi etal. (2004); (2) Sanders et al.(2004); (3) David
etal.(2001);(4)McNamaraetal.(2001)andRaffertyetal.(2005),inpreparation;(5)Dennis&Chandran
(2005); (6)Johnstone etal.(2002); (7)Ettorietal.(2002); (8)Sunetal.(2003). Wedonotgiveerrorsfor
A2199 since we were unable to obtain the observational data. We also do not give an error for the Hydra
temperaturescale-heightsincenonewasgivenintheoriginalpublication,Davidetal.(2001).
Figure1.Left:densitydataforallclustersexceptA2199forwhichweshowthebest-fit,sincewedidnothaveaccesstotheobservational
data. Right: temperature data except for A2199 and Hydra for which we give the fitted functions, since we did not have access to the
observational data.Thekeyintheright-handpanel appliestobothtemperatureanddensitydata.
where M˙ is themass flow rate, T is thegas temperature, n Consequently,themassflowrate,ataparticularradius,
isthegasnumberdensity,Λ isthecoolingfunctionandφ canberecoveredbysummingthecontributionsofthemass
rad
isthegravitationalpotential.Hereweassumeonlysubsonic deposition rate in each shell up to that radius. Henceforth,
flow of gas. we refer to the mass flow rate as theintegrated mass depo-
The mass flow rate through a spherical surface, radius sition rate.
r, is For subsonic gas flow the cluster atmosphere can be
assumed to be in approximate hydrostatic equilibrium. On
M˙ =4πr2ρvr, (10) thisbasis,weareabletoestimatethegravitationalaccelera-
where ρ is the gas density and vr is the gas velocity in the tionfromthetemperatureanddensityprofilesofthecluster
radial direction. atmosphere,
The difference between the rate at which mass enters
and leaves a spherical shell, of thickness ∆r, is called the
mass deposition rate, ∆M˙. In terms of the mass flow rate, dP dφ
=−ρ , (12)
themass deposition rate is
dr dr
dM˙
∆M˙ ≈ ∆r (11)
(cid:18) dr (cid:19) where P is the gas pressure and ρ is thegas density.
Heating Rate Profiles in Galaxy Clusters 5
The gas pressureis related todensity and temperature Therefore, the rate at which mass is deposited within
by theideal gas equation thejth shell is
P = ρkbT, (13) ∆M˙ = 4πrj2∆rnj2Λrad. (18)
µmp clas,j 5kbTj
2µmp
where µmp is the mean mass per particle. We assume that ThetotalX-rayluminosityemittedbythegaswithina
µ=16/27asappropriateforstandardprimordialabundance.
particularradiusiscalculatedbysummingthecontributions
Toavoidanomalieswhencalculatingspatialderivatives,
from all of the shells within this radius
we fit continuous analytical functions through the density
and temperature data in the previous section. This ensures
that we do not encounter any large discontinuities which LX(<r)= 5kb M˙clas(<r)T(r), (19)
may result in extremeheating rates. 2µmp
The only unknowns in equation (9) are then the inte-
grated mass deposition rate, M˙, and theheating rate, h. whereM˙clas(<r)istheintegrated mass deposition rateob-
tained by summing the contribution from each shell within
theradius, r.
If we include a heating term the expression equivalent
3.2 Calculating Mass Deposition rates to equation (15) becomes,
Intheclassical coolingflowpicture,whichassumesthatthe
5k
heating is zero everywhere, and ignoring the effect of the LX(<r)−H(<r)= b M˙coolT(r), (20)
gravitationalpotential,thebolometricX-rayluminosity,in- 2µmp
tegrated mass deposition rate and gas temperature are re-
whereM˙ istheintegratedmassdepositionrate,resulting
lated by integrating equation (9) (e.g. Fabian 1994) over a cool
fromtheexcessofcoolingoverheating.Thisintegratedmass
spherical volume. Analytically, the bolometric X-ray lumi-
depositionrateisdistinctfromtheintegratedclassicalmass
nositywithinagivenradiusissimplythevolumeintegralof
deposition rate which is only used to describe the situation
theradiative cooling rate perunit volume,
in the absence of heating.
With equation (16) becomes,
r
LX(<r)=4πZ0 n2Λradr2dr. (14) ∆(cid:0)LX,j−Hj(cid:1)= 25µkmbp(cid:0)M˙cool,j∆Tj+∆M˙cool,jTj(cid:1). (21)
Integrating the second term in equation (9) gives the
In practice, a realistic estimate of the mass deposited
reationship between bolometric luminosity, integrated clas-
into each shell, which takes into account heating, can be
sical mass deposition rate and temperature,
obtained by fitting models to the X-ray spectrum of each
shell or a larger region if desired. The spectroscopic mass
LX(<r)= 25µkmbpM˙clas(r)T(r). (15) danepionstietgioranterdatveailsuejucsatnabefitotbintaginpeadrainmeextearctilnytthhiesscaamseewanady
as for theclassical case above.
Forobservationaldatatheluminosityandclassicalmass
Generalising equation(19) to allow for heating we find
deposition rates at each radius are estimated by dividing
that if the spectroscopically determined mass deposition
the observed 2-d projection of the cluster into concentric rate, M˙ , is representative of the actual mass deposition
obs
shells. Using this method one is able to reconstruct the 3-d rate, M˙ , then onecan determine theheating rate from,
cool
properties ofa cluster.The contribution,tothetotalX-ray
luminosity,ofthejthsphericalshell,ofthickness∆r,isthen
5k 5k
calculated bydiscretizing equation (15), LX(<r)−H(<r)= b M˙obs(<r)T(r)= b M˙coolT(r).
2µmp 2µmp
(22)
∆LXj = 25µkmbp(cid:0)M˙clasj∆Tj+∆M˙clasjTj(cid:1). (16) depoIsfittihone srpateectirsoasctorpuiecarlelyprdeesetenrtmatiinveedovfatlhueerfeoarltmhaesms dases-
position rate, then the integral within the cooling radius is
In any given shell, the term involving ∆Tj is assumed simply theintegrated mass deposition rate within thecool-
tobesmallcomparedtotheterminvolving∆M˙ .Byig-
clasj ingradius,irrespectiveoftheradialdistribution.Therefore,
noring the ∆Tj term the resulting mass deposition rate is it would seem a sensible starting point for any calculations
larger than it would be if thechange in temperature across
oftheheatingratesingalaxyclusters.Toensureconsistency
the shell was taken into account. Therefore, the classical
betweentheoryandobservations,wemustemploythesame
value is the absolute maximum possible mass deposition
mass flow rate (integrated mass deposition rate) over the
rate. The luminosity of a given shell is then assumed to
sameregion,inourheatingmodels. Consequently,ifweuse
be(e.g. Voigt & Fabian 2004),
the observed spectroscopic mass deposition rate integrated
up to the cooling radius, then to be strictly accurate, the
∆LXj=4πr2∆rn2Λrad = 25µkbmTpj∆M˙clasj. (17) htheiastirnagdiruast.eswesubsequentlycalculateareonlyvalidupto
6 E.C.D. Pope, G. Pavlovski, C.R. Kaiser, H. Fangohr
3.3 Model Integrated Mass Deposition Rates 3.4 Thermal conduction
We now construct a model for the integrated mass deposi- Thermal conduction of heat from the cluster outskirts
tionrateingalaxyclustersonthebasisoftheobservational to their centres may provide the required heating of the
evidence.Althoughthemodelfunctionisessentiallyempiri- central regions without an additional energy source, like
caltherearethreemainconstraintstowhichitmustadhere. an AGN (e.g. Gaetz 1989; Zakamska& Narayan 2003;
These are: firstly, the fact that the observed spectroscopic Voigt & Fabian 2004). Of course, thermal conduction only
massdepositionrateineachshellisapproximatelyconstant transports energy from one region to another and does not
up to the cooling radius suggests linear dependence of flow lead to net heating. However, if we consider the case for
rate on radius. Secondly, the observed density profiles also which there is an infinite heat bath at large radii, then any
suggest that the mass deposition rate is negligible near the dropintemperatureat largeradii,duetotheinwardtrans-
cluster centre. Otherwise a pronounced peak would be ob- fer of thermal energy, is negligible.
served in the density profiles. Thirdly, the integrated mass Several 1-d models have been able to achieve
deposition rate at the cooling radius must be equal to the a steady-state using thermal conduction alone (e.g.
observed integrated mass deposition rate at that location. Zakamska& Narayan 2003) and the combined effects
A simple empirical form for the integrated mass deposition of thermal conduction and AGNs (e.g. Bru¨ggen 2003;
rate which is, in some way, consistent with theabove is Ruszkowski& Begelman 2002). We, therefore, compare the
requiredheatingrateswithanestimateoftheenergytrans-
portsuppliedbythermalconductiongiventhecurrentstate
of each galaxy cluster.
r 2 0.5 The volume heating rate for thermal conduction from
M˙(r)=M˙ (<r ) K+ , (23)
obs cool (cid:20) (cid:18)rcool(cid:19) (cid:21) an infinite heat bath is given by
1 d dT
where M˙obs(< rcool) is the integrated spectroscopic mass ǫcond = r2dr(cid:18)r2κdr(cid:19), (24)
deposition rate at the cooling radius, K is a constant and
where κ is the thermal conductivity and dT/dr is the tem-
r is thecooling radius.
cool
peraturegradient.
To estimate K we express it the following way, K =
(rK/rcool)2, where rK is a length scale corresponding to K. WetakethethermalconductivitytobegivenbySpitzer
(1962), but include a suppression factor, f(r), defined in
A suitable value for rK is the radius of the central galaxy,
theintroduction,totakeintoaccounttheeffectofmagnetic
roughly30kpc,sincetheeffectoftheinterstellarmediumis
fields,
likelytosignificantlyaltertheinflowofmaterial. Thisleads
to values for K ranging from 0.73, for Virgo, to 0.04, for
A478. 1.84×10−5f(r)T5/2
NotethatourexpressionforM˙ impliesthatdM˙/dr=0 κ= lnΛc , (25)
forsamllradii,meaningthatnomaterialisdeposited.How-
ever, at or near r =0 the material must be deposited since where lnΛc is theCoulomb logarithm.
For pure hydrogen the Coulomb logarithm is (e.g.
it cannot flow to negative radii. This can account for the
Choudhuri1998)
excess star formation which is observed to occur in these
regions. For reasonable gradients of the temperature distri-
b(9u)titohne,nttheendsesctoonidnfitneritmyaosnrttheendrisghtotzhearno.dTshidiseiomfpelqieusatthioant Λc =24πne(cid:18)8πkeb2Tne(cid:19)−3/2, (26)
ourmodelpredictsnegativeheatingrates,h,forsmallradii.
Inotherwords, themodelisonly valid outsidetheregion at where ne is the electron numberdensity.
theverycentresofclusters.Toindicatetherangeoverwhich Forasteady-statetoexist,theheatingbythermalcon-
ourmodelcanprovidephysicallymeaningfulpredictions,we duction must be equal tothe heating rate,
listinTable3,theminimumradiusforeachclusteratwhich
h=0.Wetakethisminimum radiusasthelowerintegration
ǫ =−h, (27)
cond
limit in our calculation of the total heating rates.
Foreach clustertheobservationsareinevitably limited where h is the required heating rate per unit volume, given
byspatialresolutionandinsomecasesthelowerintegration byequation (9).
limits,explainedabove,occuratradiiwhichareclosertothe This is essentially the same energy equation as solved
clustercentrethanthespatialresolutionlimits.Inprinciple, by (e.g. Zakamska& Narayan 2003), although we also al-
the spatial resolution presents a limit beyond which we are low for mass deposition, line-cooling and variations in the
unable to accurately describe the temperature and density Coulomb logarithm. Wethensolve equation(27),using the
of thecluster gas. However, because we havefitted analyti- observedtemperatureanddensityprofiles,todeterminethe
cal functions to the temperature and density data for each radial dependence of the suppression factor. This is differ-
cluster we can extrapolate the behaviour of the tempera- enttotheapproachof(Zakamska& Narayan2003).Intheir
tureanddensity,andhencetheheatingrates,fromthedata papertheyusethesameequationstocalculatetemperature
point nearest the cluster centre further towards the centre. anddensityprofilesthatareconsistentwithaconstantsup-
The results at r < rmin should be ignored as we have no pression factor. These derived profiles are only constrained
observational information to constrain them. byobservationsat theminimum andmaximum radii acces-
Heating Rate Profiles in Galaxy Clusters 7
sible to observations. We take the density and temperature 4.3 Required Heating rates and Thermal
profiles from the observed data and allow the suppression Conduction Suppression factors
factor to vary with radius.
Our function for the suppression factor is obtained by
integrating both sides of equation (27) overa spherical sur- The heating rates are shown in figure 2, for the form of
face and then rearranging for f, integratedmassdepositionratesgiveninequation(23).For
comparison, we also show the volume heating rate profiles
for unsuppressed thermal conduction.
r r2h(r)dr
f(r)= rmin . (28)
1.84×10R−5r2dT/drT5/2/lnΛc Figure 2 shows that the heating rates for the entire
sample of clusters exhibit similar profiles. This is because
Wenotethattheenergyfluxduetothermalconduction,
theradiativelosses most strongly dependupon thedensity,
κdT/dr, increases with radius so that, for an infinite heat
therefore so must the required heating rate. Since the ma-
bathatlargeradii,energymustalwaysbedepositedateach
jorityofthedensityprofilesaredescribedbyβ-profiles,this
smaller radius, rather than taken away.
similarity is expected.
Incomparison, itisclearthatthevolumeheatingrates
for unsuppressed thermal conduction vary from cluster to
cluster and do not share the same profile as the required
4 RESULTS:1 heating rates. Even for the clusters in which thermal con-
ductioncansupplythenecessaryenergy,thedifferentradial
4.1 Comparison of Luminosities
dependencesrequirethatthesuppressionfactorisfine-tuned
Toensurethatourestimatesarecompatiblewiththosecal- toprovidethecorrectheatingrateforallradii.Thisdemon-
culated from observations we compare the bolometric X- stratesthedifferentnatureofthephysicalprocessesinvolved
rayluminosities,withinthecoolingradius,determinedfrom in radiative cooling and thermal conduction, making a bal-
our functions fitted to the data and the observed luminosi- ance between the two hard to achieve.
ties presented in Bˆirzan et al. (2004). We assume half solar
metallicity for all clusters and use the cooling function as
Using equation (28) we calculate the thermal conduc-
given in Sutherland & Dopita (1993).
tion suppression factor, as a function of radius, for which
Table3demonstratesreasonableagreementbetweenthe therequiredheating rateisachieved.These results,plotted
results obtained using the fitted functions and those pre- in figure 3, suggest that thermal conduction could provide
sented in Bˆirzan et al. (2004). therequiredheatingratesinA2199,A478,Perseusandvery
nearly A1795. It seems impossible that thermal conduction
wouldbeabletotransportsufficientenergytowardsthecen-
tral regions for the remainder of thesample.
4.2 Integrated Mass Deposition rates
Itisevidentthatthesuppressionfactorprofilesaredif-
Since the values of the integrated mass deposition rates
ferent in almost every case, although A1795 and A478 ap-
that we use are critical to determining the heating rates,
pear to have certain features in common. The only com-
we present these first. In table 4 we compare the inte-
grated spectroscopic mass deposition rate, M˙spec(< rcool) mentraitisthattherequiredsuppressionfactortendstobe
(Bˆirzan et al. 2004), with the integrated classical mass de- largest near the centre, except for A2597 and Hydra. This
position rate, M˙ (< r ) (Fabian 1994) at the cooling suggeststhatevenintheregionswherethesuppressionfac-
clas cool
tor takes physically meaningful values (f <1), in order for
radius,andthecentralintegrated mass deposition ratepre-
thermal conduction to provide the necessary heating, the
dicted by equation (23).
parameters which determine the suppression factor require
From equation (20) it is clear that the spectroscopic
uniquefinetuningineachcluster.Wenotethatsuppression
mass deposition rates are indicative of the energy in-
factorsofgreaterthanunityareunphysicalforlaminarflows
jected into the cluster. Because of this, the spectroscopic
but are possible in mixing layers (e.g. Cho et al. 2003).
mass deposition rate should be less than the classical
value,depending upon the relative magnitudes of the heat-
ingandcooling,asisthecase.However,becauseofthetime Our suppression factors can be compared with those
taken to dissipate thisinjected energy into theambient gas found by Zakamska& Narayan (2003) who studied four of
this implies that the current spectroscopic mass deposition theclusters in oursample. They findsuppression factors or
rates are a function of the heating which has taken place 1.5, 2.4, 0.4 and 0.2 for Hydra, A2597, A2199 and A1795
overthe last few times 108 yrs. respectively. These values agree with our results in that we
The central integrated mass deposition rates predicted also find that thermal conduction must take unphysically
byequation(23)aresufficientlylowthatthetotalmassde- largevaluesforHydraandA2597andrealistic valueforthe
positedintheclustercentre,overaGyr,willnotsignificantly remaining two clusters. Voigt & Fabian (2004) also provide
alterthetotalcentralmass.Itshouldbenotedthatasmaller asuppressionfactor,roughly0.2-0.3,forA478whichiscon-
value of the constant, K, in equation (23), would result in sistent with, albeit lower than,the possible values we have
still smaller central integrated mass deposition rates. found for thesame cluster.
8 E.C.D. Pope, G. Pavlovski, C.R. Kaiser, H. Fangohr
Figure 2. Comparison of cooling and heating rate profiles for each cluster. The heating rates fall to zero at a radius which depends
uponthemassdepositionrate.Atthepointatwhichthisoccurs,wealsoterminateeachoftheothercurvessincewecannot determine
thegaspropertieswithinthisradius.
Heating Rate Profiles in Galaxy Clusters 9
Name LX(rcool)/1042ergs−1 LX(rcool)/1042ergs−1 rcool/kpc rmin/kpc
Virgo 10.7 9.8+−00..87 35 1
Perseus 550 670+−4300 102 3
Hydra 243 250+−1155 100 3
A2597 470 430+−4300 129 4
A2199 156 150+−1100 113 1
A1795 493 490+−3300 137 1
A478 1494 1220+−6600 150 3
Table 3. Comparison of bolometric luminosities derived from our functions fitted to the data (column 2)
withthosegiveninBˆirzanetal.(2004)(column3),thecoolingradius(column4)andthelowerintegration
limits(column5).
Name M˙spec(<rcool) M˙clas(<rcool) M˙spec(<rcool)rcroKol
Virgo 1.8+−10..216 10 1.3
Perseus 54+−4188. 183 4.7
Hydra 14+−97 315 1.3
A2597 59+−4400 480 3.2
A2199 2+−71.9 150 0.1
A1795 18+−1120 478 0.9
A478 150+−6608 570 6
Table4.Massdepositionrateparameters(allquantitiesinsolarmassesperyear).Theintegratedspectral
mass deposition rates are taken from Bˆirzan et al. (2004) and the classically determined values are from
Fabian (1994) and references therein. The central mass flow rates calculated using equation (23) are also
shownforcomparison.
5 COMPARISON OF REQUIRED HEATING The estimated AGN power injection rates are given in
RATES WITH OBSERVATIONS column2ofTable5.Tocomparetherequiredheatingrates
with the observational estimates of AGN heating in equa-
5.1 Comparison with AGN
tion (29) we must calculate thevolumeintegral of equation
Thetime-averaged mechanicalluminosity ofacentralAGN (9). From this we can estimate the heating luminosity re-
in each cluster is estimated using quired within a given volume to satisfy the requirementsof
theassumedsteady-state.Comparingthisvaluewiththatof
thepower output from AGNs at the centres of galaxy clus-
αPV
hL i= , (29) tersallowsustoestimatethefrequencywithwhichbuoyant
mech
t
plasmabubbles,identicaltothosecurrentlyobserved,must
where α=16. P is the ambient cluster gas pressure, V is beproducedinordertoprovidesufficientheatingwithinthe
the bubble volume and t is the estimated duty cycle for coolingradius.ThisisdefinedastheAGNdutycycle,which
an AGN. We assume initially that the duty cycle of each may be calculated using
AGN is 108 yrs. The factor 16 arises from 2 bubbles per
outburst,afactorof2fortheenergydissipatedintheshock 4π rcoolr2hdr t
expansion of thebubbleand 4for γ/(γ−1) where γ = 4/3 rmin = , (30)
RhL i τ
for relativistic gas. mech
This is simply an estimate of the rate at which energy wherer ,theupperintegrationlimit,isthecoolingradius
cool
is injected into the cluster, by the central AGN, and is not andrministheradiusatwhichtheheatingratetendstozero
related to any particular physical process by which this en- for our chosen descriptions of the mass flow rate, τ is the
ergyisdissipatede.g.theviscousdissipationofsoundwaves calculated duty cycle and t is the time defined in equation
or effervescent heating. (29).
10 E.C.D. Pope, G. Pavlovski, C.R. Kaiser, H. Fangohr
Figure3.Thermalconductionsuppressionfactorsforthesampleofsevenclusters.Thethick,solidlineshowsthemaximumphysically
meaningfulsuppressionfactor(f=1)forcomparison.
5.2 The Combined Heating by Thermal 6 RESULTS:2
Conduction and AGN
6.1 AGN Duty Cycles
Toestimatethetotalrateofenergyinjection byAGNsinto
theclustergas withinthecooling radiusweassumethatall The duty cycles (see Table 5) for Virgo and A478 are of
oftheenergyavailableinthebubblesisdissipatedwithinthe the order of 106 yrs which is very short compared to the
cooling radius. Furthermore, the integral of equation (24) predictedlifetimesofAGN(e.g.Nipoti & Binney2005,and
overasphericalsurfacegivestherateatwhichthermalcon- references therein). In contrast, the Hydra cluster requires
duction transfers heat across a radius, r, recurrent outbursts of magnitude similar to the currently
observed one only every 108 yrs, or so. The required duty
cycles for the remaining AGNs are of the order of 107 yrs.
dT Fromthisweconcludethatifthermalconductionisnegligi-
Lcond(r)=4πr2κdr. (31) bleand if this sample is representative of galaxy clusters in
general,thenmanyifnotallclusterswillprobablybeheated,
at certain points in time, by extremely powerful AGN out-
The maximum total rate at which energy can be in-
bursts. However, it is worthwhile pointing out that roughly
jectedintothisregionisthesumofthesetwoprocesses.We
71% of cD galaxies at the centres of clusters are radio-loud
comparethissumwiththeheatingraterequiredforsteady-
(Burns1990)whichislargerthanforgalaxiesnotatthecen-
state and allowing for mass deposition. Within the cooling
tres of clusters. This may suggest that the galaxies at the
radius, we can then determine the thermal conduction sup-
centres of clusters are indeed active more frequently than
pressionfactorrequiredtomaintainasteady-stategiventhe
othergalaxies.Itispossiblethatacombinationofthesetwo
currentAGNheating rate. Thissuppression factor iscalcu-
effectssatisfies theheatingrequirementswehaveidentified.
lated using
In contrast to the spectroscopically derived mass de-
position rates, the plasma bubbles, that still exhibit radio
emission and are currently observable, are probably more
L (<r)−hL i
f = heat mech , (32) indicative of the current heating rate rather than the heat-
Lcond(r) ing rate over the previous 109 years, or so. If this is true,
onlyHydraiscurrentlybeingexcessivelyheatedinoursmall
whereL (<r)istherequiredheatingrateforthespheri- sample.
heat
calvolumewithinr,hL iisthecurrentmechanicalpower Our results also show that the gas outside the cool-
mech
of the AGN assuming a dutycycle of 108yrs. ing radius requires significant amounts of energy to allow a