The polytropic approximation and X-ray scaling relations: constraints on gas and dark matter profiles for galaxy groups and clusters

1.1 MB·English

by  Pedro R. Capelo

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed30January2012 (MNLATEXstylefilev2.2)
The polytropic approximation and X-ray scaling relations:
constraints on gas and dark matter profiles for galaxy
groups and clusters
Pedro R. Capelo,1⋆ Paolo S. Coppi1,2 and Priyamvada Natarajan1,2
2 1Department of Astronomy, Yale University,PO Box 208101, New Haven, CT 06520-8101, USA
1 2Department of Physics, Yale University,PO Box 208120, New Haven, CT 06520-8120, USA
0
2
n 30January2012
a
J
ABSTRACT
7
2 The X-ray properties of groups and clusters of galaxies obey scaling relations that
provide insight into the physics of their formation and evolution. In this paper, we

] constraingasanddarkmatterparametersofthesesystems,bycomparingtheobserved
O
relationsto theoreticalexpectations,obtained assuming that the gas is in hydrostatic
C equilibriumwiththedarkmatterandfollowsapolytropicrelation.Inthisexercise,we
. vary four parameters: the gas polytropic index Γ, its temperature at large radii, the
h
darkmatterlogarithmicslope atlargeradiiζ andits concentration.When comparing
p
the model to the observed mass-temperature relation of local high-mass systems, we
-
o find our results to be independent of both the gas temperature at large radii and
r of the dark matter concentration. We thus obtain constraints on Γ, by fixing the
t
s dark matter profile, and on ζ, by fixing the gas profile. For a Navarro-Frenk-White
a dark matter profile, we find that Γ must lie between 6/5 and 13/10. This value is
[
consistent with numerical simulations and observations of individual clusters. Tak-

2 ing 6/5 . Γ . 13/10 allows the dark matter profile to be slightly steeper than the
v Navarro-Frenk-Whiteprofileatlargeradii.Uponincludinglocallow-masssystems,we
3 obtain constraints on the mass-dependence of Γ and on the value of the gas temper-
7 ature at large radii. Interestingly, by fixing Γ = 6/5 and ζ = −3, we reproduce the
5 observed steepening/breaking of the mass-temperature relation at low masses if the
5 temperature of the intercluster medium is between 106 K and 107 K, consistent with
.
1 numerical simulations and observations of the warm-hotintergalactic medium. When
1 extrapolated to high redshift, the model with a constant Γ reproduces the expected
1 self-similar behaviour. Given our formulation, we can also naturally account for the
1 observed, non-self-similar relations provided by some high-redshift clusters, as they
v: simplyprovideconstraintsonthe evolutionofΓ.Inaddition,comparingourmodelto
i theobservedluminosity-temperaturerelation,weareabletodiscriminatebetweendif-
X
ferentmass-concentrationrelationsandfind thata weak dependence ofconcentration
r onmassis currentlypreferredbydata.Insummary,this simple theoreticalmodelcan
a
still account for much of the complexity of recent, improved X-ray scaling relations,
provided that we allow for a mild dependence of the polytropic index on mass or for
a gas temperature at large radii consistent with intercluster values.
Key words:
galaxies: clusters: general – X-rays: galaxies: clusters – dark matter – methods: ana-
lytical
1 INTRODUCTION supernovæ (e.g. Riess et al. 1998; Perlmutter et al. 1999)
and the cosmic microwave background (e.g. Komatsu et al.
Clusters of galaxies are important probes of cosmology,
2009), in building thecurrently favoured “concordance cos-
as they complement other observations, including Type Ia
mology” model. As the largest gravitationally bound sys-
tems in the Universe, galaxy clusters trace the growth of
structure,which in turnstrongly dependson thecosmolog-
⋆ E-mail:[email protected]
(cid:13)c 0000RAS
2 P.R. Capelo, P.S. Coppi and P. Natarajan
ical parameters (for recent reviews on clusters in a cosmo- nosity, on the other hand, can be very helpful. If one has a
logical context, see e.g. Voit 2005; Allen, Evrard & Mantz model for the temperature profile, then one can also calcu-
2011).However,galaxyclusterscanprovideusefulcosmolog- latetheaveragetemperatureandobtainatheoreticalmass-
icalinformationonlyiftheirmass,composedpredominantly temperature(M–T)relationthatcanbeeasilycomparedto
of dark matter (DM), can be measured accurately enough. observations(e.g.Vikhlininetal.2006;Vikhlininetal.2009:
Given the elusive nature of DM, this can be achieved only V09;Sunetal.2009:S09).Thisiswhatwelargelypursuein
through indirect methods, e.g. by measuring orbital veloc- this paper. Since these average quantities are usually com-
ities of galaxies (e.g. Zwicky 1933, 1937), by gravitational puted after excising the central region of the cluster, any
lensing techniques (e.g. Zwicky 1937), or through studying uncertainty in the properties of the inner parts of the DM
the intracluster medium (see reviews by e.g. Sarazin 1986; halo becomes unimportant.On the other hand,any change
B¨ohringer & Werner2010). in the outer regions of the cluster can produce significant
Theintraclustermediumisthehot,X-rayemittinggas effectsontheaveragequantities.Forthisreason, wedonot
that to first order is in thermal equilibrium with the clus- limit ourselves to the NFW profile, but use its generalised
tergravitational potential.Oneofthemostcommonly used version by Bulbul et al. (2010: B10), in which the DM log-
models for galaxy clusters utilises a King (1962) profile to arithmic slope at large radii is not fixed. For completeness,
describethetotalgravitationalpotentialwiththeisothermal giventheuncertaintyinthemass-concentrationrelation,we
gas in thermal equilibrium. Solving thehydrostatic equilib- also let the concentration vary, thus having two DM free
rium (HE) equation,theresultingprofile, referred toas the parameters.
β-model (Cavaliere & Fusco-Femiano1976, 1978), is widely
The choice of the three primary gas quantities and the
utilised to describe radial profiles of clusters. However, de-
method for constraining them vary between authors. Ko-
tailed X-ray observations of individual clusters have shown
matsu & Seljak (2001: KS01), in their description of a uni-
clear radial gradients in their temperature profiles, hence
versalgasprofile,assumethatthegasdensityprofileisequal
calling into question the assumption of isothermality.
tothatofDMintheouterregionsoftheDMhalo(assumed
Oneofthesimplest extensions oftheisothermal model to be described byan NFW profile), obtaining a constraint
is to assume a polytropic relation, wherein gas density and on both the polytropic index and the central gas temper-
pressure(ortemperature)arerelatedbyasimplepowerlaw. ature. Their assumption is reasonable and very powerful,
This description naturally provides radial temperature gra- as it gives constraints on the polytropic index itself, which
dients.Apolytropicrelationisindeedexpectedinidealadi- has not been done in other studies. However, it predicts an
abatic processes, with the polytropic index in those cases M–T relation which is not in good agreement with recent
being equal to the ratio of specific heats at constant pres- X-ray observations. Also, the gas profile is not completely
sureandconstantvolume.Thisvalueoftheexponent,how- described, as it needs a second constraint to fix the central
ever,doesnotappeartobeconsistentwithobservations(e.g. density.Komatsu&Seljak(2002:KS02)providesuchacon-
Bode, Ostriker & Vikhlinin 2009 and references therein), straintbyrequiringthatthegasdensityintheouterregions
suggestingthatthepolytropicrelationisnotanequationof oftheDMhaloisequaltotheDMdensitymultipliedbythe
state. Although the origin of the observed polytropic rela- cosmic baryon fraction.
tionisstillapuzzle(butseeBertschinger1985forapossible
Ascasibaretal.(2003),investigatingthedifferencesbe-
explanation),HE-polytropicsolutionshavebeenconsidered
tween polytropic relations and β-models, fix instead the
since the early 1970s (e.g. Lea 1975). Hybrid descriptions
polytropic index as an output from their numerical simula-
havealsobeenconsidered,wherethepolytropicrelationand
tionsandobtainconstraintsonthecentralgastemperature
theβ-modeldescriptionforthegasdensityareemployedsi-
and density, by imposing the gas density to be zero at in-
multaneously (Markevitch et al. 1998; Ettori 2000; Sander-
finity and the baryon fraction to never exceed the cosmic
son et al. 2003; Ascasibar et al. 2003).
value. Ascasibar et al. (2006: A06), studying the origin of
Numerical prescriptions of DM have advanced to the
self-similar scaling relations, useasimilar method,with the
pointwherewearefairlycertainabouttheprofileoftheDM
slight difference that the baryon fraction at threetimes the
potential (e.g. Navarro, Frenk & White 1996: NFW). Suto,
DMscale radius must beequal to thecosmic value.
Sasaki & Makino (1998) first derived theHE-polytropicso-
lution in the case of a gravitational potential described by More recently, a series of papers (Ostriker, Bode &
an NFW profile. The concept is simple: once the DM has Babul 2005; Bode et al. 2007; Bode et al. 2009) have ex-
been completely described (in the case of NFW, by only plored in detail models with polytropic gas in HE with a
twoparameters,e.g.virialmassandconcentration),thegas DMpotential,addingadditionalphysicssuchasstarforma-
isanalyticallydescribedwithjustthreefreeparameters:the tion and feedback, in the form of simple recipes. Similar to
polytropic index and the values of two gas quantities, cho- theapproachofAscasibaretal.(2003)andA06,Ostrikeret
senamongste.g.density,pressureandtemperature,atsome al.(2005)takethepolytropicindexasagiven(fromnumeri-
givenradius(usuallyzeroorinfinity).Extractingthesethree calsimulationsandobservations)andadditionallyconstrain
gasparametersandcheckingresultswithobservationsisnot the central gas pressure and density by imposing conserva-
entirely straightforward. Radial profiles of individual clus- tionofenergyandbymatchingtheexternalsurfacepressure
ters(e.g. Pratt et al. 2007; Arnaudet al. 2010) can provide tothemomentumfluxfromtheinfallinggasatthevirialra-
somehints,butresolutionlimitsandscatterpreventusfrom dius. The second constraint arises from a sharp truncation
obtainingaccurateconstraintsonthethreeunknownscalars intheDMprofileatthevirialradius,adoptedbyOstrikeret
introducedabove.Moreover,anyuncertaintyintheDMpro- al. (2005), but not contemplated in KS01, KS02, Ascasibar
file makes theconstraining more difficult. More global scal- et al. (2003), A06, nor in this paper. By adding terms to
ing relations between cluster mass, temperature and lumi- the energy conservation equation, they are able to discrim-
(cid:13)c 0000RAS,MNRAS000,000–000
Polytropic approximation and scaling relations 3
inate between simple feedback recipes when matching the 2 POLYTROPIC GAS IN GALAXY GROUPS
resulting models to observations. AND CLUSTERS
Inthispaper,weassumethatanidealpolytropicgasis 2.1 Formulation
in HE with the DM in groups and clusters of galaxies and
exploretheconsequencesof thepolytropicsolution, consid- In this section, we model groups and clusters of galaxies as
eringdifferentwaystospecifythethreeprimarygasquanti- isolated, spherically symmetric, composite systems of stars,
tiesandthetwofreeDMparameters.Thisisdoneassuming DM and ideal gas. We describe the DM with a generalised
no prior knowledge of either the polytropic index or the NFW density profile (B10),
boundaryconditions,andonlypartialknowledgeoftheDM
profile.Thus,ourapproachdiffersfrompreviousworkinthe ρ (r)= δcρc , (1)
following ways. (i) We do not assume the DM to be NFW- DM (r/r )(1+r/r )β
s s
like at large radii; instead, we adopt a generalised version
where 1 < β ≤ 3, r is the scale radius, δ is the charac-
s c
of the NFW profile (B10), and are able to discriminate be- teristic(dimensionless)density,ρ (z)=3H2E2(z)/(8πG)is
tweendifferentvaluesofthelogarithmic slopeoftheDMat c 0
the critical density of the Universe at redshift z, E(z) ≡
large radii.(ii) Wedonotfixthevalueofthepolytropicin- H(z)/H ≡ [Ω (1+z)3+(1−Ω −Ω )(1+z)2+Ω ]1/2
0 0 0 Λ Λ
dexfromsimulations(Ascasibaretal.2003;A06;Ostrikeret and G is the gravitational constant1. Note that the loga-
al. 2005; Bode et al. 2007, 2009) norconstrain it with some
rithmic slope at small radii is the same as for the standard
assumption on the gas profile at large radii (KS01; KS02);
NFW profile, which is a particular case of this profilewhen
instead, we leave the polytropic index as a free parameter
β = 2. We further assume the gas to follow a polytropic
and, by comparing our model to observations, constrain it relation, with the pressure2 P, density ρ and temperature
asafunction of mass andredshift. (iii) Wedonot force the T related as follows: P(r) = P [ρ(r)/ρ ]Γ or, equivalently,
0 0
gas radial profile to behave in any way, either by imposing ρ(r) = ρ [T(r)/T ]1/(Γ−1), where P , ρ and T are the
0 0 0 0 0
its profile to be equal to that of DM at large radii (KS01;
centralvaluesof pressure,densityand temperature,respec-
KS02) or by requiring it to vanish at infinity (Ascasibar
tively, and d(logP)/d(logρ) ≡ Γ > 1 is the polytropic in-
et al. 2003; A06); instead, we simply require the temper- dex3,assumedtobethesameatallradii(i.e.thepolytropic
ature of the gas to never be negative. By leaving the gas
relation is complete).
temperature at large radii as a free parameter, we are able
Weassumethegastobeinglobal,thermally-supported
toconstrainthevalueofthetemperatureoftheintracluster
HEwiththetotalgravitationalpotentialφ,thatis,toobey
medium(oroftheinterclustermedium,dependingontheas-
theHEequation,∇P =−ρ∇φ,everywhere.Wealsoneglect
sumptions we make) by matching our theoretically derived
the effects of gas self-gravity, as it has been demonstrated
M–T relation to observations in the low-mass regime (i.e.
(e.g. Suto et al. 1998) that the contribution of the gas to
groups). (iv) We do not fix the concentration from simula-
the total gravitational potential does not lead to signifi-
tions (KS01; KS02; Ascasibar et al. 2003; A06; Ostriker et
cantly different results, and additionally ignore the stellar
al.2005;Bodeetal.2007,2009),butinsteadinvestigatethe
gravitational potential, as it has been shown (see Capelo,
effectsofdifferentmass-concentrationrelations;wediscrim-
Natarajan & Coppi 2010 for the low-mass case) that gas
inate between different recipes by matching our model to
profiles are affected by stars only in the very inner regions,
recently observed luminosity-temperature (L–T) relations which are not important for the results of this work4. The
(Maughanetal.2011:M11).Forthislastpartofthestudy,a
total gravitational potential is thus dominated by that of
secondconstraintwasnecessary,andweimposedthebaryon
DM, i.e. φ=φ , which is given by
DM
fraction within the virial radius to be equal to the cosmic
value.
1 (1+r/r )β−2−1
We believe the flexibility of our model to be its main φ (r)= φ0β−2(r/r )(1+s r/r )β−2 if β6=2, (2)
strength.Moregeneralassumptionsthaninpreviousstudies DM  s s
helpusobtainmoreconstraints:onthepolytropicindex;on φ0ln[1+(r/rs)]/(r/rs) if β=2,
theDMlogarithmic slopeat large radii; onthegas temper- where φ0 ≡ φDM(0) = −4πGδcρcrs2/(β −1) and we have
ature at large radii; on concentration. Also, we do extend used Equation (1) to solve Poisson’s equation, ∇2φ =
DM
the model to higher redshift and obtain constraints on pa- 4πGρ , after imposing the Dirichlet boundary condition
DM
rameter evolution. φ (+∞)=0.
DM
Throughout thepaper,weusethefollowing cosmologi-
calparameters(WMAP5;Komatsuetal.2009):Ω =0.258,
Ω = 0.0441, Ω = 0.742, H = 100h km s−1 0Mpc−1 = 1 Thisprofileandβ shouldnotbeconfusedwiththeβ-modelof
b Λ 0 Cavaliere&Fusco-Femiano(1976,1978).
70h70 kms−1 Mpc−1,h=0.719(i.e.h70 =1.027).Theval- 2 Unlessotherwisestated, pressureisthermalgaspressure,den-
ues and uncertainties of other parameters (e.g. σ8) are not sityisgasmassperunitvolume,andtemperatureisgastemper-
relevant in our study. ature.
The outline of the paper is as follows. In Section 2, we 3 In other notations, the polytropic index n is related to Γ via
n=1/(Γ−1).
formulatetheproblemandstudyindetailtheconsequences
4 To be precise, the stellar gravitational potential has a strong
of assuming hydrostatic equilibrium and a polytropic rela-
effect onthe central gas quantities (e.g. Capelo etal.2010), but
tion.InSection3,wederivegalaxygroupsandclustersscal-
notontheluminosityandaveragetemperature,whencalculated
ing relations and compare them to recent observations to with an excision of the central region (see Section 3). For the
obtainconstraintsongasandDMparameters.Weconclude same reason, we do not investigate DM profiles with different
in Section 4. logarithmicslopesatsmallradii(e.g.Mooreetal.1998).
(cid:13)c 0000RAS,MNRAS000,000–000
4 P.R. Capelo, P.S. Coppi and P. Natarajan
ThesolutiontotheHEequationisthengivenby(Suto mass M . See Appendix A for the calculation of the re-
vir
et al. 1998) lations between different overdensity masses and radii, for
both the NFW case and for the more general B10 case.
1 Moreover, some authors use equivalent but different nota-
ρ(r)=ρ0(cid:20)1− Γ−Γ 1∆gas(cid:18)1− ln(1r+/rrs/rs)(cid:19)(cid:21)Γ−1 (3) tainodns∆fovirr(tzh)e≃vi[r1ia8lπo2v+er8d2e[Ωnsmit(yz:)−M1v]ir−=394[Ωπrmv3(irz∆)−vir1Ω]2m]/(Ωz)mρ(cz/)3.
for β =2 and by The dependence of the concentration on redshift and
virial mass can beparametrized as
1
ρ(r)=ρ01− Γ−Γ 1∆gas1− β−1 2(cid:16)rr1s+(cid:16)1rr+s(cid:17)rβrs−(cid:17)2β−−21Γ−1 wcvhirer=ectAhe(cid:18)dfiDmMMenMcsivoirn(cid:19)lecsBs(p1a+ramz)ectCe,rscA,cB,cC andthesc(a7le)
(4) mass M depend on the cosmological parameters. For our
c
fiducial model and the rest of this section, we use results
for β 6=2, where the dimensionless gas parameter
from recent DM-only simulations of relaxed haloes (c =
A
9.23,cB = −0.09,cC = −0.69,Mc = 2×1012h−1M⊙; Duffy
∆ =−φ ρ0 = 4πGδcρcrs2µmp, (5) et al. 2008: D08)5, which used WMAP5 cosmological pa-
gas 0P0 kBT0(β−1) rameters, but we will consider other published relations in
k is the Boltzmann constant, m is the proton mass and Section 3.
B p
µ=0.62 is themean molecular weight of the gas (e.g. Par- Finally, onecalculates thecentraldensityρ0 byimpos-
rish et al. 2011), assumed to be completely ionized and to ing
havemean metallicity [Fe/H] =−0.5.
In both cases, we used the boundary condition ρ(0) ≡ rvir
ρ(r)4πr2dr=f M =(f −f )M , (8)
ρ , where ρ can in principle have any non-negative value gas vir b star vir
0 0 Z0
(although we will see later in this section how to con-
wheref andf arethevirial gasandstellar mass frac-
strain it). B10 imposed a more stringent boundary condi- gas star
tion, respectively. For f , we will consider two limiting
tion, T(+∞) ≡ T∞ = 0, and obtained a solution which is cases: star
a particular case of Equation (4) when (Γ−1)∆ /Γ = 1.
gas
Oursolutionisamoregeneralcasethatallows foranynon-
negativevalueofT∞,whichcanbethenfixeddependingon f =5.7×10−2 Mvir −0.26, (9)
thephysical situation (see Section 3.1.2 for more details). star 5×1013M⊙
(cid:18) (cid:19)
We now specify the problem further and consider a
following the work of Giodini et al. (2009)6, or f = 0,
system with a given virial mass M at a given redshift star
vir dependingonwhetherwewanttoquantifytheeffect(orlack
z, assuming that the virialisation was reached right before
thereof) on the mass budget of stars in low-mass systems.
the time at which we observe it (recent-formation approx-
Sinceweareneglectinggasself-gravity,thechoiceofρ (or,
imation). In this paper, unless otherwise stated, we define 0
equivalently,off )doesnotaffecttheresultsontheM–T
virial mass as the total (stars, DM and gas) mass within star
relationinSection3.1.However,itdoesaffecttheresultson
the virial radius, defined below. The DM density profile
the L–T relation in Section 3.2. For the remainder of this
given in Equation (1) can then be written, via the equal-
section, we will assume f =0 (i.e. f =f ).
ity δ =f ∆ c3 /[3f(c )], as star gas b
c DM vir vir vir All parameters but one have now been fixed. The cen-
tral pressure or, alternatively, the central temperature can
ρ (r)= MDM(rvir) rsβ−2 , (6) in principle haveany value. Inreality, there is anothercon-
DM 4πf(cvir) r(r+rs)β straint that helps usfix T0.
AgeneralattributeofthepolytropicsolutionoftheHE
wherethevirialradiusr =[3M /(4π∆ ρ )]1/3 andthe
vir vir vir c equation is that it is physically meaningful only between
concentration c = r /r are both assumed to be inde-
vir vir s zero anda maximum radiusr (Lea1975). Beyond r ,
max max
pendent of β; ∆ is the virial overdensity, described by
vir thetemperaturebecomesnegative,makingthesolution un-
∆vir(z) ≃ 18π2 +82[Ωm(z)−1]−39[Ωm(z)−1]2 (Bryan physicalinthatregion7.Theexistenceofamaximumradius
& Norman 1998); the matter fraction Ω (z) = Ω (1 +
m 0 can be explained in the same way one explains the finite
z)3/E2(z); f = M (r )/M = 1−f is the virial
DM DM vir vir b
(that is, within the virial radius) DM mass fraction; f =
b
Ωb/Ω0 is the virial baryon mass fraction, set equal to the 5 WeassumeEquation(7)tobevalidforany1<β≤3andfor
cosmicvalue,assumedconstantwith redshiftandmass;the any1012M⊙≤Mvir≤1016M⊙.
function f(x) = ln(1+ x) − x/(1+ x), when β = 2, or 6 We approximate their results for the full sample of COSMOS
f(x) = [1/(β−1)+[−x+1/(1−β)]/(1+x)β−1]/(β−2), selected groups only, extrapolating to higher masses, where the
stellarfractionbecomesnegligibleanyway,andassumingthatthe
when β 6= 2. We note that several authors use different
definitions for mass and radius. In general, one can define stellarfraction withinr500 is equal to the stellar fractionwithin
r such that M = 4πr3 ∆ ρ /3, where ∆ is the over- rvir.Thislastpointisanapproximation,sinceweknowthatthe
∆c ∆c ∆c c c c gasmassincreaseswithradius,butitdoessoonlyslightly.
density ratio. This ratio can be either a fixed number (e.g. 7 ForsomevaluesofΓ(e.g.Γ=3/2),thedensityispositivealso
200, 500, etc.) or can be replaced by the just defined ∆vir, for r > rmax, but the temperature becomes negative regardless
so that r∆c is the virial radius rvir and M∆c is the virial ofthevalueofthepolytropicindex.
(cid:13)c 0000RAS,MNRAS000,000–000
Polytropic approximation and scaling relations 5
truncation radius in classical polytropes with Γ > 6/5: a
finite r means that the gravitational potential is strong
max
enough to bind the gas in a finite region; an infinite r
max
means that the pressure of the gas is high enough to make
the gas either unbound or bound only in a limitless region.
However, theanalogy between theprofiles in this work and
thoseofclassicalpolytropescannotbetakentoofar:inclas-
sical polytropes, r =+∞ if Γ≤6/5, regardless of other
max
gas quantities; on the other hand, the polytropic solution
given by Equation (3) or (4), for a given polytropic index,
can havea finiteor infinitemaximum radius, dependingon
othergasquantities,asitisexplainedbelow.Thedifference
arisesfromthedifferentnatureofφ:inthispaper,thegrav-
itational potential is fixed and external, due to DM; in the
classical polytropecase,φisentirelyduetogasself-gravity.
BystudyingEquations(3)and(4)further,wenotethat
r , once all the other parameters (virial mass, redshift,
max
cosmologicalparameters,etc.)havebeenfixed,dependsonly
Figure1.DependenceofthepolytropicsolutiontotheHEequa-
on the gas temperature. The hotter the gas is, the larger
tion (Equations 3 and 4) on the gas central temperature, for
rmaxbecomes.Inparticular,byimposingtheweakboundary three different values of β: 3/2 (upper, red bundle of curves), 2
condition T∞ ≥ 0, there exists a threshold minimum value (NFWcase;middle,blackbundleofcurves)and5/2(lower,blue
of the central temperature, bundle of curves). The black, vertical, dot-dashed line denotes
the DM scale radius rs, whereas the vertical band denotes the
virial radius, for a range of systems at redshift z = 0 and with
T0−thr = 4πGkBδ(cβρc−rs21µ)mpΓ−Γ 1 =−φ0Γ−Γ 1µkmBp, (10) s1t0r1o2nMgl⊙yo≤ntMhevivral≤ue1o0f1β6M.W⊙.heTnhTe0t=emTp0e−rtahtrur(esoplirdoficulervdeesp),enthdes
logarithmicslopeatlargeradiid(logT)/d(logr)is≃−1forβ≥2
abovewhichthesolutionisphysicallymeaningfulatallradii,
and ≃ −β+1 for β < 2. Additionally, at very large radii there
i.e. r =+∞. This central threshold temperature, which
max is a strong dependence on T0: the dotted (dashed) curves show
depends on the system’s gravitational potential (i.e. on its thetemperatureprofilewhenthecentraltemperatureisincreased
mass) and on the steepness of the polytropic relation, is a (decreased) by1percentfromT0−thr.
characteristicquantityofthesystem:T0−thristheminimum
permitted central temperature, resulting from the require-
ment of HE and non-negative temperature at all radii. As dex. However, when Γ varies, so do T0−thr and its fractions
expected, a deeper gravitational potential requires a higher (e.g.0.99T0−thr and1.01T0−thr,showninthefigure),there-
thresholdcentraltemperature.Thereexistsalsoamaximum foreFigure 1looks exactlythesamefor anyvalueofΓ>1.
If we slightly increase or decrease the central temperature,
permittedcentraltemperature,constrainedbytheboundary
condition at large radii, usually given by the temperature the effect on T(r)/T0 is quite dramatic at very large radii,
of the intracluster medium or of the intercluster medium, whereas it is almost not observed within thevirial radius.
Wecancalculatethecentraltemperaturewhenonedoes
dependingonthetypeofsystem(e.g.groupvs.cluster;iso-
lated vs. non-isolated) under consideration. There is there- notimposermax =+∞,butimposesinsteadafinitermax =
fore, for any given Γ, a relation between the mass of a sys- κrvir, where κ is some large integer. In this case, the new
temandarangeofpermittedcentraltemperaturesofitsgas. central temperature T0−κ is given by
ThisM–T relationwillbefullyexploitedinSection3tocon-
straingasandDMparameters.IfT0 <T0−thr,rmax isfinite 1− 1 (1+κcvir)β−2−1 if β6=2,
andcanevenbemuchsmallerthanthevirialradius,depend- T0−κ = β−2κcvir(1+κcvir)β−2 (11)
ingontheotherparameters.IfT0 =T0−thr,rmax =+∞and T0−thr 1− ln(1+κcvir) if β=2.
all gas quantities at infinity vanish (and we recover the so- κcvir
lutiongiveninB10).IfT0 >T0−thr,rmax=+∞andallgas It canbe shown that the ratio T0−κ/T0−thr > 0.9 for
quantities at infinity are positive. Imposing the boundary κ>10(i.e.r >10r ),inthetypicalrange4<c <10
max vir vir
condition T(+∞)=T∞, we haveT0 =T0−thr+T∞. atz =0,forarangeofvirialmasses1012 M⊙ <Mvir <1016
In Figure 1, we show the effect of changing T0 on the M⊙, for any 2≤β ≤3.
temperature radial profile, derived from Equations (3) and In extended systems, gas quantities at large radii are
(4) and from the polytropic relation, for a few cases of β. never really zero but are equal to those of the intracluster
ThesolutionwithT0=T0−thr istheonlycaseforwhichthe medium or of the intercluster medium, depending on the
temperature becomes zero at infinity,and it does so with a type of system one considers. Assuming global HE, we can
logarithmic slope at large radii d(logT)/d(logr) ≃ −1 for computethecentraltemperaturewhenonedoesnotimpose
β ≥ 2 and ≃ −β +1 for β < 2. This strong dependence T∞ =0, but imposes instead T∞ =T0/F, where F is some
on β will be useful in Section 3.1.3, where we constrain the largenumber.Inthiscase,thenewcentraltemperatureT0−F
DMlogarithmic slopeatlargeradiiζ ≡−β−1.Noticealso is given by
that, in this particular case, the shape of the profile is in-
dependent of the polytropic index. When T0 6= T0−thr, the T0−F = F . (12)
temperature profile T(r)/T0 depends on the polytropic in- T0−thr F −1
(cid:13)c 0000RAS,MNRAS000,000–000
6 P.R. Capelo, P.S. Coppi and P. Natarajan
Figure 2. Dependence on redshift of several parameters, for a Figure3.Dependenceoftheparametersoftheclusterdescribed
cluster of virial mass Mvir = 1014 M⊙, composed of NFW DM in Figure 2 on its gas central temperature. The thick lines are
and gas with polytropic index Γ=6/5 and central temperature the same lines shown in Figure 2. The arrows and the bands
kBT0 = 2.5 keV. Top panel: we plot the temperature of the show what changes when we vary the central temperature from
clusterT0 (dotted line),itscentralthresholdtemperatureT0−thr 0.9T0 to 1.1T0. When increasing the central temperature, ∆gas
(solid line)anditstemperatureatinfinityT∞ (dot-dashed line). decreases, T∞ and rmax increaseand the threshold redshiftzthr
NoticehowkBT0−thr doesnotdependverystronglyonredshift, increases.Noticetheexistenceofaminimumcentraltemperature
varyingfrom∼2keVto∼5keVinthez=0–3range.Middle T0−min, below which the cluster cannot exist inHE at all radii,
panel:weplotthegasparameter∆gas (solid line)inrelationto atanyredshift.
theratioΓ/(Γ−1)(dottedline).Bottompanel:weplotthevirial
radiusrvir(dottedline)andthemaximumradiusrmax(solidline)
of the cluster. All panels: the dashed, vertical line denotes the
threshold redshift zthr at which T0 = T0−thr, T∞ = 0, ∆gas =
Γ/(Γ−1) and the function rmax(z) has its vertical asymptote.
When we imposeT∞ =0, z =zthr isthe only redshiftat which
the cluster can exist in HE at all radii. If we instead allow for
any T∞ ≥0, then the cluster can exist in HE at all radii inthe
z=0–zthr range.
The ratio T0−F/T0−thr . 1.01 for F & 100 (i.e. T∞ .
T /100), for any 1 < β ≤ 3. In clusters of galaxies, where
0
F >>1, we can safely assume T∞ =0. Groups of galaxies,
however, can have a much smaller F, and the assumption
of a zero temperature at infinity cannot be used. Written
in a different way: in clusters of galaxies, T∞ is negligible
with respect to T0−thr (i.e. T0 =T0−thr+T∞ ≃T0−thr). In
groupsofgalaxies,ontheotherhand,anon-zeroT∞ cannot
be neglected, and we will see in Section 3.1.2 its effects on
theM–T relation.
Consideringquantitiesatinfinity,likeT∞,ismathemat- Figure4.Dependenceoftheparametersoftheclusterdescribed
ically convenient but obviously not realistic. For example, in Figure 2 on its gas polytropic index. The thick lines are the
same lines shown in Figure 2. The arrows and the bands show
the HE assumption may break down at ∼r , where there
vir what changes when we vary the polytropic index from 0.98Γ to
mayalsobeavirialshock.Inthatcase,amorecompleteap-
proach might include the solution of the Rankine-Hugoniot 1.02Γ. When increasing the polytropic index, T0−thr increases,
T∞ andrmax decreaseandthethresholdredshiftzthr decreases.
jumpconditionsattheshockfront,inordertoobtainproper Noticetheexistenceofamaximum polytropicindexΓmax,above
boundary conditions for the polytropic solution. A similar whichtheclustercannotexistinHEatallradii,atanyredshift.
approach is required if one assumes the DM profile to have
a sharp truncation radius at r (e.g. Ostriker et al. 2005).
vir
However, note that while numerical simulations can show
letboundaryconditionhasbeenenforced,theimpositionof
virial shocks at ∼ r along particular directions from the
cluster centre (e.g. pvoirinting to filaments), when the cluster T(+∞) = T∞ can be mapped to an equivalent boundary
condition T at some finite radius ξr , with ξ≥0:
profilesaresphericallyaveragedthesimulationsdonotshow ξ vir
strongDMorgaspressureanddensitydiscontinuitiesatr
vir
(e.g.Molnar etal.2009). Moreover, giventheuniquenessof 1−(Γ−1)∆ [1−ln(1+ξc )/ξc ]/Γ
thesolutionoftheHEdifferentialequation,oncetheDirich- Tξ =T∞ 1g−as(Γ−1)∆ /Γvir vir (13)
gas
(cid:13)c 0000RAS,MNRAS000,000–000
Polytropic approximation and scaling relations 7
for β =2 and convectively unstable when (ρ/P)dP/dr < γdρ/dr, which
translates to Γ>γ, where the adiabatic exponent γ =5/3,
1− Γ−1∆ 1− 1 (1+ξcvir)β−2−1 for a completely ionized, ideal, monatomic, non-degenerate
Tξ =T∞ Γ ga1s−h (Γ−β−12)∆(ξcvir/)(Γ1+ξcvir)β−2i (14) gas.On the other hand, if we increase T , the threshold
gas 0
redshift increases. The same happens if we decrease the
for β 6=2. These formulas will be useful in Section 3.1.2. It
polytropic index of the gas, because the threshold central
isalsoworthrememberingthatobservationalquantitieslike
temperature decreases. This trend can be seen also in an-
M and T (see Section 2.4) do not depend strongly on
500 500
other way. When we decrease the polytropic index towards
the exact solution at &r , because of the low gas density
vir
its isothermal limit Γ = 1, the relation between pres-
and emissivity at large radii. lim
sure and density can be written as P(r) = [ρ(r)/ρ ]T .
0 0
In this limiting case, Equation (3) is no longer valid for
2.2 Analysing the polytropic solution β = 2, and the solution to the HE equation, using once
again the boundary condition ρ(0) ≡ ρ , becomes9 ρ(r) =
0
In this section, we show the behaviour of rmax and T0−thr ρ exp[−∆ (1−ln(1+r/r )/(r/r ))] (Makino, Sasaki &
0 gas s s
in a couple of different cases. In Figure 2, we show the de-
Suto1998). Itis clear from thissolution that thegas quan-
pendence of r as a function of redshift for a cluster of
max tities arenevernegativeat anyradius, therefore thereis no
virial mass Mvir = 1014M⊙, with gas of polytropic index minimum central temperature requirement:T0−thr =0.
Γ=6/5andcentraltemperaturek T =2.5keV,f =f ,
B 0 gas b In the example shown in Figure 2, z = 0.85 for
thr
and with DM described by an NFW profile and concentra-
Γ = 6/5. We will see in Section 3.1.1 that the polytropic
tion given by D08. We also plot the virial radius and the
indexhastypicalvaluesbetween6/5and13/10.Thismeans
threshold central temperature to allow for comparison. No-
that the threshold redshift computed for Γ=6/5 is indeed
tice how thethreshold central temperatureslowly increases
a maximum value for the example shown, for any reason-
byonlyafactorof∼2(fromkBT0−thr ∼2to∼5keV)inthe able valueof the polytropic index.If we could observe with
z =0–3 range8: for a fixed virial mass, the strength of the
enoughaccuracy thecentraltemperatureoftheclustergas,
gravitational potential only slowly increases with redshift.
we could then obtain some useful information. Using the
Finally, we also show the gas parameter ∆ , to compare
gas numbersabove,ifweforexampleobservedaclusterofvirial
it to Γ/(Γ−1), and the temperature at infinity T∞. The mass Mvir = 1014 M⊙ with a gas with polytropic index
dashed, vertical line in all panels denotes the asymptote of
Γ ≥ 6/5 and central temperature k T = 2.5 keV to be at
B 0
thefunctionr (z)andalsodenotesthethresholdredshift
max redshift z ≥ 1, then we could safely assume that the gas
zthr at which ∆gas = Γ/(Γ−1), T0 = T0−thr and T∞ = 0: is not in HE at all radii. Alternatively, if we for example
belowz ,thesolution isphysicallymeaningfulatallradii.
thr observed a cluster with the same virial mass and central
Abovez ,theradiusr 6=+∞anddecreasesasredshift
thr max temperature at redshift z ≃ 0.5 and knew by some means
increases. Assuming HE at all radii, this threshold redshift
that the gas is in global HE, then we could safely assume
istheonlyallowedredshiftforthiscluster,ifwestrictlyim-
that it has a Γ>6/5. Unfortunately, accurate observations
pose T∞ =0, or is just a maximum redshift, if we allow for of T are very difficult if not in many cases impossible, due
0
any non-negativeT∞. toresolutionlimitsofcurrentX-raytelescopes.However,the
InFigures3and4,weshowwhatchangeswhenwevary
samereasoningtranslatestotheconceptofaveragetemper-
thecentraltemperature(by±10percent)orthepolytropic
ature,definedinSection2.4.ThiswillbeevidentinSection
index (by ±2 per cent) of the gas, respectively. If we de-
3.1, where we show the dependence of the M–T relation
crease the central temperature, the threshold redshift z
thr on several parameters, including the polytropic index and
decreases, because the gas parameter ∆ increases, and
gas redshift.
there is a minimum central temperature (in this example
WenowstudythedependenceoftheHE-polytropicso-
kBT0−min ≃ 1.8 keV) under which a cluster with a given lution on the mass of the system. In Figure 5, we show
virial mass and polytropic index cannot exist in HE at all
the same parameters of Figure 2, this time as a function
radii, at any redshift. Analogously, if we increase the poly-
of virial mass, for a cluster at redshift z = 0 with NFW
tropic index, the threshold redshift decreases, because the
DM, f =f , and gas with polytropic index Γ=6/5 and
gas b
thresholdcentraltemperatureincreases,andthereisamaxi-
central temperature k T = 4 keV. Notice how the thresh-
B 0
mumpolytropicindex(inthisexampleΓ ≃13/10)above
max old central temperature increases significantly as the virial
whichaclusterwithagivenvirialmassandcentraltemper-
massincreases:forafixedredshift,thestrengthofthegrav-
aturecannotexistinHEatallradii,atanyredshift(seealso
itational potential quickly increases with virial mass. As a
Arieli&Rephaeli2003).Theexistenceofaminimumcentral
consequence,r stronglydependsonvirialmass,changing
max
temperature and of a maximum polytropic index is due to
from r ∼r to +∞ in a very small mass range.
max vir
the fact that the threshold central temperature at redshift
InFigures6and7,weshowwhatchangeswhenwevary
z=0isfiniteandisanincreasingfunctionofΓ.Noticethat
thecentraltemperature(by±10percent)orthepolytropic
Γ increases,forafixedvirialmass,ifweincreasethecen-
max index(by±2percent)ofthegas,respectively.Theeffectis
traltemperature.However,thereisanadditionalconstraint
on the polytropic index, given by the condition of convec-
tivestability.Thesolution giveninEquations(3)and(4)is
9 Intheβ6=2case,theisothermalsolutionis
8 Note that, at high redshift, some of the assumptions of our ρ(r)=ρ0exp(cid:20)∆gas(cid:18)β−1 2(r(1/r+s)r(1/r+s)rβ/−r2s)−β−12 −1(cid:19)(cid:21)>0.
model,includingHEandisolation,arelessvalid.
(cid:13)c 0000RAS,MNRAS000,000–000
8 P.R. Capelo, P.S. Coppi and P. Natarajan
Figure 5. Dependence on virialmass of several parameters, for Figure6.Dependenceoftheparametersoftheclusterdescribed
acluster atredshiftz=0, composed of NFWDMandgas with in Figure 5 on its gas central temperature. The thick lines are
polytropicindexΓ=6/5andcentraltemperaturekBT0=4keV. the same lines shown in Figure 5. The arrows and the bands
Top panel: we plot the temperature of the cluster T0 (dotted show what changes when we vary the central temperature from
line),itscentralthresholdtemperatureT0−thr(solidline)andits 0.9T0 to 1.1T0. When increasing the central temperature, ∆gas
temperatureatinfinityT∞ (dot-dashed line).NoticehowT0−thr decreases, T∞ and rmax increase and the threshold virial mass
depends strongly on virialmass, changing by more than one or- Mvir−thr increases.
der of magnitude in the Mvir = 1012–1016 M⊙ range. Middle
panel: we plot the gas parameter ∆gas (solid line) in relation
to the ratio Γ/(Γ−1) (dotted line). Bottom panel: we plot
thevirialradiusrvir (dottedline)andthemaximumradiusrmax
(solid line) of the cluster. All panels: the dashed, vertical line
denotesthethresholdvirialmassMvir−thratwhichT0=T0−thr,
T∞ =0, ∆gas =Γ/(Γ−1) and the function rmax(Mvir) has its
vertical asymptote. When we impose T∞ =0, Mvir =Mvir−thr
is the only mass that the cluster can have to exist in HE at all
radii. If we instead allow for any T∞ ≥ 0, then the cluster can
existinHEatallradiiwithavirialmassMvir=0–Mvir−thr.
similartowhatshowninFigures3and4,exceptforthefact
thatthereisnoT0−min orΓmax:foreverysmallcentraltem-
perature or large polytropic index, there is always a small
enough virial mass for which the DM gravitational poten-
tial is not strong enough to bind the gas in a finite region.
Remember, however, that Γ<γ, because of the convective
stability requirement.
Figure7.Dependenceoftheparametersoftheclusterdescribed
in Figure 5 on its gas polytropic index. The thick lines are the
2.3 Comparison to previous studies same lines shown in Figure 5. The arrows and the bands show
what changes when we vary the polytropic index from 0.98Γ to
In Section 2.1, we describe the structure of gas in clusters
1.02Γ. When increasing the polytropic index, T0−thr increases,
and groups of galaxies of known virial mass and redshift,
T∞ and rmax decrease and the threshold virial mass Mvir−thr
byassumingHEatallradiibetweenanidealpolytropicgas decreases.
and DM. If the DM potential is known, the HE-polytropic
solution is completely described once we fix its three free
parameters: Γ, ρ and T . There are several ways to “close at all radii, and by imposing (iii) that the temperature at
0 0
theproblem”,whichareallmathematicallyequivalent.Here largeradiiisnon-negative(seeEquation10andrelateddis-
we briefly summarise the assumptions we made in Section cussion).
2.1 and then contrast them with those made in prior work. KS01 (see also KS02) presented a similar description
This will help putintobettercontext ourresultsin Section fortheintraclustermediuminclusters,byalsoassuming(i)
3 and those of otherauthors. and(ii),andadditionallyrequiring(iv)thatthegasdensity
For a given polytropic index, we obtain a relation be- profileisequaltothatofDMatlargeradii,(v)theDMlarge-
tweenthegravitationalpotentialofasystemandarangeof radiilogarithmic slopeζ =−3and(vi)theconcentration is
permittedcentraltemperaturesofitsgas, bysimplyrequir- known,for a given mass and redshift.
ing (i) spherical symmetry,(ii) HE between DM (described A06(seealsoAscasibaretal.2003)alsostudiedcluster
by a generalised NFW profile) and an ideal polytropic gas modelsofpolytropicgas,byonceagainrequiring(i),(ii),(v)
(cid:13)c 0000RAS,MNRAS000,000–000
Polytropic approximation and scaling relations 9
and (vi), additionally (vii) taking the polytropic index as a The choice of the excision radius r is less straightfor-
0
given (from numerical simulations and observations), and ward. Most studies justify excising the central region be-
imposing (viii) that the temperature at infinity is always cause it is significantly affected by radiative cooling and by
zero, regardless of the system’s mass or environment. otherphysicsofwhichwedonotknowthedetails,e.g.feed-
Ostriker et al. (2005) (but see also Bode et al. 2007, back. The positive result of excision is the large decrease
2009) studied similar scenarios, by also assuming (i), (ii), in scatter in the X-ray scaling relations, at all masses (e.g.
(v),(vi) and (vii). Moreover, (ix) they constrained the cen- Markevitch1998),whichishighlyusefulforcosmologymea-
tral temperature by matching the external surface pressure surements.However,excisionalsothrowsoutpotentiallyim-
to the momentum flux from the infalling gas at the virial portant physical information relevant to understanding in
radius. detail the formation of the cluster, such as the effects of
cooling, feedback, concentration, etc. Therefore, it is useful
to study how the choice of r affects the results. In Section
2.4 Average temperature and the issue of excision 0
3.1.4,wewillseeanexampleofhowsignificantexcision can
Inthissection,weconnecttheresultsgiveninSection2.1to be.
observations,bymeansofdefiningtheaveragetemperature. Cognizant of all these considerations, in this paper
As shown in the previous sections, for a given polytropic we choose to use the integration limits of V09 (see also
index, we are able to obtain a relation between the gravi- Kravtsov, Vikhlinin & Nagai 2006) and define the average
tational potential of a galaxy group or cluster and its gas temperatureas10
central temperature, once a boundary condition (e.g. T∞)
hasbeenfixed.Naturally,thecentraltemperatureisnotan r500 n nΛ T(r)4πr2dr
ebalesytoobaspeprvlyabtlhee,bsaemcaeusreeaosforneisnogluatniodnrleimsuilttss,tboutthiteicsopnocsespit- T500 = R0.105r.5r150500r0500eneinNiΛN4πr2dr . (16)
of average temperature. We computed the average temper- WhenRstudying the effect of excision, we compute the
ature following the notation of KS01. In general, one can average temperatures T500−no−exc and Tvir−no−exc, simply
write bychanging the lower integration limit tor =0.
0
Theaveragetemperaturedescribedherewillbeusedin
r1w(r)T(r)4πr2dr Section3.1,wherewecalculatetheM–T relationforgroups
Tav = Rr0rr01w(r)4πr2dr , (15) asenrdvactliuosntse.rs of galaxies and compare it to recent X-ray ob-
wheretheRexcisionradiusr0 iseitherzeroorafractionofan
overdensityradius(e.g.r ,r ,r ,orother),thetrunca-
vir 200 500
tionradiusr issomeotherfractionof,orequalto,anover-
1 3 SCALING RELATIONS AND COMPARISON
densityradius,andw(r)istheweightfunction.Iftheweight
TO OBSERVATIONS
function w=1, T =T is the volume-weighted average
av vwa
temperature; if w = ρ, Tav = Tmwa is the mass-weighted 3.1 The M–T relation
average temperature; if w = nn Λ , where n is the num-
i e N e
In Section 2.1, we effectively obtained a relation between
ber density of electrons per unit volume, n is the number
i
thetemperatureprofileandthemassofclustersandgroups
density of ions per unit volume and Λ is thecooling func-
N
of galaxies, once the temperature at large radii was fixed.
tion, T =T is the emission-weighted average tempera-
av ewa
For a proper comparison to observations, we defined the
ture.Althoughsomeauthorsusethemass-weightedaverage
average temperature in Section 2.4. Using this quantity,we
temperature, most observers use the emission-weighted av-
canthenconstructanM–T relation thatmaybecompared
erage temperature instead, usually approximating thecool-
to observations.
ing function as Λ ∝ ρ2T1/2. Accordingly, we also use the
N
In our model, the resulting M–T relation is a function
emission-weightedtemperaturehere.However,sincewealso
of fiveparameters: thepolytropic index Γ, thetemperature
investigatethegroupregime,wheretemperaturesarelower,
we use a more general version of the cooling function, de- atlarge radiiTξ (or,equivalently,T∞),theDMlogarithmic
slope at large radii ζ (or, equivalently, β), the concentra-
scribedindetailinAppendixB.Whenusingbothdefinitions
tion c and the redshift z. To develop intuition for how
of the cooling function, we find the results to be basically vir
these parameters affect the M–T relation, in the following
identical at high temperature (as expected, since cooling is
sub-sections we will fix four of these quantities and study
dominated by Bremsstrahlung emission) and only slightly
the dependence of the M–T relation on the remaining fifth
different at low temperatures.
parameter. Comparing the predicted M–T relation to that
The choiceof r and r is an important one.The trun-
0 1
from X-ray observations, which haveimproved significantly
cationradiusr isusuallydeterminedbyobservationalcon-
1
in recentyears, leadstoconstraints on theallowable ranges
straints.Atlargeradii,bothgastemperatureanddensityare
ofparametervalues.Athoroughexaminationoftheallowed
verylow,thereforeverydifficulttoquantifyaccurately.How-
rangewould requireasimultaneousvariation ofallparame-
ever,exactlybecausetemperatureanddensityareverylow,
tersatonce,toallowforpossibledegeneracies.Suchastudy
T doesnotchangeappreciablyforradiigreaterthanr ,
ewa 500
exceptforthehighestmasssystems,anddoesnotchangeat
all for radii greater than rvir. Even when future telescopes 10 Whencomputingaveragetemperatureswithinotheroverden-
(e.g. eROSITA; Predehl et al. 2007, 2010) will permit us sityradii,wekeeptheexcisionradiusthesame.Forexample,the
to study theouter regions with better accuracy, a larger r1 integration limits for the computation of Tvir are r0 =0.15r500
should not significantly change theaverage temperature. andr1=rvir.
(cid:13)c 0000RAS,MNRAS000,000–000
10 P.R. Capelo, P.S. Coppi and P. Natarajan
is beyond the scope of this paper, although where we can,
we will note themore obvious degeneracies.
Beforeweembarkonourstudy,someconsiderationsare
in order. We determine the temperature of a galaxy group
or cluster bysolving thethermally-supported HEequation.
However, it is clear from simulations (e.g. Nagai, Vikhlinin
& Kravtsov 2007) that random gas motions and rotation
can contribute up to 10 per cent of the pressure support,
leadingtoanover-estimationofthegastemperatureneeded
for HE. On the other hand, the mass computed from X-
ray observations is the thermally-supported HE mass (e.g.
Fabricant,Lecar&Gorenstein1980),resultinginanunder-
estimationofthetruemasswithinr ofupto10percent,
500
for relaxed systems (e.g. Lau, Kravtsov & Nagai 2009; see
alsoMahdavietal.2008).SincetheM–T relationisapower
law, thesetwoeffectscancel each otherand thecomparison
betweenourmodelandobservationsismeaningful(seealso
KS01). The same reasoning applies to any other form of
Figure 8. Dependence of the theoretical M–T relation on the
non-thermalsupport,includingcosmicraysand/ormagnetic polytropic index, for systems at redshift z =0 (E(z)=1), with
fields (see e.g. Parrish et al. 2011). polytropic gas in HE with NFW DM (ζ =−3) at all radii, con-
More importantly, in our model we assume a complete centration given by D08 and T∞ = 0. The relation can always
polytropic relation (i.e. the polytropic index is the same at be well approximated by a single power law. We plot the M–T
all radii). Recent numerical studies (e.g. Shaw et al. 2010; relation for seven evenly-spaced values of 11/10 ≤ Γ ≤ 17/10.
Battaglia et al. 2011a,b) show instead a radial dependence. Theblack, solid lines aretheresultsfromourtheoreticalmodel,
whereas the red, dotted lines show the empirical approximation
Wenote,however,thatthevariationofΓwithradiusisnot
from Equation (17). Notice how the polytropic index has a sig-
very large, especially in the range 0.15r –r , where we
500 500 nificant effect only on the normalisation. For a fixed mass, the
calculate the average temperature (see Section 2.4). These
increaseofΓresultsinahigheraveragetemperature.Inthisand
same studies (Battaglia et al. 2011b) also show a slight de-
inthefollowingfigures,weassumeourmodelscalesself-similarly
pendenceofΓonredshift,andwewilladdressthispossibility withredshift.WewillseeinSection3.1.5thatthisassumptionis
in Section 3.1.5. welljustified.
3.1.1 Dependence on the polytropic index
of 11/10≤Γ≤17/10. In thisand in thefollowing sections,
In this section, we consider systems at redshift z = 0 we assume our model scales self-similarly with redshift. We
(E(z) = 1), described by an NFW DM profile (ζ = −3), willseeinSection3.1.5thatthisassumptioniswelljustified.
with concentration given by D08 and a polytropic gas with The effect of varying the polytropic index is consistent
T∞ =0, and study the dependenceof the M–T relation on withwhatwasalreadyshowninFigures4and7:foragiven
thepolytropic index Γ. virialmass,whenΓincreases,thethresholdcentraltempera-
Figure 8 shows this dependence, when assuming that tureincreases,andsodoestheaveragetemperature.WhenΓ
systems at all temperatures share the same value of Γ. We decreases, theaveragetemperaturequicklyapproaches zero
first notice that therelation can beverywell approximated proportionallyto∼[(Γ−1)/Γ]4/5,asthepolytropicrelation
by a single power law, over more than three orders of mag- tends to its isothermal limit (i.e. Γ = 1). Notice that, be-
nitude in both mass and temperature (not shown in the cause of the dependence on the Γ/(Γ−1) term, it is much
figure). Moreover, the slope of such function does not no- easiertodiscriminatebetweenlower valuesofthepolytropic
tably change with Γ: the polytropic index has a significant index than between higher values.
effect only on the normalisation. With all these considera- We now compare the theoretical M–T relation to re-
tions,wecanthenapproximatethelocalM–T relationwith cent X-ray observations of galaxy clusters and groups. The
thefitting11 formula red, dot-dashed line in Figure 9 shows the best-fitting M–
T relation calculated by V09 for a sample of seventeen,
Γ ǫ T α low-redshift, relaxed clusters: E(z)M500 = (3.02±0.11)×
E(z)M500 =M0 Γ−1 T500 , (17) 1014h−1M⊙(kBT500/5 keV)1.53±0.08, valid over the temper-
(cid:18) (cid:19) (cid:18) 0 (cid:19) ature range k T ∼ 1.5–12 keV. Since we do not have ac-
B
where ǫ = 6/5, α = 1.46, M0 = 3.9545 × 1013h−1M⊙, cess to the exact measurement and error analysis of V09,
kBT0 = 5 keV and kBT500 is in keV. This approximation we cannot make a definitive statement on the range of Γ
closelyreproducestheresultsofthemodelfortypicalvalues consistentwith V09.Assumingzerocovariancebetween the
magnitudeandtheslopeofthepublishedM–T relation,we
findthatamodelwithamass-independentpolytropicindex
11 For this and some other fits in the paper, we performed
1.22 . Γ . 1.24 is consistent with the data. Without this
a non-linear least-squares fitting with the IDL routine MPFIT
assumption, we conservatively estimate 6/5 . Γ . 13/10.
(Markwardt 2009). Curiously enough, if we plot Mvir versus
T500, we obtain a relation with a self-similar slope: Mvir = Such values of Γ are also consistent with the results ob-
M0[Γ/(Γ−1)]6/5(T500/T0)3/2,where M0 =7.09×1013h−1M⊙ tained by both observations and simulations (e.g. Bode et
andkBT0=5keV. al. 2009 and references therein).
(cid:13)c 0000RAS,MNRAS000,000–000