The Lx-T Relation and Intracluster Gas Fractions of X-ray Clusters

12 Pages·0.26 MB·English

by  Monique Arnaud

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev1.4)
L T
The − Relation and Intracluster Gas Fractions of
X
X–ray Clusters
Monique Arnaud1 and August E. Evrard2,3
1 C.E.A., DSM, DAPNIA, Service d’Astrophysique, C.E. Saclay, F-91191, Gif-Sur-YvetteCedex, France
9 2 Institut d’Astrophysique, 98bis Blvd Arago, 75014 Paris, France
9 3 Physics Department, Universityof Michigan, Ann Arbor, MI 48109–1120 USA
9
[email protected]
1
[email protected]

n
a
J
Accepted 1999January12

9
1
ABSTRACT
2
v
We re-examine the X–ray luminosity–temperature relation using a nearly homo-
3
geneousdatasetof24clustersselectedforstatisticallyaccuratetemperaturemeasure-
5
mentsandabsenceofstrongcoolingflows.Thedataexhibitaremarkablytightpower–
3
law relationbetween bolometric luminosity andtemperature with a slope 2.88±0.15.
6
0 With reasonable assumptions regarding cluster structure, we infer an upper limit
8 on fractional variations in the intracluster gas fraction h(δfgas/fgas)2i1/2 ≤ 15%. A
9 strictly homogeneous GINGA subset of 18 clusters places a more stringent limit of
/ 9%.
h
Imagingdatafromtheliteratureareemployedtodetermineabsolutevaluesoff
p gas
- within spheres encompassing density contrasts δc=500 and 200 with respect to the
o critical density. Comparing binding mass estimates based on the virial theorem (VT)
r and the hydrostatic, β–model (BM), we find a temperature–dependent discrepancy
t
s in f between the two methods caused by systematic variation of the outer slope
a gas −1 −1
: parameterβ withtemperature.Meanvalues (for Ho=50 km s Mpc )rangefrom
v f¯ =0.10 for cool (T < 4 keV) clusters using the VT at δ =500 to 0.22 for hot
gas c
Xi (T > 4 keV) clusters using the BM at δc=200. There is evidence that cool clusters
have a lower mean gas fraction than hot clusters, but it is not possible to assess the
r
a statistical significance of this effect in the present dataset. The T dependence of the
ICMdensitystructure,coupledwiththe increaseofthegasfractionwithT inthe VT
approach, explains the steepening of the L –T relation.
X
The small variation about the mean gas fraction within this majority sub–
populationofclusterspresentsanimportantconstraintfortheoriesofgalaxyformation
andsupportsargumentsagainstanEinstein–deSitteruniversebasedonthepopulation
mean gas fraction and conventional, primordial nucleosynthesis. The apparent trend
oflowergasfractionsandmoreextendedatmospheresinlowtemperaturesystemsare
consistent with expectations of models incorporating the effects of galactic winds on
the intracluster medium.
Key words: cosmology:theory–cosmology:observation–clusters:galaxies:general
– dark matter
1 INTRODUCTION temperature is determined by the total, gravitating cluster
mass,theL –T relationcanbeusedasatooltoprobevari-
X
The relation between X–ray temperature T and luminos- ations in gas fraction — theratio of ICM gas to total mass
ity LX is a sensitive diagnostic of structural regularity in — in thesesystems.
clusters of galaxies. Because the luminosity is governed by
themassofgasintheintraclustermedium(ICM)whilethe Evidenceforacorrelation betweenthesebasicobserva-
(cid:13)c 0000RAS
2 M. Arnaud and A.E. Evrard
tional quantities has existed since the early days of X–ray below this limit (Fabian et al. 1994), our analysis is of a
astronomy(Mitchelletal.1977,1979;Mushotzkyetal.1978; majority population.
Edge & Stewart 1991, David et al. 1993). Despite its rela- Table1presentsalistingofthe24clusterssatisfyingthe
tive maturity, a key characteristic of this relation was only abovecriteria.EighteenhavebothT andL determinedby
X
recently established. Fabian et al. (1994) demonstrate that theGINGAsatellite (references aregiven inthetable).Lu-
the departure of a cluster from the mean L −T relation minosities for three nearby clusters (Virgo, in particular)
X
is correlated with the strength of the emission associated are calculated from scanning data, which accounts directly
with its cooling flow core. The picture resulting from this for the extended emission outside the 1.1×2 sq deg FOV
work is of a mixed population which is likely to be time (FWHM)ofthesatelliteinstrument.Temperaturemeasure-
variable. Fabian et al. speculate that cooling flows are a re- ments for the other six clusters come from ASCA, with lu-
currentphenomenonassociated withasecularinstabilityin minositiesfromEinsteinMPC(3),EXOSAT(2),andASCA
the ICM plasma which periodically is interrupted and re- (1).
setbystrongmergingencounters.Periodic mergersofvary- Webelievethislisttobecompletewithrespecttopubli-
ing strength are a natural feature of hierarchical clustering cationsavailableendof1996.Sixteenclustersinoursample
models of structure formation, and numerical simulations are included in theX-ray fluxlimited sample of Edge et al.
suggest that cooling flow features can be erased by merg- (1990). Two more clusters (A2163 and A1413) have fluxes
ers(Garasi, Burns&Loken1997). However,thetheoretical above theEdge et al flux limit (1.7×10−11 erg s−1 cm−2).
picture remains incomplete because of the complexities of Our sample is ∼ 70% complete as compared to the Edge
the physics operating in the cluster core. Regardless of the et al. sample down to a flux of 3.4×10−11 erg s−1 cm−2
exactphysicalmechanismsresponsible,theempiricalfactis (considering from now on only weak cooling flow clusters
that thecluster population can be broadly classed into two in the Edge sample for consistency) and highly incomplete
categories bymaking a cut in cooling flowstrength M˙. below this flux (3 clusters among 20). The other clusters of
In this paper, we present and analyze the luminosity– oursamplebelowtheEdgeetal.fluxlimitareeitherdistant
temperature relation from a nearly homogeneous data set luminousclusters(A3921,A665,A2218,A370)orpoorclus-
of weak cooling flow clusters, where M˙ ≤100 M⊙ yr−1 de- ters (A2634 and AWM4). The temperature distribution of
fines weak. The data set is described in §2, with particular our sample shows a clear deficit of clusters at intermediate
attentionpaidtosystematicerrorsindeterminationofboth temperature (4−6 keV) as compared to the temperature
L and T. In §3, we present the L –T relation and show distribution of theEdge et al. sample.
X X
that these data place stringent limits on gas fraction varia- Included in Table 1 are the measured temperature and
tions within this sub–population. Imaging data taken from luminosity in the 2–10 keV energy band (cluster rest frame
theliteratureallowustoestimateabsolutevaluesofthegas and corrected for absorption). We convert the 2–10 keV
fraction at fixeddensity contrast in §4. Wecompare results luminosity to bolometric luminosities using an isothermal
fortwodifferentbindingmassestimationmethods,andem- plasmaemissionmodel(Mewe,Gronenshild&vandenOord
phasizethesystematicuncertaintyintroducedbythischoice. 1985;Mewe,Lemen&vandenOord1986)andthemeasured
Hubble constant dependencies are displayed in the pa- temperatures and abundances. The bolometric correction
pervia h50 ≡ Ho/50 km s−1 Mpc−1andweassume h50=1 factor depends mostly on the temperature; it is higher at
when quoting numerical values throughout thepaper. lowtemperature(∼3.4 at2.2 keV),decreases with increas-
ing temperature up to about 8 keV (∼ 2.1) and increases
again above(∼2.4 at 15 keV).Thestatistical errors on the
luminosities are always much smaller than the correspond-
2 THE SAMPLE ing errors on the temperature and are thus not taken into
account here.
Weanalyseanarchivaldataset,partofwhichwasassembled
and discussed by Arnaud (1994). A prime objective is to
limit both statistical and systematic errors in Lx and T, 2.2 Systematic Errors
in order to obtain an accurate assessment of the intrinsic
dispersion about the mean LX −T relation. ToestimateaccuratelytheintrinsicdispersionoftheLX−T
relation requires an attempt to understand and quantify
sources of systematic errors. Such errors can arise from
calibration uncertainties, systematic errors in background
2.1 Selection Criteria
subtraction and, for luminosities deduced from collimated
Inclusion of a cluster in the sample is based on three cri- instruments (EXOSAT, GINGA), improper correction for
teria : (i) small statistical errors in measured temperature clusterextent(loss ofefficiencyat largeradii, emission out-
∆T/T∼<10% (at 90% confidence), (ii) weak or absent cool- sidetheFOV).Furtheruncertaintiesduetotheuncertainties
ingflowM˙ ≤100 M⊙ yr−1,and(iii)kT ≥2 keV.Thevalue on the plasma emission model are negligible for clusters in
of M˙ isset bytherelative importanceof coreluminosity to the considered temperature range (see Arnaud et al. 1991;
the total, and is typically evaluated using models of multi– Arnaudet al. 1992).
phase, steady–state accretion (Thomas, Fabian & Nulsen We attempt to guage the magnitude of systematic un-
1987).Sincenearlytwo–thirds(33/51)oftheclustersinthe certaintyin thebolometric luminosity bycomparing toval-
X–ray flux limited sample of Edge et al. (1990) have flows ues derived from the ROSAT All–Sky Survey (RASS) by
(cid:13)c 0000RAS,MNRAS000,000–000
Intracluster Gas Fractions of X–ray Clusters 3
Table 1.Basicdataforthe24clustersinthesample
Name Redshift LX [2-10keV] Temperature Abund. Lbolom. β Coreradius Ref.
(ergs/s) (keV) (rel.solar) (ergs/s) (kpc)
Virgo 0.0038 2.001043 2.20±0.02 0.44 6.811043 0.46 12.6 29,29,7,7
A262 0.0164 3.701043 2.41±0.05 0.55 1.151044 0.53±0.03 95±25 34,34,12,19
A644 0.0704 1.061045 6.59±0.17 0.33 2.241045 0.70 204 16,16,20,20
A754 0.0534 1.161045 7.57±0.31 0.23 2.461045 - - 1,1
A1367 0.0215 9.091043 3.50±0.18 0.31 2.261044 0.50±0.10 390+210 16,16,29,20
−130
A1656 0.0232 1.131045 8.21±0.09 0.22 2.411045 0.75±0.03 420±24 18,18,8,8
A2256 0.0601 1.111045 7.51±0.19 0.28 2.351045 0.81±0.01 542±12 16,16,9,9
A2319 0.0564 2.241045 9.12±0.15 0.24 4.831045 0.68±0.05 410±50 34,34,19,19
A2634 0.0321 7.091043 3.07±0.11 0.38 1.891044 0.58 320 1,1,11,11
A3526 0.0109 7.001043 3.54±0.13 0.54 1.701044 0.45±0.03 93±19 32,33,23,23
A3921 0.0940 5.201044 4.90±0.55 0.26 1.151045 0.75±0.04 385±42 2,2,2,2
AWM4 0.0315 2.531043 2.43±0.28 0.93 7.521043 0.43 190 31,31,31,31
AWM7 0.0179 1.701044 3.82±0.15 0.49 3.991044 0.53±0.01 102±5 31,31,28,28
Ophui. 0.0280 1.901045 9.10±0.30 0.24 4.101045 0.62+0.01 177+46 21,21,24,24
−0.02 −93
A665 0.1820 1.501045 8.26±0.90 0.49 3.161045 0.66±0.05 379±70 17,17,5,5
A1413 0.1427 1.821045 8.85±0.50 0.19 3.921045 0.62 156 16,16,11,11
A2163 0.2010 6.001045 14.6±0.85 0.40 1.451046 0.62±0.02 305±19 14,14,14,14
A2218 0.1710 9.401044 6.70±0.50 0.20 2.001045 0.65+0.08 226+80 25,25,6,6
−0.05 −50
A370 0.3700 1.301045 [1] 8.80±0.80 0.50 2.761045 - - 4,4
A399 0.0715 7.761044 [3] 7.40±0.50 0.25 1.641045 0.52±0.05 215±35 13,15,19,19
A401 0.0748 1.471045 [3] 8.40±0.50 0.21 3.151045 0.61±0.01 285±10 13,15,19,19
A1060 0.0114 2.501043 [2] 3.10±0.20 0.40 6.601043 0.61 94. 13,30,22,22
A3558 0.0478 4.231044 [2] 5.50±0.25 - 9.061044 0.61 324 13,27,3,3
Trian. 0.0510 1.261045 [2] 10.3±0.80 0.26 2.771045 0.63±0.02 286±16 13,26,26,26
Notes: column (9) references for the data (respectively X–ray luminosity, temperature , β and core radius): 1. Arnaud
(1994);2.Arnaudetal.(1997);3.Bardellietal.(1996);4.Bautzetal.(1994);5.Birkinshaw,Hughes&Arnaud(1991);6.
Birkinshaw& Hughes (1994); 7. Bo¨hringeret al.(1994); 8. Briel,Henry&Bo¨hringer (1992); 9. Briel& Henry(1994); 10.
Buote&Canizares(1996);11.Cirimele,Nesci&Trevese(1997);12.David,Jones&Forman(1995);13.Edgeetal.(1990);
14. Elbaz, Arnaud & Bo¨hringer (1995); 15. Fujita et al. (1996); 16. Hatsukade (1989); 17. Hughes & Tanaka (1992); 18.
Hughes et al. (1993); 19. Jones & Forman (1984); 20. Jones & Forman (1990, priv. comm.); 21. Kafuku et al. (1992); 22.
Loewenstein & Mushotzky (1996); 23. Matilsky,Jones & Forman(1985); 24. Matsuzawa et al.(1996); 25. McHardyetal.
(1990); 26.Markevitch,Sarazin&Irwin(1996); 27.Markevitch&Vikhlinin(1997); 28.Neumann&Bo¨hringer(1995); 29.
Takano (1990) and Koyama (1997, priv. comm.); 30. Tamara et al. (1996); 31. Tsuru (1993); 32. Yamanaka & Fukazawa
(priv.comm.inFukazawa etal.(1994)); 33.Yamashita1992; 34.Yamashita(1997,priv.comm.)
Ebeling et al. (1996) for Abell Clusters. Eigthteen of the arelargerthantheEXOSATvaluesweusebyfactorsof1.72
clusters in our sample are in this dataset. We convert the and1.56,respectively.Bothclustersarenearbyandhaveex-
RASS band–limited, total unabsorbed flux, derived from a tentcomparableorlargerthantheEXOSATFOV(45′×45′
procedure described in the above reference, to bolometric FWHM),thevirialradius(definedbelow)ofthese2clusters
L using the same plasma emission model as that used on arerespectively at 49 and38 arcmin from thecenter. Their
X
ourown data. In addition, we include RASSdatafor Virgo flux can be underestimated due to the neglect of the loss
from B¨ohringer et al. (1994) and ROSATimaging data for of efficiency at large radii. Furthermore, both observations
Triangulum(Markevitch,Sarazin &Irwin,1996) andOphi- were significantly offset, the pointing offset being specially
uchus (Buote & Tsai 1996), from which we calculate total important for A3558 (38 arcmin, Edge 1989). Although the
luminosities by extrapolating the β–model fits given in the fluxespublishedbyEdgeetal.(1990)includecorrectionfor
references. Conversion to a bolometric measure for imaging this effect (correction as for a point source), this is likely
data is thesame as for theRASSdata. toinducefurtheruncertaintiesin thefluxestimate. For the
sake of consistency with the remaining dataset, we applied
We find good agreement between the GINGA and nocorrection to thepublished values.
ROSATbolometricluminosities;16objectsyieldameanra-
tioof1.00withstandarddeviationof0.19.Theagreementis The comparison of the GINGA and ROSAT luminosi-
notasgoodforthefivenon–GINGAmembersofoursample, ties,whichconstitutesthemajorityofoursample,indicates
but the difference is driven entirely by two clusters — Tri- thatthescatterexpectedfromsystematicuncertaintyinL
X
angulum and A3558 — for which the ROSAT luminosities is not larger than 20% (rms). Clusters with large angular
(cid:13)c 0000RAS,MNRAS000,000–000
4 M. Arnaud and A.E. Evrard
extent—inparticular,largerthanGINGA’s1.1deg×2deg
fieldofview—arelikelytoincurlargererror,butthenum-
berof such objects in oursample is small. Furthermore, we
lookedforsystematicvariationwithredshiftoftheratiobe-
tween theGINGA and ROSATluminosities. Suchadepen-
dence with redshift was noticed by Ebeling (1993) for the
ratiobetweentheluminosities(HEAO,MPCandEXOSAT
data) given by Henry & Arnaud (1991) and the RASS lu-
minosities. The observed increase of thisratio with redshift
(up to z ∼ 0.07) was interpreted as being due to an un-
derestimate of flux for nearby, extended clusters by earlier
experiments.ThiseffectisnotapparentintheGINGAdata,
due to the larger instrument FOV and the use of scanning
mode for the most extendedclusters.
We now consider systematic errors on temperature. A
detailed estimate of this error, due to calibration uncer-
tainties and background subtraction, was done by Hughes
et al. (1993) for the GINGA observation of Coma. They
found a systematic uncertainty of ±0.13 keV which, in
fact, dominates the especially small statistical uncertainty
(±0.09 keV)forthiscluster.Arnaudetal.(1992)alsofound
atypicalsystematicuncertaintyduetobackgroundsubtrac-
Figure1. Theluminosity—temperaturerelationforthe24clus-
tion of 0.14 keV for A2163 (T =14.5 keV). Although this
ters used in this analysis. Filled circles are the homogeneous
systematic error is expected to depend on cluster flux and GINGA subset while open circles denote clusters with ASCA
temperature,thesetypicalfiguresshowthat,formostofthe temperatures and EXOSAT, MPC, or ASCA luminosities. Er-
clustersconsidered here,statistical errorsstilldominatethe ror bars plotted on T are 90% confidence limits. The solid line
systematicerrors.Thisisfurtherconfirmedifonecompares isthe best–fit relation; the dashed lines areoffset byan amount
thetemperatureestimatesfromvariousinstruments.Gener- equal to the rms deviation in log(LX) about the mean relation.
ally, good agreement between MPC, EXOSATand GINGA The upper insert shows a histogram of the luminosity residuals
(index),whichmakesapparentthetwoprincipaloutliers,A1060
estimates is found (David et al. 1993), as well as between
andTriangulum.
ASCA and other high-energy instruments (Markevitch &
Vikhlinin 1997). A direct comparison between the temper-
ature estimate for the 7 clusters in our sample with both
ofthegas.Howevertheyassumed atemperatureof3.9keV
GINGA and EXOSAT measurements gives a mean ratio of
from David et al (1993), which is significantly higher than
0.97 with standard deviation of 0.06.
themorepreciseASCAvalueadoptedhere.Fromtheveloc-
ity dispersion, σ = 633km/s, determined by Girardi et al.
(1997)andthebestfitσ−T relationestablishedbyGirardi
3 THE LX −T RELATION AND etal.(1996),weactuallyexpectatemperatureof2.8keV,in
IMPLICATIONS reasonableagreementwiththeASCAvalueof3.1±0.2keV.
ThedeviationofA1060isonlysligtlydecreasedifthisvalue
3.1 The L −T Relation
X isusedinstead oftheASCAvalue.Eitherboththegalaxies
Figure 1 shows the luminosity–temperature relation for the and the gas are in a very particular dynamical state or the
sample, with 90% confidence errors plotted for T. Statisti- gas structure or content in this cluster is abnormal.
calerrorsonL aretypicallysmallerthantheplottedpoint Thedataarewellfitbyapowerlawlog(L )=(2.88±
X X
size and so are not shown. The homogeneous GINGA sam- 0.15)log(T/6 keV)+(45.06±0.03). The central values and
ple data are shown as filled circles, the remainder as open uncertainties in the fit parameters are derived from sim-
circles. The set of clusters define a remarkably tight rela- ple least squares fits (in log space) to a large number of
tion. The main outliers are A1060 and Triangulum, which bootstrap resamplings employingtheestimated T measure-
are anomalously dim (or hot) in comparison to the others. ment errors (quoted errors in slope and intercept are 1σ).
The deviation of Triangulum is significantly reduced if the Notethat since bootstrap procedure is used to estimate er-
ROSAT luminosity is employed in place of the EXOSAT rors the exact fit method is of secondary importance. The
value. As noted above, this difference is likely due to its slopeof2.88±0.15 isconsistentwiththatfoundbyEbeling
largeangularextent.ThepeculiarityofA1060wasnotedby (1993) from RASS luminosities (2.73±0.39) but is signif-
Loewenstein &Mushotzky(1996) in adirectcomparison to icantly smaller than the value of 3.36 found by David et
AWM7,aclusterofsimilartemperaturewhichisalsoinour al. (1993) using mostly MPC and EXOSAT data. This is a
sample.Girardietal.(1997) also pointedoutthedeparture direct consequence of the underestimate of luminosities for
ofthisclusterfromthegeneralL −T relation.Theynoted nearby clusters by these last two experiments (see above);
X
thatitstemperatureishighcomparedtoitsvelocitydisper- theobservedclusterswithlowluminositiesbeingonaverage
sion and suggested some anomalies in the dynamical state at smaller distance than more luminous clusters which can
(cid:13)c 0000RAS,MNRAS000,000–000
Intracluster Gas Fractions of X–ray Clusters 5
be observed up to larger distances. Our slope is also con- models and numerical simulations suggest that this edge is
sistent at better than the 2σ level with the recent value of located at density contrast δc≃200, or radius r200.
2.64±0.16 determined by Markevitch (1998) from analysis Writing the gas density in terms of the natural radial
of ASCA temperatures and ROSAT luminosities of nearby variable y≡r/r , we introducethestructure function h(y)
δc
clusters.Allen&Fabian(1998)findaslopeof2.9±0.3inthe
ρ (yr ) ≡ f δ ρ h(y), (2)
non cooling flow portion of their combined ASCA/ROSAT gas δc gas c c
sample. wheretheexplicituseofthegasfractionwithindensitycon-
The raw scatter in decimal log(LX) about the trast δc
luminosity–temperature relation is 0.14, and some of the
M (<r )
variancearisesfrommeasurementerrorsintemperature.We fgas ≡ gaMs δc (3)
estimatethiscontributiontobe0.06,andthuscalculatethe δc
intrinsicscatterinlog(L )atfixedtemperaturetobe0.13, setsthenormalizationofthestructurefunctionh(y)viathe
X
equivalent to a fractional deviation h(δLX/LX)2i1/2=0.30. condition 3 01dyy2h(y)=1 . For an ensemble of clusters,
ForthehomogeneousGINGAsubsetof18clusters,thefrac- weimagine aRset ofgasfraction values{fgas}andstructure
tional deviation reduces substantially, to 0.17. functions {h(y)}.
Considering A1060 as an outlier, the fact that there is The luminosity in equation (1) with this notation be-
nosignificanttrendtowardlargerscatterinLX atlowtem- comes
peratures implies that cooling flows do not contribute sub-
L ≡ f 2 Q(δ ,T) (4)
stantially to the total luminosity of these systems. Our se- X gas c
lection criterion ofafixedcooling flowmassdeposition rate where
isroughlyequivalenttoanabsolutelimitincoreluminosity
excess attributed to the cooling flow region. This absolute Q(δc,T) = d3y h2(y) [Λ(T(y))ρc2δc2rδc3] (5)
Z
limitrepresentsalargerfractionofthetotalluminosityfrom
smallersystems,leadingtotheconcernthatcoolingflowsof is largely determined by the structure function h(y), with
variable strength (but underour 100 M⊙ yr−1 limit) might weaker dependence on the form of the temperature distri-
inducelargervarianceatlowtemperaturesifthecoolingre- bution through T(y).
gion dominated the luminosity. This effect is not apparent The variance in luminosity at fixed temperature has
in thisdata set of modest number. contributions from gas fraction variations and structural
variations
3.2 Limits on Gas Fraction Variations δL2 = 4δf2+4δfδQ+δQ2, (6)
The small scatter in Figure 1 supports the notion that at where theδ’s are fractional variations at fixed temperature
least this sub–population of weak cooling flow clusters is
δL δf δQ
dquatitaearpepguealarraitnoidtdssstwriutchtuthraelidperoapoefrtlaiersg.eIgnaspafrraticctuiolanr,vatrhie- δL≡(cid:18) LXX(cid:19)T , δf ≡(cid:18) fggaass(cid:19)T , δQ ≡(cid:18) Q (cid:19)T. (7)
ationswithinthevirialregionsofclusters,anideahintedat The magnitude of intracluster gas variations δ can be
f
by the comparison of A1060 to AWM7 by Loewenstein & determined from theobserved scatter in luminosity δ
L
Mushotzky(1996).Wenowaddressthisissueusingaspher-
1
ically symmetricmodelasaframework for calculation. The δf2 = 4(δL2 −δQ2)−δfδQ (8)
purpose of the exercise is to provide a context for connect-
with an additional assumption about how changes in gas
ingthescatteringasfraction atfixedtemperaturewiththe
fractionandclusterstructurearecorrelated.Weassumehere
observed variance in X–ray luminosity. Theexercise is non-
thattherearenocorrelationsbetweenchangesingascontent
trivial because luminosity scatter can be due to changes in
and variations in internal structure, meaning
gas density structure as well as overall gas fraction varia-
tions. hδ δ i = 0 (9)
f Q
Consider a cluster with smooth, spherically symmetric
distributions of gas and total mass about its center. Let with theangle brackets denoting an ensemble average.
Mδc be the total mass contained in a sphere (of radius rδc With this assumption, the fact that δQ2 is positive defi-
about the center) which encompasses a mean density δ ρ , nite leads to an upper limit on gas fraction variations from
c c
whereρc≡3 Ho2/8πGisthecriticaldensityoftheuniverse equation (6)
withinwhichtheclusterisembedded.ThebolometricX–ray 1
hδ2i1/2 ≤ hδ2i1/2. (10)
luminosity of a cluster is f 2 L
and theresultant values from Figure 1 are
L = d3r ρ2 (r) Λ(T(r)) (1)
X gas
Z hδ2i1/2 ≤ 0.15 full sample, (11)
f 0.09 GINGA subsample.
whereΛ(T(r))isanappropriatelynormalizedemissivityde- n
pendent only on temperature. In practice, the integral ex- Enlarging these limits is possible if an anti–correlation
tendsnottoinfinity,buttoafiniteradiusdenotingtheedge exists between δ and δ . Such an anti–correlation would
f Q
of the relaxed, virialized portion cluster. Spherical collapse require gas poor clusters to be more centrally concentrated
(cid:13)c 0000RAS,MNRAS000,000–000
6 M. Arnaud and A.E. Evrard
than average, and vice–versa. In this way,the changein lu- assumptions does not hold in the real cluster population.
minosityduetoalower(higher)gasmasswouldbecompen- For example, if we retain bremsstrahlung emission and the
sated by an increased (decreased) central density. We show condition of virial equilibrium, then the observed slope of
below that the observational data do not support such an the L –T relation leads to a constraint on structure factor
X
anti–correlation.Inaddition,onemechanismcapableofgen- and gas fraction dependence
eratinggaslossfromclustershastheoppositeeffect.Heating
f 2(T)Qˆ(T) ∝ T0.88±0.15. (14)
due to winds from early–type galaxies leads to a less cen- gas
trallyconcentratedICMstructureandslightlydepressedgas The assumption of pure bremsstrahlung is quite accu-
fractions within the virial radius (Metzler & Evrard 1997).
rate above about 2 keV, but at lower temperatures, line
Thisimplieshδ δ i>0,aresultwhichservestostrengthen
f Q emission becomes significant unless the metallicity is very
thelimit imposed by equation (8). low. Pure bremsstrahlung emission itself does not exactly
We conclude that variations in gas fraction within the scales asT1/2 duetotheadditionalGauntfactor. However,
virial regions (overdensities δ ∼ 200) of weak, cooling
c this factors varies by less than 4% in the1–10 keV temper-
flow clusters are likely to be quite small. Fractional devi- ature range. On the other hand, the line emission reaches
ationsatfixedtemperatureδ mayhavestandarddeviation
f about18%ofthetotalemissionat2keVforatypicalabun-
smaller than 10%, particularly if galactic wind feedback, or
dance of 1/3 thesolar value. This extra line emission tends
some other, non–gravitational, entropy generating mecha- to flatten the slope of the L –T relation, as low T clusters
X
nism,hasproducedcorrelatedvariationsingasfractionand are boosted in luminosity more than their high T counter-
density structure.
parts. This effect is amplified because low T clusters tend
The limit on gas fraction variations applies to clusters tohavehigherabundances.Giventhemeasuredabundances
of similar temperature; there remains the possibility that wecomputedΛ(T)foroursample(thecorrect valueisused
the mean cluster gas fraction varies, perhaps considerably,
in all thefollowing). We found that the departure from the
overthe range of temperatures probed in our sample. T1/2 law is actually small: the best fit slope of the power
law is 0.36 instead of 0.5.
The assumption of virial equilibrium is supported by
4 TEMPERATURE DEPENDENCE OF ICM cosmological, gasdynamic simulations (Bryan & Norman
STRUCTURE 1998andreferencestherein).Therelationholdsparticularly
wellforδ ≃200−2500,with200thefiducialchoiceofvirial
In thissection, we examineour sample for evidenceof vari- c
radius and 2500 approaching the onset of the cluster core.
ationinthemeanclusterICMstructureandvirialgasfrac-
Weusethevirialtheoremasonemethodforestimatingtotal
tion with cluster temperature. Wealso discuss theorigin of
cluster masses below.
theL –T relation slope.
X
Thereremainstheissueofstructuralregularity.Recent
numericalinvestigation ofdarkmatterhalosevolvedwithin
4.1 The slope of the L –T relation hierarchical clustering cosmogonies indicates that low mass
X
halosare,onaverage,morecentrallyconcentratedthanhigh
Supposethatclusterdensityprofilesh (y)andgasfractions
T mass halos (Navarro et al. 1997). Although the magnitude
f (T)are,inthemean,temperaturedependent.Ifwefur-
gas of the change is likely to be small within the mass range
ther suppose that clusters are internally isothermal, then
probed bytheX–ray clustersin oursample, thequalitative
equation (4) can be rewritten in a form which makes clear
direction is again to flatten theslope of theL –T relation,
X
thedependenceon theshape function h (y)
T if one assumes that thegas density follows that of the dark
L (T) = f2 (T) [M (T)ρ δ Λ(T)]Qˆ(T) (12) matter.
X gas δc c c
The assumption that the gas traces the dark matter is
with Qˆ(T) = (3/4π) d3yh2T(y) is the dimensionless emis- reasonable if shock heating through gravitational infall is
sion measure. Note tRhat Qˆ(T) is equal to hρ2gasi/hρgasi2, the only entropy generating mechanism, but this assump-
with the angle brackets denoting the average over the clus- tion breaks down if extra energy sources or sinks are avail-
ter atmosphere. It is thus a structure factor which depends able to the gas. In particular, if galactic winds play an im-
solely on the gas density shape and characterises the con- portant role in heating the ICM, then the extra entropy
centration of the gas distribution. dumped into the gas can change its density structure dra-
Thetraditionalapproachistoemploythefollowing set matically (Metzler &Evrard1994). Thenetresult ofwinds
of additional assumptions : (i) pure bremsstrahlung emis- is to inflate the gas distribution, decreasing the Qˆ factor
sion (Λ(T)∝T1/2), (ii) virial equilibrium (Mδc∝T3/2), (iii) and causing diminished gas densities and, correspondingly,
structurallyidenticalclusters(Qˆ(T)=C1),and(iv)constant lowered gas fractions, at radii interior to the virial radius.
gas fraction (fgas(T)=C2). This leads to an expected scal- Becausesupernova–drivenwindsimpartroughlyafixedspe-
ing relation between luminosity and temperature of slope cificenergyintotheirsurroundings,theirimpactisfeltmore
two stronglyinshallowerpotentialwells.Thissteepensandlow-
L (T) ∝ T2. (13) ers the intercept of the LX–T relation compared to models
X without winds.
Thefactthattheslopeoftheobservedrelationissignif- The magnitude of this effect will depend on details of
icantly steeper than two implies that one or more of these the assumed wind model, but plausible models exist. The
(cid:13)c 0000RAS,MNRAS000,000–000
Intracluster Gas Fractions of X–ray Clusters 7
Figure 2 displays the best fit slope parameters against
temperature.Asnotedpreviously(Davidetal.1990;White
5 1991; Mohr & Evrard 1997), there is a clear trend toward
decreasing β toward lower T. This trend is not likely to be
4
anartifactofthefittingprocedure,asMohr&Evrard(1997)
3 seeasimilartrendwithβdefinedinanon–parametric,non–
T) azimuthally averaged fashion.
Q( Theheterogeneousnatureandlackoferrorestimatesfor
2 nearlyathirdofthesedatamakeitdifficulttodeterminethe
formal statistical significance of this effect. As a reasonable
estimate,wecalculatethebi–weightmodes(Beers,Flynn&
Gebhardt 1990) for the subsamples of 8 clusters with T <
4 keV and 14 clusters with T > 4keV. We applied a 10%
1
fractional error to each data point — slightly larger than
the rms fractional error of 7.8% for the 15 quoted error
0.8 values — and determined the expected subsample means
and standard error from bootstrap analysis. The values of
ber hβilo=0.51±0.03 and hβihi=0.66±0.03 for the low– and
et high–T subsamplesdifferatthe3.3σ level.Wesuspectthat
m
a 0.6 futureanalysiswithamoreregulardatasetwill confirmthe
ar
e p statistical significance of thistrend.
op The implication is that the ICM density structure of
sl clusters is temperature dependent, and the sense of the ef-
0.4 fect is such that Qˆ(T) is an increasing function of temper-
ature (see Figure 2). Models of cluster formation incorpo-
rating galactic winds exhibit a trend of β versus T similar
1 10 to that seen in the observational data (Metzler & Evrard
T (keV) 1997). The trend of Qˆ(T) with temperature is in the right
direction to steepen the L –T relation, but whether it is
Figure 2. Values of the gas outer slope parameter β (bottom) X
andstructurefactorQˆ (top),derivedfromβ–modelfitstoX–ray entirely responsible for this effect requires investigation of
images plotted against temperature for 22 of the 24 members of thetrend of mean gas fraction with temperature.
our sample. Point styles are the same as the previous figures.
Errorbarsavailableintheliteratureareshown.
4.3 Intracluster Gas Fraction Values
Thedetermination of absolute valuesof thegas fraction re-
numerical simulations of Metzler (1994) produce a slope of quiresamethodtoestimatethetotalclustermass.Weshow
2.96±0.05inthebolometricLX–T relation.Cavaliere,Menci here that two different methods for total mass estimation
&Tozzi(1997) createasemi–analyticmodelwhichpredicts lead to different conclusions about the behavior of the gas
aprogressivesteepeningoftheLX–T slopetowardlowtem- fraction in clusters.
perature clusters, consistent with the data of Ponman et
al. (1996). Finally, the constant central entropy model of
Evrard & Henry (1991) predicts a slope of 2.75 which is 4.3.1 Mass determination methods
consistent,within theerrors, with ourobservational sample
Estimating cluster total masses hastraditionally beendone
behaviour.
with the “β–model” (BM) approach (Cavaliere & Fusco-
Femiano 1976), for which thetotal mass is
T r (r/r )2
4.2 Gas density profiles from X–ray imaging Mβ(r) = 1.13×1015β 10 keV Mpc 1+(r/cr )2 M⊙ (16)
c
Azimuthally averaged profiles of cluster X–ray surface
This model assumes a non-rotating, isothermal gas distri-
brightness images are generally well fit by the standard,
bution in hydrostatic equilibrium, with gas density profile
β–model form
oftheform assumed toderivethesurfacebrightnessimage,
I(r) = I0 1+(r/rc)2 −3β+1/2 (15) equation(15).Thisleads toabindingmasswhich increases
linearly with radius outside the core. Estimating the mass
(cid:2) (cid:3)
derived by assuming an isothermal gas with density distri- at some density threshold δ is readily done by calculating
c
bution ρgas(r) = ρ0(1+(r/rc)2)−3β/2. Fits to this model the mean interior density as a function of radius. Outside
existforallbuttwo(A754andA370)ofthemembersofour thecore region, this radius scales as
sample.Table1listsvaluesfromtheliteratureofthebest–fit
−1/2 1/2
coreradiusrc andouterslopeparameter β.Errorsonthese r = 2.78 β1/2 δc T h−1 Mpc (17)
parameters are also given in thetable when available. δc,β (cid:18)500(cid:19) (cid:18)10 keV(cid:19) 50
(cid:13)c 0000RAS,MNRAS000,000–000
8 M. Arnaud and A.E. Evrard
An alternative method is to employ the virial theorem
(VT) at fixed density contrast, which leads to
3/2
T
MVT(rδc) = M10(δc) (cid:18)10 keV(cid:19) . (18)
Calibration of this relation with numerical experiments by
Evrard, Metzler & Navarro (1996, hereafter EMN) shows
that the intercept M10(δc) is remarkably insensitive to as-
sumptions regarding the background cosmology and to the
effectsofgalacticwinds.TheexperimentsshowthattheVT
method has intrinsically smaller variance compared to the
BM approach, because the imaging information introduces
an additional source of noise. A recent intercomparison of
12 cosmological gas dynamic techniques lends credence to
thesecalibrations.Frenketal.(1998)findagreementwithin
4% forthetotal clustermassand6% forthemassweighted
cluster temperature among codes varying substantially in
algorithmic detail, resolving power and numerical parame-
terization.
Atdensitycontrastδc=500,EMNfindM10(500)=2.2×
1015h−501 M⊙, with a corresponding physical size scaling as
r500(T)=2.5(T/10 keV)1/2h−501 Mpc. At δc=200, the mass
intercept is M10(200)=2.9×1015h−501 M⊙ and thephysical
size isr200(T)=3.7h−501 Mpc,roughly50% larger thanr500.
All values are quoted for the current epoch, and both scale Figure 3. Gas fraction versus total mass at inferred density
with redshift as (1+z)−3/2. Though typically quite small, contrasts δc =500 and 200 derived using the VT (upper) and
this redshift scaling is taken into account in our analysis. BM (lower) approaches discussed in the text. Symbol types are
Weexaminegasfractionsatthesetwocharacteristicdensity identicaltothoseinFigure1.
contrasts below. For the majority of imaging data, r500 is
comparable to or smaller than the radius used to fit the
gas modelparameters whilevaluesat r200 requireamodest efiqnudastions(16)and(18)atr500,andignoring(rc/r500)2,one
extrapolation outsidetheX–ray imaged region in nearly all
cases. Mβ(r500) = β 3/2. (19)
Todeterminethegasmass,weemployvaluesofβandrc MVT(r500) (cid:18)0.79(cid:19)
fromTable1,anddeterminethecentralgasdensitybynor-
malizing to the bolometric luminosity, assuming the latter From Figure 2, all but one of the values of β in our sample
is due to emission within r200. Though not exact, the typi- lie below 0.79. The β–model estimates are therefore consis-
cal error introduced by the use of r200 as a cutoff is small, tently lower than those of the virial theorem, and this is
since for most clusters the contribution to the luminosity clearly reflected in the overall horizontal shift of the data
outside this radius is negligible. As a check, we normalized pointsbetween theupperandlowerrows. Furthermore,the
the gas mass using r500 — an extreme choice since many systematic trend of β with temperature means the discrep-
cluster images extend beyond this radius and since r500 is ancy is larger at smaller temperatures/masses. This effect
nearly a factor 2 smaller than r200 — and thefractional in- also causes the gas fractions at constant δc in the BM to
crease in gas density was typically ∼5%, with a maximum be consistently higher than those of the VT, although the
fractionalincreaseof24%(absoluteincrease∼3%)forlowβ differenceis small at δ=200.
clusters. As detailed below, our procedure produces a pop- Insomeofthepanels,atrendofmeangasfractionwith
ulation mean gas fraction at r500 for hot clusters consistent cluster mass or temperature is evident to the eye. The lack
with the determination of Evrard (1997) from the observa- of rigorous, homogeneous, statistical error estimates for the
tional samples of White & Fabian (1995) and David et al. rc andβ valuesinoursamplemakesitdifficulttomakepre-
(1995). cisestatementsregardingthesignificanceofthesetrends.We
proceed with an approximate treatment in which we assign
a 1σ fractional uncertainty of 30% to each of the gas frac-
tionestimates.Thisapproachallows ustoexploretrendsin
4.3.2 Gas fraction estimates
thedataand providesan estimate oftheirstatistical signif-
GasfractionsandtotalmassesderivedfromtheVTandBM icance under an assumed error budget. The 30% error may
methods at δ =500 and 200 are shown in Figure 3. There begenerousgiventhatT,r andβ aretypicallydetermined
c c
are trends apparent which reflect the different behavior of to∼ 5% accuracy, and the fact that gas mass errors are
the two mass estimate methods to the input data, partic- smallerthanerrorsinβ andr alone,becausetheseparam-
c
ularly the sensitivity to the slope parameter β. Comparing etersaretightlycorrelated. However,the“cosmicvariance”
(cid:13)c 0000RAS,MNRAS000,000–000
Intracluster Gas Fractions of X–ray Clusters 9
discrepancy between A1060 and the mean of the cool sub-
sample is smaller underthe VTapproach.
The value of 16.2 ± 1.5% for the hot subsample in
the VT case is consistent with the 95% confidence region
f¯gas(r500)=17.0±1.7% (h50 =1) determined by Evrard
(1997)fromthecombinedsamplesofWhite&Fabian(1995)
and David, Jones & Forman (1995). Only 3 of the 26 clus-
tersinthecombinedstudyhaveT <4 keV(andthesehave
fairly uncertain gas fractions which make negligible contri-
butiontothesamplemean),socomparisonwithourhigh–T
subsample is reasonable. The agreement between these two
studies is not entirely trivial. Although there is nearly 50%
overlap with our subsample — 6 of our 14 are in common
Figure 4. Bi–weightmean gasfractions atdensity contrasts of
500and200fortheVTandBMmethods.Dataforthefullsample with White & Fabian — we employ here different temper-
of22clustersareshownasfilledcircles,thelowT subsampleof8 ature estimates in many cases (though all are consistent
clusters as triangles, and the high T subsample of 14 clusters as within the quoted errors) and we normalize the gas mass
filledsquares.Errorbarsare90%confidencelimits(1.65σ)deter- differently.
minedfrombootstrapresamplingandassumingafixedfractional A less obvious trend in the data is the larger variance
errorof30%onindividualclustergasfractionestimates. in gas fractions in the BM case compared to the VT. This
ismore apparent in theT–dependentsubsamples, since the
trend of mean gas fraction with temperature mimics vari-
Table 2.Bi–weightMeanGasFractions(%,h−3/2) ance in the full VT sample. The lower scatter in the VT
50
method confirms theoretical predictions that the β param-
δc=500 δc=200 eter acts as an extra source of noise in the BM mass esti-
Sample VT BM VT BM mates (EMN). An increase in scatter is also seen at larger
All 14.0±1.3 18.3±1.5 18.1±1.4 21.0±1.8 radii/lower densities, consistent with the numerical model
T <4keV 10.2±1.5 15.3±2.4 14.5±2.1 19.9±3.2 predictions and interpreted simply as reflecting the longer
T >4keV 16.2±1.5 20.0±1.9 20.1±1.6 21.6±2.2 relaxation timescales in theouter regions of clusters.
4.4 Implications
in thebinding mass estimates is believed to be ∼15−30%
at these density contrasts (EMN), and this represents the There is strong evidence from the slope of the L –T re-
X
limiting accuracy of any dataset. lation and from the behavior of the image fall-off parame-
Bi–weight mean gas fractions determined from a boot- ter β that cluster structure varies systematically with tem-
strap analysis assuming fixed 30% fractional errors in each perature. Whether the virial gas fraction varies with clus-
measurement are listed in Table 2 and summarized in Fig- ter temperature is a question whose answer is both model
ure 4. Data for the whole sample as well as cool and hot and scale dependent. The model dependency we highlight
subsamplesare provided.UncertaintiesquotedintheTable is the choice of total mass estimation method, while the
arestandarderrorsofthemean,while90%confidenceerrors chosen density contrast sets the scale. For example, the
(assuming Gaussian statistics) are plotted in theFigure. β–modelappliedatδ =200producessimilarmeanvaluesin
c
The trend of rising gas fraction with increasing radius high and low temperature subsamples — 0.216±0.022 and
(lower δ ) is apparent with both mass estimation meth- 0.199±0.032, respectively —whereas thevirial theorem at
c
ods, and reflects the more extended nature of the gas den- δ =500yieldssignificantlydifferentvaluesinthesesubsam-
c
sity compared to the assumed or inferred dark matter pro- ples—0.162±0.015 against 0.102±0.015, respectively(see
files. Gas fraction values are consistently higher in the BM Table 2 and Figure 4, all valuesscale as h50−3/2).
approach, due to the smaller, inferred total masses noted That the implications of the observed luminosity–
above. As a result, significantly different sample means can temperature relation are model dependent can be seen by
bequoteddependingon themassmethod and extentofra- considering the behavior of terms with temperature depen-
dial coverage of the dataset. Compare, for example, f¯ = dencecontained inthedefiningrelation,equation(12).The
gas
0.140±0.013fortheVTatδ =500withf¯ =0.210±0.018 structurefactorQˆ(T)appearstoincreasewithtemperature,
c gas
for the BM at δ =200. butthiseffectis notsolely responsible forthesteepeningof
c
The range becomes wider when one considers subsets the L –T relation. The dependence with T of the other
X
of the data defined by temperature. All the panels in Fig- factors depends on the total mass estimation method. In
ure (4) display a trend of decreasing gas fraction at lower the VT approach, while M (T) scales classically as T3/2,
δc
T, but only the VT method at δ =500 is deemed statis- f (T)increaseswithT.IntheBMcase,whilef isfound
c gas gas
tically significantly (2.8σ) under our 30% error hypothesis. to be nearly constant, the slope of the M (T)−kT rela-
δc
Duetotherobustnessofthebi–weightmeanestimator,this tion is steeper than 3/2, due to the extra β factor and its
conclusionisvalidwithorwithouttheinclusionoftheprin- increasewithT (equation(19)).Bothcasesresultinasteep-
cipaloutlierA1060.Itisperhapsalsoworthnotingthatthe ening of the f2 (T) M (T) term to a slope greater than
gas δc
(cid:13)c 0000RAS,MNRAS000,000–000
10 M. Arnaud and A.E. Evrard
thecanonical3/2.Thetemperaturedependenceofthisterm In wind models, gas loss is correlated with structural
contributesaboutequallywithQˆ(T)tothesteepeningofthe extension of the gas, leading to hδ δ i > 0. The inferred
Q f
L –T relation. variance in gas fraction derived from equation (8) could be
X
Allan&Fabian(1998)takeadifferentapproachinanal- substantiallysmallerthanthelimitsphrasedaboveifwinds
ysisofaclusterdatasetcombiningASCAspectroscopywith haveplayedanimportantevolutionaryrole.Numericalsim-
ROSATimaging.TheyfitASCAdatatoamulti-component ulations of clusters used by Mohr & Evrard (1997) exhibit
spectral modelwhich includesemission from acentralcool- lessthan5%fractionalvariationintheirvirialgasfractions,
ingflow(Johnstoneetal.1992),absorptionfromgalactichy- and a similar value is seen in models with galactic winds
drogen, intrinsic hydrogen absorption, and a superimposed (Metzler & Evrard 1997). The expected “cosmic variance”
backgroundplasmaattemperatureT.Comparedtosimpler in virial gas fractions is thusquitesmall.
fitswithoutcoolingflowandintrinsicabsorption,thebestfit Comparison of a number of different simulation tech-
temperatures in their complex fits are higher, and the best niquespaintsasimilarpicture.Frenketal.(1998)findagree-
fitslopeoftheLX–T relationisshallower,2.33±0.43.Allen mentatthe5%level(1σdeviation)forthevirialgasfraction
& Fabian emphasize that this slope is consistent with the ofasingleclusterdeterminedbytwelveindependentgasdy-
straightforward theoretical expectation of 2; however, note namicmethods.Thisindicatesthatsystematicuncertainties
that the range of slopes allowed in our fit does overlap the in the tested numerical simulation techniques are small for
errorrangequotedabove(derivedfromachangeδχ2 =2.71 the simplest case of a single phase ICM. However, it is im-
in their χ2 analysis). portant to note that the quoted simulations ignore galaxy
Theapproximatenatureofourerroranalysislimitsthe formation and its attendant multi–phase, gas structure. In-
conclusionswhichcanbedrawnregardinggasfraction vari- troducing galaxy formation via cooling and star formation
ations. A definitive analysis awaits uniform treatment of a is likely to increase the scatter in ICM gas fraction values,
large, homogeneous dataset. There is some supporting ev- and it will be interesting to see whether viable models can
idence for deficient gas fractions in poor groups (Hickson, do so without violating the observational limits presented
1997 and references therein), but there also appears to be here.
largerscatteratlowtemperatures(e.g.,Ponmanetal.1996) Evidence for trendsin f¯ with cluster temperature is
gas
and observational uncertainties on the ICM mass remain ambiguous. A key uncertainty lies in estimation of the to-
large in these systems. In particular the separation of the tal, binding mass. The β–model approach produces masses
ICM emission from theX–ray emission linked to individual which are characteristically smaller than the virial theorem
galaxiesisambiguousanddifficultwithpresentinstruments method as currently calibrated by numerical simulations.
(Mahdavi et al. 1997, Mulchey and Zabludoff 1998). For a cluster with β=0.6, the masses inferred at δ =500
c
differsubstantially,M =0.66M .Thisdifferentbehavior,
β VT
coupledwiththetemperaturedependenceofβ,leadstodif-
fering conclusions on whether the mean gas fraction varies
5 CONCLUDING DISCUSSION
with cluster temperature.
The narrow scatter in the L –T relation of weak cooling Forthesamereasonthereissomeambiguityintheori-
X
flow clusters (M˙ ≤ 100 M⊙ yr−1) indicates that the struc- gin of the steepening of the LX–T relation. The fact that
ture of the intracluster medium in this majority sub–class the gas is more concentrated in high-kT cluster explains at
is very regular. Undera conservative assumption of no cor- least half of theeffect,directly throughthestructurefactor
relation between virial gas fraction and internal, structural intheX–rayluminosity.Intheβ–modelapproach,thetem-
variations (hδ δ i=0), we place upper limits on the rms peraturedependenceoftheICMstructurecanfullyaccount
Q f
percentage variation in virial gas fraction of 15% for the for the LX–T relation slope, via the additional dependence
entire sample of 24 clusters, and 9% for the homogeneous of the total mass with β. On the other hand, in the VT
GINGA subset of 18 clusters. approach, the gas mass fraction increases with T and this
These limits are conservative because the X–ray imag- effect contributes about equally to theLX–T relation slope
ingdatasupportsthepointofviewthatgasinlowT clusters increase. Both effects, lower gas fraction and inflated gas
is more extended than that of high T clusters (Figure 2). distributioninlowTsystems,areexpectedinmodelsincor-
Mohr & Evrard (1997) find the same result with a non– porating theeffects of galactic winds.
parametricanalysisofX–rayimages.Theyalsoderivelimits Independent methods for estimating virial masses are
on gas fraction variations similar tothose found herebased crucial, and weak gravitational lensing provides an impor-
on the tightness of the X–ray size–temperature (ST) rela- tant approach. Allen (1997) provides evidence that weak
tion. The agreement is non–trivial because the ST analysis lensingand β–modelmass estimates areconsistent, butthe
is essentially local, being based on an differential isopho- present errors at radii between r500 or r200 are too large to
tal area, while the L –T relation analysis is global, based discriminate between the BM and VT approaches. Wesug-
X
antheintegrated emission of theentireclusteratmosphere. gest that weak lensing analysis of a small sample of mod-
Both analyses are consistent with idea that energy input erate redshift clusters selected to have β ∼< 0.6 from X–ray
from galactic winds has led to extended gas atmospheres imaging could settle the score, if individual measurement
inlowT clusters(David,Forman&Jones1990,1991; White uncertainties could be kept below a few tens of percent at
1991;Metzler&Evrard1997;Cavaliereetal.1997;Ponman δc=500.
et al. 1996). Future X–ray missions which provide simultaneous
(cid:13)c 0000RAS,MNRAS000,000–000