Regularity in the X-ray surface brightness profiles of galaxy clusters and the M-T relation

16 Pages·0.65 MB·English

by  Doris M. Neumann

A&A manuscript no.
ASTRONOMY
(will be inserted by hand later)
AND
Your thesaurus codes are:
ASTROPHYSICS
11 (11.03.1; 12.03.3; 12.04.1; 13.25.2 )
Regularity in the X-ray surface brightness profiles of galaxy
clusters and the M–T relation.
D.M. Neumann, M. Arnaud
CEA/DSM/DAPNIA/Serviced’Astrophysique,CEA-Saclay,
L’Orme desMerisiers, Bˆat. 709, F-91191 Gif-sur-Yvette, France
email: [email protected] , [email protected]
SubmittedJanuary 5, 1999
9
9 Abstract. We used archival ROSAT observations to in- 1. Introduction
9 vestigatethe X-raysurface brightnessprofilesofa sample
1 of 26 clusters in the redshift range 0.04 < z < 0.06. For There is growingobservationalevidence that the physical

n 15 of these clusters accurate temperatures (kT > 3.5 propertiesofgalaxyclustersobeyscalingrelations.Evrard
X
a keV) were available from the literature. The scaled emis- (1997)showedthatmostobservedclustersofgalaxieshave
J
sionmeasureprofileslook remarkablysimilarabove 0.2 a similar fraction of hot gas, as compared total mass,
8 timesthevirialradius(rVT200).Ontheotherhanda∼large fgas ∼ 0.060 h−3/2. This fraction sets a lower limit on
1 scatter is observed in the cluster core properties. We fit- the baryon fraction in these dark matter dominated ob-
v ted a β–model (with and without excising the central jects. Furthermore, studies focusing on the relationship
2 part) to all the ROSAT profiles to quantify the struc- betweenthe X-rayluminosityofthe hotgas(orintraclus-
9 tural variations in the cluster population, unraveling a ter medium, hereafter ICM) and its X-ray temperature
0
robust quadratic correlation between the core radius and done by Markevitch (1998), Allen & Fabian (1998) and
1
0 theslopeparameterβ.Wequantifiedtheshapeofeachgas Arnaud & Evrard (1998) have revealed a relatively tight
9 densityprofilebythevariationwithradiusofthelogarith- correlationbetweenthese twoparameters.Recently Mohr
9 mic slope, α . The bi-weight dispersion of α among the & Evrard (1997) found that there exists a strong corre-
n n
h/ clusters is less than 20% for any given scaled radii above lation between the X-ray isophotal radius and the X-ray
p x = 0.2. There is a clear minimum spread at x = 0.3, temperature of the hot ICM. The study of Hjorth, Ouk-
- which is related to the existence of a correlation between bir & van Kampen (1998), based on a sample of clusters
o
r core radius and β. These ensemble properties are insen- withgoodX-raytemperatureestimatesandmassinferred
st sitive to the exact treatment of a possible central excess from lensing observations, suggests that there is a tight
a when fitting the profiles.Onthe other hand the scatter is mass–temperature relation.
v: decreased when the radii are scaled to rVT200. Such scaling laws are expected if clusters of galaxy
i The regularity we found in the gas profiles at x> 0.2 formahomologouspopulationandareanindicationofan
X
supports the existence of an universal underlying dark underlyingstructuralregularityinthe clusterpopulation.
ar matter profile, as already predicted by theoretical works. Hence their study can give useful insight into the forma-
It suggests that non gravitational heating is negligible tion and evolution of clusters and the underlying cosmo-
for clusters with temperature above 3.5 keV. The very logical scenario. In addition they can provide an efficient
largescatterobservedinthecoreprop∼ertiesfavorscenario way to estimate some cluster properties, which are diffi-
where Cooling Flows are periodically erased by merger cultto measuredirectly.Inparticularthe M–Trelationis
events.Ourresultsareconsistentwiththeclassicalscaling ofspecialinterest,inview ofthe cosmologicalimportance
relation between Mass and Temperature (M T3/2 (1+ of determining cluster masses and the possibility to mea-
z)−3/2). Accordingly the spread in the reduce∝d mass pro- sure accurately the temperature with current and future
files derived from the hydrostatic isothermal β–model is X-ray telescopes.
small. Structuralregularityisexpectedontheoreticalgrounds
(Teyssier,Chi`eze&Alimi1997).Numericalsimulationsby
Navarro,Frenk&White(1996,1997)indicatethatCDM-
Key words:Galaxies:clusters:general–Cosmology:ob-
halos with masses spanning several orders of magnitudes
servations, dark matter – X-rays: general
follow a universaldensity profile,whoseshape is indepen-
dentofmassorcosmology.Thecorrespondingdensitypro-
file of the hot gas captured in these cluster halos can be
Send offprint requests to: D.M. Neumann reasonably well described by an isothermal β–model , as
2 D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation
shownbyEke,Navarro&Frenk(1998).TheX-raysurface 110000
brightness profile reads, for such a model:
−3β+1/2
S(θ)=S 1+(θ/θ )2 (1)
0 c
h i
which translates into the gas density distribution:
1100--11
−3β/2
n (r)=n 1+(r/r )2 (2)
g g0 c
h i ))
Theβ–modelisknownsincetheearlyEinstein observations 22nn
ii
(Jones& Forman1984)to providea goodfitto the X-ray mm
cc
profileofgalaxyclusters.HoweverMakino,Sasaki&Suto arar 1100--22
//
(1998)recentlyfoundthatthegasdensityprofilesdeduced ss
//
tt
from simulations are too steep to match with the typical (c(c
coreradiirc derivedfromtheseX-rayobservations.More- ate ate
over it is well known that there are large variations, from RR
cluster to cluster, in the observed β–model shape param- nt nt 1100--33
uu
eters, β and r . oo
c CC
Inthispaperwewanttochecktheregularityofthegas
distribution in clusters, described by a β–model . We se-
lected for this study a sample of nearby clusters observed
with ROSAT and we unravel a relation between the core 1100--44
radius and the slope parameter β. We discuss the impli-
cations of our results for the variations of the gas density
profile shape andthe existence ofa mass–temperaturere-
lation.
00..11 11 1100
The paper is organized in the following way: we de-
RRaaddiiuuss ((aarrccmmiinn))
scribeinSect.2oursample ofclustersofgalaxiesandthe
observations.Sect. 3 shows the surface brightness profiles
Fig.1. The ROSAT X-ray surface brightness profiles of
of the clusters. In Sect. 4 we present our isothermal β-
the clusters in our sample. The profiles are corrected for
fits. Sect. 5 shows the correlation between r and β. In
c vignetting effects and background subtracted. Full line:
Sect.6 we describe the consequencesofthe correlationon
clusters in the spectroscopic subsample.
the shape of the gasdensity profilesand quantify its vari-
ations and in Sect. 7 we study the mass profile and the
M–T relation. In Sect. 8 we discuss our results and give The list of the 26 clusters selected is given in Tab.1,
our conclusions. as well as the exposure times and the detector used. The
Throughoutthepaperweassume H =50km/sec/Mpc, samplecoversalargevarietyofmorphologicaltypes,from
o
Λ=0, Ω=1 (q =0.5). relaxed spherical symmetric clusters like A2107, to clus-
0
ters with substructures like A754 or A3559 and contains
non-cooling flow clusters (e.g. A119) as as well clusters
2. X-ray observations known to have strong cooling flows (e.g. A85).
For the PSPC data we only used photons in the band
We used X-ray imaging data retrieved from the ROSAT 0.5-2.0keV,fortheHRIweonlytookintoaccountchannel
databaseatMPE.Thesamplewebuiltconsistsofclusters 2-9,inordertooptimisethesignal-to-noiseratio.Foreach
ofgalaxiesfoundbyAbell,Corwin&Olowin(1989)inthe data set we calculated the exposure map, using the soft-
redshiftrangez=0.04-0.06,whichwereinthe fieldofview ware implemented in the X-ray software package EXSAS
of a ROSAT pointed observation with an off-axis angle developed at MPE for the PSPC data and the software
lessthan10arcmin.Theredshiftrangeissmallenoughto developed by S. Snowden (Snowden 1998) for the HRI
avoidtolookatmajorevolutionaryeffectswithinthesam- data.
ple and the typical cluster size for these redshifts is well We searched the literature for the emission-weighted
matchedto the PSPC centralfieldof view - mostly inside mean temperature of the clusters in our sample. We only
the rib structure of the instrument - or to the HRI field considered temperature obtained with wide energy band
of view. Furthermore we only selected clusters whose ex- experiments(Einstein/MPC,EXOSAT,GINGAorASCA)
tendedemissioncanbefittedwithanisothermalβ-model, to avoid systematic errors and exclude temperature esti-
i.e.theclustermustshowaclearcenterintheX-rayemis- mateswithstatisticalerrorsgreaterthan50%.Theadopted
sion. Finally the signal to noise ratio of the cluster must temperaturevaluesandcorrespondingreferencesaregiven
be high enough to allow accurate modeling of the data. in Tab. 1 for 15 clusters, which constitute our spectro-
D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation 3
102 102
e
r
ure 101 easu 101
s M
a
e n
M o
n ssi
o i
si 100 m 100
s E
mi d
E e
l
a
c
S
10-1 10-1
10-2 10-2
0.01 0.1 1 0.01 0.1 1
Radius (Mpc) r/r
VT200
Fig.2. The derived emission measure profile of the clus- Fig.3. The scaled emission measure profile (Eq. 7) of
ters in the spectroscopic sub sample. The dotted line theclustersinthespectroscopicsubsample.Theradiusis
shows, for comparison, the emission measure profile of normalisedtor (Eq.9).Dottedline:Abell2163pro-
VT200
Abell 2163 (z=0.201, kT=14.6 keV, Elbaz, Arnaud & file. Beyond r/r 0.1 the profiles look remarkably
VT200
∼
B¨ohringer 1995) similar.
scopic sub-sample. Note that this sub-sample does not The emissionmeasure along the line of sight at radius
contain low temperature clusters (kT >3.5keV), because r, EM(r), can be deduced from the X-ray surface bright-
only the brightest clusters of our sample have good tem- ness, S(θ):
peraturemeasurements(seeFig.1).Alltheseclustershave
4 π (1+z)4 S(θ)
flux greater than 1.7 10−11ergs/s/cm2. The spectroscopic EM(r)= ; r =d (z) θ (3)
A
Λ(T,z)
sampleis80%completeatthatfluxlimit,whencompared
to the complete sample derived by Ebeling et al. (1996) where Λ(T,z) is the emissivity in the ROSAT band, tak-
from the ROSAT All-Sky Survey. ingintoaccounttheinterstellarabsorptionandtheinstru-
ment spectralresponse,andd (z) is the angulardistance
A
at redshift z. Λ(T,z) depends, although weakly, on the
3. X-ray surface brightness profiles cluster temperature and redshift. The emission measure
is linked to the gas density n by:
g
Fig.1showsthevignettingcorrectedX-raysurfacebright-
∞ n2(R) RdR
ness profiles of all the clusters in the sample. We bin the EM(r)= g (4)
photons into concentric annuli centered on the maximum Zr √R2 r2
−
ofthe X-rayemission.Forthe PSPCwe usea widthof15 Theshapeofthesurfacebrightnessprofileisthusgov-
arcseconds per annulus and a total of 200 annuli. For the ernedbythe formofthegasdistribution,whereasitsnor-
HRI we use a width of 10 arcseconds per annulus and a malization depends also on the cluster overall gas con-
totalofonly100annuli,due tothe smallerfieldofviewof tent. If clusters formed a structurally similar population,
the HRI. We cut out serendipitous sources in the field of all the brightness profiles would appear as parallel curves
view or cluster substructures, if they show up as a local onFigure 1 (they woulddiffer by a translationin the log-
maximum. The background was subtracted using data in log plane). More precisely similarity means that the clus-
the outer part of the field of view. ters constitute a one parameter population: each cluster
4 D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation
Table 1.ROSATobservationanalysissummary.p/hstands forPSPCorHRI.Allquotederrorsare2 σ errors.The
−
redshifts are taken from NED. When the best fit β–model A and B are identical, we only give the parameters once.
model A model B
Abell β rc χ2red Rcut β rc χ2red z kpc/ exp. det. kT
name [kpc] [arcmin] [kpc] arcmin [ksec] in keV
76 0.60+0.40 413+383 1.20 0.042 68 1.7 p
−0.15 −186
85 0.53+0.05 69+4 8.26 3 0.70+0.02 282+32 1.63 0.052 83 15.9 p 6.1±0.2 m
−0.01 −4 −0.03 −33
119 0.64+0.04 463+43 1.44 0.044 71 15.2 p 5.8±0.6 m
−0.03 −40
548 0.52+0.06 132+42 1.35 0.042 68 10.9 p
−0.04 −33
754 1.04+0.08 731+32 3.29 4 0.94+0.07 853+71 2.03 0.053 85 16.8 p 7.6±0.3 a
−0.04 −30 −0.06 −67
780 0.60+0.01 62+2 4.75 2.25 0.70+0.02 148+18 1.82 0.052 83 18.4 p 3.8±0.2 m
−0.01 −2 −0.03 −23
S1101 0.57+0.01 37+3 1.61 0.5 0.59+0.02 49+5 1.37 0.058 92 47.4 h
−0.01 −2 −0.02 −5
1991 0.59+0.02 43+6 1.32 1 0.71+0.13 142+63 1.20 0.059 93 4.0 p
−0.02 −6 −0.09 −49
2107 0.60+0.02 125+18 1.72 2.75 0.76+0.12 273+74 1.11 0.041 66 8.3 p
−0.02 −17 −0.08 −63
2319 0.53+0.05 208+18 1.86 3.5 0.69+0.06 496+81 1.01 0.056 89 4.6 p 9.1±0.2 y
−0.01 −15 −0.05 −73
2589 0.60+0.02 118+12 1.45 1 0.62+0.03 135+19 1.36 0.042 68 7.3 p 3.7+2.2 d
−0.02 −11 −0.03 −18 −1.2
2626 0.54+0.06 54+19 3.55 0.057 90 27.8 h
0.04 −13
2657 0.54+0.01 112+7 4.45 4 1.24+0.27 689+137 1.76 0.040 65 18.9 p 3.7±0.3 m
−0.01 −9 −0.17 −93
2717 0.57+0.03 86+17 1.57 1 0.66+0.07 150+36 1.31 0.050 80 9.9 p
−0.02 −14 −0.05 −30
3093 0.59+0.35 120+153 1.74 0.058 92 8.1 p
−0.10 −72
3158 0.67+0.04 274+33 1.15 1 0.68+0.04 289+39 1.14 0.059 93 3.0 p 5.5±0.6 d
−0.04 −30 −0.04 −33
3223 1.00+0.45 842+355 1.53 0.060 95 7.7 p
−0.23 −224
3266 0.88+0.05 688+50 5.43 3.5 1.39+0.19 1181+140 1.91 0.059 93 13.6 p 7.7±0.8 m
−0.04 −45 −0.17 −130
3301 0.63+0.08 267+63 1.90 1.5 0.73+0.14 377+108 1.70 0.054 86 8.9 p
−0.05 −50 −0.07 −83
3391 0.54+0.04 221+37 1.35 1.25 0.57+0.05 259+53 1.29 0.053 85 6.6 p 5.7±0.7 m
−0.03 −33 −0.04 −47
3532 0.66+0.05 301+48 1.46 2.25 0.80+0.13 461+102 1.32 0.056 89 8.6 p 4.4±1.5 e
−0.05 −41 −0.09 −90
3558 0.58+0.01 228+12 9.81 2.75 0.72+0.03 405+24 2.47 0.048 77 29.5 p 5.5±0.3 m
−0.01 −10 −0.02 −23
3559 0.56+0.37 154+213 1.28 0.047 76 8.1 p
−0.10 −78
3562 0.47+0.01 102+10 2.12 4.25 0.53+0.06 245+122 1.85 0.050 80 20.2 p 3.8±0.9 d
−0.01 −9 −0.04 −151
3667 0.72+0.13 360+68 1.49 2.67 0.67+0.19 310+121 1.33 0.055 87 42.3 h 7.0±0.6 m
−0.09 −56 −0.12 −100
4059 0.58+0.02 90+9 1.42 1.5 0.64+0.04 151+28 1.14 0.046 74 5.4 p 4.1±0.3 m
−0.01 −8 −0.03 −27
m a d y e
: Markevitch 1998; : Arnaud & Evrard 1998; : David et al. 1993; : Yamashita (priv. comm.); : Edge & Stewart (priv.
comm. quoted in White, Jones & Forman 1997)
is characterizedby its physical dimension a , and the dis- tionbetweenthevirialmass(M )andradius(r )
i VT200 VT200
tribution of any given physical quantity Q is described and the overall X-ray temperature T : M /r
X VT200 VT200
∝
by a dimensionless function Q, common to all clusters: T , whereas by definition M /(4/3πρ (z)r3 ) =
X VT200 c VT200
Q (r)=Q(a )Q(x),wherexisthescaledradiusx=r/a . 200. This leads to the well known scaling relations:
i i e i
TtiohnesnboertmwaeleinzatethioenglfoabctaolrqsuQan(taiit)iedse(fien.gemthaesss,cmaleinang treemla-- rVT200 ∝ (1+z)−3/2 TX1/2 (5)
perature, luminosity ...) in the cluster population. MVT200 ∝ (1+z)−3/2 TX3/2 (6)
We firstexaminedifourdatawereconsistentwiththe The emission measure (Eq. 4) then scales as
similarityexpectedinthesimplestmodel.Inthespherical
EM(r) f2 T1/2(1+z)9/2EM(x) (7)
collapse model, the overalldensity contrast δ of virialized ∝ gas X
objects is fixed and is of the order of 200 for Ω = 1, and where x = r/r is the sgcaled radius, and EM is a
VT200
thenaturalscalingradiusisthevirialradiusdefinedasthe dimensionless function, the same for all clusters. The gas
radius containing this density contrast1. Here the density mass fraction f is not necessarily a constant bgut sim-
gas
contrast is δ = ρ¯/ρc, where ρc(z) is the current critical ilarity implies that it only depends on cluster mass (or
densityoftheUniverse,ρc(z)=3 Ho2/8πG(1+z)3,andρ¯ equivalently temperature).
isthemeanclusterdensity.Thevirialradiusisnotdirectly We thus consideredthe observedscaledemissionmea-
measurable. However, if, in addition, one assumes struc- sure profiles EM (x):
X
turalsimilarity,the virialtheoremprovidesascalingrela-
S(θ(x)) r
g VT200
EM (x)= ; θ(x)=x (8)
1 Notethatnumericalexperimentssupportthatdefinitionof X (1+z)1/2 TX1/2 Λ(T) dA(z)
g
the virial radius: the edge of the virialized part of clusters is defined from Eq. 3 and Eq. 7 ignoring possible variation
indeed found at a density contrast≃200 of fgas with TX. To compute the scaling radius rVT200 we
D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation 5
Table 2. Description of the β–model of the central region of some clusters where excess emis-
sion is observed (due to a cooling flow, a central point
β–model description
source ...). We indeed got in several cases very large χ2
A isothermal β–model fittedto the entire
values (see Tab 1). Hence, we also tried to minimise the
cluster surface brightness profile
B isothermal β–model excising thecentral region reducedχ2 byexcludingthe centralbinsfromthefit.The
when an improvement of χ2 can be achieved best fit β–model (hereafter β–model B) was determined
by increasingthe radius ofthe cut-outregion,R ,up to
cut
the radius where the reduced χ2 stopped decreasing. We
usedthenormalisationfactorobtainedbyEvrard,Metzler limited R to 350 kpc, as a too large excluded region in
cut
& Navarro (1996) from numerical simulations: the center precludes the determinationof the core radius.
T 1/2 Moreoveritis safe to assumethat the contributionofany
rVT200 =3690 X (1+z)−3/2 kpc (9) Cooling Flow excess is negligible beyond such radius.
(cid:18)10 kev(cid:19)
A brief description of β–model A and B is given in
Notethatthisnormalizationissimplyforconvenience,the
Tab 2. The best fit parameters, β and r , are listed in
exact calibration of the scaling relations does not matter c
Tab1 for both models,togetherwith the R values.For
to check similarity of the profiles. cut
about 1/4 of the clusters, excluding central bins does not
ThescaledemissionmeasuresprofilesEM (x)areshown
X
improve the fit. In that case the best fit β–models A and
onFig. 3 for the clusters in the spectroscopicsub-sample.
g B are identical and we only give the parameters once in
Theycanbecomparedtothecorrespondingunscaledpro-
the Table.
files EM (r) plotted on Fig. 2. The scaled profiles are
X
clearlynotidentical,butseveralfeaturesarestriking.First, Fortheotherclusters,excludingthecentralpartyields
we detect the emission nearly up to the virial radius. We onaveragelargerβ andcoreradiivalues(seeTab.1).This
are effectively studying the large scale density distribu- is a well known effect. If one tries to fit a β–model to the
tion of the gas, and not just the central cluster core. Sec- entireprofilewhenthereisacentralexcess,toosmallcore
ond, while the dispersion is very large at low scaled radii radiusandβ valuesarederived:decreasingrc allowsusto
(EM atthecenterspannearly2ordersofmagnitude),it fitbetterthecentralpart,whiledecreasingβ compensates
X
rapidly decreases with radius. Beyond r/r 0.1 the for the subsequent flux deficit induced at large radii. Al-
g VT200 ∼
profileslookremarkablysimilarandagreetowithinabout thoughtheβ–modelBisbetterinthatrespect,ithasalso
a factor of two. We also note that the scaling procedure its drawback.If the excludedregionis largerthanstrictly
hassignificantlydecreasedthedifferencebetweenthepro- necessarytoavoidthecentralexcess,thedeterminationof
files,butonlyinthatexternalregion(compareFig.2and the core radius is degraded: the uncertainty is increased
Fig. 3). The unscaled emission measures show a relative and the best fit value can be biased towards large values.
standard deviation of about 96% at 950 kpc and 75% at We will thus consider both models in the following. They
1500 kpc. In comparison, the relative standard deviation can be viewed as two extreme β–models , allowing to as-
of the scaled profiles is 41% at r/r = 0.3 and 39% sess the impact ofthe exacttreatmentofthe cluster core.
VT200
at r/r = 0.5, i.e the scaling has reduced the scatter This is important since the core properties vary greatly
VT200
byaboutafactoroftwo.Asthe temperaturerangeofour from cluster to cluster, as shown in the previous section.
spectroscopic sub-sample is limited (3.5<kT <9.1keV), ForA754,whichisundergoingamergerphase,asindi-
we plotted for comparison the scaled profile of A2163. It cated by many authors (Zabludoff & Zaritsky 1995; Hen-
is one of the hottest cluster known (Arnaud et al. 1992) riksen&Markevitch1996;Roettiger,Stone&Mushotzky
and is located at higher redshift than the clusters of our 1998),wevariedthecenterusedfortheconcentricbinning
sample. It fits remarkably well within our sample. andadoptedtheprofilethatprovidesfortheglobalfitthe
Insummary,thescaledemissionmeasureprofilesshow lowest reduced χ2.
indicationsforsimilaritiesinouterregions(r >0.2r ).
VT200 The most compact clusters A780 and A1991 were ob-
Onthe other handin the centersthe profilesshowa large
served with the PSPC, and their core image is somewhat
dispersion,whichwewilldiscusslateron.Tofurtherquan-
blurredbythePSF( 20-30arcsecFWHM).ForA780we
tify the density profile shape and investigate its possible ≈
estimated that the core radius is overestimated by about
variations within the cluster population, we fit, in what
10%,andforA1991by around20%.Forthe PSPCpoint-
follows, an isothermal β–model to each cluster profile.
ingofA85(theclusterwhichhasafterA780andA1991the
third smallest angular core radius) the effect of the PSF
isofthesameorderasthestatisticaluncertainties.Forall
4. Fitting the isothermal β-model
other clusters, which either have a larger core radius or
We performed two different β–model fits for each cluster. were observedwith the ROSAT HRI 2 we calculated that
First we determined the β–model that fits at best the en-
tire cluster emission(hereafter β–model A). Howeverit is 2 theHRIhasaPSFwithaFWHMof4-5arcseconds,which
wellknownthatthe overallβ–modelis apoordescription corresponds toabout 4-8 kpc
6 D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation
Fig.4.Slope parameterβ versuscoreradiusr forthe 26 Fig.5. Same as Fig. 4 only that we now display the fit
c
clusters of the sample (Tab.1). The isothermal β–model results of β–model B: isothermal β–model excising the
has been fitted to the entire ROSAT surface brightness central parts of the profile, if an improvement of reduced
profile of each cluster (β–model A). The errors bars are χ2 can be achieved (see Tab.1). Clusters belonging to the
2–σ errors. β and r are correlated. The full line shows spectroscopic subsample are marked with filled circles.
c
thebestfitparabolicrelation:β =β 1+(r /r )2 ,with
0 c s
h i
β = 0.55 and r = 885 kpc. The dotted line shows the
0 s alent plot for β–model B. A striking feature is that the
parabolic correlation obtained for the β–model B param-
data points are not distributed all over the r –β plane.
c
eters: β =0.61 and r =1020 kpc (see also Fig. 5).
0 s There is a clear trend of increasing slope parameter with
coreradius,althoughthecorrelationisnotverytight,spe-
theeffectsofthePSFaremuchsmallerthanthestatistical ciallyatlowcoreradiusvalues.Aquantitativemeasureof
errors. the correlation significance was obtained from three dif-
All clusters with largeerrorbars,such asA76,A3093, ferent tests, the Pearson’s test, the Spearman’s test, and
A3223 and A3559, have clearly visible substructures. the Kendall’sτ test (see Presset al.1993).All three tests
(applied to fit A results) give a high correlation signifi-
cance,higher than99.97%.The highestcorrelationsignif-
5. Correlation between the core radius and β icance is obtained with the Pearson’s test, with a value
of 1. 2.99 10−10. The corresponding Pearson’s rank
− ×
5.1. The β and rc values and the significance of the correlation is r =0.902.
correlation
The slope parameter β is not the same for all clusters, as
5.2. The form of the r –β relation
would be the case for perfectly similar β–model density c
profiles. The β values span a wide range, 0.47 1.04 and
−
0.52 1.4 for the β–model A and B, respectively. The The highest score obtained with the Pearson test indi-
−
distributionisnotsymmetric,itisclearlyconcentratedat cates that the correlation between rc and β is close to
lowvalues,withrarerhighvalues:themedianvalueis0.59 being linear.Onenotesthatthe slopeparameterisnearly
forA(0.68forB), 85%ofthevaluesfallinthenarrower constant at low core radii and starts to increase with rc
∼
range0.47(0.52) 0.75andthe standarddeviationisonly above rc 250kpc only, which suggests a parabolic rela-
− ∼
0.14(0.21) or 24%(30%) of the median value for model A tion between the two quantities.
(modelB).Thesamegeneralfeaturesareobservedforthe We examined which relation, linear or parabolic, bet-
core radius distribution. The median values are 143 kpc ter accounts for the data. For conciseness, the following
and 278 kpc for models A and B, respectively. discussion is based on a quantitative analysis of the pa-
Fig.4 displaysthe slopeparametersβ againstcorera- rameters of the β–model A (similar results are obtained
dius r ,obtainedwith theβ–modelA. Fig.5 isthe equiv- with β–model B).
c
D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation 7
Table 3. The best fit parameters of the β–core radius
relations
Relation β–model normalisation scale
β–rc A β0 =0.55 rs=885 kpc
β–rc B β0 =0.61 rs=1020 kpc
β–xc A β0′ =0.53 xs=0.27
β–xc B β0′ =0.59 xs=0.34
butbothχ2areveryhigh.Ifweexcludefourclusters,A85,
A780,A2319,andA3562,thereducedχ2 dropsto1.68for
the parabolicfunction.Thesefourclusterslieoffsetofthe
correlation and have extremely small statistical error on
β, less or equalto 2%.They boost therefore the χ2 value.
Theβ-modelparametersofthesefourclusterschangesig-
nificantly when one cuts out the central part (see Tab.1)
and it is likely that using the global fit for these clusters
is not a good approach. On the opposite we still obtain a
reduced χ2 of 3.36 with the linear relation after the ex-
clusion of the four principal outliers of this relation.Only
when excluding sevenclusters,we do obtaina reducedχ2
Fig.6. The correlation between scaled core radius xc = of 1.78, which is still higher than the χ2 of the parabolic
rc/rVT200 and β for the 15 clusters of the spectroscopic fit, with four clusters excluded.
subsample (see also Tab.1). The data points are for the While keeping in mind that our criteria are not rigor-
β–model B and the error bars take into account the un- ous,we concludethat aparabolicrelationbetweenr and
c
certainties in rVT200. Full line: β–xc relation for β–model β is a better description of the correlation between these
B. Dotted line: β–xc relationfor β–model A. We also dis- two quantities than a linear relation. Henceforth, we will
playthe results ofA2163locatedatz =0.201(Arnaudet only consider a parabolic relation from now on.
al. 1992; Elbaz, Arnaud & B¨ohringer 1995) and CL0016
at z =0.54 (Hughes & Birkinshaw 1998).
5.3. The best fit r –β relation
c
We considered the two parametric functions
Wedeterminedthebestfitparabolicrelationr –βforboth
β =βl 1+ r /rl (10) c
0 c s β–models A and B. The robustness of the r –β relation
c
and (cid:2) (cid:0) (cid:1)(cid:3)
mustbenoted;itisnotsensitivetotheexacttreatmentof
β =β0 1+(rc/rs)2 (11) the central part of the cluster profile. From the β–model
h i
B data, plotted on Fig. 5, we got β = 0.61 and r =
and fit them to the data. Since both quantities β and r 0 s
c
1020kpc.These values are closeto those obtainedfor the
are marred with errors and the uncertainties are highly
β–model A: β is increased by only 10% and r by
correlated, there is no well established method to define 0 s
∼ ∼
14%.Thereducedχ2issmaller:χ2 =1.92forthecomplete
thebestfitfunctionparametersandtoquantifythegood-
cluster sample;the χ2 dropsto 1.42if one excludes A780.
ness of the fit. We used the following empirical least-
Part of the improvement is certainly an artifact due to
squaresmethod.Bestfitparameterswerederivedbyomit-
ting the errors on β and r . The χ2 was minimised using the larger uncertainties on β. The two relations look very
c
much alike, as can be seen on Fig. 5.
the downhill simplex method (see Press et al. 1993). We
got βl = 0.49 and rl = 870 kpc for the linear function As β is a dimensionless quantity, while rc is not, one
0 s
would rather expect a relation between β and a scaled
andβ =0.55andr =885kpcfortheparabolicfunction
0 s
core radius. Fig. 6 shows the correlation between β and
(thecorrespondingcurveisplottedonFig.4).Tocompare
the scaled core radius x = r /r , where the virial
the goodness of the fits obtained with the two functions c c VT200
we performed a subsequent χ2-test including the error on radius rVT200 is given by Eq.9 and the data points are
from the β–model B. We fitted the parabolic relation
theβ quantityonly.Weutilisedthe2 σerrorsinsteadof
−
trheleat1e−d σweerarsosrusm.Aeds tthheisucnocuelrdtacinrutideeslyoftβakaenidntroc aacrecocuonrt- β =β0′ h1+(xc/xs)2i (12)
the two errors. totheclusterparametersofthe15clustersinthespectro-
Thereducedχ2 valuesfortheentireclustersampleare scopic subsample. We obtained β′ = 0.59 and x = 0.34
0 s
14.6 and 9.2 for the linear and parabolic function respec- for the β–model B and similar results β′ = 0.53 and
0
tively. The fit is formally better for the parabolic relation x =0.27 for the β–model A (see also Tab.3).
s
8 D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation
Table 4. The β–r relation for various cluster samples the quality ofthe data andthe cluster regionused for the
c
fit. We have already mentioned that a central excess can
Numberof Choice β–model β0 rs
yield artificially low β and core radii. At large radii the
clusters criteria
β–model tends to a simple power law. If instead the den-
26 none A 0.55 885 kpc
26 none B 0.61 1020 kpc sity profile continuously steepens, the β value obtained
13 rdet >1.45Mpc A 0.54 859 kpc from the fit will increase with the outermost radius used
13 χ2 <1.6 A 0.56 980 kpc in the fit (see for instance the numerical simulations of
13 rdet >1.45Mpc B 0.60 1050 kpc Navarro, Frenk & White 1995) and as a result probably
15 χ2 <1.5 B 0.64 1080 kpc also the derived core radius. Moreover the magnitude of
these effects will depend on the accuracy of the β–model
approximation,whichislikelytovaryfromclustertoclus-
An important question is whether β is more tightly
ter. All together this could induce a spurious correlation
related to r or to x . We cannot definitely answer this
c c between the derived parameters, similar to the one ob-
question with the present spectroscopic subsample. It is
served,i.e. an apparent increase of β with r .
too small and covers a too narrow range in temperature c
To see if this a serious issue we first investigated how
and redshift. There is only a factor 2.4 between the mini-
the β–r relationis affected by the size of the regionused
mumandmaximumtemperature.Asr onlyscalesas c
VT200
for the fit. We can first compare β–models A and B. We
√T , the effect of the scaling cannot be dramatic. How-
X
alsoderivedtherelationforthehalfsampleofthe13clus-
everthereareseveralindicationsthatthe‘scaled’relation
ters with the largest r , where r is the maximum ra-
is the most relevant one, as expected. First we compared det det
dius at which the X-ray emission is detected with a sig-
Fig.6toFig.5atlargeβ values(β >0.65),wherethede-
nificance greater than 3σ. Second we investigated if the
pendence of β with r is maximal. For the clusters in the
c
relation is sensitive to the quality of the fit obtained with
spectroscopicsubsample (filledcircles)the scatteraround
the β–model . For this purpose we selected the clusters
thebestfitrelationhasbeendecreasedbythescalingpro-
withthe lowestreducedχ2 andderivedthecorresponding
cess.SecondweconsideredtestclustersoutsidetheTand
β–r relation.Theχ2 valueisanindicatoroftheaccuracy
z range of our sample. A2163, the highest temperature c
of the β–model approximation (but also depends on the
cluster, fits well with both the β–x relation and the β–
c
quality of the data).
r relation, so this is not conclusive. On the other hand
c
The parameters of the relations derived for the vari-
Cl0016+16,the most distant cluster with good estimates
ous cluster sub-samples are given in Tab. 4. In all cases,
of β and r (β = 0.728, r = 298 kpc) fits much better
c c
a parabolic β–r relation well accounts for the data. Fur-
with the β–x (see Fig. 6). This issue is further discussed c
c
thermore foreachmodel (A orB), the parametersβ and
in Sect. 6. 0
r depend very weakly on the selection on r (0.3 3%
A summary of all fits of the β-r relation is given in s det
c −
effect)orontheselectiononthequalityofthefit(2 10%
Tab. 3. We do not give errorson the relation parameters,
−
effect). The largest discrepancy is actually seen between
β and x (or r ). We do not know of any proper method
0 s s
β–models A and B, i.e if one includes or not the central
to estimate them since the uncertainties on the related
excess regionin the fit. The effect is small as discussed in
quantitiesβ andr arehighlycorrelated.Weonlysuggest
c
theprevioussection.Fromthisobservedrobustnessofthe
that they could be of the order of the difference observed
β–r relationwe concludedthat it is unlikely that it is an
whenconsideringthetwoβ–modelAandB.Wealsowant c
artifact due to the imperfection of the β–model .
to stress that the relation parameters we determine here
are in very good agreement with the direct study of the
density profile shape presented below in Sect. 6.
5.4.2. The fitting process
5.4. Checking on spurious effects
To check the validity of the fitting procedure, we con-
Could the derived correlation be an artifact of our data struct artificial clusters, which lie outside the correlation
analysis? This question must be addressed since our re- and have exposure times and countrates similar to the
sults could in principle be affected by i) a choice of an ones in our sample. We added Poisson noise to the im-
inadequate model ii) the fitting process iii) systematic er- ages and treated these artificial clusters in the same way
rorsinthemodelingoftheinstrumentcharacteristics.We as our real data and fit the isothermal β-model to them.
now discuss each point in turn. The intrinsic values for β and r always lie well within
c
the 2 σ-errorsof the fit results of the simulated clusters
−
images. Thus, the fit procedure is correct and could not
5.4.1. The model
introduce a spurious correlation.
Theβ–modelisonlyanapproximationoftherealgasden-
sityprofile.Hence,theparametersderivedmaydependon
D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation 9
5.4.3. The instrument 6.2. The slope profiles for a theoretical parabolic β - core
radius relation
As already discussed above, only two clusters could be
For a parabolic relation between β and r (Eq. 11), the
affected by the finite instrument PSF, namely A780 and c
slope profile may be written as:
A1991.Forthe others,the PSFeffects areofthe sameor-
derthanthestatisticaluncertaintiesorsmaller.Excluding 1+(r /r )2
c s
thesetwoclustersfromthefitchangestheresultsofβ and α (r)=3 β h i (15)
0 n 0
rs by less than 3%. 1+(rc/r)2
h i
We also verified the exposure map calculation (or vi-
Similarlyforaparabolicrelationbetweenβ andthescaled
gnetting effects). We used the observations of clusters,
core radius:
thatwerenotincludedinoursamplebecausewefoundno
1+(x /x )2
prominent extended cluster emission. This is the case for c s
α (x)=3 β′ h i (16)
A195 (observing title MKN 359) observed by the PSPC, n 0 1+(x /x)2
andA1213observedbytheHRI(observingtitle1113+29). c
h i
Forbothpointingswedividedtheimageoftheactualdata The physical implications of the parabolic relation, is
by the exposure map. There is no significant variation in straightforwardfromthesetwoequations:i)ifβ isrelated
the overall photon distribution when excluding obvious to r , there is a specific radius, r , where the slope of the
c s
sources: the countrate is flat over the whole field of view gas density profile is the same for all clusters, ii) if β is
of the corresponding detector. This also validates our im- instead related to the scaled core radius, this occurs at a
plicit assumption of a flat backgroundin the FOV. specific scaled radius, x .
s
Let us first assume a relation between β and r . At
c
the specific radius r the slope is equal to 3 β , where
s 0
β is the normalisation factor of the parabolic relation.
6. The shape of the gas density profiles 0
At very large radii, r >> r , α (r) tends to 3 β - the
c n
β–model is equivalent to a power law - and there is no
A perfect similarity between the density profiles is ruled impact of the β–r relation. At low radii, α (r) roughly
c n
out by the variation of β from cluster to cluster. On the scalesas (r /r)2 andin viewof the largedynamicalrange
c
other hand the correlation found between the two shape of core radii values, it can vary by more than an order of
parametersofthedensityprofileindicatessomestructural magnitude from cluster to cluster. This is illustrated on
regularityinthegasdistribution,thatwestudyinthissec- Fig.7wherewehaveplottedwiththicklines twoextreme
tion. For that purpose we use the density profiles derived slope profiles corresponding to the best fit β–r parabolic
c
from our β–model fits of the observed brightness surface relationderivedinSect.5.3(Eq.15withparametersfrom
profiles. Tab. 3). The first one, α(1)(r), corresponds to the min-
n
imum allowed β value: β = β or r = 0. In that case
0 c
α (r) keeps constant with radius: α(1)(r) = 3 β . The
n n 0
6.1. Quantifying the shape: the slope profile secondone, α(2)(r), correspondsto the maximumβ value
n
observed in our sample. The two curves cross at r .
s
The shape of the gas density distribution may be quanti- The same properties hold for the β–xc relation, in
fied by the variationwith radiusof its logarithmic deriva- terms of the scaled radial coordinates.
tive. This derivative is also important for the estimate of
the mass profiles (see Sect. 7). For the β–model form, it
6.3. The observed slope profiles
reads:
α (r)= dlogng(r) = 3 β (13) The slope profiles αn(r) derived from model A and B are
n
− dlogr 1+(r /r)2 plotted on the left panels of Fig. 7 for all clusters in our
c
h i sample.On the rightpanels areplotted the profilesα (x)
n
or equivalently, in term of the scaled radial coordinates: against scaled radius for the spectroscopic subsample.
To quantify structural variations within the cluster
3 β
α (x)= (14) sample,weestimatedateachradius,thecenterandspread
n
1+(x /x)2 of the distribution of slopes at that radius. We used both
c
h i the classical estimators (mean value and standard devia-
We will refer to this function as the slope profile: α (r) tion)andthebi-weightestimators(bi-weightlocationand
n
is the slope of the gas distribution in the log–logplane at scale). The latter are more resistant and robust than the
radius r. It depends on the two shape parameters of the former (Beers, Flynn & Gebhardt 1990). The results are
β–model : the core radius and β. shown in Fig. 8. The “average” slope (top panels) and
10 D.M. Neumann,M. Arnaud:Regularity in thecluster surface brightness profiles and theM–T relation
33 3
AA A
11 1
n 00..88 0.8
a 00..66 0.6
00..44 0.4
00..22 0.2
33 BB 3 B
11 1
n 00..88 0.8
a 00..66 0.6
00..44 0.4
00..22 0.2
110000 11000000 0.1 1
r (kpc) r/r
VT200
Fig.7. The variation with radius of the logarithmic slope of the gas density distribution. Top-left panel: the slope
profilesα (r)= dlog(n (r)/dlogr versusradiusr derivedfromthebest-fitβ–modelA,forallclustersofthesample.
n g
−
Theboldlinescorrespondtotheextremeslopevariationsallowedbytheβ–r correlation(Eq.11withparametersfrom
c
Tab. 3 and Eq.15): the horizontal line is for β = β = 0.55, r = 0 and the other line is for β = 1.4, r = 1100 kpc.
0 c c
Top-rightpanel:theslopeprofilesα (x)= dlog(n (x)/dlogxversusscaledradiusx=r/r forthespectroscopic
n g VT200
−
subsample (β–model A). Bottom panels: Same as top panels but for β–model B. In that case the profiles are plotted
only beyond the excised central region.
the “relative dispersion aroundthe mean”, defined as the within the two extreme curves α(1)(r) and α(2)(r) corre-
n n
spread normalised to the average value (bottom panels) sponding to the best fit β–r relation and the scatter is
c
are plotted as a function of radius (left) for the whole very large among the cluster population (Fig. 7, bottom-
sampleandscaledradius(right)forthespectroscopicsub- left panel). This scatter diminishes with radius until it
sample.TheresultsofmodelA(dottedlines)andmodelB reaches a minimum at the specific radius r = 1020 kpc
s
(full lines) are provided. We also computed the bi-weight (Fig. 7 and Fig. 8 bottom-left panel). At that radius the
means of the core radii and cut-out radii R , the loca- average α value agrees well with the best fit 3 β value
cut n 0
tions of which are indicated by arrows in the figure. (Fig.8top–leftpanel).Theclassicalandbi-weightestima-
tors give the same results, the slope distribution at that
radiusbeingclosetoGaussian.Theresidualscatteronα
n
6.4. General properties at r - the slope is not exactly the same for all clusters -
s
cannaturallybe explainedby the scatter aroundthe best
The characteristics of the observed profiles are those ex- fitβ–r relation.Abover the scatterincreasesagain,but
c s
pectedfromthebestfitβ–coreradiusrelationswederived. saturate at a value of the same order as the variations of
Letusconsiderforinstancetheslopeprofilesα (r)derived 3β,theasymptoticvalueoftheslope.Forinstance,at2.5
n
frommodelB.Atsmallradiithe observedprofileslie well Mpc the bi-weight mean slope is 1.96 and the bi-weight