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A&A517,A94(2010) Astronomy
DOI:10.1051/0004-6361/201014566 &
(cid:2)c ESO2010 Astrophysics
(cid:2)
The clusters Abell 222 and Abell 223: a multi-wavelength view
F.Durret1,2,T.F.Laganá3,C.Adami4,andE.Bertin1,2
1 UPMCUniversitéParis06,UMR7095,Institutd’AstrophysiquedeParis,75014Paris,France
e-mail:[email protected]
2 CNRS,UMR7095,Institutd’AstrophysiquedeParis,75014Paris,France
3 IAG,USP,R.doMatão1226,05508-090.SãoPaulo/SP,Brazil
4 LAM,Pôledel’ÉtoileSitedeChâteau-Gombert,38rueFrédéricJoliot-Curie,13388MarseilleCedex13,France
Accepted30March2010/Received7May2010
ABSTRACT
Context.TheAbell222 and223 clustersarelocatedat anaverageredshift z ∼ 0.21 andareseparatedby 0.26deg. Signaturesof
mergershavebeenpreviouslyfoundintheseclusters,bothinX-raysandatopticalwavelengths,thusmotivatingourstudy.InX-rays,
they arerelativelybright, and Abell223 shows adouble structure. A filament hasalso beendetected between theclustersboth at
opticalandX-raywavelengths.
Aims.Weanalyse theopticalproperties ofthesetwoclustersbasedon deepimaging intwobands, derivetheir galaxyluminosity
functions(GLFs)andcorrelatethesepropertieswithX-raycharacteristicsderivedfromXMM-Newtondata.
Methods.TheopticalpartofourstudyisbasedonarchiveimagesobtainedwiththeCFHTMegaprime/Megacamcamera,coveringa
totalregionofabout1deg2,or12.3×12.3Mpc2ataredshiftof0.21.TheX-rayanalysisisbasedonarchiveXMM-Newtonimages.
Results.The GLFs of Abell 222 in the g(cid:4) and r(cid:4) bands are well fit by a Schechter function; the GLF is steeper in r(cid:4) than in g(cid:4).
ForAbell 223, theGLFsinbothbands requireasecond component atbright magnitudes, added toaSchechter function; theyare
similarinbothbands. TheSerna&Gerbalmethod allowstoseparatewellthetwoclusters.Noobvious filamentarystructuresare
detectedatverylargescalesaroundtheclusters,butathirdclusteratthesameredshift,Abell209,islocatedataprojecteddistanceof
19.2Mpc.X-raytemperatureandmetallicitymapsrevealthatthetemperatureandmetallicityoftheX-raygasarequitehomogeneous
inAbell222,whiletheyareveryperturbedinAbell223.
Conclusions.TheAbell222/Abell223systemiscomplex.Thetwoclustersthatformthisstructurepresentverydifferentdynamical
states. Abell 222 is a smaller, less massive and almost isothermal cluster. On the other hand, Abell 223 is more massive and has
most probably been crossed by a subcluster on its way to the northeast. As a consequence, the temperature distribution is very
inhomogeneous. Signs of recent interactions are also detected in the optical data where this cluster shows a “perturbed” GLF. In
summary,themultiwavelengthanalysesofAbell222andAbell223areusedtoinvestigatetheconnectionbetweentheICMandthe
clustergalaxypropertiesinaninteractingsystem.
Keywords.galaxies:clusters:individual:Abell222–X-rays:galaxies:clusters–galaxies:clusters:individual:Abell223–
galaxies:clusters:intraclustermedium
1. Introduction Bouéetal.(2008),thisslopecanstronglyvaryfromonecluster
toanother,andfromoneregiontoanotherinagivencluster.
Galaxy evolution is known to be influenced by environmen-
Supersedingthe dynamicalanalyses of Proust et al. (2000)
tal effects, which are particularly strong in galaxy clusters, all
and Girardi & Mezzetti (2001), Dietrich et al. (2002) have ob-
the more when they are undergoingmerging events. The anal-
tained a redshift catalogue for 183 galaxies in the region of
ysis of galaxyluminosityfunctions(GLF), and in particular of the cluster pair Abell222/223,outof which153belongto one
their faint-end slopes is a good way to trace the history of the
of the two clusters. Their analysis was also based on photom-
faint galaxypopulationand the influence of mergingeventson
etry in the V and R bands. They have estimated redshifts of
this population. As summarized for example in Table A.1 of 0.2126±0.0008and0.2079±0.0008,andvelocitydispersionsof
1014and1032kms−1,forAbell222andAbell223respectively.
(cid:2) BasedonobservationsobtainedwithMegaPrime/MegaCam,ajoint AsindicatedbyDietrichetal.(2002)thesevelocitydispersions
project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii are somewhathigherthanthose expectedfromthe X-raylumi-
Telescope(CFHT)whichisoperatedbytheNationalResearchCouncil nosities.Abell222appearsrelativelyrelaxed,withagalaxyve-
(NRC)ofCanada,theInstitutNationaldesSciencesdel’Universofthe locityhistogramconsistentwithaGaussian.Ontheotherhand,
Centre National de la Recherche Scientifique (CNRS) of France, and theadaptivekernelgalaxydensitymapofAbell223showstwo
the University of Hawaii. This work has also made use of data prod-
peaks,thoughsurprisinglytheDressler-Schectmantest(Dressler
uctsproducedatTERAPIXandtheCanadianAstronomyDataCentre
& Schectman1988) doesnotfindthese two peaksandtheDIP
aspartoftheCanada-France-Hawaii TelescopeLegacySurvey,acol-
statistics cannotrejectthe nullhypothesisofa unimodaldistri-
laborative project of NRC and CNRS. The X-ray analysis is based
bution, implying that a Gaussian parent population cannot be
on XMM-Newton archive data. This research has made use of the
NASA/IPACExtragalacticDatabase(NED)whichisoperatedbytheJet excluded.Dietrichetal.(2002)derivedrespectivemasstolight
Propulsion Laboratory, CaliforniaInstituteof Technology, under con- ratiosof202±43and149±33h70 M(cid:5)/L(cid:5)andluminosityfunc-
tractwiththeNationalAeronauticsandSpaceAdministration. tionsintheR banddowntoabsolutemagnitudesof−19.5,and
ArticlepublishedbyEDPSciences Page1of15
A&A517,A94(2010)
Table1.Summaryoftheobservations.
Filter g(cid:4) r(cid:4)
Numberofcoaddedimages 5 8
Exposuretime(s) 2000 4000
Seeing(arcsec) 0.81 0.73
Limitingmagnitude(5σ) 26.7 26.4
found thata bridgeof galaxiesseems to exist between the two
clusters.Notethatsuchbridgesofgalaxiesjoiningtwoclusters
arestillquiterare,anotherexampleatacomparableredshiftbe-
ingthefilamentdetectedbetweenAbell1763andAbell1770by
Faddaetal.(2008).
A weak lensing analysis of the Abell 222/223 system was
performed by Dietrich et al. (2005), who found evidence for
a possible dark matter filament connecting both clusters. They
interpret the difference in redshift between Abell 222 and
Abell 223 as implying a separation along the line of sight of
15±3h−1 Mpc. Their weak lensing surface mass density con-
70
toursagainshowevidenceforadoublestructureinAbell223.
Theexistenceofa1.2Mpcwidefilamentbetweenbothclus-
terswasalsofoundinX-raysbyWerneretal.(2008). Fig.1.Centralsurfacebrightnessinther(cid:4)bandasafunctionofr(cid:4)mag-
nitude. Thehorizontal andoblique fullredlinesseparate thegalaxies
We presenthere a muti-wavelengthopticalandX-ray anal-
(abovetheselines)fromthestars(belowtheselines).Theobliquedot-
ysisoftheclustersAbell222andAbell223.Theopticalpartis ted red line shows the limit for stars computed for 17.25 < r(cid:4) < 22
basedonimagingintheg(cid:4)andr(cid:4)bandsobtainedwiththeCFHT (see text). The two vertical linescorrespond to r(cid:4) = 17.25 where the
Megaprime/Megacamcameraandretrievedfromthe Megapipe starsstopbeingsaturated,andr(cid:4)=22,wherethefittocalculatethestar
image stacking pipeline at CADC (Gwyn 2009). As seen in sequencewaslimited(seetext).
Table 1, the r(cid:4) band data are about 0.55 mag deeper than the
g(cid:4)banddata(bytakinganaveragecolorg(cid:4)−r(cid:4) =0.85),andwill
begivenpriorityinthemorphologicalanalysis;theg(cid:4) datawill We did not use the catalogues available for these images,
be used to select brightcluster membersfromcolor-magnitude because they were made without masking the surroundings of
diagrams,andalsobecausetheg(cid:4) bandismoresensitivetostar bright stars, so we preferred to build masks first, then to ex-
formationthanther(cid:4) band.Theseimagesaredeeperthanthose tract sourceswith SExtractor (Bertin & Arnouts1996). Due to
ofDietrichbyatleast0.5mag(ourrawgalaxycountsstartde- theditheringpattern,thetotalareacoveredbyther(cid:4) imagewas
creasingforr(cid:4) > 24.5whileDietrichetal.saythattheircounts slightlylargerthantheMegacam1deg2field:1.196deg2.
intheRbandstopfollowingapowerlawforR>24),andthere- Objectsweredetectedandmeasuredinthefullr(cid:4)image,then
fore allow to probe the GLF to somewhat fainter magnitudes. measured in the g(cid:4) image in double image mode. Magnitudes
OurX-rayanalysisisbasedonarchiveXMM-Newtondata. areintheABsystem.Theobjectslocatedinthemaskedregions
Wewillconsiderhereafterthatthetwoclustershaveacom- werethentakenoutofthecatalogue,leadingtoafinalcatalogue
mon redshiftof 0.21,for whichWright’s cosmologycalculator of 223414 objects. All these objects have measured r(cid:4) magni-
(Wright 2006) givesa distance of 1035Mpc, a spatial scale of tudes,and214669alsohavemeasuredg(cid:4)magnitudes.
3.427 kpc/arcsecand a distance modulusof 40.07(assuming a Since the r(cid:4) catalogueis deeper,and the seeing is also bet-
flatΛCDMcosmologywith H = 70kms−1 Mpc−1,Ω = 0.3 ter in this band (see Table 1), we will perform our star-galaxy
0 M
andΩΛ =0.7). separationinther(cid:4)band.
The paper is organised as follows. We describe our optical
analysis in Sect. 2, and results concerningthe galaxy luminos-
2.2.Star-galaxyseparation
ity functionin Sect. 3. A search for substructuresin bothclus-
ters based on the Serna-Gerbalmethod is described in Sect. 4. Inordertoseparatestarsfromgalaxies,weplottedthemaximum
The X-ray data analysis andresults, includingtemperatureand surface brightness μ in the r(cid:4) band as a function of r(cid:4). The
max
metallicitymaps,arepresentedinSect.5.Anoverallpictureof resultisshowninFig.1.
thisclusterpairisdrawninSect.6. The bestfit to the star sequencevisible in Fig. 1 calculated
for 17.25 < r(cid:4) < 22 is μ = 0.996∗r(cid:4) −0.247, with stan-
max
darddeviationsontheslopeandconstantof0.002and0.047re-
2. Opticaldataandanalysis spectively. The point-source (hereafter called “star”) sequence
is clearly visible for r(cid:4) < 22, with the star saturation show-
2.1.Theopticaldata
ing well for r(cid:4) < 17.25. We will define galaxies as the ob-
We have retrieved from the CADC Megapipe archive (Gwyn jects with μ (r(cid:4)) > 17.25 for r(cid:4) < 17.25, and as the objects
max
2009) the reduced and stacked images in the g(cid:4) and above the line of equation μ = 0.996 ∗ r(cid:4) − 0.247 + 0.3,
max
r(cid:4) bands (namely G002.024.493-12.783.G and G002.024.493- or μ = 0.996∗ r(cid:4) + 0.053. Stars will be defined as all the
max
12.783.R) and give a few details on the observations in otherobjects(seeFig.1).Thesmallcloudofpointsobservedin
Table 1. Observations were made at the CFHT with the Fig. 1 under the star sequence is in fact defects, but represents
Megaprime/Megacamcamera,whichhasapixelsizeof0.186× lessthan2%ofthenumberofstars.Wethusobtainastaranda
0.186arcsec2. galaxycatalogue.
Page2of15
F.Durretetal.:Abell222/223
As a checkto see up to whatmagnitudewe couldtrustour
star-galaxyseparation,we retrievedthe star cataloguefromthe
BesançonmodelforourGalaxy(Robinetal.2003)ina1deg2
regioncenteredonthepositionoftheimageanalysedhere.Such
a catalogue is in AB magnitudes(as ours) and is corrected for
extinction. In order for it to be directly comparable to our star
catalogue,wecorrectedourstarcatalogue(andourgalaxycata-
logueaswell,forlaterpurposes)forextinction:0.071maging(cid:4)
and 0.052 mag in r(cid:4) (as derived from the Schlegel et al. 1998
maps).
Ther(cid:4) magnitudehistogramoftheobjectsclassifiedasstars
inourr(cid:4)imageroughlyagreeswiththeBesançonstarcatalogue
for r(cid:4) ≤ 20, though in some bins our counts tend to be some-
what higher than those of the Besançon model even for bright
magnitudes.Forr(cid:4) >20,wedetectmorestarsthanpredictedby
the Besançon model (the difference is about 35% at r(cid:4) ∼ 21).
Thiscouldimplythatourstar-galaxyseparationisnotvalidfor
r(cid:4) > 20;however,assuggestedbyFig. 1,thisisaratherbright
limit, as confirmed by simulations adapted to match compara-
ble data, where we found that the star-galaxy separation deter-
minedwiththismethodwasreliableatleastuptor(cid:4) =21(Boué Fig.2.Positionsoftheobjects(starsandgalaxies)intheAbell222(ma-
genta) and Abell 223 (cyan) catalogues. The objects used to estimate
etal.2008).Inreality,thedisagreementbetweenourstarcounts
backgroundcounts(seetext)areshowninblueforthenorthrectangle
and those predicted by the Besançon model is most probably
andinredfortheannulussurroundingtheclusters.Notethatthefigure
duetothefactthatourfieldislocatedinthedirectionofoneof covers1.2×1.2deg2,andisslightlylargerthantheimages,whichcover
the densest regionsof the Sagittarius stream (Ibata et al. 2001; 1.04×1.15deg2.
Yannyetal. 2009).Itisthereforenormalto observemorestars
thanpredictedbytheBesançonmodel,whichdoesnottakethis
streamintoaccount.Notethattheremayalsobeasmallcontri-
butionduetoquasarsandAGN,sincetheirnumberatr(cid:4) ∼ 20.5 the CCD images and then attempt to recover them by running
SExtractor again with the same parameters used for object de-
isexpectedtobeabout20persquaredegree(seeRichardsetal.
tectionandclassificationontheoriginalimages.Inthisway,the
2006,Fig.13).
For r(cid:4) > 20, we will conservatively consider that the star- completenessismeasuredontheoriginalimages.
Inpractice,weextractfromthefullfieldofviewtwosubim-
galaxy separation may be wrong, and we will compute galaxy
ages, each 1500×1500pixels2, correspondingto the positions
counts by counting the total number of objects (galaxies plus
oftheAbell222andAbell223clustersontheimage.
stars) per bin of 0.5 mag, and considering that the number of
Ineachsubfield,andforeach0.5magbinbetweenr(cid:4) = 20
galaxiesisequaltothetotalnumberofobjectsminusthenumber
and27,wegenerateandaddtotheimageonestarthatwethen
ofstarspredictedineachbinbytheBesançonmodel.Asmen-
try to detectwith SExtractor,assumingthe same parametersas
tionedbelow,wehavecheckedthatthegalaxynumbercountsin
the20<r(cid:4) <22rangeestimatedbybothmethods(i.e.ourstar- previously. This process is repeated 100 times for each of the
twofieldsandbands.
galaxyseparationandthesubtractionoftheBesançoncountsto
thetotalnumberofobjects)areconsistentwithinerrorbars. Such simulations give a completeness percentage for stars.
Since the clusters are quite distant, they do not cover the This is obviously an upper limit for the completenesslevel for
wholefield,soweextractedfromthestarandgalaxycatalogues galaxies,sincestarsareeasiertodetectthangalaxies.However,
two catalogues as large as possible corresponding to the two wehaveshowninapreviouspaperthatthismethodgivesagood
clusters. We defined the J2000.0 positions of Abell 222 and estimate of the completenessfornormalgalaxiesif we applya
Abell223ascoincidingwiththe brightestgalaxyofeachclus- shift of ∼0.5 mag (see Adami et al. 2006). Results are shown
ter,i.e.24.3921,−12.9912and24.5099,−12.7575respectively inFig.3.
(indegrees).Notethatthesearenotexactlytheclusterpositions Fromthesesimulations,andtakingintoaccountthefactthat
givenbytheNEDdatabase.Themaximumpossibleradiustoob- resultsareworseby∼0.5magformeangalaxypopulationsthan
tainindependentcataloguesforthetwoclusterswas0.1290deg, forstars,wecanconsiderthatourgalaxycatalogueiscomplete
or1.6Mpcataredshiftof0.21. tobetterthan80%forg(cid:4) ≤25andr(cid:4) ≤24.5.
FortheAbell222andAbell223clustersseparately,wethus
obtainedthreecomplementarycatalogues(withg(cid:4) andr(cid:4) mag):
2.4.Galaxycounts
objects classified as galaxies (classification valid at least for
r(cid:4) ≤ 20),objectsclassifiedasstars,andacompletecatalogueof The surfacescoveredby the Abell222and223catalogues(af-
galaxies+starswhichwillbeusedforr(cid:4) > 20.Thepositionsof terexcludingmaskedregions)are0.0429deg2and0.0475deg2
thegalaxiesintheregionsofthetwoclustersareshowninFig.2. respectively. Galaxy counts were computed in bins of 0.5 mag
normalizedtoasurfaceof1deg2.
For r(cid:4) ≤ 20, galaxy counts were derived directly by com-
2.3.Cataloguecompleteness
puting histograms of the numbers of galaxies in the Abell 222
Thecompletenessofthecatalogueisestimatedbysimulations. andAbell223catalogues.Forr(cid:4) > 20,webuiltforeachcluster
For this, we add “artificial stars” (i.e. 2-dimensional Gaussian histogramsofthetotalnumbersofobjects(galaxies+stars)and
profiles with the same Full-Width at Half Maximumas the av- obtainedgalaxycountsbysubtractingthenumbersofstarspre-
erage image Point Spread Function) of different magnitudesto dictedbytheBesançonmodel.Theresultinggalaxycountswill
Page3of15
A&A517,A94(2010)
Fig.3.Pointsourcecompletenessasafunctionofmagnitudeinpercent-
agesforAbell222(top)andAbell223(bottom)ing(cid:4)(left)andr(cid:4)(right)
forpoint-likeobjects(seetext).
beusedinthenextsectiontoderivetheGLFsforbothclusters
inbothbands.
As a test, we considered the galaxy counts in the four
0.5 mag bins for 20 < r(cid:4) < 22 computed for both clusters
with the two methods (i.e. first method: considering that the
star-galaxy separation is valid, and second method: consider-
ingthetotalnumberofobjects(galaxies+stars)andsubtracting
the number of stars predicted by the Besançon model to ob-
tain the number of galaxies). In all cases, the differences are
smaller than 15% (and in most cases they are only a few per-
cent). Therefore, the limit (between r(cid:4) = 20 and 22) at which
weconsiderthatourstar-galaxyseparationisnotvalidanymore
willnotstronglyinfluenceourresults.
Note that no k-correction was applied to the galaxy
magnitudes.
Fig.4.(g(cid:4)−r(cid:4))vs.r(cid:4) colour−magnitudediagramsforAbell222(top)
3. Results:colour–magnitudediagramsandgalaxy and Abell 223 (bottom) for objects classified as galaxies from the
luminosityfunctions μmax-magnitude relation. The vertical dashed lines indicate the mag-
nitude interval where the colour−magnitude relation was computed.
Inorderto computethe galaxyluminosityfunctionsofthetwo The long oblique line shows the colour−magnitude relation, while
clusters,itisnecessarytosubtracttothetotalgalaxycountsthe the two short oblique lines indicate an interval of ±3σ around the
numbercountscorrespondingtothecontaminationbythefore- colour−magnituderelation.Thefilledcirclesshowthegalaxiesbelong-
groundandbackgroundgalaxies. ingtotheclustersaccordingtotheirspectroscopicredshifts.
For galaxies brighter than r(cid:4) = 20 we will select galaxies
with a high probability to belong to the clusters by drawing
colour-magnitudediagramsandselectinggalaxieslocatedclose beabsolutelyidentical).Wetheneliminatedthegalaxieslocated
tothisrelation.Afewspiralsmaybemissedinthisway,buttheir morethan3σawayfromthisrelationandcomputedtheg(cid:4)−r(cid:4)
numberinanycase isexpectedtobe small,asexplainedatthe vs.r(cid:4)relationsagain.
end of Sect. 3.1 (also see e.g. Adamiet al. 1998). For galaxies The equations of the colour–magnituderelations are found
fainterthanr(cid:4) =20wewillsubtractgalaxycountsstatistically. tobe:g(cid:4)−r(cid:4) =−0.0545r(cid:4)+2.091with3σ=0.21forAbell222,
andg(cid:4)−r(cid:4) =−0.0474r(cid:4)+1.945with3σ=0.21forAbell223.
Forr(cid:4) < 20,we willconsiderhereafterthatallthegalaxies
3.1.Colour–magnitudediagrams
locatedwithin3σoftheserelations(i.e.betweenthetwoblack
Themeanvaluesforg(cid:4)−r(cid:4)are0.85and0.87forAbell222and linesofFig.4)belongtotheclusters.Withthiscondition,there
Abell 223. The g(cid:4) − r(cid:4) vs. r(cid:4) colour−magnitude diagrams are are141and144galaxiesbelongingtoAbell222andAbell223
showninFig.4forthetwoclusters.Asequenceiswelldefined respectivelywithr(cid:4) ≤20.
forgalaxiesinthemagnituderange18<r(cid:4) <22inbothclusters. We also plot in Fig. 4 the galaxies belonging to the clus-
We computed the best fit to the g(cid:4) − r(cid:4) vs. r(cid:4) relations in this ters according to their spectroscopic redshifts, i.e. galaxies in
magnituderangebyapplyingalinearregression(separatelyfor the[0.195,0.215]redshiftinterval(seeSect.4).Wecanseethat
thetwoclusters,sincetheircolour-magnituderelationsmaynot theirpositionsagreewellwiththecolour−magnitudeselection.
Page4of15
F.Durretetal.:Abell222/223
Forbothclusters,wehaveestimatedthenumberofblueclus-
tergalaxieslostbyselectinggalaxiesintheredsequenceinterval
inthefollowingway.First,wecomputedhistogramsofnumbers
of galaxies in the red sequence and below the red sequence in
bins of 1 absolute magnitudein the r(cid:4) band. The bins of inter-
esthereare Mr(cid:4) =−21.5and−20.5(seeFig.4),roughlycorre-
spondingtor(cid:4) = 18.5and19.5.Thenweestimatedthenumber
offoregroundgalaxiesexpected.Sincethecomovingvolumeat
z = 0.21is2.622Gpc3 (Wright2006),andeachofourclusters
coversan areaof0.0522deg2 onthesky,thevolumeinthe di-
rectionofeachclusteris3318Mpc3.ByusingtheRbandlumi-
nosityfunctionbyIlbertetal.(2005)inthe[0.05−0.20]redshift
bin (see their Fig. 6 and Table 1), we find that the percentages
of“lost”galaxiesareoftheorderof10%−25%forMr(cid:4) =−20.5
andof10%forMr(cid:4) =−21.5.
3.2.Comparisonfield
In order to perform a statistical subtraction of the background
contributionforr(cid:4) >20,weconsideredseveralpossibilities.
The most obvious solution would be to extract the back- Fig.5.Galaxycountsintheg(cid:4)(left)andr(cid:4)(right)bandsformagnitudes
groundinanannulussurroundingtheclusters.Thelargestouter r(cid:4) > 20wherethebackground mustbesubtractedstatistically,inlog-
arithmicscale.ThecountsinAbell222and223aredrawninmagenta
circlethatcanbeextracted,centeredonthemiddlepositionbe-
tweenthetwoclusters(24.4510,−12.8744J2000.0indegrees), andcyanrespectively. Theblackdashedlinesshowthegalaxycounts
fromtheCFHTLSDeepsurvey,andthefullgreenlineshowstheaver-
hasaradiusof0.4472deg,or5.55Mpc.Wecanthentakeanin-
ageoftheDeep1,Deep2andDeep3fieldcounts. Theblueandred
nercircleofradius0.4028deg,or5Mpc.Thisannulusisdrawn
lines correspond to the “local” backgrounds extracted from the same
in Fig. 2. However,the number countsin this circle are higher imagesasourclusters(red:intheannulusbetween thetwocirclesof
thantheCFHLTSDeepsurveycounts(seebelow),probablybe- Fig.2,blue:inthenorthrectangle,seetextandFig.2).Errorbarsare
cause the annulus is too close to the clusters and still includes Poissonianandarenotplottedforclarity.
clustergalaxies,sowewillnotusethesecountsasabackground.
The Canada-France-Hawaii Telescope Legacy Survey
theGLFsthusderivedforAbell222and223.Thiswillillustrate
(CFHTLS)hasbeentakenwiththesametelescope,cameraand
thedifficultytodetermineGLFparametersunambiguously.
filters as the data we are analyzing. The Deep survey explores
a solid angle of 4 × 1 deg2 of the deep Universe, in four in- We can note from Fig. 5 that galaxy counts are notably
dependentfields(http://www.cfht.hawaii.edu/Science/ higherinAbell223thaninAbell222,implyingthattheformer
CFHLS/).Observationsarecarriedoutinfivefilters(u∗,g(cid:4),r(cid:4),i(cid:4) clusterisricherthanthelatter.
andz(cid:4))providingcatalogsofsourcesthatare80%completeup
toiAB =26.0(seehttp://terapix.iap.fr/cplt/oldSite/ 3.3.Galaxyluminosityfunctions
Descart/CFHTLS-T0005-Release.pdf). We only consider
the Deep survey here, because we are interested in the GLFs The Galaxy Luminosity Functions (GLFs) of Abell 222 and
downtofaintmagnitudes.Forthefourdeepfields,wecomputed Abell223werecalculatedinbinsof0.5magandnormalizedto
the galaxynumbercountsin binsof0.5mag,ing(cid:4) andr(cid:4), nor- 1deg2,asdescribedabove.Wesubtractedthebackgroundcon-
malizedtoa surfaceof1 deg2. Thegalaxycountsinthese four tribution using as backgroundgalaxy counts: 1) the average of
fields are drawn in Fig. 5. As discussed before (see e.g. Boué thethreeDeep1,Deep2andDeep3fields;2)the“local”counts
etal.2008,andreferencestherein),thecountsinthesefourfields inthenorthbluerectangleofFig.2.
somewhat differ, due to cosmic variance, and in particular the The GLFs are displayed for Abell 222 and Abell 223 in
countsintheDeep4fieldarehigherthanintheotherthree.We Figs.6and7respectively.Theerrorbarsdrawninthesefigures
will thereforetake as backgroundgalaxycountsthe averageof weretakentobe4timesthePoissonianerrorsongalaxycounts,
theDeep1,Deep2andDeep3fieldcounts. as suggested by detailed simulations previously performed by
ourteamforsimilardata(seeBouéetal.2008,Fig.5).
For comparison, we also extracted galaxy counts from a
TheGLFs(asafunctionofabsolutemagnitude)werefitby
region assumed to be representative of the background in our
aSchechterfunction:
image. This region was chosen to be a rectangle north of the
clusters, in an area devoidof brightstars. It is shownas a blue S(M)=0.4 ln10φ∗yα+1e−y
rectangleinFig.2andcoversaneffectiveareaof0.06825deg2.
The corresponding counts (hereafter the “local” background withy=100.4(M∗−M).
counts) appear to be notably lower than the CFHTLS-Deep TheparametersoftheSchechterfunctionfitsoftheGLFsare
counts(Fig. 5); this may be the case if this regioncorresponds respectivelygiveninTables2and3 forthetwo differentback-
toavoidlocatedbetweenthelargescalefilamentsthatconverge groundsubtractions:theaverageoftheCFHTLSDeepfieldsand
towardstheclusters. the “local” background.The absolute magnituderange consid-
AlthoughweconsiderthattheCFHTLS-Deepcountsproba- eredisindicatedforeachfit.ThefitsaredrawninFigs.6and7
blyrepresentbetterthegalaxybackgroundcounts,wewillalso forAbell222andAbell223respectively.
fit the GLFs with these “local” backgroundcounts, keeping in If we compare the results given in Tables 2 and 3, we see
mindthefactthatthese“local”countsarelikelytooverestimate thattheΦ∗ and M∗ parametersdependlittle onthebackground
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A&A517,A94(2010)
Table2.Schechterparametersforgalaxyluminosityfunctions(backgroundgalaxycountstakenfromtheCFHTLSDeepfieldcounts).
Cluster Filter Range Φ∗ M∗ α
Abell222 g(cid:4) [−22.5,−16.0] 2891±134 −20.04±0.06 −0.60±0.05
r(cid:4) [−23.5,−16.0] 937±64 −22.01±0.07 −1.43±0.01
Abell223 g(cid:4) [−25.5,−15.5] 1017±72 −21.05±0.06 −1.30±0.02
r(cid:4) [−26.0,−15.0] 1107±62 −22.05±0.06 −1.29±0.01
Table3.Schechterparametersforgalaxyluminosityfunctions(“local”backgroundgalaxycounts).
Cluster Filter Range Φ∗ M∗ α
Abell222 g(cid:4) [−22.0,−16.0] 2269±133 −20.32±0.06 −0.94±0.03
r(cid:4) [−23.0,−16.0] 1132±63 −21.85±0.06 −1.42±0.01
Abell223 g(cid:4) [−25.0,−15.0] 819±55 −21.20±0.05 −1.42±0.01
r(cid:4) [−26.0,−15.0] 1042±58 −22.10±0.06 −1.34±0.01
Fig.6. Galaxy luminosity functions for Abell 222 in the g(cid:4) (top) and Fig.7.SameasFig.6forAbell223.
r(cid:4) (bottom)bands,inlogarithmicscale.Thepoointscorrespondtothe
subtraction of the background counts taken from the CFHTLS.Error
barsare4timesthePoissonianerrorsongalaxycounts(seetext).The
ForAbell222,theGLFisquitewellfitbyaSchechterfunc-
bestSchechterfunctionfitsaredrawninredforthesubtractionofback- tionin ther(cid:4) band,witha faintendslopeα = −1.43(thesame
ground counts taken from the CFHTLSand in blue for “local” back-
forbothbackgroundsubtractions).Ontheotherhand,theGLF
groundcounts(seetext).
is surprisinglyflat in g(cid:4) (slope α = −0.60or −0.94,depending
onbackgroundsubtraction).
A Schechter function is obviously not sufficient to fit the
galaxy counts chosen. On the other hand, some variations are GLFsofAbell223,bothing(cid:4) andr(cid:4):Abell223hasmorevery
found in the faint end slope α, as expected since it is at faint brightgalaxiesthanAbell222,andasecondcomponentatbright
magnitudesthatthebackgroundgalaxysubtractionhasastrong magnitudesisobviouslyrequired.Atfaintmagnitudes,theGLF
influence. slopesing(cid:4) andr(cid:4) arecomparableforAbell223.Theyvaryby
Page6of15
F.Durretetal.:Abell222/223
3.4.Morphologicalsegregation
We applied a new tool developed in SExtractor that calculates
foreachgalaxytherespectivefluxesinthebulge(spheroid)and
xtractor
disk. This new experimental SE feature fits to each
galaxyatwo-dimensionalmodelcomprisedofadeVaucouleurs
spheroid(bulge)andanexponentialdisk.Briefly,thefittingpro-
cess is very similar to that of the GalFit package (Peng et al.
2002)andisbasedonamodifiedLevenberg-Marquardtminimi-
sation algorithm.The modelis convolvedwith a supersampled
modelofthelocalPointSpreadFunction(PSF),anddownsam-
pled to the final image resolution. The PSF model used in the
x
fitswasderivedwiththePSFE software(Bertinetal.2010,in
preparation)fromaselectionof5436(g(cid:4))/6922(r(cid:4))pointsource
images.ThePSFvariationswerefitusinga6thdegreepolyno-
mialofxandyimagecoordinates.
Themodelfittingwascarriedoutindependentlyintheg(cid:4)and
r(cid:4) bands.Thebulgetototalratiosdiscussedherewereextracted
fromtheredchannel,limitedtor(cid:4) =22,sincebeyondthismag-
nituderesultsmaybecomeunreliable.
We applied this tool to look for evidence for morphologi-
cal segregation, by computing the fraction of galaxies with a
spheroid to total flux smaller than 0.3, as a function of radius
Fig.8.Fractionofdisk-dominated galaxiesasafunctionofradiusfor (i.e.inconcentriccirclesofstep0.02deg).Theresultisplotted
Abell222(inmagenta)andAbell223(incyan).
inFig.8.Ineachbin,1σerrorswerecomputedasfollows:ifn
isthenumberofgalaxieshavingaspheroidtototalfluxsmaller
than 0.3 and N the total number of galaxies, we assume that n
and (N − n) are independent variables; in this case, the error
onn/N is:
smallquantities(0.12and0.05respectively)betweenonegalaxy
(cid:2)
backgroundsubtractionandtheother.However,theseslopesare
(n/N) (1/n−1/N).
only indicative, since as seen in Fig. 7 the GLFs show strong
wigglesforabsolutemagnitudesbetween−20and−17inr(cid:4)and
between −20 and −16 in g(cid:4). Since Abell 223 is itself a double Wecanseethatthereisageneralincreaseofthefractionofdisk-
cluster (Dietrich et al. 2002), it is not surprising to find that it dominated galaxies with radius as expected (see e.g. Biviano
hasa“perturbed”GLF.Thisisalsothecaseforexampleforthe etal.1997).
Coma cluster, known to have a two-component GLF (Biviano Inordertoseeiftherewasamorphologicalsegregationthat
etal.1995). could be linked to the presence of cosmological filaments, we
IfwecomparetheGLFsofthetwoclusters,wecanseethat computedthatsamefractioninanglesof45degallaroundeach
theyare moreor lesscomparablein ther(cid:4) band(exceptforthe cluster.TheresultisshowninFig.9,whereangularsectorsare
wigglesin Abell223),whilein the g(cid:4) bandthe GLFis notably numbered from 1 to 8 clockwise from east (see Fig. A.1). No
steeperin Abell223thanin Abell222.Anexplanationforthis strong trend is foundin this plotwithin error bars, as expected
can be that both clusters have comparable old galaxy popula- if there is no obvious cosmological filament linking the clus-
tions,butthatthenumberoffaintstarforminggalaxiesishigher ters with the surrounding cosmic web. We can note however
in Abell 223, where star formation can be triggered by recent that for both clusters there is a dip in the fraction of bulge-
interactions. dominated galaxies in sector 6, which corresponds to the line
joiningthetwoclusters;thefilamentalreadydetectedbyWerner
If we compare these GLFs to those found by other au- et al. (2008) using wavelet-decomposition in the [0.5−2] keV
thors, we can note that the bright part of the GLF Schechter
energybandislocatedalongthissamedirection.Inthiszonethe
fits (r(cid:4) < 18, or Mr(cid:4) <−18.9) for both clusters are very simi- gasisnotyethotenoughtoinfluencethegalaxies.Ifwelookat
larinshapetotheGLFsrecentlyobtainede.g.byAndreonetal.
Fig.9fromLaganáetal.(2009),2keVisthelowerlimitwhere
(2008);theseauthorsanalyzedtheGLFsofasampleofclusters
ram-pressure plays an important role in galaxy clusters. Thus
atvariousredshifts,limitedtorelativelybrightabsolutemagni-
wewouldexpecttodetectatrendonlyifthefilamentwashotter
tudes:M <−19.
V than2keV.
TheαslopesofthefaintendsoftheGLFsderivedhere(ex-
Another possibility for this dip in the fraction of bulge-
cept for that of Abell 222 in the g(cid:4) band) are within the broad
dominatedgalaxiesisahighernumberofbluegalaxiesinwhich
rangeofvaluesestimatedbypreviousauthorsfordifferentclus-
starformationwastriggeredduetoongoingcollisions.Poggianti
ters,clusterregionsandphotometricbands(seee.g.thecompi-
et al. (2004) arguedthat there is a striking correlationbetween
lationinTableA.1ofBouéetal.2008).
thepositionsoftheyoungandstrongpost-starburstgalaxiesand
NotethatAbell222andAbell223are atredshift0.21,and substructure in the hot intra-cluster medium (ICM) identified
few GLFs are available for clusters at such redshifts. Andreon from XMM-Newton data, with these galaxieslying close to the
etal.(2005)foundsomewhatshallowerslopesforthreeclusters edgesoftwoinfallingsubstructures.Thisresultsuggeststhatthe
atredshifts∼0.3,butintheK band,sothecomparisonwithour interactionwiththedenseICMcouldberesponsibleforthetrig-
resultsisnotstraightforward. geringofthestarformation.
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A&A517,A94(2010)
Fig.11. Positionsof the galaxieswith measured redshifts and magni-
tudesintheregionscoveredbyAbell222andAbell223(blackcrosses).
Thegalaxiesbelongingtothetwomaingravitationallyboundsystems
Fig.9.Fractionofdisk-dominatedgalaxiesinsectorsof45degwithin
identifiedwiththeSGmethodareshownincolours(seetext).
each cluster,withthesamecolours asintheprevious figure.Angular
sectorsarenumberedfrom1to8clockwisefromeast(seeFig.A.1).
4. Cluster-scaletolarge-scalestructure
WesearchedtheNEDdatabaseforgalaxieswithredshiftsavail-
ableinaverylargeregionof3degradiusaroundAbell222/223
and found 980 galaxies. This radius corresponds to 37 Mpc at
thedistanceofthestudiedclusters,i.e.tothetypicaldiameterof
a“bubble”orofavoidintheuniverse(seee.g.Hoyle&Vogeley
2004).Thiscatalogueofgalaxieswithmeasuredredshiftswillbe
usedfirsttosearchforsubstructuresinthezonescoveredbyour
clusters,andsecondtosearchforstructures(suchasfilaments)
atalargerscale.
4.1.Searchforgravitationallyboundstructures
Within the regions of our images covered by Abell 222 and
Abell223,246galaxieshavemeasuredredshifts,and210galax-
ies haveboth measuredredshiftsandmagnitudes.Outof these
210galaxies,173haveredshiftsinthe[0.18,0.24]interval.The
positionsofthe210galaxieswithmeasuredredshiftsandmag-
nitudes are displayed in Fig. 11. It is interesting to note that,
as traced by these galaxies, both clusters appear to be strongly
elongated, with elongationposition angles that differ from one
cluster to the other. This strongly suggests that the clusters are
Fig.10.Averager(cid:4)magnitudeasafunctionofradiusforAbell222(in notindynamicalequilibrium,inagreementwithpreviousstud-
magenta)andAbell223(incyan). ies (see above, and also Zabludoff et al. 1995; Roettiger et al.
1996;Boschinetal.2004;Girardietal.2006).
Thecorrespondingredshifthistogram,zoomedonthecluster
redshiftinterval,isshowninFig.12.
3.5.Luminositysegregation We applied to the catalogue of 210 galaxies with redshifts
andmagnitudestheSerna&Gerbal(1996)algorithm(hereafter
We searched for luminosity segregationin both clusters by se- SG), which allows to search for substructures, separate galax-
lecting galaxies brighter than r(cid:4) = 24 and within the red se- ies forming gravitationally bound structures, and estimate the
quence defined in Fig. 4, to increase the probability that they masses of these substructures. The dendogram thus obtained
belongtotheclusters.Themeanr(cid:4)magnitudewascalculatedin is shown in Fig. 13. It clearly shows two dynamically dis-
eachoftheradialbinspreviouslydefinedandresultsareshown tinctsubstructures,correspondingtothetwoclusters:Abell222
inFig.10.Weobserveanincreaseofthemeanmagnitudewith in the top half and Abell 223 in the bottom half. The two
radius: it is about 21.3 in the central bin, and then increases maingravitationallyboundstructurescorrespondingtothesetwo
withradiusfromvaluesaround22to22.6. clusters respectively include 55 and 64 group members. Their
Page8of15
F.Durretetal.:Abell222/223
Fig.12.Redshifthistogramsforallthegalaxieswithmeasuredredshifts
intheregionscoveredbyAbell222andAbell223(inblack),andfor
the galaxies in the gravitationally bound structures of Abell 222 and
Abell 223, in magenta and cyan respectively (the positions of these
galaxiesareshowninFig.11)withthesamecolourcoding.
corresponding velocity dispersions are 1014 and 1170 km s−1
respectively. The virial masses derived for these two substruc-
tures (assuming M/L = 400 solar units) are 1.9 × 1011 and
4.2×1013 M(cid:5).
ForAbell222,andprobablyalsoforAbell223,thesemasses
are obviouslyunderestimated,particularlyfor Abell 222, since
Fig.13.DendogramshowinghowAbell222andAbell223canbedy-
such a valuedoes notevenreach that of a cD galaxy.We have
namicallyseparatedwhenapplyingtheSerna&Gerbalalgorithm.The
triedtoanalyzethetwoclustersseparately,butthismodifiesthe
abscissaisthenegativebindingenergywhilethecataloguenumbersof
resulting masses only slightly. The fact that we find too small thevariousgalaxiesareshownalongtheordinate(thetopofthedendo-
cluster masses can be due to severalreasons. First, the redshift gramistruncated,butwouldonlyshowaconstantgravitationalbinding
sample we are using is taken from the NED data base, and is energy).
obviouslynotcomplete,evenatbrightmagnitudes.Wehavees-
timatedthecompletenessoftheredshiftcataloguebycomparing
thenumbersofgalaxieswithredshiftsinthe[0.18,0.24]redshift simulations that for merging systems where the subcluster has
intervaltothetotalnumberofgalaxies,inmagnitudebinsof0.5 amasslargerthanonefourththatofthelargecluster,thevirial
inther(cid:4) band.Wefindthatfor17<r(cid:4) <19.5thecompleteness masscanbelargerthantwicetherealmass.Thismostprobably
is around 50%, then rapidly decreases for r(cid:4) > 19.5, confirm- appliesatleasttotheAbell223cluster.
ing that the cluster masses based on such a redshift catalogue We can also apply the mass-temperaturerelation estimated
are probably unreliable. Second, as pointed out by the referee, from X-raydata by Arnaudet al. (2005), with the overalltem-
the SG method relies on the unlikely assumption that galaxies peratures given in Table 5; this relation gives masses M of
200
inclustersarethemasscarriers,andadditionallyrequiresanas- 3.8×1014 and 4.7×1014 M(cid:5) for Abell 222 and Abell 223 re-
sumptionabouttheM/Lratio,heretakentobe M/L=400;this spectively(alsowithH =70kms−1Mpc−1).Thesevaluesmay
0
valuemaybereasonableforaclustertakenasawhole,butistoo beclosertotherealvaluesthanthoseestimatedfromthegalaxy
largeforindividualgalaxies(thedarkmatterhalosofindividual velocitydispersions,sincee.g.Takizawaetal.(2010)foundthat
galaxies would be overlapping in the dense central regions of for merging clusters X-ray derived masses were usually more
clusters,andstrongmasssegregationwouldresultbydynamical reliablethanvirialmassestimations.However,asexplainedfor
friction,whichisnotobserved). example by Nagai et al. (2007), although the total ICM mass
Better estimates of the cluster masses can be obtained by can be measuredquite accurately (to better than ∼6%) in clus-
applying other methods. For example, the relation between ters, the hydrostaticestimate of the gravitationallyboundmass
massandvelocitydispersioncomputedbyBivianoetal.(2006) isbiasedlowbyabout5−20%throughoutthevirialregion,pri-
leads to masses of 1.2 × 1015 and 1.4 × 1015 M(cid:5) (with marilydueto additionalpressuresupportprovidedbysubsonic
H = 70 km s−1 Mpc−1) for Abell 222 and Abell 223 re- bulkmotionsintheICM,ubiquitousinoursimulationsevenin
0
spectively,basedonrespectivevelocitydispersionsof1014and relaxedsystems.Anothersourceoferroristhefactthattheclus-
1070km s−1. Note howeverthat since the clustersare notviri- tertemperatures,particularlythatofAbell223(seeFig.18)are
alized these values are probably overestimates. For example, nothomogeneous.Forexample,Rasiaetal.(2004)haveshown
Takizawa et al. (2010) have recently shown from numerical thatanincompletethermalisationofthegasduetoanincomplete
Page9of15
A&A517,A94(2010)
Table4.InformationonX-raydata.
Name Filter t (ks) n (1020cm−2)
exp H
MOS1 MOS2 pn
Abell222 Thin1 11.814 18.005 10.226 2.26
Abell223 Thin1 23.479 26.069 10.612 2.26
excluded substructure PN
-12:48:00.0
-13:00:00.0 12.5 arcmin
Fig.14. Distributionofthegalaxieswithavailableredshiftsinalarge
regionsurroundingtheclusters.Bluepoints:allgalaxieswithmeasured
rTseohduesthhtwiwftoess,ctr:leuAdstbpeerolslin2ut0ns9:d.egrTalshatexuipdehysyawsrieictahclirrsecidzleesdho.iffAttshtiihnsimrtdhaecp[liu0ss.14te84r,.70ap×.2p44e4]a.ri7nstMteorpvtcah2le.. 12:00.0
9:12.0 48.0 24.0 1:38:00.0 37:36.0 37:12.0 48.0 24.0 36:00
Fig.15.Annuluswherethebackgroundwasextractedforsubtraction.
virialisationoftheclustercouldleadtoanunderestimateofthe
mass.Ontheotherhand,iftheX-raygasisheatedbyacollision,
themasscouldbeoverestimated. Table5.OverallX-raytemperaturesandmetallicities.
WecanalsonotethatMamon(1993)hascalculatedthebias
onthe M/Lratiowhenthevirialtheoremiswronglyappliedto Cluster r kT Z
200
agroupwhichisstillevolvingdynamically(seehisFig.11).He (Mpc) (keV) (solar)
found that in some cases the mass could be underestimatedby Abell222 1.28±0.11 3.77±0.15 0.23±0.06
a factor reaching 100. Therefore the masses estimated for our Abell223 1.55±0.15 4.38±0.16 0.23±0.08
clustersassumingthattheyarevirializedareclearlynotvalid.
Unfortunately, the SG method does not allow to separate
thetwosubstructuresinAbell223discoveredbyDietrichetal.
v8.0andcalibrationdatabasewithallupdatesavailablepriorto
(2002)basedongalaxydensitycontours(seetheirFig.6).
November 2009. The initial data screening was applied using
Note that Abell223 is probablyclose to the intersection of
recommendedsetsofeventpatterns,0−12and0−4fortheMOS
twoormorefilaments,sinceitismassiveandsubstructured,and
andPNcameras,respectively.
itsGLFiscomparabletothate.g.ofComa(seeSect.3.3).Onthe
Thelightcurvesare notconstantandlargevariationsin in-
otherhand,Abell222isprobablyonlyfollowingitspathalong
tensity are visible (flares). To improve the signal-to-noise ratio
afilament.
wediscardedperiodsofflaresandthecleanedlightcurvesinthe
energyrangeof [1−10]keV exhibitedstable meancountrates;
4.2.Large-scalestructure exposuretimesaregiveninTable4.Weconsideredeventsinside
thefieldofview(FOV)andexcludedallbadpixels.
Out of the 980 galaxies found in NED in a 3 deg radius re-
ThebackgroundwastakenintoaccountbyextractingMOS1,
gion, 310 have redshifts in the [0.18,0.24] interval (i.e. about
±9000 km s−1 around the mean cluster velocities). Their spa- MOS2 and pn spectra from the publicly available EPIC blank
skytemplatesofAndyRead(Read&Ponman2003).Theback-
tialdistributionisshowninFig.14.Aclusterisvisibletowards
ground was normalized using a spectrum obtained in an annu-
the southwest: Abell 209. It has a redshift similar to that of
lus(between12.5−14arcmin)wherethecluster emissionisno
Abell 222 and Abell 223 and is located about 19.2 Mpc away
longer detected. In one of the pointings, the A223 cluster falls
fromAbell222inprojectiononthesky.Nosignificantfilamen-
tarystructureisdetectedtowardstheAbell222/223clusters. exactlyinside theannuluswherewe determinethe background
contribution. We thus masked this region, avoiding any cluster
contamination. In Fig. 16 we show a comparison between the
5. X-rayanalysis observedandReadbackgroundsforthethreedetectors,aswell
astheresidualsbetweenbothdeterminations.
5.1.Datareduction
Inordertodeterminetheglobalpropertiesoftheseclusters,
The pair of clusters Abell 222/223 was observed with weadoptedthevirialradiir determinedbytheweaklensing
200
XMM-Newton in two different pointings(revolutions1378 and analysis (Dietrich et al. 2005). These values, together with the
1380)with a totalexposuretime of144ks. We useddatafrom mean temperaturesandmetallicitiesare givenin Table5. They
all EPIC cameras (MOS1, MOS2 and pn). The data were re- werecalculatedfixingthehydrogencolumndensitytoitsgalac-
duced with the XMM-Newton Science Analysis System (SAS) ticvalueofn =1.56×1020cm−2.
H
Page10of15