Reconstructing the projected gravitational potential of Abell 1689 from X-ray measurements

0.16 MB·English

by  Celine Tchernin

Astronomy&Astrophysicsmanuscriptno.23242˙FINAL (cid:13)c ESO2015
January14,2015
Reconstructing the projected gravitational potential of Abell 1689
from X-ray measurements
Ce´lineTchernin1,CharlesL.Majer2,3,SvenMeyer2,EleonoraSarli2,DominiqueEckert1,MatthiasBartelmann2
1AstronomicalObservatoryoftheUniversityofGeneva,ch.d’Ecogia16,1290Versoix,Switzerland,
e-mail:[email protected],
2Universita¨tHeidelberg,Zentrumfu¨rAstronomie,Institutfu¨rTheoretischeAstrophysik,Philosophenweg12,69120Heidelberg,
3Germany,DivisionofMedicalPhysicsinRadiology,GermanCancerResearchCenter(DKFZ),69120Heidelberg,Germany.
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Preprintonlineversion:January14,2015
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0
ABSTRACT
2

n Context.Galaxyclusterscanbeusedascosmologicalprobes,buttothisend,theyneedtobethoroughlyunderstood.Combiningall
a clusterobservablesinaconsistentwaywillhelpustounderstandtheirglobalpropertiesandtheirinternalstructure.
J Aims.Weprovideproofoftheconceptthattheprojectedgravitationalpotentialofgalaxyclusterscandirectlybereconstructedfrom
3 X-rayobservations. Wealsoshow thatthisjointanalysiscanbeusedtolocallytestthevalidityoftheequilibriumassumptionsin
1 galaxyclusters.
Methods.We used a newly developed reconstruction method, based on Richardson-Lucy deprojection, that allows reconstructing
] projectedgravitationalpotentialsofgalaxyclustersdirectlyfromX-rayobservations.Weappliedthisalgorithmtothewell-studied
O cluster Abell 1689 and compared the gravitational potential reconstructed from X-ray observables to the potential obtained from
gravitationallensingmeasurements.WealsocomparedtheX-raydeprojectedprofilesobtainedbytheRichardson-Lucydeprojection
C
algorithmwiththefindingsfromthemoreconventionalonion-peelingtechnique.
h. Results.Assumingsphericalsymmetryandhydrostaticequilibrium,thepotentialsrecoveredfromgravitationallensingandfromX-
p rayemissionagreeverywellbeyond500kpc.OwingtothefactthattheRichardson-Lucydeprojectionalgorithmallowsdeprojecting
- each line of sight independently, this result may indicate that non-gravitational effects and/or asphericity are strong in the central
o regionsoftheclusters.
r Conclusions.We demonstrate the robustness of the potential reconstruction method based on the Richardson-Lucy deprojection
t
s algorithmandshowthatgravitationallensingandX-rayemissionleadtoconsistentgravitationalpotentials.Ourresultsillustratethe
a powerofcombininggalaxy-clusterobservablesinasingle,non-parametric,jointreconstructionofconsistentclusterpotentialsthat
[ canbeusedtolocallyconstrainthephysicalstateofthegas.

1 Keywords.galaxies:cluster:general-X-rays:galaxies:clusters-gravitationallensing:strong-gravitationallensing:weak
v
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1. Introduction vature of the projected gravitationalpotentialof these clusters.
0
3 Ourstudyisbasedontheideathatasimultaneouscombination
0 Thecolddarkmatter(CDM)modelfavoredbysubstantialtheo- ofasmanygalaxy-clusterobservablesaspossiblemayleadtoa
. retical and observational evidence (e.g., Komatsuetal. 2011) consistentmodelfortheprojectedclusterpotentialbecauseun-
1 predicts the formation of small structures at early times and derequilibriumassumptions,alltheseobservablescanbecom-
0
theirlatermergingintolargersystems.IntheCDMframework, binedonthegroundsofthegravitationalpotential.
5
galaxyclustersareexpectedtobethelatestgravitationallybound
1
systemstoform.
: Inadditiontoimprovingthereconstructionoftheprojected
v Galaxy clustersaremainlycomposedof threecomponents: cluster potential beyond gravitational lensing, combining as
i
X galaxies,hotgas,anddarkmatter.Thedarkmatterisbydefini- manyclusterobservablesaspossiblewillalsoallowustocover
tion impossible to observedirectly, but it is expected to follow a wider range of angular or radial scales. The joint analysis
r
a auniversaldensityprofileapproximatedbytheNavarro-Frenk- of weak and strong gravitational lensing allows us to map the
White profile (NFW, Navarroetal. 1997). However, observa- dark-matter distribution within the entire cluster. Furthermore,
tions of the intracluster medium (ICM) and of cluster galaxies because the emissivity depends on the square of the gas den-
canconstrainthedark-matterdistributionitself.AstheICMfills sity, X-ray flux and temperature profiles are most reliable in
thedeepgravitationalpotentials,its distributionisgovernedby and near the central region of the cluster (Ettori&Molendi
thedepthandshapeofthedark-matterpotentialwell(see,e.g., 2011), while the thermalSZ effecthasa shallower dependence
Ettorietal.2013,forareview),hencetheobservablesprovided on the local gas density because the thermal SZ signal is pro-
bythethermalplasmacarryinformationonthegravitationalpo- portional to the gas pressure integrated over the line of sight
tential. (PlanckCollaborationetal. 2013; Eckertetal. 2013). Hence,
ThephysicalpropertiesoftheICMcanbedirectlystudiedin owing to the different dependencesof differentobservable sig-
X-raysthroughthermalbremsstrahlungandlineemissionandat nals on angular scales, weak-lensing and thermal-SZ measure-
millimetrer wavelengths through the Sunyaev-Zel’dovich (SZ) mentsextendto relativelylarge distancesfrom the cluster cen-
effect.Ontheotherhand,measurementsofgravitationallensing ter, while strong-lensingand X-ray observationsare more sen-
inclustersconstrainthegravitationaltidalfieldandthusthecur- sitive to regions near the cluster center. In combination, more
1
C.Tcherninetal.:GravitationalpotentialofAbell1689fromX-raymeasurements
reliablereconstructionsofclusterpotentialscanbeexpectedon binedweak-lensingwithX-raydatafromtheChandrasatellite.
allscales. ConsideringthecentralpartoftheclusterouttoR∼1000kpc/h,
Anotheradvantageofthisjointanalysisis thatthe assump- theyfoundthatatradiilargerthanR∼200kpc/h,theX-rayand
tionsmadeonthephysicalstateofthegascanbeverified.The lensingmassesareconsistent,whereastheyfoundadiscrepancy
lensing potential can be recoveredwithout any equilibrium as- betweenthemassesatsmallerradii.Theseauthorsexplainedthis
sumptions,while the gravitationalpotentialreconstructedfrom discrepancybya projectioneffect.Thisconclusionagreeswith
X-ray and thermal-SZ observations is recovered assuming hy- theresultsofSerenoetal.(2013).Inthisstudy,theauthorscon-
drostaticequilibrium.Therefore,theprojectedgravitationalpo- strainedtheshapeandorientationoftheclusterinajointanalysis
tentialrecoveredfromgravitationallensingobservationscanbe oftheICMdistributioninthecluster.UsingX-rayandSZmea-
comparedwiththegravitationalpotentialrecoveredfromthehot surements,theclusterappearstohaveatriaxialshapeelongated
ICMobservations,andtheresultcanbeusedtodeterminewhich alongthelineofsight.Accordingtotheseauthors,thismayex-
clusterssatisfytheassumptionsandwhichdonot.Thiswillhelp plainthemassdiscrepancy.
ustodistinguishthedifferentprocessesoccurringinclusters.
Finally,in a joint analysis of X-ray surface brightnesswith
We here focus on reconstructing the projected potential
strong- and weak-lensing measurements, Morandietal. (2011)
of galaxy clusters using X-ray observations. However, other
inferred the fraction of nonthermal pressure required for the
observables may also be used, such as galaxy kinematics
clustertobeinhydrostaticequilibrium,takingintoaccountthe
(Sarlietal. 2013) or the thermal SZ effect (Majeretal. 2013).
triaxialshapeofthecluster.Theauthorsfoundthat20%ofthe
Our final goalis to set up one commonlikelihood functionfor
totalpressureshouldbenonthermal,mostprobablycontributed
thegravitationalpotential,joininginformationfromallavailable
byturbulentgasmotiondrivenbyhierarchicalmergerevents.
observables. As a proof of concept, in this study the projected
potentialofthewell-studiedclusterAbell1689isreconstructed To summarize, a mass discrepancy has been observed in
fromX-rayobservations. many studies, and different explanations have been suggested.
To understand the origin of this discrepancy, we here propose
the followingmethod:1) reconstructthe three-dimensionalgas
The cluster Abell 1689 is located at redshift 0.183 in the
distributionusingadeprojectionmethodthatoperatesindepen-
Virgo constellation. As a well-known strong-lensing cluster, it
dently on each line of sight: the Richardson-Lucydeprojection
has been the subject of many studies. The multiwavelength
algorithm (R-L, Lucy 1974, 1994); and 2) substitute the mass
observations performed on this object extend from the mid-
withalocalquantity,thatis,thegravitationalpotential.
infrared,forinstance,usingISOCAM onboardthe ISOsatellite
(Faddaetal.2000),totheX-rayband,forinstance,usingSuzaku Thisprocedure,appliedheretoajointanalysisofX-rayand
(Kawaharadaetal. 2010), ROSAT (Allen 1998), Chandra1 lensingobservations,isexpectedtoallowustodirectlycompare
(Xue&Wu 2002), and XMM-Newton (Andersson&Madejski the dark matter and gas distributions and to test the validity of
2004; Snowdenetal. 2008). In visible light, the optical signal theequilibriumassumptionsateachprojectedradius.Usingthe
hasbeenobservedusingtheSubaru/Suprime-Camtelescopeand gravitationalpotentialshouldrenderthecomparisonwithgravi-
theHubbleSpacetelescope(Broadhurstetal.2005).Thislistis tationallensingmeasurementmorestraightforwardbecausethe
notcomprehensive. gravitational potential is directly observable from the lensing
The main result of these observations is that Abell 1689 measurements.
is a dynamically active cluster: kinematics and X-ray stud-
To recover the cluster potential from X-ray measurements,
ies indicate a merger aligned with the line of sight (see,
we first deprojectthe X-raydata, then convertthe X-ray three-
e.g., Andersson&Madejski 2004, and references therein).
dimensionalprofileintothethree-dimensionalgravitationalpo-
Combinations of different observations in joint analyses have
tential using equilibrium assumptions, and project it again to
been performed by many authors and have shown that the hy-
compareitwiththetwo-dimensionalgravitationalpotentialob-
drostatic mass is lower than the mass estimated from gravita-
tained by gravitationallensing. Applyingthis methodwith dif-
tional lensing measurements in the central regions (see, e.g.,
ferentgeometries(sphericalandtriaxial)isexpectedtoremove
Pengetal.2009;Morandietal.2011;Miralda-Escude&Babul
the degeneracy between the different possible explanations for
1995;Łokasetal.2006;Serenoetal.2013).
the mass discrepancy. We test this algorithm here for the first
Miralda-Escude&Babul (1995) combined strong-lensing
timeonarealclusterassumingsphericalsymmetry.Thegener-
and X-ray surface brightness measurements to test the physi-
alizationtoaspheroidalshapeisongoing.
cal state ofthe gasnearthe cluster center. The authorsshowed
thattheobservedtemperatureinthecentralregionsislowerthan This paper is structured as follows: In Sect. 2 we list the
that expectedfrom hydrostaticequilibrium. As a result, a non- assumptions of the method. In Sect. 3 we apply the R-L de-
negligible nonthermal pressure component should be present projection algorithm on the cluster Abell 1689 and compare
andactagainstgravity.Accordingtotheseauthors,thisnonther- thiswiththeonion-peelingdeprojectionmethod(O-P,see,e.g.,
mal pressure support could be associated with the merger pro- Fabianetal. 1981; Krissetal. 1983, for a detailed overviewof
cess.Inacomplementaryanalysis,Łokasetal.(2006)measured this method), which is the standard deprojection method in X-
thevelocitydistributionofthegalaxiesinthefieldandidentified rayastronomy.InSect.3.1,O-PandR-Ldeprojectionmethods
severalmatterclumpsalignedwiththelineofsightthroughthe are directly compared for different cluster observables, and in
cluster, which do not interact with the cluster dynamicallyand Sect. 3.2 the advantage of the R-L method with respect to the
might affect the lensing mass estimates without modifying the O-P method is discussed. In Sect. 3.3, the reconstructed pro-
X-ray mass estimate. This provides an alternative explanation jected gravitational potential obtained using the method devel-
for the mass discrepancy observed between the two methods. opedinKonradetal.(2013)iscomparedtothelensingpotential
In a different analysis of Abell 1689, Pengetal. (2009) com- oftheclusterasrecoveredbyJ.Merten(privatecommunication).
Finally,theresultsarediscussedinSect.4,andweconcludein
1 seealsotheChandracatalogofCavagnoloetal.(2009). Sect.5.
2
C.Tcherninetal.:GravitationalpotentialofAbell1689fromX-raymeasurements
2. Assumptionsofthemethod in the 0.4−2 keV band. For the details of the analysis proce-
dure,werefertoEckertetal.(2011).
The general reconstruction formalism outlined in Konradetal.
(2013)isbasedonthefollowingassumptions: In this section, we aim at comparingthe results of the R-L
deprojectionmethodwiththoseobtainedwiththeO-Pmethod.
– TheICMisinhydrostaticequilibrium,
Forthispurpose,weextractedtheemissivity,thedensityprofile,
and the temperature profile from the X-ray surface brightness
∇P=−ρ∇Φ, (1)
usingbothmethods.
where Φ denotes the Newtonian gravitational potential,
whilePandρarethegaspressureanddensity.
– Thegasfollowsthepolytropicrelation 3.1.TestingtheR-Ldeprojectionmethod
P ρ γ
= , (2) The R-L algorithm is an iterative method that tends to depro-
P0 ρ0! jectthefeaturesoftheobservedprojectedfunctiononallscales.
To avoid reproducing the instrumental fluctuations and other
where the suffix 0 denotes fiducial values for the pressure
sources of noise in the deprojected profile, Lucy (Lucy 1994)
andforthegasdensityatanarbitraryradiusr andwhereγ
0 suggested to add a regularization term to a likelihood function
isthepolytropicindex.
whose scale andamplitudewere controlledbytwo parameters,
– Thegasisideal,
ρ the regularization parameter α and the smoothing scale L (see
P= kBT , (3) Konradetal.2013,formoredetails).Inthefollowing,weadapt
m¯
theseparameterstotheX-raydataobtainedbytraditionaltech-
whereT isthegastemperature,k isBoltzmann’sconstant,
B niques.
andm¯ isthemeanmassofagasparticle.
– ThebremsstrahlungemissionisassumedtodominatetheX-
rayobservations.Thisassumptionisjustifiedaslongasthe
3.1.1. Three-dimensionalemissivity
temperatureofthe plasma doesnotfall below∼ 2keVand
theFeXXV-FeXXVIlinecomplexat6−7keVisavoided
The mean value and the error bars of the recovered emissivity
(Peterson&Fabian 2006). The bolometric bremsstrahlung
wereobtainedusinga MonteCarlo (MC)methodbasedonthe
emissivity, j ,canbewrittenas
x R-L deprojection method as follows: The value of the surface
j =CT1/2ρ2 , (4) brightness profile was randomized within each radial ring, ac-
x
cordingtoaGaussiandistributionwhosemeanwastheobserved
whereCisanamplitudeformedbyallrelevantphysicalcon- valueandwhosewidthwasgivenbytheerrorbars.After10000
stants. We integrated the emissivity over the energy band simulations,thedeprojectedvalueanditserroraregivenbythe
[0.4−2]keV(j[0.4;2])insteadofovertheentireelectromag- meanand1σdeviationoftheresultingdistribution.
neticspectrum.Introducingfiniteboundariesfortheenergy
InFig.1,weshowtheresultsoftheX-raysurface-brightness
bandsmodifiesthe temperaturedependenceof the emissiv-
deprojection using the O-P and R-L methods. To obtain these
ity,butnotitsdependenceondensity.Sincethedensitypro-
results,theparametersoftheR-Lmethodweresettoα = 0.01
fileissteeperthanthetemperatureprofile,andtheemissivity
and L = 300kpc. These parameter values are lower than the
dependsonthesquareofthedensity,theemissivityprofileis
expectations(seeKonradetal.2013,formoredetails)andimply
dominatedbythedensityprofileanduncertaintiesinthetem-
thattheregulationtermintroducedintheR-Lmethodisnotvery
peraturedependencecanbeconsiderednegligiblefornow.
effectiveinthecaseofAbell1689.
– Except for the natural constants and the averaged parame-
ters,allquantitiesintroducedhereareassumedtodependon More detailed investigations with simulated clusters have
thecluster-centricradius.Thisdependenceisnotwrittenex- shown that the results of the reconstruction depend very little
plicitlytosimplifythenotation.Theeffectiveadiabaticindex on the values chosen for the regularizationparameters.Testing
is assumedto be constantacross the cluster. While this has wide ranges for both α and L, typically covering an order of
been shown not to be true in general, it can be a good ap- magnitude,changestherecoveredpotentialsroutinelybymuch
proximationaslongasthecoreoftheclusterisavoided(see, lessthanthestatisticaluncertainties.
e.g,.Tozzi&Norman2001;Capeloetal.2012;Eckertetal.
We emphasize that the regularization term is controlled by
2013,andthediscussionsection).
the difference of the potential reconstructions between subse-
– Theclusterisassumedtobesphericallysymmetric.
quentiterationssteps,whichvanishesprogressivelyastheitera-
tionproceeds.Theregularizationisthusgraduallyswitchedoff
3. DeprojectionoftheclusterAbell1689 towardtheendoftheiteration.Wefixedthenumberofiterations
to10,whichcorrespondstothenumberofiterationsatwhichthe
To construct the surface-brightness profile of the cluster,
algorithmconverges,becausenosignificantchangesintheemis-
we used an archival 13.4 ks observation of Abell 1689
sivityprofilewereobservedwhenmoreiterationswereused.
with ROSAT/PSPC. This choice was motivated by the low
background of this instrument, which allowed us to ex- This is endorsed by our result, which shows that the emis-
tract high-quality surface-brightness profiles out to large radii sivity profiles obtained with the two methods agree very well
(Vikhlininetal. 1999; Eckertetal. 2011). We reduced the data withintheerrorbars.However,theoutermostpointsofthepro-
using the ESAS software (Snowdenetal. 1994). The non-X- filearedifferentinthetwotechniques,whichcanbeattributedto
raybackgroundwasmodeledandsubtractedfromtheobserved edge effects (see, e.g., McLaughlin1999). The large error bars
image, and a vignetting correction was applied. The surface- intheoutermostpointsarecausedbytheweaknessofthesignal
brightnessprofile was then extracted from the correctedimage atlargeradii.
3
C.Tcherninetal.:GravitationalpotentialofAbell1689fromX-raymeasurements
1e-07 0.1
O-P Method O-P Method
1e-08
R-L method R-L method
3] 1e-09 0.01
-c
1 kp 1e-10 -3m]
-sivity [cts s 11ee--1121 density [ce 0.001
s n
mi 0.0001
e 1e-13
1e-14
1e-05
1e-15
102 103 102 103
R[kpc] R[kpc]
Fig.1. Emissivity obtained by applying the R-L (this analysis) Fig.2. Electron-density reconstructed profile obtained from
andtheO-Pmethods(Eckertetal.2012)totheROSATsurface- the R-L deprojection method (this analysis) compared to the
brightness profile. The reconstructed emissivity using the R-L electron-density profile recovered from the O-P deprojection
method has been normalized to the O-P data points within a method (Eckertetal. 2012). For both deprojection methods,
spherical shell delimited by the radii in the range [183; 2700] Eq. [5] has been used. The reconstructed density using the R-
kpc. L method has been normalizedto the O-P data points within a
spherical shell delimited by the radii in the range [183; 2515]
kpc.
3.1.2. Three-dimensionaldensityprofile
The temperature of the cluster Abell 1689 lies between 2keV 3.2.AdvantagesoftheR-LmethodcomparedtotheO-P
and 10keV (Kawaharadaetal. 2010). This means that the soft method
energyband[0.4;2]keVisbelowthebremsstrahlungcutoffen-
In observations, the only information available is integrated
ergy, and the emissivity only weakly depends on the tempera-
alongthelineofsight.Intheframeworkofthisstudy,wedefine
ture.Inthisparticularcase,thedensityprofilecanbeexpressed
aslocaladeprojectionmethodthatallowsindependentlydepro-
asafunctionoftheemissivity, j ,as
[0.4;2]
jectingthisprojectedinformationforeachlineofsight,meaning
thatitisnotnecessarytoknowthevalueofthedeprojectedquan-
ρ(r)∝ j[0.4;2](r). (5) tityinonelineofsighttoderivethedeprojectedvalueinanother
p andthatregionsofdifferentprojectedradiusarenotmixeddur-
InFig.2wecomparethedensityprofileobtainedusingEq.[5] ingthedeprojectionprocedure.
for the two deprojection methods: the O-P method (in blue, InSect.3.1,wesawthattheR-LandO-Pdeprojectionmeth-
Eckertetal. 2012) and the R-L method (in red, this analysis). ods lead to consistent results. We now explain why we aim at
Again,thedensityprofileiswellreproducedusingtheR-Lpro- usingtheR-LmethodinsteadoftheO-Pprocedure.
cedure. Atthispointoftheanalysis,itisnecessarytobrieflyreview
the concept of the O-P method. We refer to Krissetal. (1983)
formoredetails.TheO-Pmethodisageometricaldeprojection
3.1.3. Temperatureprofile
techniqueinitiallyintroducedusingtheassumptionofspherical
Usingthepolytropicrelation(Eq.[2])andthefactthatwecon- symmetry(Krissetal.1983),butitcanbegeneralizedtoclusters
sidered only the soft energy band [0.4;2] keV, the temperature withaxialsymmetry(see,e.g.,Buote&Humphrey2012a).This
profilecanbeexpressedas method assumes that the cluster has an onion-like shell struc-
ture,withauniformemissivityineachoftheconcentricspheri-
calshells.Thethree-dimensionalprofileoftheemissivityisthen
T(r)∝ρ(r)γ−1 ∝ j (r)(γ−1)/2 . (6)
[0.4;2] recovered from the surface brightness map in an iterative way
fromringtoring,progressingfromtheoutskirtstothecenterof
Figure 3 shows the temperature profile reconstructed with a thecluster.
polytropic index of γ = 1.19±0.04 and the temperature pro- Tocarryoutthisiterativeprocedure,theO-Pmethodrequires
files obtained using different techniques. The polytropic index the errors to propagate from ring to ring as the observed pro-
γ=1.19±0.04resultsfromfittingtheobservedpressureprofile jectedprofileisgenerated.Whenaquantityisoverestimatedina
(PlanckCollaborationetal.2013)withthereconstructeddensity radialbin,thesamequantityneedstobeunderestimatedinanad-
profile(obtainedinSect.3.1.2),usingEq.[2]. jacentbin,causingthemethodtoproducefluctuationsinthede-
For completeness, we also fitted Eq. [6] to the joint XMM- projectedprofile(see,e.g.,Ameglioetal.2007).Thus,theerrors
Newton and Suzaku data and obtained γ = 1.17±0.03 for the associatedwitheachbinarecorrelated.Thestatisticaluncertain-
polytropicindexwith1σerror,whichisconsistentwithourear- tiesassociatedwiththereconstructedthree-dimensionalprofiles
lierresult. are usuallyestimated using MonteCarlo simulations(see, e.g.,
Thedecreaseinthetemperatureprofilesupportsthevalidity Buote2000),whichbythesamemeanssmoothesthesystematic
ofthepolytropicassumption(Eq.[2]). errorsduetothefluctuationsinadjacentbins.
4
C.Tcherninetal.:GravitationalpotentialofAbell1689fromX-raymeasurements
bodycanbetakenintoaccountbyasuitable(de)projectionker-
18
nel in the R-L method. As the mixing between regions of dif-
16 R-L method for γ=1.19 +/- 0.04 ferent projected radius is a consequence of the symmetries as-
R-L method for γ=1.19
14 sumptionandnotofthedeprojectionprocedure,testingdifferent
SZ effect + O-P Method clustersymmetriesshouldhelpusunderstandtheeffectofthese
eV] 12 Suzaku mixingsontheresultingprofiles.
e [k 10 XMM This feature of local deprojection is most relevant for our
atur study since the information provided by gravitational lensing
er 8 measurements that we use here is also local. In the SaWLens
p
m
e 6 methoddevelopedbyMertenetal.(2009),estimatesofthelens-
T
ing potentialare obtainedlocally in all cells of a gridcovering
4
the region considered on the sky, and therefore, the observa-
2 tionsofweakgravitationallensinggiveinformationonthelocal
galaxydensity.
0
102 103
R[kpc]
3.3.Reconstructingthelensingpotential
Fig.3.TemperatureprofileobtainedfromtheR-Lreconstructed
The reconstructed projected potential ψ is the projection along
emissivityprofileviaEq.[6]withγ=1.19±0.04,comparedto
the line of sight of the gravitationalpotentialφ. In our formal-
temperature profiles obtained using different techniques. Dark
ism,thegravitationalpotentialisexpressedintermsoftherecon-
blue:profilederivedfromcombinedX-rayandthermal-SZmea-
structedemissivityby(seeKonradetal.2013,forthederivation)
surements(PlanckCollaborationetal.2013)usingtheidealgas
assumption (Eckertetal. 2013); light blue: spectral fitting of 2(γ−1)
XMM observations (Snowdenetal. 2008); pink: spectral fit- φ(r)∝ jx(r)η , η= 3+γ . (7)
tingofSuzakuobservations(Kawaharadaetal.2010);red:mean
value and errors on the reconstructed profile for γ = 1.19; Thethree-dimensionalgravitationalpotentialreconstructedthis
gray: mean value and errors on the reconstructed profile for wayisthensimplyprojectedalongthelineofsight.Inthisequa-
γ=1.19±0.04.Themeanandtheerrorbarswereobtainedwith tion,weusethebolometricemissivity(j )andnottheemissivity
x
a Monte Carlo method based on the R-L deprojection.The re- restrictedtotheenergyband[0.4;2]keV.Torecoverthebolomet-
constructedtemperatureprofilehasbeennormalizedtothedata ricemissivityfromtheenergy-restrictedemissivity(seeFig.1),
points from Eckertetal. (2013) within a spherical shell delim- weusedthetemperatureprofile(seeFig.3)andthethin-plasma
itedbytheradiiintherange[183;2515]kpc. code APEC (Smithetal. 2001, fixing the metallicity to 0.3) to
computetheratiobetweenthebolometricfluxandtherestricted
flux for each radius. This factor was then used to recover the
In the R-L method, the error propagation and the fluctua- bolometricemissivityfromtherestrictedemissivity.
tionsduetothe deprojectionofthe instrumentalnoiseare con- The lensing potentialis recoveredfrom observationsof the
trolledbyaregularizationterm(Lucy1994).Thechoiceofthis distortedimagesofdistantbackgroundgalaxies.Thedistortion
regularization term is adapted to the data themselves and van- (shear and magnification) of these images is related to linear
ishes as the deprojection converges. Furthermore, simulations combinations of second derivatives of the lensing potential of
show that large-scale features are reproduced quickly, while a foregroundcluster. This distortion is due to the gravitational
small-scale structures are recovered slowly (Reblinsky 2000; tidalfieldofadeeppotentialwellbetweentheobserverandthe
Puchwein&Bartelmann 2006). Therefore, the R-L method al- backgroundgalaxies.Inthecaseofweak-lensingmeasurements,
lowscontrollingthefluctuationsthatarisefromdeprojectingthe thedistortionofindividualgalaxiesistooweaktobeobserved,
instrumentalnoise(Lucy1994). but the distortion averaged over sufficiently many neighboring
Animportantdifferencebetweenthetwomethodsisthatthe galaxies is used to estimate the gravitational tidal field of the
O-Pmethodisnonlocalinthesenseintroducedabove:torecon- foregroundgalaxycluster.Inalargeenoughsample,therandom
structthethree-dimensionalshellatagivenradius,theresultsof intrinsic orientation of the background galaxies is expected to
the deprojectionof all shells at larger radiiare needed, and re- averageout,and the ellipticity inducedby lensing canbe mea-
gionscorrespondingto differentprojectedradiiare mixed.The sured.Thelensingpotentialdatausedherewererecoveredwith
R-Ldeprojectionmethod,however,islocalinthesensethatthe SaWLensinanonparametricwayfromweak-lensingdata.
two-dimensionalprofileisindependentlydeprojectedalongeach In Fig. 4, we show the reconstructed, two-dimensional po-
lineofsight(see,e.g.,Reblinsky2000;Puchwein&Bartelmann tentialassumingapolytropicindexγ=1.19±0.04.Thisprofile
2006),suchthatthedeprojectedfunctionisindependentlyeval- iscomparedwiththelensingpotentialobtainedfromweaklens-
uatedateachimagepointandthatthisalgorithmcanbeapplied ing.Atradii&500kpcourreconstructedpotentialprofileagrees
to data sets with incomplete coverage. However, the relations wellwiththereconstructedlensingprofilewithintheerrorbars,
between different projected radii are controlled by the symme- eventhoughthepotentialreconstructedfromtheX-rayemission
tryassumptions,andtheinformationcontainedindifferentlines appearstobeslightlylesscurved.Thediscrepancyatsmallradii
ofsightmaybemixed,evenifthedeprojectionprocedureislo- andtheslightcurvaturechangeatlargerradiimaybecausedby
cal.Forinstance,assumingsphericalsymmetry,theinformation the lackof resolutionin the weak-lensingmeasurementandby
collectedatsameprojectedradiusismixedforanylineofsight. theotherassumptionsmade(sphericity,polytropicandequilib-
Nevertheless,asfortheO-Pmethod,itispossibletogeneral- riumassumptions).Thesepointsarediscussedinmoredetailin
izetheR-Ldeprojectionmethodtointrinsicallyspheroiralclus- thenextsection.
ter bodies (see, e.g, Reblinsky 2000; Puchwein&Bartelmann As we have mentionedin the introduction,studyingthe in-
2006). In this formalism, the intrinsic shape of the projected tracluster gas may help us understandthe dark-matterdistribu-
5
C.Tcherninetal.:GravitationalpotentialofAbell1689fromX-raymeasurements
lensingis flatter thanthe one reconstructedfromthe X-rayob-
servations.Thisflatteningcanbeproducedbythelargeresolu-
reconstructed for γ=1.19 +/- 0.04
tionlimitpresentintheweak-lensingmeasurements(∼100kpc,
recovered from lensing see,e.g., Bartelmann2003,forareview),whichcansmooththe
0.01
data so that the lensing potential recovered from weak-lensing
al measurementsappearsflatter than the potentialrecoveredfrom
nti
e X-ray observations. This discrepancy is expected to vanish if
ot
P morepotentialobservablesaretakenintoaccount.Forinstance,
g
sin acombinedstrong-andweak-lensingreconstructionisexpected
n
e to yield a better match in the region close to the cluster center
L
becausethestronglensingissensitivetothisregion.
At small radii (R . 500kpc), however, the two pro-
0.001 jected potentials differ. Such a discrepancy in the cen-
102 103
tral region of the cluster was also found by Pengetal.
s[kpc]
(2009), Andersson&Madejski (2004), Morandietal. (2011),
Fig.4. Projected gravitationalpotential reconstructedusing the Miralda-Escude&Babul (1995), Łokasetal. (2006), and
method described in Konradetal. (2013) compared to the po- Serenoetal. (2013), as mentioned in the introduction. This
tential recoveredfrom weak gravitationallensing as a function could be explained by a modification of the balance between
of the projectedradius s. The projectedpotentialreconstructed the physical processes in the inner part of the cluster. It may
from the X-ray emission has been obtained assuming a poly- indicatethatsomeoftheassumptionsmadeshouldnotbeused
tropicindexγ=1.19±0.04(seethetextfordetails).Thepoten- inthecentralpartofthecluster.
tial recovered by lensing is shown in blue; the mean projected Forinstance,thepolytropicassumptionwithconstantpoly-
potentialobtainedfromX-rayobservationswithγ=1.19±0.04 tropic index may not be valid in the central part of cool-core
isshowninred,theuncertaintiesofthelatterareshowningray. clustersbecausetheradiativecoolingnearthecenterrendersthe
The mean and errors have been obtained using a Monte Carlo coolingtime shorterthanthe Hubbletime (e.g., Fabian1994).
method to randomize the algorithm described in Konradetal. Coolingaltersthepolytropicequationbyloweringthetempera-
(2013).Thereconstructedlensingpotentialhasbeennormalized tureof the gas,which impliesthatthe temperatureprofiledoes
tothelensingdatawithinacircularshellregiondelimitedbythe not increase toward the center as the gas density does, but in-
radiiintherange[92;1734]kpc. stead decreases. Therefore, the assumption of a constant poly-
tropic index is not expected to be correct at the cluster cen-
tre.However,theassumptionthatthegasfollowsthepolytropic
tion in the cluster becausethe ICM fills the dark-matterpoten- equationseemstobevalidinAbell1689,withafittedpolytropic
tialwell.AsaresultofthelocalityoftheR-Ldeprojection,our indexofγ=1.19±0.04(Fig.3).
methodopensawayforclusterpotentialstocomplementthein-
Ontheotherhand,theassumptionofhydrostaticequilibrium
formation provided by lensing and joins two entirely different
islikelyvalidinthecentralregionsofcool-coreclustersbecause
observablesonthegroundsofthegravitationalpotential,which
of the deep gravitational potential (see, for instance, Fig. 2 of
providesanewtoolforconstrainingthephysicalstateofthegas
Lauetal.2009).Thismaynotbethecaseinnon-cool-coreclus-
locally.
ters.AshasbeenshownbyMahdavietal.(2013),forinstance,
departure from the equilibrium assumption is expected in un-
relaxed clusters. The authors compared the lensing mass with
4. Discussion
themassobtainedunderhydrostaticequilibriumassumptionsfor
We used the R-L deprojection method to recover the three- a sample of clusters at R . R500 (∼ 1.34Mpc in the case of
dimensional gas distribution in Abell 1689 from its X-ray sur- Abell1689)andconcludedthatthedeparturefromtheequilib-
face brightness. We then converted this gas distribution into a riumassumptionisdifferentbetweenrelaxedandmergingclus-
three-dimensional gravitational potential that we subsequently ters:whileforcool-coreclusters,almostnobiasisobservedbe-
projected for comparison with the lensing potential. Using the tween the lensing and the hydrostatic mass, a bias of 15-20%
R-L deprojectionmethodis advantageousbecauseit allowsin- is present for non-cool-core clusters. Abell 1689 is known to
dependentlydeprojectingtheX-rayobservablesalongeachline experience a merging event aligned with the line of sight, and
of sight as well as treating data sets with incompletecoverage. therefore, the gas is not expected to be in hydrostatic equilib-
This therefore allows excluding regions where the equilibrium rium near the cluster center (see also, e.g., Nelsonetal. 2012;
assumptionsmaynothold.As a result,a substantialamountof Morandietal.2011).Thismaycontributetothediscrepancyat
informationbecomesavailable by the direct comparisonof the smallradiiobservedinFig.4.
lensingpotentialwiththeprojectedpotentialreconstructedfrom Furthermore,anydeparturefromsphericalsymmetrycould
X-rayemission(Fig.4). biastheestimateoftheclustermass,andifwherethemajoraxis
For example, the agreementbetween the two reconstructed of the cluster is preferentially aligned with the line of sight, it
potentialsatradiifrom500−2000 kpcindicatesthattheassump- shouldleadtoanoverestimationofthelensingmasswithrespect
tions underlying the derivation of the projected potential from to the X-ray mass. This is consistent with the conclusions of
X-rayobservations,namely hydrostaticequilibrium,polytropic Lee&Suto(2003,2004),whoshowedthatthedarkmatterpro-
stratification, and the ideal-gas equation of state [Eqs. (1)-(4)] fileinnonsphericalclustersisalwaysmoresensitivetotriaxiality
arenotsubstantiallyviolatedintheclusterAbell1689withinthis than the gas distribution from X-ray measurements. This weak
radialrange.Nevertheless,we canobservea slightdiscrepancy dependenceoftheX-raymeasurementsontheclustershapewas
between the potentials curvature: the potential recovered from also observed in Buote&Humphrey (2012a,b), where the au-
6
C.Tcherninetal.:GravitationalpotentialofAbell1689fromX-raymeasurements
thorscomputedthebiasesintheobservablesobtainedbyapply- withtheprofilesfromRichardson-Lucydeprojection.Figures1-
ingasphericalO-Pdeprojectionmethodonclusterswithsignif- 3showthebremsstrahlungemissivity,electrondensity,andtem-
icanttriaxiality.Theyconcludedthatsphericalaveragingbiases perature profiles obtained using the two deprojection methods.
theobservablesbyasmallfactor(<1%)withrespecttoatriax- Allprofilesareconsistentwithintheerrorbars.
ialdeprojection. Figure 4 shows the projected gravitational potential recon-
Therefore,thenonsphericityoftheclustercouldexplainthe structed from X-ray data following the procedure described in
discrepancy at small radii between the X-ray and lensing pro- Konradetal.(2013).Thisfigureshowsthattheprojectedpoten-
jectedpotentialprofilesforAbell1689,butinterestingly,itdoes tial obtained directly from X-ray observations agrees with the
notseemtostronglyaffectthereconstructionofthepotentialin lensingpotentialobtainedfromweak-lensingmeasurements.
theregion500−2000 kpc. Therefore,for sufficiently precise data sets, the R-L depro-
jection method is expected to help us constrain the validity of
the assumed equilibriumassumptionsatall projectedradii. We
At large radii (R & R ), simulations tend to show that the
500 obtained these results using a spherical symmetry assumption.
equilibrium assumptionsare not valid any more due to mixing
However,apossiblebiascomingfromthisassumptioncanalso
oftheICMwiththeinfallingmaterialfromthelarge-scalestruc-
contributetothediscrepancybetweentheX-rayandlensingpo-
ture(see, e.g,Reiprichetal. 2013,fora review).Thedeviation
tentials at small radii (Fig. 4). To remove the degeneracy be-
fromthehydrostaticequilibriumcanbeduetoresidualgasmo-
tweenthedifferentexplanations(breakdownofhydrostaticequi-
tions or incomplete thermalization of the ICM (Mahdavietal.
libriumorprojectioneffects),wearenowworkingongeneraliz-
2013).Suchphenomenaareexpectedintheoutskirtsofclusters,
ingthisdeprojectionmethodusingatriaxialkernel.
where matter is accreted and substructures such as clumps are
more likely to be found(Nagai&Lau 2011; Vazzaetal. 2013;
Zhuravlevaetal.2013),sothatourreconstructionmethodisnot Weprovidedageneralproofofconceptofpotentialreconstruc-
expected to be valid at radii larger than R (∼ 1.339Mpc, for tion combining X-ray data with weak-lensing measurements.
500
Abell 1689). However, as the lensing data used in our study The next step now consists of combining all observables pro-
do not extend to much larger radii than R , we cannot test videdbygalaxyclusters:strongandweakgravitationallensing,
500
thisstatement.Whenweextrapolatedthelensingdatapointsto X-ray surface-brightness and temperature, galaxy kinematics,
larger radii, we did not find significant differences between X- andthethermalSunyaev-Zel’dovicheffectintoajointconstruc-
rayandlensinginformationinthisradialrange. tionofoneconsistentmodelforthe projectedclusterpotential.
Thiscombinationwillprovideasubstantialamountofinforma-
tionandwillhelpustoquantifythevalidityoftheassumptions
Some systematic effects arise from the different assumptions
made. If the assumptions are valid, the combination of all ob-
we made in treating the X-ray observations. In Eq. [4], we as-
servables will produce the best constrained galaxy-cluster po-
sumedthattheemissivityisdominatedbybremsstrahlungemis-
tentials.
sion.Thisapproximationrepresentsa differenceofatmost8%
in the normalization compared to the normalization we would
Acknowledgements. WearegratefultoJulianMertenforhismanyvaluablein-
obtaintakingintoaccountthecontributionoftheemissionlines sightsandforthelensingdataheprovidedforthisanalysis,andtoSaraKonrad
for a metallicity of 0.3. Then, the conversion from 0.4-2 keV for her contribution in developing this reconstruction method. This project
tothebolometricemissivity,computedassumingatemperature was supported in part by the Baden-Wu¨rttemberg Foundation under project
“InternationaleSpitzenforschungII/2”andbytheCollaborativeResearchArea
profile that follows Eq. [6] with γ = 1.19, implies systematic
TR-33 “The Dark Universe” as well as project BA 1369/17 of the Deutsche
errorsofabout4%atmost.Thedeviationsfromaflatfieldwith ForschungsgemeinschaftandinpartbytheSwissNationalScienceFoundation.
the ROSAT satellite have been computed to introduce system-
aticeffectsontheorderof6%ofthebackgroundinEckertetal.
(2012).Thisvaluecorrespondstothescatterintheresidual,re- References
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