Discrepant Mass Estimates in the Cluster of Galaxies Abell 1689
AAccepted to ApJ: June 20, 2009
Preprint typeset using L A TEX style emulateapj v. 04/20/08
DISCREPANT MASS ESTIMATES IN THE CLUSTER OF GALAXIES ABELL 1689
E.-H. Peng , K. Andersson , M. W. Bautz , G. P. Garmire Accepted to ApJ:
June 20, 2009
ABSTRACTWe present a new mass estimate of a well-studied gravitational lensing cluster, Abell 1689, fromdeep
Chandra observations with a total exposure of 200 ks. Within r = 200 h − kpc, the X-ray massestimate is systematically lower than that of lensing by 30-50%. At r > h − kpc, the mass densityprofiles from X-ray and weak lensing methods give consistent results. The most recent weak lensingwork suggest a steeper profile than what is found from the X-ray analysis, while still in agreementwith the mass at large radii. Fitting the total mass profile to a Navarro-Frenk-White model, wefind M = (1 . +0 . − . ) × h − M (cid:12) with a concentration, c = 5 . +1 . − . , using non-parametricmass modeling. With parametric profile modeling we find M = (0 . +0 . − . ) × h − M (cid:12) and c = 6 . +0 . − . . This is much lower compared to masses deduced from the combined strong andweak lensing analysis. Previous studies have suggested that cooler small-scale structures can biasX-ray temperature measurements or that the northern part of the cluster is disturbed. We find thesescenarios unlikely to resolve the central mass discrepancy since the former requires 70-90% of the spaceto be occupied by these cool structures and excluding the northern substructure does not significantlyaffect the total mass profiles. A more plausible explanation is a projection effect. Assuming that thegas temperature and density profiles have a prolate symmetry, we can bring the X-ray mass estimateinto a closer agreement with that of lensing. We also find that the previously reported high hard-band to broad-band temperature ratio in A1689, and many other clusters observed with Chandra ,may be resulting from the instrumental absorption that decreases 10-15% of the effective area at ∼ Subject headings: galaxies: clusters: individual: Abell 1689 — X-rays: galaxies: clusters INTRODUCTION
Abell 1689 is a massive galaxy cluster with the largestknown Einstein radius to date, θ E = 45 (cid:48)(cid:48) for z s = 1(e.g., Tyson et al. 1990; Miralda-Escude & Babul 1995;Broadhurst et al. 2005a,b), located at a moderately lowredshift of z = 0 .
187 (Frye et al. 2007). It has a regularX-ray morphology, indicating that the cluster is likely inhydrostatic equilibrium, but the mass derived from theX-ray measurement is often a factor of 2 or more lowerthan that from gravitational lensing at most radii. Us-ing
XMM-Newton observations, Andersson & Madejski(2004, A04 hereafter) find an asymmetric temperaturedistribution and a high redshift structure in A1689, pro-viding evidence for an ongoing merger in this cluster.Saha et al. (2007) confirm the existance of substruc-tures, using different sets of lensed images. This is alsoseen in other lensing work (e.g., Broadhurst et al. 2005a;Diego et al. 2005; Zekser et al. 2006; Halkola et al. 2006;Limousin et al. 2007). Though these clumps are clearlyidentified, they only contribute (cid:39)
7% of the total masswithin 250 h − kpc and are likely to be line-of-sight fila-ments rather than distinct merging groups. Furthermore,(cid:32)Lokas et al. (2006) used the redshift distribution of galax-ies to conclude that A1689 is probably surrounded by afew structures superposed along the line of sight thatdo not interact with the cluster dynamically, but would Electronic address: [email protected] MKI, Massachusetts Institute of Technology, Cambridge, MA02139, USA Department of Astronomy and Astrophysics, PennsylvaniaState University, PA 16802, USA affect lensing mass estimates.A recent joint
Chandra , HST/ACS, and Sub-aru/Suprime cam analysis by Lemze et al. (2008a,L08 hereafter) suggested that the temperature of A1689could be as high as T = 18 keV at 100 h − kpc, almosttwice as large as the observed value at that radius. Thederived 3D temperature profile was based on the X-raysurface brightness, the lensing shear, and the assumptionof hydrostatic equilibrium. From the disagreement be-tween the observed X-ray temperature and the deducedone, L08 concluded that denser, colder, and more lumi-nous small-scale structures could bias the X-ray temper-ature.In another study of 192 clusters of galaxies from the Chandra archive, Cavagnolo et al. (2008) find a veryhigh hard-band (2/(1+ z )-7 keV) to broad-band (0.7-7keV) temperature ratio for A1689, 1 . +0 . − . comparedto 1 . ± .
10 for the whole sample. They also find thatmerging clusters tend to have a higher temperature ra-tio, as predicted by Mathiesen & Evrard (2001) wherethis high ratio is attributed to accreting cool subclus-ters lowering the broad-band temperature by contribut-ing large amounts of line emission in the soft band. Thehard-band temperature, however, should be unaltered bythis emission. The simulations of Mathiesen & Evrard(2001) show an increase of temperature ratios of ∼ Chandra data (Riemer-Sørensen et al. 2009) claim that the cluster harbors a coolcore and thus is relaxed based on a hardness-ratio map a r X i v : . [ a s t r o - ph . C O ] J un analysis. They further calculate a mass profile from theX-ray data and conclude that the X-ray and lensing mea-surements are in good agreement when the substructureto the NE is excluded.In this work, we examine the possibility of an extraspectral component in the X-ray data and derive an im-proved gravitational mass profile, including a recent 150ks Chandra observation. § §
3, we explore the physi-cal properties of the potential cool substructures under atwo-temperature (2T) model and examine if they can beused to explain the high hard-band to broad-band tem-perature ratio. In §
4, assuming that the temperatureprofile derived by L08 is real, we investigate what thisimplies for the required additional cool component. In §
5, we derive the mass profile under both one and twotemperature-phase assumptions, using both parametricand non-parametric methods. Finally, we discuss our re-sults in § § H = 100 h − kms − Mpc − , Ω m = 0 .
3, and Ω λ = 0 .
7, which gives 1 (cid:48)(cid:48) =2.19 h − kpc at the cluster redshift of 0.187 (Frye et al.2007). Abundances are relative to the photospheric solarabundances of Anders & Grevesse (1989). All errors are1 σ unless otherwise stated. DATA REDUCTION
ChandraChandra data were processed through CIAO 4.0.1 withCALDB 3.4.3. Since all of the observations had gonethrough Repro III in the archive, reprocessing data wasnot needed. Updated charge-transfer inefficiency andtime-dependent gain corrections had already been ap-plied. For data taken in VFAINT telemetry mode, addi-tional screening to reject particle background was used.Events with bad CCD columns and bad grades were re-moved. Lightcurves were extracted from four I-chipswith cluster core and point sources masked in the 0.3-12 keV band and filtered by lc clean which used 3 σ clipping and a cut at 20% above the mean. Finally, make readout bg were used to generated Out-of-Timeevent file. These events were multiplied by 1.3% and sub-tracted from the images or the spectra to correct read-out artifacts. For spectral anlysis, emission-weighted re-sponse matrices and effective area files were constructedfor each spectral region by mkacisrmf and mkwarf . Background Subtraction and Modeling
Blank-field data sets were used to estimate the back-ground level. After reprojecting the blank-sky datasets onto the cluster’s sky position, the background wasscaled by the count rate ratio between the data and theblank-field background in the 9.5-12 keV band to accountfor the variation of particle induced background. Below1 keV, the spatial varying galactic ISM emission (Marke-vitch et al. 2003) could cause a mismatch between thereal background and the blank-field data. By analysingthe spectra in the same field but sufficiently far from thecluster, tailoring this soft component can be made using http://asc.harvard.edu/cal/Acis/Cal_prods/bkgrnd/acisbg . no r m a li z ed c oun t s s − k e V − ObsID 6930+7289 ObsID 5004 ObsID 540+16631 2 5−0.100.1 r e s i dua l s Energy (keV)
Fig. 1.—
The 0.6-9.5 keV Chandra spectrum of A1689 from thecentral 3 (cid:48) region. The upper panel shows the data, plotted againstan absorbed VAPEC model (solid line) with each element’s abun-dance and absorption column density as free parameters. The lowerpanel shows residuals. an unabsorbed T ∼ . r > (cid:48) .The background normalization factors used for eachobservation are listed in Table 1 XMM-Newton
The data from two MOS detectors were processed withthe XMMSAS 6.1.0 tool, emchain . Background flareswere removed by a double-filtering method (Nevalainenet al. 2005) from
E >
10 keV and 1-5 keV light curves.Only events with pixel PATTERNs 0-12 were selected.Since
XMM data were only used to crosscheck the re-sult of the multi-component analysis of
Chandra spec-tra, extracted from the central region where backgroundmodeling is relatively unimportant, we used the simplerlocal background, taken from 6 (cid:48) -8 (cid:48) . Spectral responsefiles were created by rmfgen and arfgen . We did notinclude PN data because the measured mean redshift,0 . ± . XMM
MOS or
Chandra data. This could indicate a possiblegain offset for PN detector, although A04 did not findany evidence for that.
Systmatic Uncertainies
L08 pointed out some issues about previous
Chandra observations (ObsID 540, 1663, and 5004). The columndensity from
Chandra data is much lower than the Galac-tic value, 1 . × cm − (Dickey & Lockman 1990),iscrepant Mass in A1689 3 TABLE 1
Chandra
Observation Log
ObsID Data Obs. Date Exposure Background NormalizationMode (ks) I0 I1 I2 I3540 FAINT 2000-04-15 10.3 1.06 1.09 1.03 1.111663 FAINT 2001-01-07 10.7 1.00 0.98 0.99 1.045004 VFAINT 2004-02-28 19.9 0.93 0.89 0.89 0.946930 VFAINT 2006-03-06 75.9 1.21 1.18 1.18 1.287289 VFAINT 2006-03-09 74.6 1.19 1.20 1.19 1.27 −20−10010 ∆ χ −10010 ∆ χ ObsID 6930+7289ObsID 5004ObsID 540+1663 ∆ χ Energy (keV)
Fig. 2.—
Fit residuals, showing each channel’s contribution to thetotal χ . Top : an absorbed VAPEC model fit to central 3 (cid:48) spec-trum.
Middle : same as the above, ignoring data in 1.7-2 keV.
Bottom : adding an absorption edge with E thresh =1.77 keV and τ =0.12. which is also supported by the ROSAT data (Andersson& Madejski 2004). The temperature difference can be ashigh as 1.3 keV depending on the choice of column den-sity. In the high energy band, the data is systematicallyhigher than the model prediction. With two long Chan-dra observations, ObsID 6930 and 7289, we clearly seean unusual feature in the datasets which may give cluesto problems mentioned before. Fig. 1 shows an absorbedAPEC model (Smith et al. 2001) fit to the central 3 (cid:48) spec-trum. The prominent residual at ∼ χ (See Fig. 2). This residual cannot be eliminated by adjusting individual abundances inthe cluster or in the absorbing column (the cluster is athigh galactic latitude). Because the residual around 1.75keV is an order of magnitude larger than the background,it is not likely related to the background subtraction. Inaddition to this absorption, the residuals are systemat-ically rising with the energy from negative to positivevalues. This trend is not changed when fitting the spec-trum with data between 1.7-2.0 keV excluded (Fig. 2).We found that multiplying a XSPEC Edge model cancorrect the residual at ∼ TABLE 2Absorption edge parameters
Model fit range E thresh τ (keV)1T 2.5 (cid:48) × (cid:48) . +0 . − . . +0 . − . r < (cid:48) . +0 . − . . +0 . − . r < (cid:48) , ignore 1.75-1.85 keV 1 . +0 . − . . +0 . − . a r < (cid:48) , ignore 1.75-1.85 keV 1 . +0 . − . . +0 . − . b r < (cid:48) , ignore 1.75-1.85 keV 1 . +0 . − . . +0 . − .
1T 0.2 (cid:48) < r < (cid:48) . +0 . − . . +0 . − .
1T 0.2 (cid:48) < r < (cid:48) , ignore 1.75-1.85 keV 1 . +0 . − . . +0 . − . T = 8 . +0 . − . keV, T = 34 +12 − keV. b T = 9 . +0 . − . keV, Γ = − . +0 . − . . Since the spectrum was extracted from a very large re-gion, we averaged the position-dependent response ma-trices and effective area functions by the number ofcounts at each location. It is possible that the absorp-tion feature is caused by improper weighting of thoseresponse files, or that this peculiarity only exists at cer-tain regions. To dispel those doubts, we separated thecentral 2.5 (cid:48) × (cid:48) area into 12 square regions and simul-taneously fit these spectra with one spectral model (Weonly used data from ObsID 6930 and 7289 to simplifythe fitting procedure). All parameters, except for thenormalization, were tied together. The residuals fromthe single temperature fit are shown in Fig. 3. Althoughthe fit is now acceptable with a χ /dof = 3448 . / . +0 . − . ) × cm − (90% confidence level), ap-pears low. When adding an absorption edge to the singletemperature model, the derived parameters of this edge, E thresh = 1 . +0 . − . keV and τ = 0 . +0 . − . , are consis-tent with results from the integrated spectrum. In fact, E thresh and τ do not strongly depend on how we modelthe cluster spectrum. We list fitted values of E thresh and τ from different cluster models and spectral extrac-tion regions in Table 2. Similar values are also found inother Chandra datasets (see the Appendix).The low column density can be explained by the ab-sorption at ∼ − − Δ χ Energy (keV)
Fig. 3.—
Residuals from an absorbed APEC model fit to 12spectra extracted from central 2.5 (cid:48) × (cid:48) region of ObsID 6930 and7289. We simultaneously fit these spectra and tied all parameters,except the normalization, together. E thresh = 1 .
74 keV and τ = 0 .
14, fit with a single tem-perature (1T) model, and compared with the observa-tions. The spectral normalization is increasing with ris-ing column density as we exclude more data around 1.75keV. Meanwhile, the cluster temperature and abundanceare slightly decreasing. The changes of those parame-ters from different bandpass used in the fitting matchperfectly to what are seen in the real data.The CCD calibration around the Si-edge for ACIS-Idetectors is a known issue (N. Schulz, private communi-cation). However, it is unknown whether a correction likean edge model is needed, or if we should simply ignorethe data around the Si edge. If the former is true, re-sults from the multiple-component analysis or the hard-band/broad-band temperature measurement without ap-plying this edge model beforehand are very questionable.As seen in Table 3, including the edge model can makethe hard-band temperature 30% hotter than the that ofthe broad-band. This temperature ratio depends on thecluster temperature and the quality of the data. On theother hand, if the latter is true, the spectrum implies thatan additional component which is much harder than 10keV emission is definitely required. Though, the fit isnot as good as that with an edge model. From the factthat the absorption depth is sufficiently far from the zero,even if we exclude data around 1.75 keV (Table 2), theintensity jump around this energy indeed exists.A1689 is a very hot cluster that unfortunately will beseriously affected by the calibration uncertainty aroundthe Si edge if that can be modeled by something similarto an edge model. Lacking the knowledge that correctlytreats the systematic residuals seen in the data, we pro-vide both models, applying an absorption edge or simplyignoring the data around 1.75 keV, as our best guess tothe thermal state of this cluster.
Spectral fitting
Spectra were fit with XSPEC 12.3.1 package (Arnaud1996). We adopted χ statistic and grouped the spec-tra to have a minimum of 25 counts per bin. However,when fitting background dominated spectra, χ statis-tic is proven to give biased temperature (Leccardi &Molendi 2007). Another choice available in XSPEC is us-ing Cash statistic with modeled, rather than subtractedbackground. Since modeling the background needs manycomponents: cosmic ray induced background (brokenpower-laws plus several Gaussians), particle background(broken power-laws), cosmic X-ray background (power-law), galactic emission (thermal), etc, the whole spectral model will be very complicated for analysis like depro-jection, which simultaneously fits all of the spectra ex-tracted at different radii. We decided to use χ statisticbut with a different grouping method to bypass the dif-ficulty in background modeling.As shown in Table 4, we simulated 500 Chandra spectrawith N H = 1 . × cm − , Z = 0 . T = 9 , , = 1 . × , . × , . × ,respectively. The spectral normalization was chosen tomatch the observed flux at r = 6.5 (cid:48) -8.8 (cid:48) , where the back-ground is ∼
90% of the source in 0.9-7.0 keV band. Spec-tra were generated based on the response files of ObsID6930 and 7289 with a total exposure time of 150 ks.When data are binned to have a minimum of 25 totalcounts (background included) per channel, a 9 keV gaswill be measured to be 5 keV. Raising the threshold canlessen this bias. However, even with 400 counts per bin,which greatly reduces the spectral resolution by a factorof 10, the temperature is still being underestimated by ∼ T = 9 keV gas. We found out that binning datato have at least 2 counts above the background can re-cover the true temperature, though this minimum haveto be adjusted according to the background contribu-tion. Spectra at large radii were binned by this groupingscheme. SPECTRAL ANALYSIS
Single temperature model
We first determined the general properties of the clus-ter, using the spectrum extracted from the central 3 (cid:48) (395 h − kpc) region and fitting it with a single temperatureVAPEC model. The Ne, Mg, Si, S, Ar, Ca, Fe, Ni abun-dances and the redshift were free to vary. The columndensity was fixed at the Galactic value. The best-fit pa-rameters are listed in Table 5. For Chandra data we fixedthe Si abundance at 0.4 Z (cid:12) , since the residual at 1.75 keV( § xiv K α line (2.01 keV, rest-frame).Table 5 shows that a single-temperature model is mod-erately adequate for XMM-Newton
MOS data but notfor
Chandra when the absorption edge is not modeled.In addition to this difference, the
Chandra temperatureis ∼ XMM-Newton . Thistemperature disagreement is likely related to the cross-calibration problems, as noted in other studies (e.g., Ko-tov & Vikhlinin 2005; Vikhlinin et al. 2006; Snowdenet al. 2008, L. David ), but it could also be caused by in-correct cluster modeling. A two-temperature model willbe investigated in § Chandra data also have a higher Fe and a much higherNi abundance, resulting an unusually high Ni/Fe ratioof 7 . ± . . ± . (cid:12) /Fe (cid:12) with and withoutan absorption edge correction, respectively, in contrastto the XMM-Newton value of 1 . ± . (cid:12) /Fe (cid:12) . Our XMM-Newton
MOS result is in agreement with that ofde Plaa et al. (2007), 0 . ± . (cid:12) /Fe (cid:12) , obtained fromMOS and PN spectra from the r < . (cid:48) region with a dif-ferential emission measure MEKAL-based model, wdem (Kaastra et al. 2004). Such a high Ni/Fe ratio greatly Spectral normalization, Norm = − π ((1+ z ) D A ) R n e n H dV ,where n e and n H are in cm − , V in cm , and D A in cm. http://cxc.harvard.edu/ccw/proceedings/07_proc/presentations/david iscrepant Mass in A1689 5 TABLE 3Summary of r < (cid:48) spectral fits and 1T simulations Chandra observations Simulated 200 ks spectra a fit range T Z N H Norm χ /dof T Z N H Norm(keV) ( Z (cid:12) ) (10 cm − ) (10 − ) (keV) ( Z (cid:12) ) (10 cm − ) (10 − )0.6-9.5 keV b . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . ignore 1.75-1.85 keV 10 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . ignore 1.7-2.0 keV 10 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . ignore 1.7-2.5 keV 10 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . ignore 1.7-3.0 keV 10 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . Note . — Errors are 1 σ or 68% CL for 100 simulations. a The absorption edge is at E thresh = 1 .
74 keV with τ = 0 .
14. We used T = 10 . Z = 0 . Z (cid:12) , z = 0 . . × − (correspondingto S X [0 . − . keV ] = 2 . × − erg cm − s − ), N H = 1 . × cm − . b Multiplied by an absorption edge at E thresh = 1 .
74 keV with τ = 0 . TABLE 4Summary of 1T simulations of 150 ks
Chandra spectra at r = (cid:48) -8.8 (cid:48) . Min counts T (keV) T med (keV) dof mean (1) (2) (3) (4)25 (tot) 9 4 . +2 . − . . +1 . − . . +0 . − . . +5 . − . . +3 . − . . +1 . − . . +7 . − .
407 6 . +4 . − .
405 4 . +1 . − .
402 (net) 9 8 . +7 . − . . +4 . − . . +1 . − . Note . — (1) minimum (total or net) counts perchannel, (2) input temperature, (3) median temper-ature, (4) median dof . Errors are 68% CL for 500simulations. exceeds the yield of typical SN Ia models (Iwamoto et al.1999), which range from 1.4-4.8 Ni (cid:12) /Fe (cid:12) . Since there isa known temperature discrepancy between
Chandra and
XMM-Newton that would affect elemental abundance de-terminations, direct Fe and Ni line measurements will beconducted in § xvi K α line at 2.62 keV (rest-frame), should be accurately mea-sured for XMM-Newton
EPIC since it suffers little sys-tematic uncertainty (Werner et al. 2008). However, with-out an absorption edge correction, there is basically noS detection for
Chandra data, which strongly contradictsthe
XMM-Newton result. This shows the great impactof the absorption at 1.75 keV. When including an edgemodel into the fit, we have consistent S abundances forboth instruments.
Two temperature model
To get some clues to the nature of the claimed coolsubstructures in A1689, a simple two temperature modelwas fit to the spectrum extracted from the r < (cid:48) (395 h − kpc) region where the quality of the data was highenough to test it. We used two absorbed VAPEC mod-els, with variable normalization but linked metallicitiesbetween the two phases. The column density was fixed atthe Galactic value. To reduce the uncertainty on measur-ing metallicities, we tied the abundances of α -elements(O, Ne, Mg, Si, S, Ar, and Ca) together and fixed theremaining abundances at the solar value, except for Feand Ni. Since the hotter phase temperature, T hot , washarder to constrain, it was frozen at a certain value abovethe best-fit single temperature fit, T . We changed thisincrement from 0.5 to 50 keV to explore the whole pa-rameter space.Fig. 4 shows the temperature of the cooler gas, T cool , and the fractional contribution of the cooler gas,EM cool /EM total , as a function of T hot . As T hot increases, T cool and EM cool /EM total increase as well. T cool eventu-ally becomes T once T hot is greater than 20 keV andvery little gas is left in the hot phase, which is also sup-ported by the XMM-Newton data. For T hot ≈
18 keV,there has to be 30%, 60% of the cool gas at the tempera-ture of 5, 8 keV inferred from
Chandra and
XMM-Newton data, respectively.
Chandra absorption corrected datashow similar results as
XMM-Newton data do at this tem-perature. Although there is some inconsistency between
Chandra and
XMM-Newton data, both indicate that thecool component, if it indeed exists, is not cool at all. T = 5 keV is the typical temperature of a medium sizedcluster with a mass of M = 2 . × h − M (cid:12) (Vikhlininet al. 2006).To quantify how significant the detection of this ex-tra component was, we conducted an F -test from thefits of 1T (the null model) and 2T models. However,because the 2T model reduces to 1T when the normal-ization of one of the two components hits the parameterspace boundary (ie, zero), the assumption of F -test is notsatisfied (see Protassov et al. 2002). Therefore, we simu-lated 1000 1T Chandra spectra and performed the sameprocedure to derive the F -test probability, P F , based onthe F distribution. Fig. 4 shows the distribution of P F from simulated data at the 68, 90, 95, and 99 percentileoverplotted with P F from Chandra and
XMM-Newton data. We plot P F in Fig. 4 rather than the F statistic,since P F is a scaler that does not depend on the degrees TABLE 5Best-fit VAPEC parameters T (keV) z Ne Mg Si S Ar Ca Fe Ni χ /dof Chandra a . +0 . − . . +0 . − . . +0 . − . . +0 . − . . f < . < . < .
12 0 . +0 . − . . +0 . − . b . +0 . − . . +0 . − . < .
18 1 . +0 . − . . f . +0 . − . < . < .
09 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . < .
57 1 . +0 . − . . +0 . − . . +0 . − . Note . — Al, O fixed at 0.4 solar and He, C, N at 1 solar. For the elements whose abundances reach the lower bound, zero, only the upper limits areshown. Errors are 1 σ . a without an absorption edge and ignoring data at 1.75-1.85 keV. b with an absorption edge. Edge parameters are determined from the data with E thresh = 1 . +0 . − . keV and τ = 0 . +0 . − . . Fig. 4.—
The temperature of the cooler gas T cool , the emissionmeasure ratio EM cool /EM total , and the F-test probability P F areplotted as a function of T hot . The shaded region represents 68%,90%, 95%, and 99% CL from 1000 simulated T = 10 . Chandra spectra. The P F from Chandra data without the absorption edgecorrected (circles) is multiplied by 10 . of freedom of the fits and is ideal to compare observationsthat have different data bins. For T hot <
20 keV, boththe edge-corrected
Chandra data and the
XMM-Newton data are within the 95 percentile of the simulated 1Tmodel and we conclude that a 2T model is possible butnot necessary to describe the data.
Hard-band, broad-band temperature
In addition to multiple-component modeling, measur-ing the temperature in different band-pass is anotherway to demonstrate the presence of multiple components.Cavagnolo et al. (2008) reported a very high hard-band tobroad-band temperature ratio for A1689, 1 . +0 . − . , fromanalysis of 40 ks of Chandra data, suggesting that thiscould relate to ongoing or recent mergers. Following theconvention in Cavagnolo et al. (2008), we fit the spectrumin the 0.7-7.0 keV (broad) and 2.0/(1+ z )-7.0 keV (hard) Fig. 5.—
The hard-band to broad-band temperature ratio T . − . /T . − . of simulated Chandra
2T spectra (circles) plot-ted against T hot . The shaded regions show the observed temper-ature ratios from Chandra and
XMM-Newton
MOS data. Alsoshown is the temperature ratio from 40 ks
Chandra data by Cav-agnolo et al. (2008). band with a single-temperature model. In contrast toC08, we do not use the r < R region with the coreexcised, but simply take the spectrum from the wholecentral 3 (cid:48) (395 h − kpc) region. The hard-band to broad-band temperature ratio from Chandra data, 1 . ± . XMM
MOS, 1 . ± . Chandra spectrum ( § τ = 0 . − .
10. As a consistencycheck, we simulated spectra according to the best-fit 2Tmodels (from
Chandra data) from § Chandra data. Thus we conclude that there is no evidence fromthis ratio of the presence of multiple components or merg-ing activity. Furthermore, Leccardi & Molendi (2008) donot find any discrepancy between the hard band (2-10keV) and broad band (0.7-10 keV) temperature profiles,except for r < . r , for a sample of ∼
50 hot, inter-mediate redshift clusters based on
XMM-Newton obser-vations. The high hard-band to broad-band temperatureratio seen in A1689, as well as in many other clusters ob-served with
Chandra (Cavagnolo et al. 2008), might bedue to the aforementioned calibration uncertainty.
Emission line diagnostics
When fitting the whole spectrum, the temperature ismainly determined by the continuum due to the lowiscrepant Mass in A1689 7
TABLE 6
Line Energy Centroid a Width a (keV) (keV) (eV)Fe xxv K α xxv K β b xxvi K α c xxvii K α xxviii K α a Emissivity-weighted center and one standard deviation. The lineemissivity is calculated at T = 10 keV from Chandra
ATOMDB1.3.1. b including Fe xxvi K β , xxv K γ , and xxv K δ . c including Fe xxvi K δ and xxvi K γ . amount of line emission at the temperature of A1689.In order to extract the emission line information, whichcan provide an additional temperature diagnostic, wefit the 4.5-9.5 keV spectrum with an absorbed ther-mal bremsstrahlung model plus Gaussians. There are42 lines whose emissivity is greater than 10 − pho-tons cm − s − at kT = 10 keV from ions of Fe xxv ,Fe xxvi , Ni xxvii , Ni xxviii , according to Chandra
ATOMDB 1.3.1. Considering the CCD energy resolu-tion, we grouped those lines into seven Gaussians andused the emissivity-weighted centroid and one standarddeviation as the line center and width, respectively. TheNi xxvii K α line is ∼
80 eV away from the Fe xxv K β line, not separable under CCD resolution unless we haveextremely good data quality. Since we obtained an un-usually high Ni/Fe ratio of ∼ (cid:12) /Fe (cid:12) from a VAPECmodel fit to the whole spectrum ( § xxvii K α and Fe xxv K β lines individually. Fig. 6 shows thespectrum and the best-fit model. The modeled lines arelisted in Table 6.Strictly speaking, using fixed values of line centroidsand widths is not correct because those quantities changewith temperature. In addition, we approximated the linecomplex as a Gaussian whose line centroid and widthcalculated from the model may not be the same afterbeing convolved with the instrument response. To prop-erly compare our fit results with the theory, we simulatedspectra and fit them the same way we fit the real data.Fig. 7 shows the observed line ratios and results fromsimulated VAPEC spectra with 9 Ni (cid:12) /Fe (cid:12) . 100 spectrawere produced at each temperature and the flux was keptat the same level as that of the data. From the goodmatch of fitted results from simulations to the directmodel prediction, we confirmed that the fitting is accu-rate enough to measure the line flux, though only Fe xxv K α and Fe xxvi K α lines are precise enough for temper-ature determination. Table 7 shows the temperature andabundances, inferred from a single-temperature APECmodel. The iron line temperature is in very good agree-ment with the continuum temperature for both Chandra and
XMM-Newton data. All the
Chandra and
XMM-Newton observed line fluxes, except Fe xxv K β , are con-sistent with each other (after an overall 9% adjustmentto the flux). Using Fe xxv + xxvi K α and Ni xxvii K α line flux, we obtain accordant Fe and Ni abundancesfrom both instruments. The larger Fe and Ni abun-dances found in § Chandra data are likely due to
TABLE 7Summary of line analysis
Chandra XMM
MOSContinuum T (keV) 10 . +2 . − . . +0 . − . Emission lines T a (keV) 9 . +0 . − . . +0 . − . Ni/Fe b † (Ni (cid:12) /Fe (cid:12) ) 8 . +3 . − . . +1 . − . Ni/Fe c † (Ni (cid:12) /Fe (cid:12) ) 5 . +3 . − . . +1 . − . Fe d † (Z (cid:12) ) 0 . ± .
02 0 . ± . e † (Z (cid:12) ) 1 . +0 . − . . +0 . − . from Fe xxvi K α /Fe xxv K α . b from (Ni xxvii K α +Fe xxv K β )/Fe xxvi K α . c from Ni xxvii K α /Fe xxvi K α . d from (Fe xxvi K α + xxv K α )/continuum. e from Ni xxvii K α /continuum. † assuming T = 10 keV. the higher temperature determined by the broad-bandspectrum and the much stronger Fe xxv K β line.As discussed previously, the 2T analysis of Chan-dra data suggested that another spectral component isneeded if no absorption edge modeling is applied. Fig. 8shows the line ratios predicted by the best-fit modelsfrom § Chandra spec-trum is not sensitive to the hot phase temperature of the2T model once it exceeds 15 keV (Fig. 4). With the goodconstraint from the Fe xxvi K α /Fe xxv K α line ratio,models with T hot >
20 keV, which are composed of greatamounts of cooler gas, are rejected. Meanwhile, the ra-tio of higher energy states (Ni xxviii K α , Fe xxvi K β ,Fe xxv K γ , K δ ) to the well-measured Fe xxvi K α linesuggests that models with lower T hot are preferable.As for the 2T models based on Chandra with an ab-sorption edge model and
XMM-Newton broad-band spec-tra, predicted line ratios all agree with the observedvalue. In fact, models with T hot >
20 keV from
XMM-Newton data are essentially a one temperature model,since the normalization of the hot component in thesemodels is zero. Adding the fact that an additional tem-perature component does not significantly improve the χ of the fit for those spectra and the remarkably goodagreement on the temperature measured by the contin-uum and the iron lines from both Chandra and
XMM-Newton , we conclude that the simple 1T model is ad-equate to describe the X-ray emission from the central3 (cid:48) region of A1689. DEPROJECTION ANALYSIS
Assuming that the hotter phase gas has the 3D temper-ature profile of L08, the radial distribution of the coolergas can be derived. We extracted spectra from concentricannuli up to 8.8 (cid:48) (1.2 h − Mpc). The emission from eachshell in three-dimensional space was modeled with anabsorbed two-temperature APEC model with T hot fixedat the value of L08 and then projected by the PRO-JCT model in XSPEC. Because of the complexity of thismodel, we used coarser annular bins than those used in −3 no r m a li z ed c oun t s s − k e V − ObsID 6930+7289 ObsID 5004 ObsID 540+16635 6 7 8 9−10−505 Δ χ Energy (keV)
Fig. 6.—
The 4.5-9.5 keV
Chandra spectrum of the central3 (cid:48) region. The spectrum is modeled with an absorbed thermalbremsstrahlung plus the seven Gaussian lines listed in Table 3.4.
L08. Data of L08 were binned using the weighting schemeof Mazzotta et al. (2004) to produce a spectroscopic-liketemperature. T cool , abundance, and the normalizationof both components were free to vary. The outermosttwo annuli were background dominated, so spectra werebinned to have at least 15 net counts per bin at r =4.8 (cid:48) -6.5 (cid:48) (625-852 h − kpc) and 2 net counts at r = 6.5 (cid:48) -8.8 (cid:48) (852-1161 h − kpc) (see § h − kpc, and that tempera-ture was slightly below the observed one. Therefore, weallowed T hot to change in the last two bins. The coldcomponent was removed and the abundance was fixed at0.2 solar in these regions in order to constrain the rest ofthe parameters better.Assuming two phases in pressure equilibrium, the vol-ume filling fraction of the i th component can be obtainedfrom f i = Norm i T i (cid:80) j Norm j T j (1)(e.g., Sanders & Fabian 2002). Once f i is determined,the gas density ρ gi = µ e m p n ei can be derived fromNorm i = 10 − π ((1 + z ) D A ) (cid:90) n ei n Hi f i dV, (2)where n H / n e and µ e are calculated from a fully ionizedplasma with the measured abundance (He abundanceis primordial, and others are from Anders & Grevesse1989). For Z = 0 . Z (cid:12) , n H / n e = 0 .
852 and µ e = 1 . h − kpc is occupiedby the ”cool” component with a temperature of ∼ Chandra absorption edge corrected data,and this gas constitutes 90% of the total gas mass.Kawahara et al. (2007) show that local density andtemperature inhomogeneities do not correlate with eachother in simulated clusters, which undermines the as-sumption of two phases in thermal pressure equilibrium.However, other cosmological simulations find that gas
TABLE 8Total mass profile r M M M ( h − kpc) (10 h − M (cid:12) ) (10 h − M (cid:12) ) (10 h − M (cid:12) )32 +21 − . +0 . − . . +0 . − . . +0 . − . +72 − . +0 . − . . +0 . − . . +0 . − . +158 − . +0 . − . . +0 . − . . +0 . − . +181 − . +0 . − . . +1 . − . . +0 . − . +133 − . +6 . − . . +4 . − . . +3 . − . Note . — 2T assumption is only held within 625 h − kpc.The upper and lower limits of r indicate the radii ¯ r of twocontiguous rings used to calculate the mass. See text for defi-nitions of r and ¯ r . motions contribute about 5-20% of the total pressuresupport (e.g., Faltenbacher et al. 2005; Rasia et al. 2006;Lau et al. 2009). If the pressure balance is off by 20%, itwill not significantly change the gas mass fraction ( (cid:46) (cid:46) MASS PROFILE
Given the 3D gas density and temperature profiles, thetotal cluster mass within a radius r can be estimatedfrom the hydrostatic equilibrium equation (e.g., Sarazin1988), M ( r ) = − kT ( r ) rGµ m p (cid:18) d ln ρ g ( r ) d ln r + d ln T ( r ) d ln r (cid:19) , (3)For Z = 0 . Z (cid:12) , µ = 0 . ρ g , T replaced by ρ g hot , T hot , respec-tively. Nonparametric method
To evaluate the derivatives in Eq. 3, we took the dif-ferences of deprojected temperature and the gas densityin log space. The radius of each annulus was assigned at¯ r such that F D (¯ r ) 4 π (cid:0) r out − r in (cid:1) = (cid:90) r out r in F D ( r ) 4 πr dr, (4)where F D is the deprojected flux density from a finelybinned surface brightness profile, and r in ( r out ) is theinner (outer) radius of the annulus. The radius r outsideof the brackets of Eq. 3 is taken at the geometric mean(i.e. the arithmetic mean in log scale) of the radii oftwo adjacent rings, r = √ ¯ r i ¯ r i +1 , and the temperatureis linearly interpolated at this radius. Because errorsfrom e.g. T and dT /dr are not independent, standarderror propagation is not easily applied. Uncertaintiesare estimated from the distribution of 1000 Monte-Carlosimulations of T and ρ g profiles. Fig. 9 shows the totalmass profile from both 1T and 2T models and the resultsare listed in Table 8. Two-temperature modeling, basedon the T hot of L08, increases the total mass by 30-50% forall radii within 625 h − kpc. Beyond that radius, the 2Tassumption is not held because of the lack of constrainton T hot .Although the inclusion of an absorption edge in thespectral model greatly changes the derived compositioniscrepant Mass in A1689 9 Fig. 7.—
The predicted 1T plasma line ratio (dotted line) as a function of temperature, for various lines. The observed ratio and its 1 σ confidence are shown as a solid line and shaded region. The circles show the fitted results of 100 simulated Chandra spectra drawn from aVAPEC model with 9 Ni (cid:12) /Fe (cid:12) . Fig. 8.—
The predicted line ratio from the best-fit 2T (VAPEC) models ( § T hot .The solid line and shaded region shows the observed ratio and its 1 σ error. The x-axis is in log scale. of the multi-phase plasma, it does not affect the massmeasurement much. This is because we use a fixed T hot profile. Once the temperature is determined, the totalmass only depends on the logarithmic scale of the gasdensity, which produces ∼
13% difference at most.
1T parametric method
If the temperature does not vary dramatically on smallscales, we can obtain a mass profile with higher spatialresolution since the gas density can be measured in de-tail from the X-ray surface brightness with the assump-tion of a certain geometry of the cluster. To achievethis, modeling of the temperature and the gas densityis necessary. Following the procedure of Vikhlinin et al.(2006), we project the 3D temperature and the gas den- sity models along the line of sight and fit with the ob-served projected temperature and the surface brightnessprofiles. A weighting method by Mazzotta et al. (2004),Vikhlinin (2006) is used to predict a single-temperaturefit to the projected multi-temperature emission from 3Dspace. This method has been shown (Nagai et al. 2007)to accurately reproduce density and temperature profilesof simulated clusters.The gas density model is given by n p n e = n ( r/r c ) − α (1 + r /r c ) β − α/ r γ /r sγ ) ε/γ + n (1 + r /r c ) β , (5)0 T dep r o j ( k e V ) Lemze et al. 08T hot (+ edge)T cool T cool (+ edge)T (+ edge) E M c oo l / E M t o t
10 100 100000.20.40.60.8 r ( h −1 kpc) f c oo l −3 −2 −1 n e ( h . c m − ) n e hot n e cool n e hot (+edge)n e cool (+edge) M ga s ( h − . M s un ) M gas hot M gas tot M gas hot (+edge)M gas tot (+edge)
10 100 100010 r ( h −1 kpc) M D ( h − M s un ) Fig. 9.—
Temperature, emission measure ratio of the cool component EM cool /EM tot , volume filling fraction of the cool component f cool , gas number density n e , cumulative gas mass M gas , and cumulative total mass M D profiles from the 2T deprojection analysis withan absorption edge correction (squares) and without the correction (circles). Also shown is the 1T analysis (diamonds) and results fromLemze et al. (2008a) (asterisks). T hot of the first 4 annuli was fixed at the value derived from lensing and X-ray brightness data (Lemzeet al. 2008a), which were grouped into fewer bins. The cool component of the last 2 bins was frozen at zero. The 2T assumption is heldwithin 625 h − kpc. X-data points of the 1T and 2T models have been shifted by +10% and -10% for clarity, and their error bars are alsoomitted. which originates from a β model (Cavaliere & Fusco-Femiano 1978) modified by a power-law cusp and a steep-ening at large radii (Vikhlinin et al. 1999). The secondterm describes a possible component in the center, espe-cially for clusters with small core radius. The tempera-ture model is given by T ( r ) = T ( r/r cool ) a cool + T min /T r/r cool ) a cool ( r/r t ) − a (1 + ( r/r t ) b ) c/b , (6)which is a broken power law with central cooling (Allenet al. 2001). Best-fit parameters for the gas density andtemperature profiles are listed in Tables 9 and 10, respec-tively. Errors are estimated from the distribution of thefitted parameters of 1000 simulated projected tempera-ture and surface brightness profiles generated accordingto the observed data and their measurement uncertain-ties. Since parameters are highly degenerate, some of thebest-fit values are not covered by the upper or lower lim-its with the quoted confidence level (upper/lower boundsare for one parameter). The observed temperature andsurface brightness profiles, the best-fit model, and thesurface brightness residual are shown in Fig. 10. Themodel describes the data very well ( χ /dof =154.3/155).The best-fit T and n e models are shown in Fig. 11. Also plotted are the profiles from the spectral deprojectionfitting ( § T and n e can avoid flucutations from the di-rect spectral deprojection, which is a common problemas the deprojection tends to amplify the noise in the data(see Appendix in Sanders & Fabian 2007).Although the second break of the first term in Eq. 5was designed to describe the steepening at r s > . r (Vikhlinin et al. 1999, 2006; Neumann 2005), we foundthat if the initial guess for r s is not big enough, r s tendsto converge to a relatively small value, ≈ h − kpc,compared to the typical value of 400-3000 h − kpc fornearby relaxed clusters (Vikhlinin et al. 2006). It is pos-sible to use the first core radius r c or the core radius ofthe second component r c to account for the sharpeningat 200 h − kpc. This consequently yields a more reason-able r s at ≈ h − Mpc. Both cases, small (Model 1)and large (Model 2) r s , give acceptable fits with χ /dof of 153.4/155 and 154.3/155, respectively. However, large r s is harder to constrain. This makes the mass estimatemore uncertain at large radii than the small r s case.Comparing the surface brightness profile of the north-eastern (NE) part to the southwestern (SW), Riemer-Sørensen et al. (2009) found that the NE part is 5-15%brighter outside 350 h − kpc and 25% under-luminous atiscrepant Mass in A1689 11 T p r o j ( k e V ) best−fit model90% confidence bound −6 −5 −4 −3 S X ( c t s / c m / s / a r c m i n )
10 100 1000−4−2024 r ( h −1 kpc) S X r e s i dua l ( σ ) Fig. 10.—
Projected temperature and surface brightness profileswith the best-fit model (solid lines) and its 90% confidence bounds(dashed lines). Bottom panel: residual between the surface bright-ness and the model. This fit gives a χ /dof of 154.3/155. T D ( k e V ) −4 −3 −2 r ( h −1 kpc) n e ( h . c m − ) best−fit model90% confidence bound Fig. 11.—
Best-fit T and n e models (solid lines) and 90%confidence bounds (dashed lines). Also shown are unparameterizedresults (diamonds) from § h − kpc than the SW. To see if this asymmetry canaffect the mass estimate, we fit a symmetric model tothe image and iteratively removed any part of the clus-ter that deviates significantly from the azimuthal mean,mainly the northern clump at 460 h − kpc, the south-ern less luminous region at 330 h − kpc, and possiblysome point sources not completely removed beforehand.We did not exclude these regions from our temperaturemeasurement since they were unlikely to bias the aver-age temperature much for such a hot cluster, as shown inFig. 4 that at least 10-20% of the total emission measurefrom another spectral component was needed in order thechange the spectroscopic temperature by 1 keV. Best-fitgas density and temperature for these models are listedin Tables 9 and 10, labelled with Model 3 (small r s ) and4 (large r s ).The total mass profiles from these analytic gas densityand temperature models are given in Table 11. We listthe total mass at the radii where masses from the non-parametric method are evaluated (Table 8). The last en-try of Table 11 shows the total mass at the boundary ofthe ACIS-I chips, 12 (cid:48) (1.6 h − Mpc ≈ r ), where S X isdetected at (cid:46) σ . Removing asymmetric parts from theimage or restricting r s to be greater than 350 h − kpc in-creases the total mass estimate with (cid:46) dT /dr , at a cer-tain radius, so the uncertainty associated with the posi-tion is not included in the error on the mass, σ M , butseparately shown on the radius. Therefore, σ M appearssmaller if data are binned more coarsely. For the para-metric method, the dependency of σ M on the data bin-ning is weaker. The departure from the model for anydata point is assumed to be random noise and is filteredout through the fitting. Hence, σ M reflects only the un-certainty of the fitted function and it depends stronglyon the modeling. Comparison with other studies
The total mass profiles of A04, based on
XMM-Newton ,and L08, a joint X-ray, strong and weak lensing study arealso shown in Fig. 12. Our result is in good agreementwith A04, but disagrees with L08 around ∼ h − kpc.To compare our mass estimate with other lensing works,we derived the total mass density and integrated it alongthe line-of-sight. The total mass density, ρ , is obtainedthrough the hydrostatic equation,4 πGρ = − kµ g m p ( ∇ T + T ∇ ln ρ g + ∇ ln ρ g · ∇ T ) . (7)For the nonparametric method, we evaluated Eq. 7 ina similar fashion as we did in § T and ρ g profiles. Fig. 13 shows the surface mass densityprofiles from both parametric and nonparametric meth-2 TABLE 9Best-fit parameters for the gas density (Eq.5) n r c r s α β ε n r c β γ − h cm − h − kpc 10 h − kpc 10 − h cm − h − kpc(1) 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +2 . − . . +1 . − . (2) 0 . +0 . − . . +0 . − . . +17 . . . +0 . − . . +0 . − . . +3 . − . . +0 . − . . +0 . − . . +0 . − . . +7 . − . (3) 2 . +1 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . (4) 0 . +0 . − . . +0 . − . . +27 . − . . +0 . − . . +0 . − . . +2 . − . . +0 . − . . − . − . . − . − . . +8 . − . Note . — (1) small r s , (2) large r s , (3) small r s with northern clumps removed, (4) large r s with northern clumps removed. Errors are 95% CLfor one parameter from 1000 Monte-Carlo simulations. Since parameters are highly degenerate, some of the best-fit values are not covered by theupper and lower limits at this confidence level. a parameters hit the hard limit. TABLE 10Best-fit parameters for the temperature (Eq.6) T T min /T r cool r t a b c d keV 10 h − kpc 10 h − kpc(1) 12 . +6 . − . . +0 . − . . +1 . − . . +20 . − . . +0 . − . . +0 . − . . +1 . − . . +2 . − . (2) 12 . +7 . − . . +0 . − . . +1 . − . . +21 . − . − . +0 . − . . +0 . − . . +2 . − . . − . − . (3) 14 . +5 . − . . +0 . − . . +1 . − . . +0 . − . . +0 . − . . +1 . − . . +1 . − . . +3 . − . (4) 11 . +8 . − . . +0 . − . . +1 . − . . +23 . − . − . +0 . − . . +0 . − . . +1 . − . . − . − . Note . — (1) small r s , (2) large r s , (3) small r s with northern clumps removed, (4) large r s with northernclumps removed. Errors are 95% CL for one parameter from 1000 Monte-Carlo simulations. TABLE 11Parametric total mass profile r ( h − kpc) M ( r ) (10 h − M (cid:12) )(1) (2) (3) (4)32 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − .
94 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − .
264 2 . +0 . − . . +0 . − . . +0 . − . . +0 . − .
559 4 . +0 . − . . +0 . − . . +0 . − . . +0 . − .
855 5 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +2 . − . . +0 . − . . +3 . − . Note . — (1) small r s , (2) large r s , (3) small r s with north-ern clumps removed, (4) large r s with northern clumps re-moved. Errors are 68% CL from 1000 Monte-Carlo simula-tions. ods, along with the HST/ACS strong lensing analysisof Broadhurst et al. (2005a), and the combined Subarudistortion and depletion data by Umetsu & Broadhurst(2008). Since it requires at least 3 points to calculate thesecond derivative, ρ at the boundary is unknown. Thiswill introduce additional systematic errors to the innerand the outer projected profile. To demonstrate how thismay affect our nonparametric result, we insert two arti-ficial points at 1 (cid:48)(cid:48) (2 h − kpc) and 13 (cid:48) (1.7 h − Mpc) tothe nonparametric T and ρ g profiles with their values es-timated from the parametric model. The projected den-sity derived this way is shown in red filled diamonds inFig. 13. The X-ray data are consistent with those fromthe weak lensing, but disagree with the strong lensinganalysis. Although the nonparametric data appears toagree with the strong lensing estimate at r = 80 h − kpc,this is probably due to the temperature fluctuation men-tioned in §
10 100 100010 M D ( h − M s un ) r ( h −1 kpc) This workAndersson & Madejski 04Lemze et al. 08
Fig. 12.—
The parametric mass profile (solid line) compared tothe unparameterized result (diamonds). Dashed lines show the 95%confidence bounds. Also shown are
XMM-Newton result from A04(crosses) and combined X-ray, strong and weak lensing analysis ofL08 (asterisks). The mass profile of L08 is mainly determined bythe lensing data. the cumulative projected mass profiles, M D , shown inFig. 14. The weak lensing M D profile of Umetsu &Broadhurst (2008) includes the integration of the dataof Broadhurst et al. (2005a) in the inner region. Un-certainties are from Monte-Carlo simulations of the con-vergence profiles. The last 3 data points of Umetsu &Broadhurst (2008) (1-2.3 h − Mpc) are discarded sinceonly the upper limits are available. Also shown areparametric strong lensing profiles (Halkola et al. 2006;Limousin et al. 2007), and other X-ray analyses (A04;iscrepant Mass in A1689 13
10 100 100010 r ( h −1 kpc) Σ ( h M s un / M p c ) This workBroadhurst et al. 05Umetsu & Broadhurst 08
Fig. 13.—
Surface mass density profiles from non-parametric(open diamonds) and parametric X-ray model (solid and dashedlines, 95% CL), compared to HST/ACS strong lensing analysis ofBroadhurst et al. (2005a) (triangles), and combined Subaru distor-tion and depletion data by Umetsu & Broadhurst (2008), based ona maximum entropy method (circles). Filled diamonds show themass from the nonparametric T D and n e profiles that include es-timations from the parametric result at 1 (cid:48)(cid:48) (2 h − kpc) and 13 (cid:48) (1.7 h − Mpc).
Riemer-Sørensen et al. 2009). To convert M D to M D ,A04 assume that the last data point reached the clus-ter mass limit, which unavoidably leads to underestima-tions especially at large radii. Riemer-Sørensen et al.(2009) use only the SE part of the cluster and four of the Chandra observations (excluding ObsID 540) and derive M D based on a best-fit NFW model fit to the M D profile. Their mass profile is generally lower than ourestimate at most radii. This is contradictory to mostfindings that claim that the hydrostatic mass is underes-timated in unrelaxed systems (e.g., Jeltema et al. 2008).Using such reasoning, and removing the NE part, pre-sumably disturbed according to Riemer-Sørensen et al.(2009), should increase the overall mass estimate. TheX-ray M D is 25-40% lower than that of lensing within200 h − kpc, corresponding to a ∼ . × h − M (cid:12) dif-ference in the total projected mass. NFW profile parameters
The total mass profile M D was fit to the NFW model(Navarro et al. 1997) to obtain the mass and the con-centration parameter. To fit the nonparametric data, weweighted each point according to its vertical and hori-zontal errors, given by, σ = σ M + σ r (cid:18) dMdr (cid:19) , (8)where σ r is assigned to be 68% of the width of the hori-zontal error bar and dM/dr is iteratively evaluated fromthe NFW model until it converges. In the parametric ap-proach, a NFW model was fit to the parametrized massprofile that evaluated only at the radii where the pro-jected temperature was measured with errors estimatedfrom the standard deviation of a sample of mass profilesconstructed from the simulated T proj and S X profiles de-scribed in §
10 100 100010 r ( h −1 kpc) M D ( h − M s un ) This workAndersson & Madejski 04Riemer−Sorensen et al. 08Broadhurst et al 05Umetsu & Broadhurst 08Halkola et al. 06Limousin et al. 07
Fig. 14.—
Projected mass profiles from non-parametric (opendiamonds) and parametric analyses (solid and dashed lines, 95%CL), compared to
XMM-Newton result from A04 (crosses),
Chan-dra result by Riemer-Sørensen et al. (2009) (squares), HST/ACSand Subaru results by Broadhurst et al. (2005a) (triangles) andUmetsu & Broadhurst (2008) (circles). We integrated the lensingsurface mass profile (shown in Fig. 13) and estimated its uncer-tainties from Monte-Carlo simulations. Also shown are parametricstrong lensing profiles of Halkola et al. (2006) and Limousin et al.(2007) (shaded regions, 68% CL). Riemer-Sørensen et al. (2009)used only SW part of the X-ray data and converted M D to M D with a NFW profile. A04 assumed that the last data point reachedthe cluster mass limit. Filled diamonds, same as Fig. 13. mass profiles in the sample. Resulting NFW parameterswere used to estimate the uncertainty.Table 12 lists the best-fit NFW parameters, M and c , for the total mass from both methods and fromother studies, all converted to the adopted cosmology.Compared to other X-ray studies, our derived M is30-50% higher, closer to weak lensing results. The differ-ences between our NFW parameters and those of A04from XMM-Newton are primarily attributed to theirslightly lower but yet consistent mass at the last datapoint (Fig. 12). This demonstrates that the accuratemass measurement at large radii, where systematic errorsare usually the greatest, is crucial to the determinationof NFW parameters.Our results are consistent with weak lensing measure-ments, but with a lower concentration than what recentweak lensing studies seem to suggest (Umetsu & Broad-hurst 2008; Corless et al. 2009). When these analysesare added with strong lensing information, a very tightconstraint on the concentration parameter can be ob-tained, giving C = 9 . +0 . − . (Umetsu & Broadhurst2008), which hardly can be reconciled with our value,5 . +1 . − . . However, if the gas emission is modeled withtwo spectral components with T hot from L08, the X-rayderived concentration is in a closer agreement to those ofcombined strong and weak lensing studies, but this alsoimplies that the majority of the gas is in the cool phaseand occupies most of the intracluster space ( § Gas mass fraction
The cumulative gas fraction f gas = M gas /M total , de-rived from our best-fit T and n e model, is 0 . +0 . − . TABLE 12Comparison of best-Fit NFW Parameters
Method Instrument M c χ /dof Reference(10 h − M (cid:12) )Spherical modelX-ray (1T+edge) Chandra 1 . +0 . − . . +1 . − . T , n e ) Chandra 0 . +0 . − . . +0 . − . this workX-ray (2T a ) Chandra 1 . +0 . − . . +1 . − . a +edge) Chandra 1 . +0 . − . . +0 . − . . ± .
36 7 . +1 . − . d . +1 . − . Broadhurst et al. (2005a)SL ACS 2 . ± .
32 5 . ± . . ± .
13 7 . ± . . ± .
17 13 . ± . . ± .
14 10 . +4 . − . . ± .
13 10 . +1 . − . . ± .
11 9 . +0 . − . S X ) ACS+Subaru+Chandra 1.42 9 . +0 . − . b ACS+Subaru 1 . +0 . − . . +1 . − . c CFHT 0 . ± .
16 12 . ± . Note . — see Comerford & Natarajan (2007); Umetsu & Broadhurst (2008); Corless et al. (2009) for a more complete compilation. a with T hot from L08 b under a flat prior on the axis ratios. c under a prior on the halo orientation that favors the line-of-sight direction. d converted from best-fit parameters, ρ = 7 . × M (cid:12) kpc − , r s = 174 kpc ( h = 0 .
7, Ω m = 0 .
28, and Ω λ = 0 . c , the concentration at r where the enclosed mean density is 200 times the critical density (private communication). h − . at r (493 +11 − h − kpc), ∼
20% higher than whatis found using
XMM-Newton data (A04). In spite of thisseemingly large difference, the data agree that f gas doesnot converge at r . Much like in the case of A1689,the low- z relaxed cluster A1413 does not have a strongcooling core and also has a steadily rising f gas profileout to r (Pratt & Arnaud 2002). Comparing the f gas profile of A1413 with another nearby prominent coolingcore cluster, A478, Pointecouteau et al. (2004) specu-late that the flatter f gas profile of A478 is related to thepresence of a cooling core. Our f is 11% lower thanthe mean gas fraction of Allen et al. (2008) derived from42 relaxed clusters observed with Chandra , but our f ,0 . ± . h − . , agrees within 1% of the M − f gas rela-tion of Vikhlinin et al. (2009). DISCUSSION
Nagai et al. (2007) show that following the data analy-sis of Vikhlinin et al. (2006), the hydrostatic mass is un-derestimated by 14 ±
6% within estimated r for sim-ulated clusters visually classified as ”relaxed”. Based onthe X-ray morphology, A1689 is likely to be categorizedas a relaxed cluster. The X-ray centroid is within 3 (cid:48)(cid:48) ofthe lensing and optical centers (Andersson & Madejski2004), with a very minimal centroid shift or asymmetry(Hashimoto et al. 2007). At the X-ray estimated r of493 h − kpc, we derive an enclosed hydrostatic mass of(4 . ± . × h − M (cid:12) , ≈
30% lower than the lensingmass from L08. At r = 200 h − kpc, this becomes a 50%difference (see Fig. 12). Such a strong bias is not seen inthe relaxed cluster sample of Nagai et al. (2007), assum-ing that the lensing mass is unbiased, although this is not TABLE 13Comparison of M Method M r (10 h − M (cid:12) ) ( h − Mpc)parametrized T , n e . +1 . − . . +0 . − . M − T X † . ± . . ± . M − Y X † a . +0 . − . . +0 . − . † Scaling relations from Vikhlinin et al. (2009) withindices fixed to self-similar theory values. Errors onlyreflect the measurement uncertainties. Dispersions ofthe relation is not included. T X = 10 . ± . r = 1 . (cid:48) − . (cid:48) ( ≈ . r − r ). a By solving Eq. 14 of Vikhlinin et al. (2009). The fi-nal Y X = T X × M gas determined at r is (5 . +0 . − . ) × h − . M (cid:12) keV. unusual for ”unrelaxed” clusters, referring to those withsecondary maxima, filamentary structures, or significantisophotal centroid shifts.Table 13 shows the comparison of measured M withothers derived from the M − Y X and M − T X rela-tions of Vikhlinin et al. (2009), calibrated from 49 low- z and 37 high- z with (cid:104) z (cid:105) = 0 . Chandra and
ROSAT . A very good agreement has beenachieved between these estimates. Since the M − Y X relation is insensitive to whether the cluster is relaxedor not (Kravtsov et al. 2006) and merging clusters tendto be cool for their mass (Mathiesen & Evrard 2001),consistency among these mass estimates indicates thatA1689 is relaxed in the sense that it behaves like other”relaxed” clusters on the scaling relation.iscrepant Mass in A1689 15On the other hand, projection effects, such as triaxialhalos or chance alignments, always have to be taken intoaccount when comparing projected (lensing) and three-dimensional (X-ray) mass estimates. From kinematics ofabout 200 galaxies in A1689, (cid:32)Lokas et al. (2006) sug-gest that there could be a few distant, possibly non-interacting, substructures superposed along the line ofsight. Lemze et al. (2008b), based on a 0 . × . VLT/VIMOS spectroscopic survey from Czoske (2004)which includes ∼
500 cluster members, disagree with thisprojection view. They conclude that only one identifi-able substructure at +3000 km/s, 1.5 (cid:48) to the NE (TheX-ray clump is at ∼ . (cid:48) NE). This background groupis seen in the strong lensing mass analysis (Broadhurstet al. 2005a), but is determined not to be massive ( < ∼ ∼ −
60% larger concen-trations than other clusters with similar masses and red-shifts. Gavazzi (2005) demonstrates that using a prolatehalo with axis ratio ∼ .
4, they were able to explainthe mass discrepancy between the lensing and X-ray es-timates of cluster MS2137-23. This cluster has a welldefined cool core (e.g., Andersson et al. 2009), thus pre-sumably relaxed, and yet a factor of 2 difference in themass is not lessened with a multiphase model for the coreregion (Arabadjis et al. 2004). In contrast, triaxial mod-eling not only solves the mass inconsistency, but also thehigh concentration problem and the misalignment be-tween stellar and dark matter components in MS2137-23(Gavazzi 2005).To see how the triaxiality changes our mass measure-ments, we modeled T and ρ g with prolate profiles, byreplacing r in Eq. 5 and 6 with ( x /a + y /b + z /c ) / ,where we assumed a = b < c and the major axis, z -axis, is perfectly aligned along the line-of-sight. Follow-ing the same analysis outlined in § T and ρ g profiles. The derived mass profiles under various axisratios a/c are shown in Fig. 15. The uncertainties onΣ( r ) and M D ( r ) are similar to those in Figs. 13 and14. We integrated the density from z = − . h − Mpc to+4.5 h − Mpc ( ≈ r for a/c=1) for all the cases. Theuncertainties of T and ρ g profiles at large radii ( (cid:38) r = r and increasing further afterward) does notsignificantly change the projected mass at smaller radii( (cid:46)
3% within 500 h − kpc).The total mass enclosed within a sphere of radius r , M D ( r ), and the spherically averaged mass density, ρ ( r ),are basically unchanged under different assumptions oftriaxiality, considering the typical measurement uncer-tainty. The same conclusion was drawn by Piffarettiet al. (2003) and Gavazzi (2005), though they assumeda β or a NFW model with gas isothermality. For the M D (r) ( h − M s un ) M D (r) ( h − M s un ) a/c=1.0a/c=0.8a/c=0.6a/c=0.5a/c=0.30.1 1 10r (arcmin)10 100 100010 ρ (r) ( h M s un / M p c ) r ( h −1 kpc) 10 100 100010 Σ (r) ( h M s un / M p c ) r ( h −1 kpc) Fig. 15.—
Best-fit mass profiles for various axis ratios a/c fromModel 1 (dash-dot line), 2 (dotted line), 3 (solid line), and 4(dashed line).
Top left : total mass enclose within a sphere of radius r , M D ( r ). Top right : spherically average mass density ρ ( r ). Bot-tom left : azimuthally average surface mass density Σ( r ). Bottomright : projected mass within a cylinder of radius r , M D ( r ). azimuthally averaged surface mass density Σ( r ) or theprojected mass within a cylinder of radius r , M D ( r ),a factor of 2 or more difference can be easily made byincreasing the ellipticity. An axis ratio of 0.6, giving M D ( < (cid:48)(cid:48) ) = 1 . × h − M (cid:12) (by a factor of 1.6 in-crease), can resolve the central mass discrepancy, butoverpredicts the mass by ∼
40% at large radii. For aratio of 0.7, the X-ray mass estimate data agrees withthose of strong and weak lensing within 1% ( − σ ) and25% (+1 σ ), respectively. Since the gas distribution isrounder than that of the DM, a larger axis ratio thanthe finding of Gavazzi (2005) is expected.Not only does the projected mass increases with thetriaxiality, but also does the steepness of the profile.This explains a higher than X-ray derived concentra-tion from the lensing data ( § h − kpc. This can not be characterized as the6properties of a cool core cluster. In fact, as shown inAndersson et al. (2009), A1689 is an intermediate stagecluster in terms of central baryon concentration with aminimal core temperature drop. This, does not neces-sarily provide evidence that the cluster is disturbed butwe do not either expect the properties of a cool corecluster. Hardness-ratio maps are very sensitive to ac-curate background subtraction, especially for high en-ergy splittings. We suspect that the hardness ratio map( S/H = E [0 . − . /E [6 . − . (cid:48) . This is an extremely low ratio for anyreasonable cluster temperature and it is in disagreementwith the observed temperature profile. For comparison,a background-free spectrum from an isothermal clusterat 10 keV would exhibit a count-ratio of ∼
47 in ACIS-Igiven the energy bands mentioned above. The usage ofunsubtracted hardness-ratios in these bands shows thatthe high-energy band has a significant fractional back-ground contribution and hence, is more spatially flatcompared to the low energy band. This does not provideinformation about the spatial distribution of gas temper-atures in the ICM. SUMMARY
We have investigated a deep exposure of Abell 1689using the ACIS-I instrument aboard the Chandra X-raytelescope. In order to study the discrepancy of the grav-itational mass from estimates from gravitational lensing,to that derived using X-ray data, we test the hypoth-esis of multiple temperature components in projection.The result of a two-temperature model fit shows that itis very important to take into account all details of thecalibration of the instrument. We detect an additionalabsorption feature at 1.75 keV consistent with an ab-sorption edge with an optical depth of 0.13. In analyzingmultiple additional datasets, we find similar parametervalues for this edge.If the edge is not modeled, fitting the cluster datawithin 3 (cid:48) strongly favors an additional plasma compo-nent at a different temperature. However, when this ab-sorption feature is modeled, the second component doesnot improve the statistic significantly and the fit resultsis in better agreement with the XMM MOS data. In allcases, a second component has to have
T > S X profiles. This contradictsthe assertion that cool clumps are biasing the X-ray tem-perature measurements since these substructures wouldnot be cool at all. We also find that, if the temperatureprofile of the ambient cluster gas is in fact that of Lemzeet al. (2008a), the ”cool clumps” would have to occupy70-90% of the space within 250 kpc radius, assuming thatthe two temperature phases are in pressure equilibrium.In conclusion, we find the scenario proposed by Lemzeet al. (2008a) unlikely.Further studying the ratio of Fe xxvi K α and Fe xxv K α emission lines, we conclude that these show no signsof a multi-temperature projection and the best fit of thisratio implies a single temperature consistent with thecontinuum temperatures from both XMM-Newton
MOSand the
Chandra data when the absorption edge is mod-eled.The discrepancy between lensing and X-ray mass es-timates remains, particularly in the r < h − kpc re-gion. Our X-ray mass profile shows consistent resultscompared to those from weak lensing (e.g., Broadhurstet al. 2005b; Limousin et al. 2007; Umetsu & Broad-hurst 2008; Corless et al. 2009). Strong lensing massprofiles from different studies generally give consistentresults (e.g., Broadhurst et al. 2005a; Halkola et al. 2006;Limousin et al. 2007), but none of them agrees with thosederived from X-ray observations (Xue & Wu 2002; An-dersson & Madejski 2004; Riemer-Sørensen et al. 2009).Using a simple ellipsoidal modeling of the cluster withthe major axis along the line of sight, we find that theprojected mass, as derived from the X-ray analysis, in-creases by a factor of 1.6 assuming an axis-ratio of 0.6.We conclude that the mass discrepancy between lensingand X-ray derived masses can be alleviated by line ofsight ellipticity and that this also can explain the highconcentration parameter found in this cluster.We thank the anonymous referee for valuable sugges-tions on the manuscript. Support for this work was pro-vided by NASA through SAO Award Number 2834-MIT-SAO-4018 issued by the Chandra X-Ray ObservatoryCenter, which is operated by the Smithsonian Astrophys-ical Observatory for and on behalf of NASA under con-tract NAS8-03060. EP sincerely thanks John Arabadjisfor his support during early phases of this project. REFERENCESAllen, S. W., Rapetti, D. A., Schmidt, R. W., Ebeling, H.,Morris, R. G., & Fabian, A. C. 2008, MNRAS, 383, 879Allen, S. W., Schmidt, R. W., & Fabian, A. C. 2001, MNRAS,328, L37Anders, E., & Grevesse, N. 1989, Geochim. Cosmochim. Acta, 53,197Andersson, K., Peterson, J. R., Madejski, G., & Goobar, A. 2009,ArXiv e-printsAndersson, K. E., & Madejski, G. M. 2004, ApJ, 607, 190Arabadjis, J. S., Bautz, M. W., & Arabadjis, G. 2004, ApJ, 617,303Arnaud, K. A. 1996, in Astronomical Society of the PacificConference Series, Vol. 101, Astronomical Data AnalysisSoftware and Systems V, ed. G. H. Jacoby & J. Barnes, 17–+ Broadhurst, T., Ben´ıtez, N., Coe, D., Sharon, K., Zekser, K.,White, R., Ford, H., Bouwens, R., Blakeslee, J., Clampin, M.,Cross, N., Franx, M., Frye, B., Hartig, G., Illingworth, G.,Infante, L., Menanteau, F., Meurer, G., Postman, M., Ardila,D. R., Bartko, F., Brown, R. A., Burrows, C. J., Cheng, E. S.,Feldman, P. D., Golimowski, D. A., Goto, T., Gronwall, C.,Herranz, D., Holden, B., Homeier, N., Krist, J. E., Lesser,M. P., Martel, A. R., Miley, G. K., Rosati, P., Sirianni, M.,Sparks, W. B., Steindling, S., Tran, H. D., Tsvetanov, Z. I., &Zheng, W. 2005a, ApJ, 621, 53Broadhurst, T., Takada, M., Umetsu, K., Kong, X., Arimoto, N.,Chiba, M., & Futamase, T. 2005b, ApJ, 619, L143Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2008,ApJ, 682, 821Cavaliere, A., & Fusco-Femiano, R. 1978, A&A, 70, 677Comerford, J. M., & Natarajan, P. 2007, MNRAS, 379, 190 iscrepant Mass in A1689 17
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INSTRUMENTAL ABSORPTION IN
CHANDRA
DATA
To see whether the absorption feature found in A1689 ( § Chandra
ACIS-I observations. Objects that have high-quality data and are relatively easy to model are pulsar windnebula G021.5-00.9, elliptical galaxy NGC 4486 (M87), and the Coma cluster. Unfortunately, due to the high galacticabsorption in G021.5-00.9, it is not suitable to use those observations to verify the instrumental absorption. Details ofthe datasets we used are listed in Table 14. These observations have low N H ( < × cm − ), low background level ( <
2% in 1.7-2.0 keV band), and high signal-to-noise ratios. Although the central part of M87 has very complex structuresproduced by the AGN (e.g., Forman et al. 2007), the
XMM-Newton observation indicates that the intracluster mediumis likely to be single-phase in nature outside those regions (Matsushita et al. 2002). We extracted the spectra from r = 6 (cid:48) − . (cid:48) of M87 and r < (cid:48) of Coma cluster for each ACIS-I chip and fit with an absorbed single-temperature APECmodel, multiplied by an absorption edge. The column density was fixed at the Galactic value (Dickey & Lockman1990). The redshift and all the elemental abundances, except Al, were free to vary. Parameters for this absorption edgeare listed in Table 15. For M87 whose emission is dominated by lines, these parameters are sensitive to the choice ofthe plasma model. Results from the latest MEKAL-based model, SPEX (version 2.0; Kaastra & Mewe 2000), are alsoshown in Table 15. In general, an absorption at ∼ .
75 keV with an optical depth of 0.1-0.15 is seen in the datasets.8
TABLE 14
Chandra
Observation Log
Name N H a z ObsID Data Obs. Date Exp. Background Norm. Region f B b S/N c (10 cm − ) Mode (ks) I0 I1 I2 I3 (%)M87 2.59 0.00423 5826 VFAINT 2005-03-03 125.5 1.03 1.00 1.04 1.07 6 (cid:48) -7.5 (cid:48) < (cid:48) a Dickey & Lockman (1990) b Background fraction in 1.7-2.0 keV band. c Signal-to-noise ratio in 1.7-2.0 keV band.
TABLE 15Absorption edge parameters
APEC SPEXName ObsID ccd E thresh τ z E thresh τ z(keV) (keV)M87 5826 0 1 . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (5 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (12 . +0 . − . ) × − . +0 . − . . +0 . − . (8 . +1 . − . ) × − . +0 . − . . +0 . − . (5 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +1 . − . ) × − . +0 . − . . +0 . − . (5 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (2 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +0 . − . ) × − . +0 . − . . +0 . − . (2 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (5 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (8 . +0 . − . ) × − . +0 . − . . +0 . − . (7 . +0 . − . ) × − . +0 . − . . +0 . − . (8 . +0 . − . ) × − . +0 . − . . +0 . − . (7 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (1 . +3 . − . ) × − . +0 . − . . +0 . − . (5 . +0 . − . ) × − . +0 . − . . +0 . − . (2 . +0 . − . ) × − . f . +0 . − . (6 . +0 . − . ) × − . f . +0 . − . (1 . +1 . − . ) × − Coma 9714 0 1 . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +0 . − . ) × − . +0 . − . . +0 . − . (2 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +0 . − . ) × − . +0 . − . . +0 . − . (3 . +0 . − . ) × − . +0 . − . . +0 . − . (4 . +0 . − . ) × − . +0 . − . . +0 . − . (2 . +0 . − . ) × − A1689 540 13 1 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . However, for ObsID 7212, the absorption depth is determined less than 0.05 (1 σσ