Mass, velocity anisotropy, and pseudo phase-space density profiles of Abell 2142
aa r X i v : . [ a s t r o - ph . C O ] A p r Astronomy & Astrophysicsmanuscript no. A2142 c (cid:13)
ESO 2018October 9, 2018
Mass, velocity anisotropy, and pseudo phase–space densityprofiles of Abell 2142
Munari, E. , Biviano, A. , , and Mamon, G. A. Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11, I-34131 Trieste, Italye-mail: [email protected] INAF / Osservatorio Astronomico di Trieste, via Tiepolo 11, I-34131 Trieste, Italye-mail: [email protected] Institut d’Astrophysique de Paris (UMR 7095: CNRS & UPMC), 98 bis Bd Arago, F-75014 Paris, Francee-mail: [email protected]
October 9, 2018
ABSTRACT
Aims.
We aim to compute the mass and velocity anisotropy profiles of Abell 2142 and, from there, the pseudo phase–space densityprofile Q ( r ) and the density slope - velocity anisotropy β − γ relation, and then to compare them with theoretical expectations. Methods.
The mass profiles were obtained by using three techniques based on member galaxy kinematics, namely the caustic method,the method of Dispersion - Kurtosis, and MAMPOSSt. Through the inversion of the Jeans equation, it was possible to compute thevelocity anisotropy profiles.
Results.
The mass profiles, as well as the virial values of mass and radius, computed with the di ff erent techniques agree with oneanother and with the estimates coming from X-ray and weak lensing studies. A combined mass profile is obtained by averagingthe lensing, X-ray, and kinematics determinations. The cluster mass profile is well fitted by an NFW profile with c = . ± .
5. Thepopulation of red and blue galaxies appear to have a di ff erent velocity anisotropy configuration, since red galaxies are almost isotropic,while blue galaxies are radially anisotropic, with a weak dependence on radius. The Q ( r ) profile for the red galaxy population agreeswith the theoretical results found in cosmological simulations, suggesting that any bias, relative to the dark matter particles, in velocitydispersion of the red component is independent of radius. The β − γ relation for red galaxies matches the theoretical relation only inthe inner region. The deviations might be due to the use of galaxies as tracers of the gravitational potential, unlike the non–collisionaltracer used in the theoretical relation. Key words.
1. Introduction
The measure of the mass of cosmological objects, such as clus-ters of galaxies, has proven to be an important tool for cosmo-logical applications. The mass is not a direct observable, andmany techniques have been developed to infer it by measuringobservable quantities. Two methods that are widely used to inferthe mass profile of galaxy clusters are the X-ray and the lens-ing techniques. The former makes use of the observations of theX-ray emission of the hot intracluster plasma (ICM hereafter).The lensing technique makes use of the relativistic e ff ect of dis-tortion of the trajectories of light emitted by distant backgroundgalaxies caused by the mass of the observed cluster. These twomethods have some limitations either way. In the case of X-raytechnique, the limitation comes from the usual assumption thatthe plasma of the cluster is in hydrostatic equilibrium, and thecluster approximately spherically symmetric (Ettori et al. 2002)with no important recent merger activity (Böhringer & Werner2010). As for the lensing technique, its limitation is that it onlyallows computing the projected mass, and this includes all theline-of-sight ( los ) mass contributions. The complementarity ofthe di ff erent techniques is a strong advantage for reliably con-straining the mass of a cluster.In this article, we use another kind of information that comesfrom the kinematics of the galaxies belonging to the observedcluster. In fact, the potential well of the cluster, due to the mass, is the main driver of the orbital motion of the galaxies, which inthe absence of mutual interactions, can be treated as test parti-cles in the gravitational potential of the cluster. The kinematicsof galaxies therefore carries the information about the mass con-tent of the cluster. The motion takes place in a six-dimensionalphase space, but the observations are able to capture only threeof these dimensions, namely two for the position and one for the los velocity. This is one of the most important limitations of amass estimate via observation of the kinematics of galaxies. Toovercome this problem, most methods assume spherical symme-try. A spherically symmetric density profile following the univer-sal relation provided by Navarro, Frenk, & White (1996, 1997)(NFW hereafter) has often been adopted in these analyses. Sucha profile is characterized by its “scale radius” parameter, whichis the radius where the logarithmic slope of the density pro-file is equal to −
2. With the advent of simulations with in-creasingly higher resolution, the universality of the NFW den-sity profile has been questioned (see e.g. Navarro et al. 2004;Vogelsberger et al. 2011; Ludlow et al. 2013). While the self-similarity of the density profiles of DM-only haloes may nothold as well as initially thought, another physical parameter ap-pears to have a quasi-universal radial profile, the pseudo phase–space density (PPSD hereafter) Q ( r ) = ρ/σ , where ρ is thetotal matter density profile and σ the 3D velocity dispersion of Article number, page 1 of 13 & Aproofs: manuscript no. A2142 the tracers of the gravitational potential (Taylor & Navarro 2001;Ludlow et al. 2010). Still, some doubts have been raised about itsuniversality (Ludlow et al. 2011). The use of the radial velocitydispersion instead of the total one has proven to be a valid androbust alternative for computing the PPSD, in this case called Q r ( r ). The link between these two formulations of the PPSD isconstrained by the velocity anisotropy (hereafter, anisotropy) ofthe system, which plays a non trivial role in shaping the structureof a system. The density profile and the anisotropy profile arein fact found to correlate. An empirical relation is provided byHansen & Moore (2006) and Ludlow et al. (2011), linking thelogarithmic slope of the density profile γ = d ln ρ/ d ln r andthe anisotropy β ( r ) = − ( σ t /σ r ) , where σ r and σ t are thevelocity dispersions of the radial component and of one of thetwo tangential components, respectively. Hereafter we refer toanisotropy as β or the equivalent σ r /σ t = / p − β . We alsodenote the relation between anisotropy and logarithmic slope ofthe density profile as the β − γ relation.In this article, we study Abell 2142 (A2142 hereafter), arich galaxy cluster at z ∼ .
09. The large number of galaxymembers allows us to derive the total mass profile, to test dif-ferent models, as well as to perform dynamical analyses in or-der to derive the anisotropy of the orbits of galaxies that al-lows to compute the pseudo phase–space density profile andthe β − γ relation. This cluster shows evidence of some recentmergers. In fact, the X-ray emission appears to have an ellip-tical morphology elongated in the north–west south–east direc-tion (Markevitch et al. 2000; Akamatsu et al. 2011). The merg-ing scenario is also supported by the presence of substructuresof galaxies lying along the direction of the cluster elongation, asfound in the SZ maps by Umetsu et al. (2009), lensing analysisby Okabe & Umetsu (2008), and analysis of the distribution oflos velocities of Owers et al. (2011). However, after analysing XMM-Newton images to investigate the cold fronts of A2142,Rossetti et al. (2013) argue that the mergers have intermediatemass ratios rather than major ones.Throughout this paper, we adopt a Λ CDM cosmology with H =
70 km s − Mpc − , Ω = . Ω Λ = .
7. The virial quanti-ties are computed at radius r .
2. The data
The photometric information has been obtained from the SDSSDR7 database after searching for the galaxies that have238 ◦ . < RA < ◦ . ◦ . < DEC < ◦ .
834 andpetroMag r ′ <
22. The spectroscopic information has been pro-vided by Owers et al. (2011). The full sample is composed of1631 galaxies with both photometric and spectroscopic informa-tion. The cluster centre is assumed to coincide with the X-raycentre provided by De Grandi & Molendi (2002).Two algorithms have been used to select clustermembers, those of den Hartog & Katgert (1996) and ofMamon, Biviano, & Boué (2013), hereafter dHK and clean ,respectively. Both identify cluster members on the basis oftheir location in projected phase-space : R , v rest , using the spec-troscopic values for the velocities. We adopt the membership r ∆ is the radius within which the mean density is ∆ times the criticaldensity of the Universe. http: // cas.sdss.org / astro / en / tools / chart / chart.asp R is the projected radial distance from the cluster centre. We assumespherical symmetry in the dynamical analyses. The rest-frame velocityis defined as v = c ( z − z ) / (1 + z ). The mean cluster redshift z is redefined at each new iteration of the membership selection, until conver-gence. Fig. 1.
Distribution of the galaxies of Abell 2142 in the projected phase-space of projected radii and los rest-frame velocities. Cluster members,as identified by both dHK and clean algorithms, are denoted by bluefilled dots. The red diamond is the galaxy identified as member by dHKbut not by the clean algorithm. The purple solid lines are the caustic,described in Sect. 3. The vertical dashed line locates the virial radius ofthe combined model (see Sect. 4). determination of dHK, resulting in 996 members. In fact, the clean algorithm removes one more galaxy that is very closeto the distribution of selected members and therefore seemsunlikely to be an interloper. Anyway, this galaxy is ≈ ff erence inthe analysis here. Fig. 1 shows the location of galaxies inprojected phase-space with the identification of cluster membergalaxies using the two methods. We use the method describedin Appendix B of Mamon et al. (2013) to obtain a preliminaryestimate of the virial radius from the velocity dispersion of thecluster members. The value we obtain is 2 .
33 Mpc. This is usedlater as an initial–guess value for the virial radius, only to besuccessively refined with more sophisticated techniques (seeSect. 4.2).The cluster mean redshift and los velocity dispersion, aswell as their uncertainties, have been computed using the bi-weight estimator (Beers et al. 1990) on the redshifts and rest–frame velocities of the members: h z i = . ± . σ los = + − km / s We identify the red sequence iteratively by fitting the g ′ − r ′ vs. r ′ colour-magnitude relation of galaxies with r ′ < . g ′ − r ′ > .
7, then selecting galaxies within ± σ of the foundsequence (where σ is the dispersion around the best fit relation).We refer to the cluster members within ± σ of the red sequence,and those above this range, as red sequence galaxies, and to thecluster members more than 2 σ below the red sequence as bluegalaxies, as shown in Fig. 2. Owers et al. (2011) found some substructures in A2142, proba-bly groups that have been recently accreted by the cluster. Thesesubstructures can alter the kinematics of the system since theystill retain memory of the infall kinematics. For this reason, wecompute the mass profile of the system excluding the galaxies
Article number, page 2 of 13unari, E. et al.: Mass, velocity anisotropy, and pseudo phase–space density profiles of Abell 2142
Fig. 2.
Colour magnitude diagram g ′ − r ′ vs. r ′ . Red (blue) points are rel-ative to red (blue) member galaxies. Black points are galaxies for whichwe have photometric information, that are not identified as members.The red solid line locates the red sequence. Table 1.
Coordinates with respect to the cluster center, radii, and num-ber of galaxies of the three main substructures, as found by Owers et al.(2011). x c [Mpc] y c [Mpc] r [Mpc] N g al S2 0.600 0.763 0.467 49S3 2.007 1.567 0.700 54S6 2.327 –0.180 0.812 53belonging to these substructures. In particular we consider thelargest substructures in this cluster, namely S2, S3, and S6, fol-lowing the nomenclature of Owers et al. (2011). Therefore, weremove galaxies inside circles, the centres and radii of which arereported in Table 1.
Some of the techniques (described in Sect. 3) that we use tocompute the mass profile of the cluster rely upon the assump-tion of equilibrium of the galaxy population. Red galaxies aremore likely an older cluster population than blue galaxies, proba-bly closer to dynamical equilibrium (e.g. Moss & Dickens 1977;van der Marel et al. 2000). For this reason, red galaxies consti-tute a better sample for such techniques. Among red galaxies,those outside substructures (see Sect. 2.2) are the most likely tobe in dynamical equilibrium. We therefore use these galaxies fordetermining the mass profile.The three samples that are used hereafter are as follows. Werefer to the sample made of all the member galaxies to as theALL sample. BLUE is the sample made of blue galaxies, andRED is the sample made of red galaxies not belonging to thesubstructures described in Sect. 2.2. See Table 2 for a summaryof the number of galaxies belonging to each sample. The ALLand BLUE samples do contain substructures.
Table 2.
Number of galaxies in the three samples.
Sample n tot n ALL 996 706RED 564 447BLUE 278 162
Notes.
For each sample, the total number of member galaxies and thenumber of member galaxies within r are shown, the latter being thevalue of the combined model (see Sect. 4).
3. The techniques
In this section, we briefly describe the main features of the threedi ff erent techniques used in this work to compute the mass pro-file of A2142. Besides the virial values of radius and mass, weobtain estimates of the mass scale radius, which is where thelogarithmic slope of the total density profile is equal to −
2, fromwhich it is possible to recover the cluster mass profile. Thesemethods all assume spherical symmetry.
DK: The dispersion kurtosis technique, hereafter shortened toDK, first introduced by Łokas (2002), relies upon the jointfit of the los velocity dispersion and kurtosis profiles ofthe cluster galaxies. In fact, fitting only the los velocitydispersion profile to the theoretical relation coming fromthe projection (see Mamon & Łokas 2005b for single inte-gral formulae for the case of simple anisotropy profiles) ofthe Jeans (1904) equation (see e.g. Binney & Mamon 1982;Binney & Tremaine 1987) does not lift the intrinsic degener-acy between mass profile and anisotropy profile determina-tions (as Łokas & Mamon 2003 showed for the Coma clus-ter). This technique assumes dynamical equilibrium of thesystem, and it allows us to estimate the virial mass, the massscale radius and the value of the cluster velocity anisotropy,considered as a constant with radius. MAMPOSSt: The MAMPOSSt technique, recently developedby Mamon et al. (2013), performs a maximum likelihood fitof the distribution of galaxies in projected phase space, as-suming models for the mass profile, the anisotropy profile,the projected number density profile and the 3D velocitydistribution. In particular, for our analysis we used di ff erentNFW models for the mass and the projected number densityprofiles, either a simplified Tiret profile (Tiret et al. 2007) ora constant value for the anisotropy profile and a Gaussianprofile for the 3D velocity distribution. As in the DK method,MAMPOSSt assumes dynamical equilibrium of the system.By this method we estimate the virial mass, the scale radiusof the mass density profile, and the value of anisotropy of thetracers.Caustic: The caustic technique, introduced byDiaferio & Geller (1997), is di ff erent from the othertwo methods because it does not require dynamical equi-librium. As a result, this technique also provides themass distribution beyond the virial radius. In projectedphase space (see Fig. 1), member galaxies tend to lie ina region around v los = − . Measuring the velocityamplitude A of the galaxy distribution gives information Richardson & Fairbairn (2013) have recently extended the DKmethod to more general anisotropy profiles.Article number, page 3 of 13 & Aproofs: manuscript no. A2142 about the escape velocity of the system. In turn, the es-cape velocity is related to the potential, hence the massprofile: M ( r ) = M ( r ) + (1 / G ) R rr A ( s ) F β ( s ) ds , where F β ( r ) = − π G (3 − β ) / (1 − β ) r ρ ( r ) / Φ ( r ) (Diaferio 1999).Because F β is usually approximated with a constant value(Diaferio 1999; Serra et al. 2011), it is customary to call it a“parameter”.Since the DK and MAMPOSSt techniques make use of theassumption of dynamical equilibrium of the system, the use ofthe RED sample allows a more correct application of those tech-niques, since this sample is likely to be the most relaxed sam-ple. In fact these methods just need a tracer that obeys the Jeansequation. As long as we consider a collisionless tracer, sphericalsymmetry, no streaming motions, and a stationary system, DKand MAMPOSSt are able to reliably recover the mass content ofthe cluster. On the other hand, we use the ALL sample for thecaustic technique.As discussed in Sect. 1, some studies suggest an ellipticalmorphology of the system, with evidence of some recent in-termediate mass-ratio mergers. Although this might violate theassumptions of both spherical symmetry and equilibrium re-quired in DK and MAMPOSSt, and the spherical symmetryalone for the Caustic technique, these methods are not stronglya ff ected by this. In fact, they have been tested on Λ CDM haloesextracted from simulations. Although these haloes are neitherspherical nor fully relaxed, and they present substructures, theDK (Sanchis et al. 2004), MAMPOSSt (Mamon et al. 2013) andCaustic (Serra & Diaferio 2013; Gi ff ord et al. 2013) techniquesprovide reliable estimates of halo masses. As we see below (Sect.4), the fairly close results of these dynamical methods with thosefrom the weak lensing analysis (which does not assume equilib-rium) of Umetsu et al. (2009) suggest that this cluster cannot befar from dynamical equilibrium.In all three methods, we consider the scale radius of thegalaxy distribution and the scale radius of the mass distributionas two separate and independent parameters. To compute the parameter values with the MAMPOSSt tech-nique, we have considered the galaxies of RED sample withinthe “first guess” virial radius, presented in Sect. 2. As discussedin Mamon et al. (2013) (in particular see their Table 2), MAM-POSSt does not critically depend on this choice. We then per-formed a Markov Chain Monte Carlo (MCMC) procedure (seee.g. Lewis & Bridle 2002), using the public CosmoMC code ofA. Lewis. In MCMC, the parameter space is sampled followinga procedure that compares the posterior (likelihood times prior)of a point in this space with that of the previous point, and de-cides whether to accept the new point following a criterion thatdepends on the two posteriors. We use the Metropolis-Hastingsalgorithm. The next point is chosen at random from a hyperel-lipsoidal Gaussian distribution centred on the current point. Thisprocedure ensures that the final density of points in the param-eter space is proportional to the posterior probability. MCMCthen returns probability distributions as a function of a singleparameter, or for several parameters together. Here, the errors on However, beyond ≈ . r , the infall streaming motions are impor-tant enough that the usual Jeans equation is inadequate for determiningthe radial velocity dispersion (Falco et al. 2013). http: // cosmologist.info / cosmomc Fig. 3. F β parameter as a function of clustercentric distance for an NFWmodel. Black solid line refers to the isotropic case, while red dashed linerefers to an ML anisotropy (Mamon & Łokas 2005b) with r anis = r s .Blue dash dotted line refers to the F β by Biviano & Girardi (2003). Thedotted vertical line locates the virial radius of the combined model (seeSect. 4). a single parameter are computed by marginalizing the posteriorprobabilities over the other two free parameters.For the caustic technique, we use the ALL sample, since theequilibrium of the sample is not required, also considering thegalaxies beyond the virial radius. To apply the caustic technique,the F β parameter (Diaferio 1999) must be chosen. The choiceof the parameter is quite arbitrary, so we tested three di ff erentchoices: the constant value 0.5, as first suggested in Diaferio(1999); the constant value 0.7 as suggested in Serra et al. (2011);and the profile described in Biviano & Girardi (2003). The lastis a smooth approximation of the F β ( r ) derived from numeri-cal simulations by Diaferio (1999). The actual values of F β arenot likely to be very di ff erent from these we decided to test.In fact, Fig. 3 shows that an NFW model leads to F β = . r = r s ≃ r for isotropic orbits, while for orbits withML (Mamon & Łokas 2005a) anisotropy, it produces F β = . r = r s ≃ r . As a comparison, in Fig. 3 the profile byBiviano & Girardi (2003) is shown. It has higher values in thecentre, but rapidly falls in the outer regions. The value 0.5 al-lows us to take both the innermost region, where the values of F β are very low, and the outer part, where the values are largerand closer to 0.7 into account.When using F β = . F β = . r s < r < r s ≈ r one can approximate F β ≃ csttypically to ±
11% accuracy, the mass profile returned by thecaustic method changes normalization but not the shape for dif-ferent values of F β . Therefore this method turns out to be veryuseful for constraining the mass profile shape, since it does notassume a parametric profile like an NFW, so that we can checkwhether the assumption of NFW for the mass profile is a goodone. We adopt r =
0, which relieves us from the choice of amass at some finite radius r . Once we have computed the mass Article number, page 4 of 13unari, E. et al.: Mass, velocity anisotropy, and pseudo phase–space density profiles of Abell 2142 profile, we fit it with an NFW profile to obtain an estimate of themass scale radius.
The NFW scale radius of the galaxy distribution is used as in-put for the DK and MAMPOSSt analyses, therefore it has beencomputed for the RED sample. The number density profile ofthe spectroscopic sample is a ff ected by the incompleteness is-sue. We need to known the distribution of tracers along the los .Assuming spherical symmetry, we can adopt the deprojection ofthe tracer surface density profile, but we must first correct forspectroscopic incompleteness. Owers et al. (2011, see their Fig.2) have measured their spectroscopic incompleteness in variousmagnitude bins. Since their incompleteness depends rather littleon magnitude, we adopt their cumulative incompleteness mea-sured for R ≤ .
5. This completeness has then been correctedin order to take into account the artificial reduction of the numberof galaxies due to the presence of a bright star in the cluster field.Also, since we do not wish to consider galaxies inside substruc-tures, we also have to correct the completeness to account forthe removal of the substructures. We divided the cluster in radialbins and counted the galaxies inside each bin. In the bins wherethe presence of the star and the removal of substructures causesa lack of detection, the area of the bin is artificially reduced, andthe mean density of galaxies is computed in the remainder of thebin. This value is then assigned to the whole bin.The RED galaxy number density profile is well fitted bya projected NFW profile (Bartelmann 1996; Łokas & Mamon2001). The fit is an MLE fit performed on all RED membersthat provide a scale radius 0 . ± .
14 Mpc. The ALL and BLUEsamples are less concentrated, the values of the scale radius be-ing 1 . ± .
25 Mpc for the ALL sample and 16 ±
11 Mpc forthe BLUE sample. A KS test (e.g. Press et al. 1993) provides anestimate of the reliability of these fits. The probabilities of ob-taining greater discrepancy by chance for the RED and BLUEsamples are P = .
95 and 0 .
20, respectively, indicating that themodel adequately fits the data. However, for the ALL sample, thecorresponding probability is only P = .
05, indicating that themodel only marginally fits the data. In Fig. 4 the surface numberdensity profiles for the di ff erent samples are shown. The scaleradius for the BLUE sample is very high and is due to a very flatdistribution of these galaxies.
4. Mass profiles
We used the velocities of the galaxies within the “first guess”virial radius (see Sect. 2) to compute the mass profile of A2142.In Figure 5, the velocity dispersion profiles are shown, alongwith the best-fit profiles coming from the DK and MAMPOSStanalyses.The DK technique assumes a constant value for theanisotropy, while we have chosen two profiles for the anisotropymodel in MAMPOSSt, a constant value and a Tiret profile β ( r ) = β + ( β ∞ − β ) r / ( r + r anis ). Here, we set β = r anis to the scale radius of the galaxy’s number density profile.The maximum values of the likelihoods are similar when usingthe two anisotropy models, therefore for the sake of simplicitywe consider only the case of a constant velocity anisotropy. InSect. 5, we compute the anisotropy profile for the RED sampleand find that indeed it is compatible with a constant value. Fig. 4.
Surface number density profiles for the ALL, RED, andBLUE samples, along with their best-fit projected NFW profiles. Thedashed vertical line locates the virial radius of the combined model (seeSect. 4).
We also tried to assume di ff erent mass profiles and velocityanisotropy models in MAMPOSSt, namely a Burkert (Burkert1995), a Hernquist (Hernquist 1990) and a softened isothermalsphere profile (e.g. Mamon 1987; Geller et al. 1999), all withboth constant and Tiret anisotropy profiles. However our data-setis not large enough to allow us to distinguish between these dif-ferent models. All provide acceptable fits. As a consequence, theresulting estimates of virial mass and mass profile concentrationare very similar to the case of NFW mass profile with constantanisotropy, with di ff erences of very few percent. We thereforeonly considered the NFW model for the mass profile.The results are summarized in Table 3. Figure 6 shows thedetailed results of our MAMPOSSt MCMC analysis. The massscale radius is not well constrained by MAMPOSSt. This doesnot a ff ect the subsequent analysis, since in Sect. 4.2, we performa weighted mean of the results from the di ff erent methods.In Fig. 7, we show the mass profiles obtained from the dif-ferent methods, along with the virial values of mass and radius.The results coming from the X-ray (Akamatsu et al. 2011) andweak lensing (Umetsu et al. 2009, WL hereafter) analysis arealso shown. Article number, page 5 of 13 & Aproofs: manuscript no. A2142 s ss Fig. 6.
Parameter space and probability distribution functions for the virial radius, mass profile scale radius, and velocity anisotropy, as found byMAMPOSSt. The coloured regions are the 1,2,3 σ confidence regions, while the red stars and the red arrows locate the best-fit values. These arebased upon an MCMC analysis with 6 chains of 40 000 elements each, with the first 5000 elements of each chain removed (this is the burn-in phase that is sensitive to the starting point of the chain). The priors were flat within the range of each panel, and zero elsewhere. Table 3.
Virial quantities of Abell 2142 obtained from di ff erent techniques Method Sample M [10 M ⊙ ] r [Mpc] r s [Mpc] c σ r /σ t β Caustic ( F β = .
5) ALL 1 . + . − . . + . − . . + . − . . ± . . + . − . . + . − . . + . − . . ± . . + . − . . + . − . MAMPOSSt RED 1 . + . − . . + . − . . + . − . . + . − . . + . − . . ± . . + . − . . ± .
14 0 . ± .
17 3 . ± . . + . − . . + . − . . ± .
31 2 . ± . . + . − . . ± .
10 0 . ± .
08 4 . ± . . ± .
13 2 . ± .
08 0 . ± .
07 4 . ± . Notes.
Values of virial mass, virial radius, mass scale radius, concentration, and two measures of the velocity anisotropy, for di ff erent techniques.Also shown are the average value of the kinematical techniques after symmetrizing the errors and the value of the combined model, obtained asthe result of the average of all the values coming from the di ff erent techniques (see Sect. 5 for the average procedure). X-ray values come fromAkamatsu et al. (2011), weak lensing (WL) from Umetsu et al. (2009). Both for X-ray and WL we had the values and the errors of the virial radiusand the concentration: we have symmetrized these errors and propagated them to obtain the estimates of the errors on the mass scale radii.Article number, page 6 of 13unari, E. et al.: Mass, velocity anisotropy, and pseudo phase–space density profiles of Abell 2142 Fig. 5.
Velocity dispersion profiles for the ALL, RED, and BLUE sam-ples. For the RED sample we also show the best-fit profile coming fromthe DK analysis (black), and the profile computed after the MAMPOSStanalysis (dashed red). The dashed vertical line locates the virial radiusof the combined model (see Sect. 4).
We combined the constraints from the di ff erent mass modellingmethods to build a combined mass profile, assuming again anNFW density profile. We attempted to give the same weight tokinematics, X-ray, and WL in the final estimate of the param-eters, so we computed single values coming from kinematicaltechniques for the mass scale radius and virial radius. For this,we took the mean of the values r s and r of the di ff erent meth-ods, inversely weighting by the symmetrized errors. Since themeasures of these two quantities by the various methods are notindependent (as they are based on essentially the same data sets),we multiplied the error on the average by √
3, 3 being the num-ber of values used to compute the average. In fact, the usual er-ror on the weighted average decreases like the square root of thenumber of values.The mean value and its error are shown in the left panelsof Fig. 8. In the right panels of Fig. 8, we plot the values ofmass scale and virial radius obtained from the three indepen-dent methods: kinematics, X-ray, and WL. The average error-weighted value and its error, this time computed without multi-
Fig. 7.
Mass profiles computed from the di ff erent methods. The blackdash-dotted line and the triangle with error bars refer to DK technique,the dashed blue line and blue square to the caustic method, the solid redline and red point to MAMPOSSt. The symbols with error bars refer tothe virial mass and radius. The purple asterisk with error bars and thepurple dash triple dotted line are the result of the X-ray analysis, whilethe orange diamond with a long dashed line is the one coming fromweak lensing analysis. The shaded area is the 1 σ confidence region ofthe mass profile according to the MAMPOSSt results. Fig. 8.
Virial (top panels) and mass scale (bottom panels) radius forall the methods.
Left panels: blue diamonds are values obtained fromthe caustic technique, red ones for MAMPOSSt, and black ones for DK(from left to right, respectively). The average value and its error are thesolid and dashed lines, respectively. See the text for the computation ofthe error.
Right panels: values obtained from the kinematical analysis,X-ray and WL (from left to right, respectively). The average value andits error are the solid and dashed lines, respectively. plication factor (since the three measures are independent), are r = . ± .
08 Mpc, r s = . ± .
07 Mpc.
5. Velocity anisotropy profiles
Although with DK and MAMPOSSt we have assumed somemodels for the velocity anisotropy profile, we now wish todetermine it in a non-parametric way using the Jeans equa-tion. For this, we use the mass profile we obtained by com-bining the information coming from the three dynamical meth-ods, X-ray and WL. The Jeans equation contains four unknown
Article number, page 7 of 13 & Aproofs: manuscript no. A2142 quantities, therefore to solve it we need three other relations,namely the Abell integrals to relate the projected number den-sity and velocity dispersion to the real ones and assume amass profile for the cluster. This anisotropy inversion was firstsolved by Binney & Mamon (1982), but several other authorshave provided simpler algorithms. We follow the approach ofSolanes & Salvador-Solé (1990), and we tested the results bycomparing them with those obtained following the approach ofDejonghe & Merritt (1992). Once the mass profile is specified,this procedure is fully non–parametric. In fact, instead of fittingthe number density profile, we binned and smoothed it with theLOWESS technique (see e.g. Gebhardt et al. 1994). We then ob-tained the 3D number density profile by using Abel’s equation(e.g. Binney & Mamon 1982). In the same way, we smoothedthe binned σ los profile. This procedure requires the solution ofintegrals up to infinity. Mamon et al. (2010) show that a 3 σ clip-ping removes all the interlopers beyond 19 virial radii. There-fore, an extrapolation up to such a distance is enough to solvethe integrals having infinity as the integration limit. We used 30Mpc as the maximum radius of integration, and extrapolate thesmoothed profiles up to this limit. A factor–2 change in the up-per limit of integration does not a ff ect our results in a significantway.The result of the anisotropy inversion is shown in Fig. 9. Theconfidence levels were obtained by estimating two error contri-butions. One contribution comes from the uncertainties in thesurface density and σ los profiles. Since 80% of the relative un-certainty of the product Σ σ comes from the uncertainty of σ los (Trilling et al. 2014), we only considered the error contributionfrom the latter. It is virtually impossible to propagate the errorson the observed σ los through the Jeans inversion equations to in-fer the uncertainties on the β profile solution. We then proceededto estimate these uncertainties the other way around. We modifythe β profile in two di ff erent ways: 1) β ( r ) → β ( r ) + S + T r , and 2) β ( r ) → J β ( r ) + Y , using a wide grid of values for the constants,respectively ( S , T ) and ( J , Y ). Using the mass and anisotropyprofiles, it is then possible to determine σ r ( r ) and then the σ los profile (e.g. Mamon & Łokas 2005b). The range of acceptable β profiles is determined by a χ comparison of the resulting σ los profiles with the observed one.In addition, another source of uncertainty on the β profilesolution comes from the uncertainty in the mass profile. Thisis estimated by running the anisotropy inversion for four di ff er-ent mass profiles corresponding to the combination of allowedvalues of virial and mass scale radii within 1 σ . The profiles ob-tained modifying the mass profile (not shown) lie within the con-fidence interval of the main result, so that the confidence intervalrepresents the uncertainty on the anisotropy profile well.The ALL sample β ( r ) depends weakly on radius: the inner-most region is compatible with isotropy, while the anisotropy isincreasingly radial at large radii. The RED sample is compatiblewith isotropy at all radii. The di ff erence between the two sam-ples is almost entirely due to the BLUE galaxies, the anisotropyof which is compatible with isotropy in the centre, then becomesrapidly radially anisotropic, and finally flattens at radii > β obtained from theanisotropy inversion with the best-fit results of DK and MAM-POSSt. In these techniques, we assumed a constant value of theanisotropy for the RED sample, which appears to be a good as-sumption given the results of β after the inversion. The valueestimated by both DK and MAMPOSSt is β = .
0, consistentwithin the uncertainties with the β profile shown in Fig. 9. Fig. 9.
Velocity anisotropy profiles for the ALL, RED, and BLUEsamples. The solid line is the result of the inversion of the Jeans equa-tion, while the dotted lines are the 1 σ confidence intervals. The verticaldashed line locates the virial radius. Q ( r ) and β − γ relations Since our anisotropy inversion provides us with the radial varia-tions of σ ( r ) = q σ r + σ t , β ( r ) (from which σ r ( r ) follows), wecan take advantage of the results just found for the galaxy popu-lations of A2142 to test the PPSD profile and the relation linkingthe logarithmic slope of the density profile and the anisotropy β ( r ).Both the PPSD and β − γ relations were derived from dissi-pationless single-component simulations. It is therefore not clearwhether the power-law PPSD and the linear beta-gamma rela-tion, both found for the particles of single component dark mat-ter (DM)-only simulations, will be obtained when using galax-ies to measure the velocity dispersion or velocity anisotropy andwhether one should use the total density or the galaxy numberdensity in these two relations. We discuss this further in Sec-tion 7. We begin by adopting the total density profile ρ ( r ). We computeboth the PPSD profile Q ( r ) = ρ/σ and its radial counterpart Q r ( r ) = ρ/σ r . In Fig. 10, we show, for the di ff erent tracers (ALL, Article number, page 8 of 13unari, E. et al.: Mass, velocity anisotropy, and pseudo phase–space density profiles of Abell 2142
Fig. 10.
Radial profiles of Q (left columns) and Q r (right columns)within the virial radius, and the 1 σ confidence regions (shaded areas),for di ff erent types of member tracers: green for the ALL sample (toppanels), red for the RED sample (middle panels), and blue for the BLUEsample. The shaded areas represent the propagation of the errors asso-ciated with ρ , σ and σ r . The dashed lines are the power-law relations Q ( r ) ∝ r − . and Q r ( r ) ∝ r − . found by Dehnen & McLaughlin (2005)on numerically simulated haloes. The vertical dotted lines locate thevirial radius of the combined model (see Sect. 4). RED, BLUE), the radial profile of Q ( r ) (left panels) and Q r ( r )(right panels) within the virial radius. To compute the errors onthe best-fit slope parameters, we have assumed that the numberof independent Q and Q r values are the same as those of theobserved velocity dispersion profile (see Fig. 5).Assuming a power-law behaviour of the PPSD profile, assuggested by Dehnen & McLaughlin (2005), we fit the profilesof both Q ( r ) and Q r ( r ) in two ways, either keeping the exponentfixed to the values found for haloes in Λ CDM simulations byDehnen & McLaughlin (2005) or considering it as a free param-eter. In both cases the normalization is left as a free parameter.In Table 4 the results of such fits are shown. The Q ( r ) profilefor the RED sample is consistent within less than 2 σ with the r − . relation by Dehnen & McLaughlin (2005). The fit of theprofile with a linear relation in the log-log plane is compatiblewith the theoretical value − .
84 within 1 . σ . On the other hand,for the BLUE sample, the slope of the PPSD is steeper than thetheoretical expectation.The σ r profile is a ff ected by larger uncertainties than the σ profile, because the former combines the uncertainties fromthe latter and β ( r ), which are the parameters produced bythe anisotropy inversion algorithm of Solanes & Salvador-Solé(1990). It is therefore not surprising that, within the uncer-tainties, the Q r profiles of all three samples appear consistentwith the theoretical expectation for simulated Λ CDM haloes(Dehnen & McLaughlin 2005), Q r ∝ r − . . We note, however,that the agreement is quite remarkable (within 0 . σ ) for theRED sample.Ludlow et al. (2010) warn against fitting the pseudo phase–space density profile outside the scale radius, because of the up-turn they find in the Q ( r ) profile in the outer regions. However,for our three samples, none of the Q ( r ) and Q r ( r ) profiles showsignificant curvature in log-log space.In Fig. 11, we show the anisotropy - density slope relation.The β − γ relation of the ALL sample matches the one foundby Hansen & Moore (2006) closely in single-component dissi-pationless simulations (cosmological and academic); however, Fig. 11.
Velocity anisotropy versus logarithmic slope of the total den-sity profile. The samples are ALL galaxies (top), RED (middle), andBLUE galaxies (bottom panel). The shaded areas are the 1 σ confidenceregions. The β − γ relation found by Hansen & Moore (2006) for single-component dissipationless simulations is shown as the dotted lines. Thedashed line is the limit below which the relation by Ciotti & Morganti(2010) holds. The vertical dot-dashed line locates the value of γ at thevirial radius. the β − γ relation for the RED sample shows curvature, withlower values of β at the steeper slopes (larger radii) than foundin simulations by Hansen & Moore (2006).It can be proven that all multicomponent spherical systemswith positive phase-space distribution function, for which 1)the density of a component is a separable function of totalgravitational potential and radius, 2) β (0) ≤ / β ( r ) < − γ ( r ) /
2, where the velocity anisotropy β and the loga-rithmic slope of density γ are for that component, as shown inCiotti & Morganti (2010) (see also Van Hese et al. 2011). It isnot clear whether the galaxy components of clusters of galaxieshave such separable densities. As we discuss at length in Sect.7, it is not obvious that oneshould use the total mass density profile rather than the tracernumber density profile in evaluating the PPSD and the β − γ relations, when we want to compare them to those found in nu-merical simulations. As a result, we now repeat our analyses ofthe PPSD and the β − γ relations, replacing the total mass densitywith the number density of the tracer of the sample.In Fig. 12, we show the PPSD computed using the galaxynumber density profile instead of the total matter densityone. These PPSDs are either consistent with the relation ofDehnen & McLaughlin ( Q ( r ) for BLUE sample) or only slightlyshallower, but not less consistent with that relation than foundfor the PPSDs computed with the mass density profile. In Fig. 13, we show the β − γ relation computed using thegalaxy number density profile instead of the total matter densityone. The behaviour does not change significantly from the caseof the β − γ relation computed using the total matter density pro-file: the overall shapes of the profiles are similar, but the BLUEsample now presents a noisier profile, while ALL and RED pro- Items in red in the main text are the consolidated version includingthe Corrigendum of Munari et al. (2015) Article number, page 9 of 13 & Aproofs: manuscript no. A2142
Table 4.
Best-fit parameters of the PPSD profile Q ( r ) Q r ( r ) A B A B [M ⊙ Mpc − km − s ] [M ⊙ Mpc − km − s ]Fixed slopeALL 5534 ± − .
84 25071 ± − . ± − .
84 38484 ± − . ± − .
84 3998 ± − . ± − . ± .
11 29175 ± − . ± . ± − . ± .
09 38881 ± − . ± . ± − . ± .
50 5413 ± − . ± . Q ( r ) GAL Q r ( r ) GALA B A B [10 − Mpc − km − s ] [10 − Mpc − km − s ]Fixed slopeALL 3 . ± . − .
84 17 . ± . − . . ± . − .
84 13 . ± . − . . ± . − .
84 0 . ± . − . . ± . − . ± .
10 17 . ± . − . ± . . ± . − . ± .
09 14 . ± . − . ± . . ± . − . ± .
48 0 . ± . − . ± . Notes.
The PPSD profile is parametrized as Q ( r ) = A r B . The first panel at the top shows the results of the fit of Q ( r ) and Q r ( r ) for the di ff erentsamples, both when keeping the exponent fixed to the values suggested by Dehnen & McLaughlin (2005), and when considering the exponent as afree parameter. In the bottom panel (identified by Q ( r ) GAL and Q r ( r ) GAL ), the same quantities are shown, but they refer to the PPSD computedusing the galaxy number density profile instead of the total matter density profile. files are shifted towards higher values of γ , reflecting the shal-lower trend of the galaxy number density profile with respect tothe matter density one. We discuss these results below.
7. Discussion
Munari et al. (2013) report the scaling relation between the virialmass of clusters and the velocity dispersion of the member galax-ies within the virial sphere. Using the most realistic (“AGN”)hydrodynamical simulation at their disposal, they find σ D = h ( z ) M / M ⊙ ] . for the galaxies within the virialsphere, where σ D is the total 3 D velocity dispersion within r ,divided by √
3. The analysis was carried out in the 6D phasespace, so is immune to projection e ff ects. The statistical natureof their relation suggests that it should hold for real, observed,and relaxed systems. As a test, we checked the consistency ofthe velocity dispersion – mass relation found by Munari et al.(2013) with our findings for A2142. The values of virial massobtained with this relation are: 1 . × M ⊙ for the ALL sam- ple, 1 . × M ⊙ for the RED sample, and 2 . × M ⊙ forthe BLUE sample. The values obtained for the ALL and REDsamples agree, within the uncertainties, with the combined valueof the mass of A2142. This seems to indicate that RED clustermembers are in, or very close to, equilibrium. The large di ff er-ence obtained for the BLUE cluster members warns against us-ing the blue galaxy los velocity dispersion as a proxy for thecluster mass.A glance at Table 3 indicates that our di ff erent estimates ofthe mass concentrations are bimodal: the caustic and weak lens-ing have values ≃
4, while those for the DK, MAMPOSSt, andX-ray methods are <
3. Could these lower mass concentrationsfound by methods based upon internal kinematics be a sign thatA2142 is out of dynamical equilibrium? The substructures foundby Owers et al. (2011) and the results by Rossetti et al. (2013) onthe importance of the mergers undergone by A2142 suggest thatfull relaxation is to be excluded. On the other hand, the agree-ment on the virial radius amongst the di ff erent methods and withthe results from X-ray and lensing (the latter does not requireequilibrium) suggests that A2142 is not far from dynamical equi-librium. Article number, page 10 of 13unari, E. et al.: Mass, velocity anisotropy, and pseudo phase–space density profiles of Abell 2142
Fig. 12.
Same as Fig. 10, but now using the radial profiles of galaxynumber density instead of total mass density to estimate the PPSD.
Fig. 13.
Same as Fig. 11, but now using the radial profiles of galaxynumber density of the three samples instead of total mass density toestimate the slope.
In Fig. 14, the concentration – mass relation for A2142 isshown along with theoretical relations by Bhattacharya et al.(2013) and De Boni et al. (2013) based on cosmological N–body and hydrodynamical simulations, respectively. The valueof the combined model [ M = (1 . ± . × M ⊙ and c = . ± .
5] agrees within 1 σ with the “relaxed” case ofDe Boni et al. (2013), while it is in excellent agreement withboth the relations by Bhattacharya et al. (2013). This strength-ens the scenario of A2142 being very close to equilibrium. Previous studies based on the kinematics of galaxies in clus-ters have shown that galaxy populations have similar con-centrations to those of the total matter, or slightly smaller,blue galaxies being instead much less concentrated (see, e.g.Biviano & Girardi 2003; Katgert et al. 2004). On the other hand,Biviano & Poggianti (2009) found in the ENACS clusters thatthe red galaxy population has a concentration that is as muchas 1.7 times lower for the total matter density profile. Here, wefind that the scale radius for the RED galaxy number densityprofile (0.95 Mpc) is 1.8 times greater than for the total mass
Fig. 14.
Concentration – mass relation, with respect to an overden-sity 200 times the critical one. Purple asterisk refers to the X–ray val-ues by Akamatsu et al. (2011), orange diamond to the WL values byUmetsu et al. (2009), small red circle refers to the values obtained byMAMPOSSt, blue square by the caustic method, black triangle by DK,the big black circle to the values of the combined model. Lines arethe theoretical predictions, and in black the relations by De Boni et al.(2013) when considering all (solid) and relaxed (dotted) clusters. Ingreen the relations by Bhattacharya et al. (2013) when considering all(solid) and relaxed (dotted) clusters. density profile from our combined model, which agrees withthe ENACS result. Collister & Lahav (2005) found, on a stackedsample from the 2dFGRS, values of galaxy concentration com-parable to ours, when considering objects as massive as A2142(see their Fig. 7), although with uncertainties that are too largeto distinguish between red and blue samples.The scale radius of the BLUE population in Abell 2142 ap-pears unusually high, leading to concentrations (using our com-bined virial radius) of 0.16 (best) or 0.39 ( + σ ), which are muchlower than expected from previous studies. Blue galaxies withinthe virial cones of clusters are more prone to projection e ff ectsthan red galaxies: Mahajan et al. (2011) analyzed clusters andtheir member galaxies in the SDSS, using los velocities and cos-mological simulations to quantify the projection e ff ects. Theyconclude that 44 ±
2% of galaxies with recent (or ongoing) star-bursts that are within the virial cone are outside the virial sphere.Since galaxies with recent star formation have blue colours, ourBLUE sample includes this recent-starburst subsample, plus per-haps some more galaxies with more moderate recent star for-mation. Moreover, an analysis of cosmological simulations byMamon et al. (2010) indicates that there is a high cosmic vari-ance in the fraction of interlopers within the DM particles insidethe virial cone. This suggests that the unusually low concentra-tion of the blue galaxy sample could be a sign of an unusuallyhigh level of velocity interlopers with low rest–frame velocitiesin front of and behind Abell 2142.Wojtak & Łokas (2010) find a virial radius that correspondsto r = . + . − . Mpc, in excellent agreement with our di ff er-ent estimates of the virial radius (Table 3). On the other hand,they find a mass scale radius r s = . + . − . Mpc not compatiblewith our value of the combined model, although in agreementwith the results of the DK, MAMPOSSt, and X-ray analyses.Wojtak & Łokas assumed that the DM and galaxy scale radiiwere equal. Such an unverified assumption may have biased hightheir scale radius for the mass distribution. On the other hand,the values of the mass scale radii that we found from DK and
Article number, page 11 of 13 & Aproofs: manuscript no. A2142
MAMPOSSt (0.93 and 0.83 Mpc, respectively, see Table 3) areconsistent with that of the RED galaxy population used as thetracer (0.95 Mpc), as is in Wojtak & Łokas, both within the un-certainties.
The velocity anisotropy profile for the ALL sample in the cen-tre is compatible with the one found by Wojtak & Łokas (2010).In the outer part, at ≃ σ r /σ θ found byWojtak & Łokas (2010) is higher and o ff set from ours by 1 . σ .An analysis of a stacked sample of 107 nearby ENACS clusters(Biviano & Katgert 2004; Katgert et al. 2004) shows that the or-bits of ellipticals and S0s (hence red) galaxies are compatiblewith isotropy, while those of early and late-type spirals have ra-dial anisotropy. At slightly higher redshifts, van der Marel et al.(2000) also find red galaxies close to isotropy. The velocityanisotropy profile for our BLUE sample presents behaviour thatlies in between the profiles found in Biviano & Katgert for theearly spirals and the late spirals together with emission linegalaxies, suggesting agreement between their findings and ours.The anisotropy profile we found for the ALL sample appearsto be consistent with those measured by Lemze et al. (2012)and Mamon et al. (2013) in simulated Λ CDM haloes. In sim-ulations, data are usually stacked or averaged, and the scatterin the anisotropy profiles is considerable (see e.g. Lemze et al.2012; Mamon et al. 2013) and this reflects the variety of config-urations of galaxy clusters. A2142 does not present strong de-viations from the general trend, because its anisotropy profile iscompatible with this scatter. β − γ relation Biviano et al. (2013) analyzed the pseudo phase–space densityon MACS1206, a cluster at z = .
44. They find a Q ( r ) pro-file with a slope for the blue galaxies in agreement with thepredictions of Dehnen & McLaughlin (2005), at odds with ourfindings. We speculate that this di ff erent behaviour might pro-vide a hint of the dynamical history of clusters. In fact, a clusterthat has only recently undergone the phase of violent relaxationmight present a population of blue galaxies in rough dynamicalequilibrium. On the other hand, a cluster that has undergone theviolent relaxation phase a long time ago, should have had timeto transform its blue galaxies into red ones. Therefore the bluegalaxy population would be mainly composed of only recentlyaccreted galaxies, hence not in dynamical equilibrium.While galaxies are biased tracers of the DM velocity dis-persion (Munari et al. 2013), if the velocity dispersion profile ofthe galaxy component is proportional to that of the DM compo-nent at all radii (i.e. no velocity bias relative to the DM), thenthe PPSDs built from the galaxies should have the same slopeas the one built from the DM. On the other hand, if the ve-locity bias of the galaxy component is a function of radius, asfound by Wu et al. (2013) in cosmological hydrodynamical sim-ulations of clusters, then the PPSD built from the galaxies willbe di ff erent from the one built with the DM component (afterproper normalization). Since the DM component dominates thegravitational potential of clusters, we infer that the consistencyof the PPSD, built with the total density and the velocity disper-sion of the RED galaxy component, suggests that the velocitybias of the component of red galaxies outside of substructures isroughly independent of radius. At all radii, the RED galaxy sample shows somewhat lower β for given γ (measured with total mass density) than found insimulated haloes. However, the β − γ relations have been de-rived using DM-only simulations, which do not take the e ff ectsof the presence of baryons into account. Now, if the tangentialand radial components of the velocity dispersion of the galaxypopulation are proportional to those of the DM, then the veloc-ity anisotropy of the galaxy population, written as A = σ r /σ θ should be proportional to that of the DM, but the non-linear func-tion of A , β = − / A , measured for the galaxies, will not nec-essarily be proportional to the analogous β for the DM. There-fore, any radial variation of A , hence β , will lead to a bias in the β − γ relation. Finally, the β − γ relation may vary from clusterto cluster (Ludlow et al. 2011).
8. Conclusions
We have computed the mass and velocity anisotropy profiles ofA2142, a nearby ( z = .
09) cluster, using the kinematics of clus-ter galaxies. After a membership algorithm was applied, we con-sidered the sample made of all members (ALL sample), as wellas two subsamples, consisting of blue member galaxies (BLUEsample) and red member galaxies that do not belong to substruc-tures (RED sample).We made use of three methods based on the kinematicsof galaxies in spherical clusters: DK, MAMPOSSt and Caus-tic (see Sect. 3). The mass profiles, as well as the virial valuesof the mass and the radius, are consistent among the di ff erentmethods, and they agree with the results coming from the X-ray (Akamatsu et al. 2011) and the weak lensing (Umetsu et al.2009) analyses. Serra et al. (2011) find that the caustic techniquetends to overestimate the value of mass in the central region of acluster. Our results appear consistent with this finding, becausethe caustic mass profile increases more rapidly with radius inthe inner part with respect to the profiles coming from DK andMAMPOSSt.The parameters describing the mass profile are then used toinvert the Jeans equation and compute the velocity anisotropyfor the three di ff erent samples considered. Despite large uncer-tainties, the β ( r ) profile for the full set of cluster members iscompatible with isotropy, becoming weakly radially anisotropicin the outer regions. The behaviour of the RED sample is dif-ferent. Although compatible within 1 σ with isotropy at all radiiwithin r , it is suggestive of a decreasing slope, starting slightlyradially anisotropic in the centre and becoming slightly tangen-tially anisotropic at large radii. The di ff erence between the β ( r )profiles for the ALL sample and the RED sample is mainlydue to the behaviour of the BLUE sample, which shows radialanisotropy at all radii except in the centre where it is isotropic.With the information obtained on A2142, we were able totest some theoretical relations regarding the interplay betweenthe mass distribution and the internal kinematics of a cluster.We investigated the radial profile of the pseudo phase–spacedensity (PPSD) Q ( r ), as well as its radial counterpart Q r ( r ).When we considered the total density profile to compute Q and Q r , we found that the profiles for A2142 are weakly consistentwith the theoretical expectations (Dehnen & McLaughlin 2005;Ludlow et al. 2010) when considering the ALL sample, but agood agreement is observed in the RED sample. This strength-ens the scenario of blue galaxies being a population of galaxiesrecently fallen into clusters, which have had no time to reachan equilibrium configuration yet, or are heavily contaminated byinterlopers. Article number, page 12 of 13unari, E. et al.: Mass, velocity anisotropy, and pseudo phase–space density profiles of Abell 2142
We estimated the PPSD profile of the total matter, making theassumption that the galaxy velocity dispersion is a good proxyfor the total matter dynamics. The PPSDs computed replacingthe total mass density by the number density of the tracer forwhich we compute the velocity dispersion are consistent withthose computed with the mass density profile.The velocity anisotropy configuration of the internal kine-matics reflects the formation history of the cluster. Therefore weexpect a relation between the velocity anisotropy and the poten-tial of the cluster. A relation linking the β ( r ) profile and γ ( r ),the logarithmic slope of the potential, has been analysed andcompared to the theoretical results provided by Hansen & Moore(2006), resulting in weak agreement. A correlation between the β and γ appears to hold out to γ ≃ − . ≃ . r ≃ Λ CDM haloes also follow theHansen & Moore relation out to slopes of γ ≈ − . β − γ relation using the loga-rithmic slope of the number density profile of galaxies instead ofthe total matter density profile.Before reaching any conclusion, we must keep in mind thatthe present theoretical studies of the β − γ and PPSD relationslack the influence of baryonic physics, as well as the dynamicalprocesses acting on galaxies but not on DM particles. This mightinduce the di ff erences when comparing the theoretical predic-tions with the observational results.When we have better control of these properties, the PPSDmight provide a powerful tool for the study of structure forma-tion. As an example, the PPSD of the blue galaxies in A2142appears very di ff erent from what has been found for the bluegalaxies in another cluster, MACS J1206.2–0847 at z = . Acknowledgements.
We thank Luca Ciotti, Colin Norman, Barbara Sartoris, andRadek Wojtak for useful discussions, an anonymous referee for helpful com-ments, and Anthony Lewis for building the public CosmoMC Markov ChainMonte Carlo code. We acknowledge partial support from “Consorzio per laFisica - Trieste” and from MIUR PRIN2010–2011 (J91J12000450001). AB andEM acknowledge the hospitality of the Institut d’Astrophysique de Paris.
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