Dark matter and Modified Newtonian Dynamics in a sample of high-redshift galaxy clusters observed with Chandra
DDark matter and Modified Newtonian Dynamics in asample of high-redshift galaxy clusters observed withChandra
Carl Blaksley a , Massimiliano Bonamente b a Department of Physics, University of Alabama, Huntsville, AL, USA andArnold Sommerfeld Center For Theoretical Physics, Faculty of Physics,Ludwig-Maximilians Universit¨at, M¨unchen, Germany b Department of Physics, University of Alabama, Huntsville, AL, USA andNASA National Space and Science Technology Center, Huntsville, AL, USA
Abstract
We compare the measurement of the gravitational mass of 38 high-redshiftgalaxy clusters observed by
Chandra using Modified Newtonian Dynamics(MOND) and standard Newtonian gravity. Our analysis confirms earlierfindings that MOND cannot explain the difference between the baryonicmass and the total mass inferred from the assumption of hydrostatic equi-librium. We also find that the baryon fraction at r using MOND isconsistent with the Wilkinson Microwave Anisotropy Probe (WMAP) valueof Ω B / Ω M . Key words: galaxies: clusters: generalcosmology: dark matter95.35.+d95.85.Nv
1. Introduction
Since the introduction by Milgrom in 1983 (Milgrom, 1983), MONDhad success in explaining galaxy rotation curves using only the mass-to-light ratio as a free parameter, and was able to predict the applicability ofthe Tully-Fisher relation to low surface brightness galaxies before dynamicinformation on them was available (Scarpa, 2006). These successes are notsurprising, given the fact that MOND was created as a phenomenologicaltheory in order to eliminate the need for dark matter in galaxies.While there have been numerous applications of MOND to individualgalaxies, it is also important to study MOND on those scales where dark
Preprint submitted to New Astronomy April 13, 2019 a r X i v : . [ a s t r o - ph . C O ] J u l atter is believed to dominate, in particular that of galaxy clusters. Therehave been far fewer studies of MOND on this scale, with previous work onthe subject including Sanders (1999), Aguirre et al. (2001), Pointecouteau& Silk (2005) and Angus et al. (2008). These results indicate that MONDdoes not eliminate the need for dark matter in galaxy clusters. In this paperwe use MOND to calculate gravitational and baryonic masses for a sampleof 38 galaxy clusters. The motivation of this work is the need to confirm theprevious results by using a larger sample of massive clusters at high redshift(z=0.14-0.89).We begin with a brief overview of the data and the models used, foundin Section 2. Section 3 describes the calculation of the gas mass using X-rayobservation, the derivation of the MOND acceleration, and the calculationof total masses from hydrostatic equilibrium. We then present our resultsin Section 4 and our conclusions in Section 5. The cosmological parameters h = 0 .
7, Ω M =0.3 and Ω Λ =0.7 are used throughout this work.
2. Chandra X-ray Data And Data Modeling
We analyze
Chandra
X-ray data from 38 clusters in the redshift range z =0.14—0.89 (Table 1). The Chandra observations and the data model-ing with the isothermal β model are presented in Bonamente et al. (2006);LaRoque et al. (2006); Bonamente et al. (2008). Here those aspects of thedata modeling and analysis that are relevant to the present investigation.The electron density model is based on the isothermal spherical β -model(Cavaliere & Fusco-Femiano, 1976, 1978), which has the form n e ( r ) = n e (cid:18) r r c (cid:19) − β/ , (1)where n e is the central electron number density, r c is a core radius, and β is a power-law index. The radial profile of the X-ray surface brightness isobtained via integration along the line of sight, and results in the followinganalytical expression: S x = S x (cid:18) θ θ c (cid:19) (1 − β ) / . (2)Best-fit model parameters and confidence intervals for all model param-eters are obtained using a Markov chain Monte Carlo (MCMC) methoddescribed in detail by Bonamente et al. (2004) and LaRoque et al. (2006).For each cluster, the Markov chain constrains the parameters S x , β , θ c ,2 e , and the chemical abundance (see LaRoque et al. 2006 for best-fit val-ues). We use the cosmological parameters h = 0 .
7, Ω M =0.3 and Ω Λ =0.7 tocalculate each cluster’s angular diameter distance D A (e.g., Carroll et al.,1992).All of our calculations are done out to a maxium radius of r , theradius at which the cluster mass density is 2500 times the critical density,which is also approximately the radius out to which our Chandra data aresensitive without any extrapolation of the models. In LaRoque et al. (2006)we have compared masses obtained using the simple isothermal β modelwith those obtained using a more complex non-isothermal β model, andshown that the two methods yield the same ratio of baryonic to total mass.The non-isothermal model included a double β model distribution for thedensity (Equation 7 in LaRoque et al. 2006), and an arbitrary temperatureprofile that was constrained by assuming that the plasma is in hydrostaticequilibrium with an NFW potential (Equation 8 in LaRoque et al. 2006).We therefore expect no significant bias in the study of MOND masses usingthis simple model, which has the advantage of analytical expressions for theobservables and the masses.
3. Calculation of gas, Newtonian and MOND Masses
For the isothermal β model, the gas mass enclosed within a given clusterradius is given by (e.g. LaRoque et al., 2006) M gas ( r ) = A (cid:90) r/D A (cid:16) θ θ c (cid:17) − β/ θ dθ, (3)where A = 4 πµ e n e ◦ m p D A , µ e is the mean electron weight (calculated fromthe Chandra data, with typical value of µ e = 1 . D A is the clusterdistance. The central electron density n e ◦ is calculated from the X-raysurface brightness (LaRoque et al., 2006; Birkinshaw et al., 1991) as n e ◦ = (cid:34) S x ◦ π (1 + z ) ( µ H /µ e )Γ(3 β )Λ eH D A π / (3 β − / θ c (cid:35) / , (4)in which µ H is the mean hydrogen weight (typical value µ H = 1 . eH is the emissivity of the plasma. The gas mass in Equation (3) is measureddirectly from observables and does not depend on the law of gravity in anyway. 3or the total mass, M total , we solve the hydrostatic equilibrium equationusing Newtonian gravitation and the ideal gas law, to obtain the Newtonianmass: M total ( r ) = − kr Gn e ( r ) µ tot m p (cid:34) T e ( r ) dn e ( r ) dr + n e ( r ) dT e ( r ) dr (cid:35) , (5)in which µ tot is the mean molecular weight (typical value µ tot = 0 . M total ( r ) = 3 βkT e Gµm p r r c + r . (6)Starting with the most basic form of MOND (Pointecouteau & Silk,2005; Scarpa, 2006), we derive the gravitational mass based on the MONDlaw of gravity. The MOND acceleration ( a ) is related to the Newtonianacceleration ( a N ) by the relation GM ( r ) r = a N = a · µ ( aa ◦ ) . (7)where a ◦ is a critical acceleration that is meant to be a new physical constantand represents the acceleration below which Newtonian gravity is no longervalid (Milgrom, 1983; Scarpa, 2006); the function µ is any interpolatingfunction that provides a transition between the two regimes. We take thecritical acceleration to be a =1 . e − cm/s as used in Angus et al. (2008);this value is derived from galaxy rotation curve fits Scarpa (2006). In Section4 we also discuss the results obtained using a variable value of the criticalacceleration.The interpolating function µ ( x ) is any function which yields the appro-priate asymptotic behavior, µ ( x ) = (cid:26) x (cid:29) x for x (cid:28) µ ( x ) = x √ x . (8)4sing Equation (8) in Equation (7) and solving for a , gives us the relation a = a N (cid:32)
12 + 12 (cid:115) a ◦ a N (cid:33) . (9)Equation (9) is the MOND acceleration as a function of the Newtonianacceleration and the critical acceleration parameter (see also Pointecouteau& Silk, 2005). We use the hydrostatic equilibrium equation according toMOND, − ρ g ( r ) dP g dr = a ( r ) , (10)( ρ g ( r ) is the gas density and P g its pressure) in order to determine theMOND equation for total mass analogous to Equation (6) for Newtoniangravity. Using Equations (9) and (10) with a N = GM ( r ) /r leads to theelimination of the Newtonian acceleration a N : GM ( r ) r = (cid:32) − ρ ( r ) dPdr (cid:33) (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) a ◦ + (cid:32) − ρ ( r ) dPdr (cid:33) (cid:33) . (11)For the β model profile of density we obtain: M MOND ( r ) = r G (cid:32) βKTµm p rr c + r (cid:33)(cid:118)(cid:117)(cid:117)(cid:116) a ◦ + (cid:32) βKTµm p rr c + r (cid:33) . (12)This equation can be further simplified when one realizes that the numera-tor is the total mass according to Newtonian dynamics, Equation (6). Thissimplification gives us a final equation for the total mass based on the hy-drostatic equilibrium condition and using MOND gravity. M MOND ( r ) = M Newton ( r ) (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) a ◦ µm p βKT ( r c + r ) r (cid:33) (13)This equation displays the appropriate asymptotic behavior, namely thatfor a critical acceleration of zero we recover the Newtonian M total .5 . Results Having found an equation for the MOND total mass ( M MOND ) we cal-culate M gas , M total and M MOND for each of the 38 galaxy clusters in oursample out to a radius of r . In each case M gas was calculated usingEquation (3), M total by means of Equation (6), and M MOND using Equa-tion (13). For each cluster, the MOND gravitational mass is lower than thethe Newtonian gravitational mass by just ∼ − M gas (see Table 1). The radial profiles ofthe masses are shown for all clusters in Appendix A.In order to calculate the baryonic mass we approximate the stellar con-tribution by M ∗ = 0 . M gas , as is done in Pointecouteau & Silk (2005).We therefore estimate the total baryonic mass as M baryon = M gas + M ∗ .We find that the average f gas ( r ) for MOND is 14.2 ± f baryon ( r ) is 16.4 ± f baryon =20.2 ± r . We find that, even withthe inclusion of M ∗ , the baryon mass is still significantly lower than thetotal MOND mass for every cluster, confirming the fact that MOND isincapable of eliminating the need for dark matter in clusters. The mean f gas and f baryon in Table 2 are calculated as the weighted mean of all mea-surement, in which the weight is the standard error of each measurement.We also calculate the un-weighted means and their root-mean-square er-rors as f gas =0 . ± .
003 (Newton) and f gas =0 . ± .
004 (MOND), and f baryon =0 . ± .
004 (Newton) and f baryon =0 . ± .
005 (MOND).We show the average distributions of f baryon and M total /M MOND as func-tion of physical radius in Figure 1. The distribution of f baryon shows thatthe dynamical mass is always significantly higher than the baryonic mass.The distribution of M total /M MOND shows that the difference between theNewtonian and MOND masses are largest near the cluster core and at thelargest radii, where the gravitational accelerations are lowest and thereforethe MOND correction largest. Typical values of r are between 0.5 and 1Mpc, therefore we do not extrapolate these plots beyond 1 Mpc. The Chan-dra data presented in this paper were fit with a 100 kpc core cut (LaRoqueet al., 2006), and therefore we do not extrapolate our models below thisradius either.The cluster baryon fraction calculated via the standard Newtonian dy-namics in this paper ( (cid:104) f baryon (cid:105) = 0 . β -Model Cluster z r D A M gas M total M Mond ( arcsec ) ( Gpc ) (10 M (cid:12) ) (10 M (cid:12) ) (10 M (cid:12) )Abell1413 . . . . . . . . . . 0 .
143 201 . +5 . − . .
52 2 . +0 . − . . +0 . − . . +0 . − . Abell1689 . . . . . . . . . . 0 .
18 217 . +5 . − . .
63 5 . +0 . − . . +0 . − . . +0 . − . Abell1835 . . . . . . . . . . 0 .
25 171 . +5 − . .
81 5 . +0 . − . . +0 . − . . +0 . − . Abell1914 . . . . . . . . . . 0 .
17 226 . +4 . − . . +0 . − . . +0 . − . . +0 . − . Abell1995 . . . . . . . . . . 0 .
32 133 . +4 . − . .
96 3 . +0 . − . . +0 . − . . +0 . − . Abell2111 . . . . . . . . . . 0 .
23 140 . +9 . − . .
76 2 . +0 . − . . +0 . − . . +0 . − . Abell2163 . . . . . . . . . . 0 . . +3 − . .
68 8 . +0 . − . . +0 . − . . +0 . − . Abell2204 . . . . . . . . . . 0 .
15 256 . +13 − .
54 4 . +0 . − . . +0 . − . . +0 . − . Abell2218 . . . . . . . . . . 0 .
18 190 +5 . − . .
63 3 . +0 . − . . +0 . − . . +0 . − . Abell2259 . . . . . . . . . . 0 .
16 172 . +8 . − . .
57 1 . +0 . − . . +0 . − . . +0 . − . Abell2261 . . . . . . . . . . 0 .
22 148 . +6 . − . .
73 3 . +0 . − . . +0 . − . . +0 . − . Abell267 . . . . . . . . . . . 0 .
23 131 . +8 . − . .
76 2 . +0 . − . . +0 . − . . +0 . − . Abell370 . . . . . . . . . . . 0 .
38 97 . +4 . − . .
07 2 . +0 . − . . +0 . − . . +0 . − . Abell586 . . . . . . . . . . . 0 .
17 181 . +8 − . . +0 . − . . +0 . − . . +0 . − . Abell611 . . . . . . . . . . . 0 .
29 110 . +3 . − . . . +0 . − . . +0 . − . . +0 . − . Abell665 . . . . . . . . . . . 0 .
18 160 . +3 . − . .
63 2 . +0 . − . +0 . − . . +0 . − . Abell68 . . . . . . . . . . . . 0 .
26 153 +10 − . .
83 3 . +0 . − . . +0 . − . . +0 . − . Abell697 . . . . . . . . . . . 0 .
28 133 . +5 − . .
88 4 . +0 . − . . +0 . − . . +0 . − . Abell773 . . . . . . . . . . . 0 .
22 148 . +5 . − . .
73 2 . +0 . − . . +0 . − . . +0 . − . CLJ0016+1609 . . . . . 0 .
54 79 . +3 − .
31 4 . +0 . − . . +0 . − . . +0 . − . CLJ1226+3332 . . . . . 0 .
89 66 . +7 . − . . . +0 . − . . +2 − . . +1 . − . MACSJ0647.7+7015 0 .
58 91 . +6 . − . .
36 4 . +0 . − . . +1 . − . . +1 . − MACSJ0744.8+3927 0 .
69 58 . +3 . − . .
47 3 . +0 . − . . +0 . − . . +0 . − . MACSJ1149.5+2223 0 .
54 70 . +3 . − . .
31 3 . +0 . − . . +0 . − . . +0 . − . MACSJ1311.0-0310. 0 .
49 73 . +7 . − . .
25 2 . +0 . − . . +0 . − . . +0 . − . MACSJ1423.8+2404 0 .
55 65 . +2 . − .
32 2 . +0 . − . . +0 . − . . +0 . − . MACSJ2129.4-0741. 0 .
57 72 . +4 . − . .
35 3 . +0 . − . . +0 . − . . +0 . − . MACSJ2214.9-1359. 0 .
482 91 . +5 . − . .
23 3 . +0 . − . . +0 . − . . +0 . − . MACSJ2228.5+2036 0 .
41 83 . +4 . − . .
12 2 . +0 . − . . +0 . − . . +0 . − . MS0451.6-0305 . . . . . 0 .
55 82 . +3 . − . .
32 4 . +0 . − . . +0 . − . . +0 . − . MS1054.5-0321 . . . . . 0 .
83 33 . +7 . − . .
57 0 . +0 . − . . +0 . − . . +0 . − . MS1137.5+6625 . . . . 0 .
78 41 . +3 . − . .
54 1 . +0 . − . . +0 . − . . +0 . − . MS1358.4+6245 . . . . 0 .
33 113 . +5 . − . .
98 2 . +0 . − . . +0 . − . . +0 . − . MS2053.7-0449 . . . . . 0 .
58 54 . +5 . − . .
36 0 . +0 . − . . +0 . − . . +0 . − . RXJ1347.5-1145 . . . . 0 .
45 122 . +3 . − . .
19 8 . +0 . − . . +0 . − . . +0 . − . RXJ1716.4+6708 . . . 0 .
81 45 . +4 . − . .
56 1 . +0 . − . . +0 . − . . +0 . − . RXJ2129.7+0005 . . . 0 .
24 128 . +5 . − . .
78 2 . +0 . − . . +0 . − . . +0 . − . ZW3146 . . . . . . . . . . . . 0 .
29 131 . +2 . − . . . +0 . − . . +0 . − . . +0 . − . Cluster f { gas } f { baryon } Newton MOND Newton MONDAbell1413 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell1689 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell1835 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell1914 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell1995 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell2111 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell2163 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell2204 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell2218 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell2259 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell2261 . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell267 . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell370 . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell586 . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell611 . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell665 . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell68 . . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell697 . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Abell773 . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . CLJ0016+1609 . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . CLJ1226+3332 . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ0647.7+7015 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ0744.8+3927 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ1149.5+2223 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ1311.0-0310. 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ1423.8+2404 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ2129.4-0741. 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ2214.9-1359. 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MACSJ2228.5+2036 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MS0451.6-0305 . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MS1054.5-0321 . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MS1137.5+6625 . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MS1358.4+6245 . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . MS2053.7-0449 . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . RXJ1347.5-1145 . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . RXJ1716.4+6708 . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . RXJ2129.7+0005 . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . ZW3146 . . . . . . . . . . . . 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Weighted Mean 0 . ± .
001 0 . ± .
002 0 . ± .
001 0 . ± . f baryon and M total /M MOND for the 38clusters; the dashed lines represent the root-mean-square uncertainty of the38 measurements at each radius.value of Ω B / Ω M =0 . ± .
02 (Bennett et al., 2003), indicating the presenceof undetected baryons in clusters. Within MOND, the results of this paper( (cid:104) f baryon (cid:105) = 0 . B / Ω M . We therefore speculatethat MOND can in principle reconcile the cluster baryon fraction with thecosmic value of Ω B / Ω M , without the need for additional undetected baryonsin clusters. We also consider the possibility of a being a free parameter of theMOND theory, and use our data to determine it. To this end, we beginwith Equation (13) for the MOND total mass, and calculate the value of thecritical acceleration at each radius such that M MOND ( r ) = M baryon ( r ), M baryon ( r ) = M MOND ( r ) = M Newton ( r ) (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) a ◦ µm p βKT r c + r r (cid:33) . (14)We then solve equation (14) for a ◦ and find that a ◦ ( r ) = 3 βKTµm p rr c + r (cid:115) M Newton M baryon − , (15)9here M Newton is given by equation (6). The radial distribution of a ◦ isshown for all clusters in Appendix B.In Figure 2 we show the value of the critical acceleration calculated at r for all 38 clusters, with a mean value of a ◦ = 1 . ± . × − cm/s .The main result is that the value of the ”free” a ◦ at r differs from thatmeasured from galaxy rotation curves (e.g., Scarpa, 2006) by one order ofmagnitude. Moerover, we find statistically significant differences in the mea-surement of a ◦ at different radii for a given cluster (see Appendix B), andfor different clusters at the same radius of r (Figure 2).Figure 2: Calculation of a ◦ ( r ). The horizontal line is the mean of a ◦ and its root-mean-square error, calculated as the un-weighted mean of the38 measurements. The weighted mean of the 38 measurements is a ◦ =1 . ± . × − cm/s .We also show the average values of the best-fit critical acceleration a ◦ for the entire sample, as function of radius, in Figure 3. At all radii, thecritical acceleration required to explain the hydrostatic MOND mass is atleast one order of magnitude larger than the canonical value obtained fromgalaxy rotation curves (e.g., Scarpa, 2006).
5. Conclusions
In this paper we have measured gravitational and gas masses at r for a sample of 38 high-redshift galaxy clusters using standard Newtoniangravity and Modified Newtonian Dynamics. This is the largest sample ofgalaxy clusters to which the MOND theory of gravity is applied to date.10igure 3: Average distribution of a ◦ for the 38 clusters; the dashed linesrepresent the root-mean-square uncertainty of the 38 measurements at eachradiusWe initially used a fixed value of the MOND critical acceleration a ◦ =1 . e − cm/s and measured an average (cid:104) f baryon (cid:105) = 0 . a ◦ . Thevalue of a ◦ required to achieve a baryon fraction of unity varies both withcluster radius, and between clusters; moreover, the mean value of this free a ◦ is one order of magnitude larger than that required by galaxy dynamics.We therefore conclude that X-ray observations of the hot cluster plasma dorequire strong presence of dark matter even when MOND is used.This analysis also finds that the f baryon at r calculated using MONDis in statistical agreement with the WMAP value of Ω b / Ω M , similar to theresults of Pointecouteau & Silk (2005).11 ppendix A In this appendix we provide the radial profiles of the gas, Newtonian andMOND masses for all clusters, calculated between 100 kpc and r .Figure 4: Distribution of M gas , M total , and M MOND as a function of radius(arcsec) forall 38 clusters. The symbols are used as follows: (cid:13) for M total , × for M MOND , and (cid:52) for M gas (1) Abell1413 (2) Abell1689 (3) Abell1835(4) Abell1914 (5) Abell1995 (6) Abell2111(7) Abell2163 (8) Abell2204 (9) Abell2218
10) Abell2259 (11) Abell2261 (12) Abell267(13) Abell370 (14) Abell586 (15) Abell611(16) Abell665 (17) Abell68 (18) Abell697
19) Abell773 (20) CLJ0016+1609 (21) CLJ1226+3332(22) MACSJ0647.7+7015 (23) MACSJ0744.8+3927 (24) MACSJ1149.5+2223(25) MACSJ1311.0-0310 (26) MACSJ1423.8+2404 (27) MACSJ2129.4-0741
28) MACSJ2214.9-1359 (29) MACSJ2228.5+2036 (30) MS0451.6-0305(31) MS1054.5-0321 (32) MS1137.5+6625 (33) MS1358.4+6245(34) MS2053.7-0449 (35) RXJ1347.5-1145 (36) RXJ1716.4+6708(37) RXJ2129.7+0005 (38) ZW3146 ppendix B In this Appendix we provide the radial distribution of the best-fit valuesof the critical acceleration a , for all clusters. The plot extend between100 kpc and r .Figure 6: The critical acceleration a ◦ as a function of radius, under the condition M MOND = M baryon for all 38 clusters . (1) Abell1413 (2) Abell1689 (3) Abell1835(4) Abell1914 (5) Abell1995 (6) Abell2111(7) Abell2163 (8) Abell2204 (9) Abell2218
10) Abell2259 (11) Abell2261 (12) Abell267(13) Abell370 (14) Abell586 (15) Abell611(16) Abell665 (17) Abell68 (18) Abell697
19) Abell773 (20) CLJ0016+1609 (21) CLJ1226+3332(22) MACSJ0647.7+7015 (23) MACSJ0744.8+3927 (24) MACSJ1149.5+2223(25) MACSJ1311.0-0310 (26) MACSJ1423.8+2404 (27) MACSJ2129.4-0741
28) MACSJ2214.9-1359 (29) MACSJ2228.5+2036 (30) MS0451.6-0305(31) MS1054.5-0321 (32) MS1137.5+6625 (33) MS1358.4+6245(34) MS2053.7-0449 (35) RXJ1347.5-1145 (36) RXJ1716.4+6708(37) RXJ2129.7+0005 (38) ZW3146 eferenceseferences