Quantifying the Collisionless Nature of Dark Matter and Galaxies in A1689

Abstract

We use extensive measurements of the cluster A1689 to assess the expected similarity in the dynamics of galaxies and dark matter (DM) in their motion as collisionless "particles" in the cluster gravitational potential. To do so, we derive the radial profile of the specific kinetic energy of the cluster galaxies from the Jeans equation and observational data. Assuming that the specific kinetic energies of galaxies and DM are roughly equal, we obtain the mean value of the DM velocity anisotropy parameter and the DM density profile. Since this deduced profile has a scale radius that is higher than inferred from lensing observations, we tested the validity of the assumption by repeating the analysis using results of simulations for the profile of the DM velocity anisotropy. Results of both analyses indicate a significant difference between the kinematics of galaxies and DM within r 0.3r vir. This finding is also reflected in the shape of the galaxy number density profile, which flattens markedly with respect to the steadily rising DM profile at small radii. Thus, r ~ 0.3r vir seems to be a transition region interior to which collisional effects significantly modify the dynamical properties of the galaxy population with respect to those of DM in A1689.

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arXiv:1008.2758v1 [astro-ph.CO] 16 Aug 2010
Draft version August 18, 2010
Preprint typeset using LATEX style emulateapj v. 11/10/09
QUANTIFYING THE COLLISIONLESS NATURE OF DARK MATTER AND GALAXIES IN A1689
Doron Lemze1, Yoel Rephaeli1,2, Rennan Barkana1, Tom Broadhurst1,3,4, Rick Wagner2,5& Mike L. Norman2,5
Draft version August 18, 2010
ABSTRACT
We use extensive measurements of the cluster A1689 to assess the expected similarity in the dynamics
of galaxies and dark matter (DM) in their motion as collisionless ‘particles’ in the cluster gravitational
potential. To do so we derive the radial profile of the specific kinetic energy of the cluster galaxies from
the Jeans equation and observational data. Assuming that the specific kinetic energies of galaxies and
DM are roughly equal, we obtain the mean value of the DM velocity anisotropy parameter, and the
DM density profile. Since this deduced profile has a scale radius that is higher than inferred from
lensing observations, we tested the validity of the assumption by repeating the analysis using results of
simulations for the profile of the DM velocity anisotropy. Results of both analyses indicate a significant
difference between the kinematics of galaxies and DM within r ? 0.3rvir. This finding is reflected
also in the shape of the galaxy number density profile, which flattens markedly with respect to the
steadily rising DM profile at small radii. Thus, r ∼ 0.3rvirseems to be a transition region interior to
which collisional effects significantly modify the dynamical properties of the galaxy population with
respect to those of DM in A1689
Subject headings: clusters: galaxies – clusters: individual: A1689
1. INTRODUCTION
The properties of dark matter (DM), which is be-
lieved to be mostly cold and collisionless, have been ex-
tensively explored in dynamical simulations.
touted result from these simulations is the “universal”
density profile (Navarro, Frenk, & White 1997, hereafter
NFW; Moore et al. 1998) of galaxies and galaxy clusters.
As the main mass constituent of galaxy clusters, DM
largely self-gravitates and dominates the hydrodynam-
ics of intracluster (IC) gas and the dynamics of member
galaxies. Cluster DM density profiles deduced from X-
ray observations (Pointecouteau, Arnaud, & Pratt 2005;
Vikhlinin et al. 2006; Schmidt & Allen 2007; Arnaud,
Pointecouteau, & Pratt 2008), galaxy velocity distribu-
tion (Diaferio, Geller, & Rines 2005), SZ measurements
(Atrio-Barandela et al. 2008), and strong and weak lens-
ing measurements (Broadhurst et al. 2005a, hereafter
B05a; Broadhurst et al. 2005b, hereafter B05b; Limousin
et al. 2007; Medezinski et al. 2007; Lemze et al. 2008,
hereafter L08; Broadhurst et al. 2008; Zitrin et al. 2009,
2010; Umetsu et al. 2010) are broadly claimed to be con-
sistent with NFW profiles. However, the shape of the
profile in the inner cluster region where it is predicted to
have a characteristic radial slope of −1 has been deduced
to be shallower in some studies (Kravtsov et al. 1998;
Ettori et al. 2002; Sanderson et al. 2004) and steeper,
around −1.5, in others (Fukushige & Makino 1997, 2001,
2003; Moore et al. 1999; Ghigna et al. 2000; Klypin et
al. 2001; Navarro et al. 2004; Limousin et al. 2008).
A much
1School of Physics and Astronomy, Tel Aviv University, Tel
Aviv, 69978, Israel; doronl@wise.tau.ac.il
2Center for Astrophysics and Space Sciences, University of
California, San Diego, La Jolla, CA 92093, USA
3Department of Theoretical Physics, University of Basque
Country UPV/EHU,Leioa,Spain
4IKERBASQUE, Basque Foundation for Science,48011, Bil-
bao,Spain
5Physics Department, University of California, San Diego, La
Jolla, CA 92093, USA
A more complete description of the clustering proper-
ties of DM necessitates characterizationof its phase space
distribution. Whereas the DM density profile can be de-
termined directly from analysis of X-ray, lensing, and SZ
measurements, deducing the DM velocity distribution is
considerably more challenging. On the other hand, the
dynamical properties of member galaxies can be studied
directly in terms of their density and velocity profiles,
including a study of the radial behavior of the velocity
dispersion. Also, the location of the velocity ‘caustics’
can now be studied both in individual massive clusters
where the data quality is high (Lemze et al. 2009, here-
after L09) and in composite surveys for which only lower
quality measurements are available but statistical results
can be derived (e.g., Biviano & Girardi 2003).
The dynamical evolution of the galaxy population in
a cluster is presumed to be largely collisionless follow-
ing an initial phase of mean-field (‘violent’) relaxation of
the main sub-cluster progenitors that merged to form the
cluster. This collisionless behavior is expected particu-
larly outside the central cluster region. As such, the basic
dynamical characteristics of cluster galaxies are expected
to resemble those of DM, which is strictly collisionless.
For example, in the cluster 1E0657-56 - the ”bullet clus-
ter” - despite a recent collision of two massive clusters,
the spatial distributions of DM and galaxies are quite
similar. A conical shaped shock front is visible indicat-
ing two clusters have passed through each other with
an obvious collisionally merged gas distribution, but the
galaxies and the lensing mass are largely intact, implying
straightforwardly that the DM and galaxies are collision-
less (Markevitch et al. 2002; Clowe et al. 2004; Bradaˇ c
et al.2006). In the colliding cluster A520, on the other
hand, a massive dark core is claimed to coincide with
the central X-ray emission peak, but the region is largely
devoid of galaxies (Mahdavi et al. 2008), though this de-
pends on the way the weak lensing analysis is formulated
(Okabe & Umetsu 2008).

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With increasingly extensive and precise data, such as
we have acquired for A1689, it is now possible to assess
the collisionless nature of galaxies and DM by measuring
the degree of consistency between the measured galaxy
and DM density profiles and the profile of the DM veloc-
ity anisotropy. A1689 seems well relaxed (with possibly a
small deviation from a relaxed state; Andersson & Made-
jski 2004). It has a centrally located cD galaxy, and an
X-ray emission region that is spherically symmetric (Xu
& Wu 2002; L08; Riemer-Sorensen et al. 2008). The
cluster has well-defined galaxy velocity caustics with no
major infall of matter close to the virial radius (L09),
and only a low level of substructure (Broadhurst et al.
2005a,b; Umetsu & Broadhurst 2008).
We have previously determined the DM and gas den-
sity profiles in A1689 from a combined analysis of lens-
ing and X-ray measurements (L08), using an approach
that we refer to as model-independent, since we did not
assume particular functional forms for the profiles. Ad-
ditionally, we extended the analysis by including photo-
metric and spectroscopic measurements of a very large
number of galaxies in the A1689 field, from which we de-
duced the positions and radial velocities of 476+27
members. These results made it possible to deduce the
galaxy velocity anisotropy profile, which was found to
exhibit the expected behavior, varying between predom-
inantly radial orbits at large radii towards more tangen-
tial orbits near the center (L09).
Here we show that with the above information we can
infer the DM velocity anisotropy which we compare with
the galaxy velocity anisotropy profile and with results
from simulations. The paper is organized as follows; in
§ 2 we describe our method for determining the DM ve-
locity anisotropy. In § 3 we compare the DM and galaxy
density profiles (§ 3.2), derive the DM velocity anisotropy
and compare it with the galaxy velocity anisotropy pro-
file and results from simulations (§ 3.1), and estimate
the collisionless profile of cluster galaxies (§ 3.3). We
conclude with a summary and discussion in § 4.
−43cluster
2. METHODOLOGY
In this section we present the procedure for deriving
the DM velocity anisotropy using results from our pre-
vious analyses of the galaxy dynamics (L09) and the
total mass density profile (L08) of A1689. In L08 we
combined lensing and X-ray measurements to determine
model-independent profiles of the gas and total mass den-
sity profiles (i.e., without assuming particular functional
forms). In the second stage of the work (L09) the galaxy
surface number density and the projected velocity disper-
sion were included and analyzed using the Jeans equa-
tion, from which we obtained profiles of the 3D galaxy
number density and the galaxy velocity anisotropy.
The dynamics of a collisionless gas are governed by the
Jeans equation (Binney & Tremaine 1987)
1
ρi
d
dr
?ρiσ2
i,r
?+2βiσ2
i,r
r
= −GM
r2
, (1)
where i =DM, gal, and ρi(r) is the density of element i.
The velocity anisotropy profile βi(r) is
β(r) ≡ 1 −σ2
t(r)
σ2
r(r), (2)
where σr(r) is the radial velocity dispersion, and σt(r) =
σθ(r) = σφ(r) is the (1D) transverse velocity dispersion.
Using eq. 1 for the galaxies, the degeneracy between σgal,r
and βgal can be removed with sufficient spectroscopic
data (L09). This procedure is obviously irrelevant in the
case of DM, for which we have to adopt an alternative
approach.
The orbit of a test particle in a collisionless gravita-
tional system is independent of the particle mass. This
would presumably imply that once hydrostatic equilib-
rium is attained, most likely as a result mixing and mean
field relaxation, DM and galaxies should have the same
mean specific kinetic energy, i.e.,
σ2
DM,tot(r) = σ2
gal,tot(r) ,(3)
where
σ2
i,tot(r) = σ2
i,r(r)+σ2
i,θ(r)+σ2
i,φ(r) = σ2
i,r(r)(3 − 2βi(r)) .
(4)
Additionally, it is expected that the total specific kinetic
energy of DM particles is proportional to that of the gas
(e.g., Mahdavi 2001; Host et al. 2008), T ∝ σ2
servational evidence for this scaling relation comes from
combined X-ray and optical observations of groups and
clusters for which the mean emission-weighted gas tem-
perature scales roughly as the second power of the total
galaxy velocity dispersion (Mulchaey & Zabludoff 1998;
Xue & Wu 2000).
The temperature and density profiles of IC gas can
be deduced from X-ray spectral and surface brightness
measurements. These profiles can then be used to de-
termine the total mass distribution from a solution of
the hydrostatic equilibrium equation. We have recently
developed a model-independent joint lensing/X-ray anal-
ysis procedure (L08) to examine the consistency of X-ray
temperature and emission profiles with the lensing based
mass profile, finding that the cluster temperature profile
is systematically ∼ 30-40% lower than expected when
solving the equation of hydrostatic equilibrium using the
lensing-deduced mass profile. This discrepancy may re-
flect in part the ambiguity in deriving 3D temperatures
from projected spectral X-ray data, stemming from the
sampling of a range of gas temperatures along any given
line of sight (Mazzotta et al. 2004; Vikhlinin 2006). This
could also be partly related to the small-scale structure of
the gas [possibly including relatively dense cooler clouds
found in simulations (Kawahara et al. 2007)] which may
result in a significant downward bias in temperature es-
timates from spectral X-ray observations. The inferred
temperature is also sensitive to instrumental effects, as
has recently been deduced in the analysis of Chandra ob-
servations of A1689 (Peng et al. 2009). Other reasons for
temperature biases can be deviations from equilibrium
and non-thermal pressure (Molnar et al. 2010). Even in
the best-case scenario the temperature can be used as a
reliable tracer for σtotonly in the inner part of the clus-
ter, where hydrostatic equilibrium applies, and also only
at radii larger than about 0.1rvir, since at smaller radii
the specific energy of the gas and DM may be different
(Rasia, Tormen, & Moscardini 2004).
For our analysis we assume an NFW-like profile for the
DM density
tot. Ob-
ρDM(r) ∝ [(r/rs)(1 + r/rs)α]−1, (5)

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3
and take the DM velocity anisotropy to be constant, i.e.,
βDM = const, since the data is not sufficient to mean-
ingfully constrain more than one free parameter in this
quantity. We then relate our model parameters to the
DM radial velocity dispersion, σDM,r, using the Jeans
equation, where the (total) mass profile is taken from
L08. Here we allow for the possibility of a difference be-
tween the total density profile (directly measured by lens-
ing) and the profile of just the DM. From the DM velocity
anisotropy and radial velocity dispersion we can then de-
duce the DM total specific kinetic energy. Best-fit values
of ρDM(r) and βDM are determined by fitting the DM
total specific kinetic energy to the galaxy total specific
kinetic energy. More specifically, βDM and ρDM(r) are
determined by minimizing χ2=?
where V (ri) = σ2
variance matrix of the measured σ2
total specific kinetic energy itself is not a direct measure-
ment. It was derived as shown in eq. 4 from σgal,rand
βgal(r). Since the values of σgal,totat the various radial
positions ri are derived from underlying parameterized
expressions (as given below for βgal(r)), their uncertain-
ties are correlated.
The derivation of βgal(r) and σgal,tot(ri) is carried out
by following the same procedure as in L09, except for
allowing a higher freedom of βgal(r) at large radii. N-
body simulations for a variety of cosmologies show that
the velocity anisotropy has a nearly universal radial pro-
file (Cole & Lacey 1996; Carlberg et al. 1997). In accord
with the work of L09, βgal(r) is taken to have the follow-
ing form:
iVT(ri)·C−1·V (ri),
gal,tot(ri) and C is the co-
gal,tot(ri). The galaxy
DM,tot(ri) − σ2
βgal(r) = (β0+ β∞)
(r/rc)2
(r/rc)2+ 1− β0, (6)
where we note that β∞= 1 was adopted by L09. Now,
the total number of σgal,tot(ri) bins cannot exceed the
total number of free parameters in the expressions for
σgal,rand βgal(r), which is 6, since a larger number would
cause a complete degeneracy among the various values of
σgal,tot(ri). Even taking 6 bins resulted in an unphysi-
cal correlation matrix (i.e., one having a negative eigen-
value), which still indicates a near-degeneracy. There-
fore we adopted 5 radial bins of σgal,tot(ri), which was
the maximum number for which degeneracy is not sig-
nificant and error estimates are reasonable.
3. RESULTS
3.1. Velocity anisotropy profiles
The DM velocity anisotropy, βDM, was determined as
described in § 2, with the constraint that the DM total
specific kinetic energy must satisfy σ2
sulting acceptable fit, with χ2/dof = 3.5/(5−3), is shown
in figure 1. The best-fit parameters of the analytical
expressions for the DM density and velocity anisotropy
profiles are given in table 1. It is important to note that
while the errors on the parameters are rather large, in
this fit we have constrained the DM parameters based
only on the fit to equation 3, without assuming the DM
density profile to be similar to the total mass density
profile. Thus, the results allow a largely independent
comparison between the consequences of assuming equa-
tion 3 and the results of other observational probes of
DM,tot≥ 0. The re-
TABLE 1
The values of the parameters of DM density and velocity
anisotropy. The errors are 1-σ confidence.
Parameter
rs
α
βDM
Value
1330+1210
−605
2.79+1.27
−0.76
0.49+0.13
−0.27
[h−1kpc]
050010001500
0
2
4
6
8
10
12
14
16x 10
6
Radius [h−1 kpc]
σ2
tot
Fig. 1.— Profile of the total specific kinetic energy of the galaxies
(blue circles and 1–σ errorbars) and the fitted total specific kinetic
energy of the DM (red line).
the cluster.
The corresponding galaxy and DM velocity anisotropy
profiles are plotted in figure 2 (top panel) together with
their respective 1–σ uncertainties.
panel) we compare the derived DM velocity anisotropy
value to the profile derived from simulations. The current
data only allow us to determine an overall, typical value
of βDMin the cluster, and given the rather large uncer-
tainties, there is fair agreement between this value and
the typical value of βgalin this cluster, and also with the
typical values of βDM seen in simulated clusters. Note
from the figure that while we allowed β∞ to be a free
parameter for the galaxies, the best-fit value came out
quite close to unity.
We briefly describe the derivation of the DM veloc-
ity anisotropy profile from results of an ENZO simula-
tion (for a more complete description of this particular
simulation, please refer to Hallman et al. 2007). Since
A1689 is a moderately-distant (z=0.183), high mass clus-
ter, Mvir∼ 1.5 · 1015h−1M⊙(Broadhurst et al. 2005a;
Oguri et al. 2005; Limousin et al. 2007; L08; Umetsu
& Broadhurst 2008; L09; Umetsu et al. 2009; Corless
et al. 2009; Coe et al. 2010), we derive the DM veloc-
ity anisotropy profiles for a sample of high-mass clus-
ters with Mvir > 1015h−1
ters were drawn from a cosmological adaptive mesh re-
finement (AMR) simulation performed with the ENZO
code developed by Bryan & Norman (1997) and Norman
& Bryan (1999) assuming a spatially flat CDM model
(very similar to the concordance model). The AMR sim-
ulation assumed adiabatic gas dynamics (i.e., neither ra-
diative heating, cooling, star formation or feedback were
included). The box size was 512 h−1Mpc comoving on
a side with 5123DM particles, and DM mass resolution
of about 1011h−1
grid cells, and the grid was refined by a factor of two,
In figure 2 (lower
0.7M⊙ at z = 0.2. The clus-
0.7M⊙. The root grid contained 5123

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020040060080010001200 14001600 18002000
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Radius [h−1 kpc]
β
00.20.40.60.81
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Radius [rvir]
βDM
Fig. 2.— Top panel: comparison between βDMand βgal. Shown
are the velocity anisotropy profiles of galaxies (blue dotted lines)
and DM (black solid lines). Bottom panel: comparison between the
DM velocity anisotropy inferred from data (black solid lines) and
that derived from simulations (blue dot-dashed lines). For each set
the central line shows the best fit and the other two lines show the
± 1–σ uncertainty range.
up to seven levels, providing a spatial resolution of 6.5
kpc h−1at z = 0.2. For each halo out of the total num-
ber of 51, we derive the DM velocity anisotropy profile,
which is averaged over all halos, with an uncertainty that
is taken to be the standard deviation as determined by
Lemze et al. (2010, in preparation). The derived profile
is in agreement with those found in other studies (e.g.,
Colin, Klypin, & Kravtsov 2000; DM04; Mamon & Lokas
2005; Valdarnini 2006).
3.2. Density profiles
To further explore phase space occupation of DM and
galaxies, we compare the deduced DM density profile to
the galaxy and to the total density profile. The shape of
the galaxy density profile is represented with a core, and
that of the total density with either a model-independent
profile, a ‘universal’ NFW form, or a cored form.
figure 3 we compare our results for these profiles, i.e.,
we compare the DM density from our current fitting to
equation 3 (green dot-dashed curves showing the best-fit
result as well as the 1–σ uncertainty region) to our previ-
ous results for the galaxy density (blue solid curves) and
the total mass profile, whose various versions are shown
by the points with error bars (model-independent fit),
dashed black curve (NFW), and dotted red curve (core).
In order to include all these in the same figure, and allow
a comparison of the relative shapes, we arbitrarily scaled
the profiles so that they match at 700 h−1kpc ∼1
In
3rvir
10
1
10
Radius [h−1 kpc]
2
10
3
10
0
10
1
10
2
10
3
10
4
10
5
Density profiles
Fig. 3.— The galaxy density (blue solid curve with upper and
lower solid curves marking the 1–σ uncertainty) is compared to the
deduced DM density (green dot-dashed curves showing the best-fit
and and 1–σ uncertainty) and to the total matter density for the
model-independent fit (points with error bars), the NFW fit (black
dashed curve), or the core fit (red dotted curve). We show relative
density profiles, all scaled to match at 700 h−1kpc (except for the
galaxy 1–σ lines). The left and right black vertical lines indicate
1
3rvirand rvir, respectively. Note that the galaxy density line at
low radii is an extrapolation due to lack of data in this region (and
therefore no error bars are shown).
(with arbitrary units in the y-axis).
While the DM scale radius derived here from the ve-
locity dispersion fit has a large error, its best-fit value
is significantly higher than that derived previously (L08,
table 4) for the total (mostly DM) density. This is re-
flected by the slower rise of the DM density at small radii
compared to the total density (figure 3). This may indi-
cate that our assumption of eq. 3 invalid at small radii.
Another perspective on the assumption that DM fol-
lows the same dynamics as the galaxies can be seen from
the steepness of the directly measured density profiles.
To assess the steepness of the fitted profiles we plot in
figure 4 the distribution of their power-law indices,
γ(r) =dlog[ρ(r)]
dlog[r]
. (7)
The power-law index of the galaxy profile is shown by
the blue lines, and that of the total mass profile by the
black (NFW) and red (core) lines. We did not plot also
the power-law slope of the model-independent fit, since
it is clear from figure 3 that it would look very similar to
the results for the NFW and core profiles. Since the DM
profile should be rather similar to the total mass profile,
the conclusion from these figures here again is that the
galaxies and the DM have consistent density profiles for
r ?
3rvir, but the profiles are significantly different at
smaller radii.
1
3.3. The collisionless profile of cluster galaxies
In § 3.1 we derived the DM velocity anisotropy profile
assuming that both galaxies and DM are collisionless,
finding that the best-fit value of DM density scale radius
is higher than inferred from direct lensing observations
for the total mass (L08). As was mentioned above, this
may indicate that eq. 3 is not valid at all radii. To quan-
tify differences between σ2
DM,totand σ2
gal,totwe use the

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5
100100020003000
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
Radius [h−1 kpc]
γ
100100020003000
0
0.5
1
1.5
Radius [h−1 kpc]
γgal/γtotal
Fig. 4.— Top panel: power-law indices of the galaxy and total
mass profiles. The index of the galaxy profile is shown by the
blue solid line, with upper and lower solid lines indicating 1–σ
uncertainty, and that of the total mass is shown by the black dashed
(NFW) and dotted red (core) lines, with 1–σ uncertainties. Bottom
panel: ratio between the power-law index of the galaxy density to
that of the total matter (NFW - black dashed with upper and 1–σ
uncertainty; core - red solid with 1–σ uncertainty). The left and
right black vertical lines mark1
3rvirand rvir, respectively.
ratio
fcoll(r) ≡σ2
DM,tot(r)
σ2
gal,tot(r)
,(8)
which we expect to be very close to unity if both compo-
nents are fully collisionless (or if both deviate comparably
from the purely collisionless limit).
We have thus far assumed that fcoll(r) must equal
unity and tried to obtain acceptable fits under this as-
sumption, deriving other results in the process. Here
we try a different approach, where we start by assum-
ing that the DM velocity anisotropy profile in A1689
matches the profile derived from simulations (see § 3.1).
We used our previously-determined DM mass density,
ρDM≃ ρtot− ρgas, derived by assuming profiles for the
total and gas density profiles. Two kinds of profiles were
assumed, a model-independent and a model-dependent
one, where in the model-dependent case we assumed an
NFW and double beta model for the total and gas den-
sity profiles, respectively. After assuming a profile for
the total and gas density profile, we can then solve the
Jeans equation to determine σ2
DM,tot. In figures 5 and 6
05001000 1500
0
0.5
1
1.5
2
2.5
Radius [h−1 kpc]
fcoll
050010001500
0.5
1
1.5
2
2.5
3
3.5
Radius [h−1 kpc]
b
Fig. 5.— Profiles of fcoll(top panel, blue solid curve) and the
velocity bias b (lower panel, blue solid curve) taking the total and
gas density model independent profiles from L08 along with their
1–σ uncertainty regions (marked by blue dot-dashed and dashed
lines, for the case when the uncertainty of DM velocity anisotropy
from simulations is included, and the case when this uncertainty
is not included, respectively). The vertical black line marks1
and the horizontal dotted line indicates the expected value if both
DM and galaxies are purely collisionless.
3rvir,
we show the resulting ratio fcoll(r), or equivalently the
corresponding velocity bias of the galaxies relative to the
DM
b(r) =
?
1/fcoll(r) ,(9)
for the model-independent and model dependent pro-
files, respectively. Our results are consistent with both
the galaxies and the DM being purely collisionless at
r ?
3rvir, but there is some evidence for a significant
deviation from this limit at smaller radii (except for the
very smallest radii, where the observational constraints
are weak).
1
4. DISCUSSION
The work reported here is a continuation of our com-
prehensive study of the dynamical properties of DM and
galaxies, and the hydrodynamical properties of IC gas
in the well-observed cluster A1689 (L08, L09). In L09
we derived the galaxy density and velocity distributions,
from which we deduced the specific kinetic energy of the
galaxies. In the work reported here we assumed that if
DM and galaxies are fully collisionless they should have
the same average specific kinetic energy (as manifested