The collision velocity of the bullet cluster in conventional and
  modified dynamics
Mon. Not. R. Astron. Soc. 000, 1–8 (2007) Printed 27 October 2018 (MN LATEX style file v2.2)
The collision velocity of the bullet cluster in conventional
and modified dynamics
G. W. Angus1⋆, S. S. McGaugh2†
1SUPA, School of Physics and Astronomy, University of St. Andrews, Scotland KY16 9SS
2Department of Astronomy, University of Maryland, College Park, MD 20742-242 USA
Accepted ... Received ... ; in original form ...
ABSTRACT
We consider the orbit of the bullet cluster 1E 0657-56 in both CDM and MOND using
accurate mass models appropriate to each case in order to ascertain the maximum
plausible collision velocity. Impact velocities consistent with the shock velocity (∼
4700 km s
) occur naturally in MOND. CDM can generate collision velocities of at
most ∼ 3800 km s−1, and is only consistent with the data provided that the shock
velocity has been substantially enhanced by hydrodynamical effects.
Key words: gravitation - dark matter - galaxies: clusters: individual (1E 0657-56)
1 INTRODUCTION
Many lines of observational evidence now oblige us to be-
lieve that the universe is filled with a novel, invisible form of
mass that dominates gravitationally over normal baryonic
matter. In addition, a dark energy component which exerts
negative pressure to accelerate the expansion of the universe
is also necessary (Chernin et al. 2007). Though this ΛCDM
paradigm is well established, we still have only ideas about
what these dark components might be, and no laboratory
detections thereof.
One possible alternative to ΛCDM is the Modified New-
tonian Dynamics (MOND; Milgrom 1983a,b,c). This hy-
pothesis has been more successful than seems to be widely
appreciated (McGaugh & de Blok 1998; Sanders & Mc-
Gaugh 2002), and has received a theoretical boost from the
introduction of generally covariant formulations (Bekenstein
2004; Sanders 2005; Zlosnik, Ferreira, & Starkman 2006,7).
The dark matter and alternative gravity paradigms are rad-
ically different, so every observation that might distinguish
between them is valuable.
ΛCDM is known to work well on large scales (e.g.,
Spergel et al. 2006) while MOND is known to work well in in-
dividual galaxies (Sanders & McGaugh 2002). This success,
incorporating the tight correlation between dark and lumi-
nous mass in the DM framework (McGaugh 2005; Famaey
et al. 2007b) extends over five decades in mass (Fig.1) rang-
ing from tiny dwarfs (e.g., Milgrom & Sanders 2007) through
spirals of low surface brightness (de Blok & McGaugh 1998),
our own Milky Way (Famaey & Binney 2005) and other high
⋆ email: gwa2@st-andrews.ac.uk
† email: ssm@astro.umd.edu
surface brightness (Sanders 1996; Sanders & Noordermeer
2007) to massive ellipticals (Milgrom & Sanders 2003). The
recent observations of tidal dwarf galaxies by Bournaud et
al. (2007) provides a severe challenge to CDM but is natu-
rally explained in MOND with zero free parameters (Mil-
grom 2007; Gentile et al. 2007). Having said that, MOND
persistently fails to completely explain the mass discrepancy
in rich clusters of galaxies. Consequently, clusters require
substantial amounts of non-luminous matter in MOND.
That rich clusters contain more mass than meets the
eye in MOND goes back to Milgrom’s original papers (Mil-
grom 1983c). At the time, the discrepancy was very much
larger than it is today, as it was not then widely appreci-
ated how much baryonic mass resides in the intra-cluster
medium. Further work on the X-ray gas (e.g., Sanders 1994,
1999) and with velocity dispersions (McGaugh & de Blok
1998) showed that MOND was at least within a factor of a
few, but close inspection revealed a persistent discrepancy
of a factor of two or three in mass (e.g., Gerbal et al. 1992;
The & White 1998; Pointecouteau & Silk 2005, Buote &
Canizares 1994). Weak gravitational lensing (Angus et al.
2007a; Takahashi & Chiba 2007; Famaey, Angus et al. 2007)
provides a similar result.
To make matters worse, the distribution of the unseen
mass does not trace that of either the galaxies or the X-
ray gas (Aguirre et al. 2001; Sanders 2003; Angus et al.
2007b; Sanders 2007). In Fig. 1 we plot the baryonic mass
of many spiral galaxies and clusters against their circular
velocity together with the predictions of MOND and CDM.
MOND is missing mass at the cluster scale. CDM suffers an
analogous missing baryon problem on the scale of individual
galaxies.
The colliding bullet cluster 1E-0657-56 (Clowe et al.
c© 2007 RAS
http://arxiv.org/abs/0704.0381v2
2 G. W. Angus and S. S. McGaugh
2004,2006, Bradac et al. 2006, Markevitch et al. 2004,
Markevitch & Vikhlinin 2007) illustrates in a spectacular
way the residual mass discrepancy in MOND. While cer-
tainly problematic for MOND as a theory, it does not con-
stitute a falsification thereof. Indeed, given that the need for
extra mass in clusters was already well established, it would
have been surprising had this effect not also manifested it-
self in the bullet cluster. The new information the bullet
cluster provides is that the additional mass must be in some
collision-less form.
It is a logical fallacy to conclude that because extra
mass is required by MOND in clusters, that dark matter
is required throughout the entire universe. While undeni-
ably problematic, the residual mass discrepancy in MOND
is limited to groups and rich clusters of galaxies: these are
the only systems in which it systematically fails to remedy
the dynamical mass discrepancy (see discussion in Sanders
2003). Could we be absolutely certain that we had accounted
for all the baryons in clusters, then MOND would indeed
be falsified. But CDM suffers an analogous missing baryon
problem in galaxies (Fig.1) in addition to the usual dynam-
ical mass discrepancy, yet this is not widely perceived to be
problematic. In either case we are obliged to invoke the ex-
istence of some dark mass which is presumably baryonic (or
perhaps neutrinos) in the case of MOND. In neither case
is there any danger of violating big bang nucleosynthesis
constraints. The integrated baryonic mass density of rich
clusters is much less than that of all baryons; having the
required mass of baryons in clusters would be the proverbial
drop in the bucket with regards to the global missing baryon
problem.
A pressing question is the apparently high relative ve-
locity between the two clusters that comprise the bullet
cluster 1E 0657-56 (Clowe et al. 2006, Bradac et al. 2006,
Markevitch et al. 2004, Markevitch & Vikhlinin 2007). The
relative velocity derived from the gas shockwave is vrel =
4740+710
−550 kms
−1 (Clowe et al. 2006). Taken at face value,
this is very high, and seems difficult to reconcile with ΛCDM
(Hyashi & White 2006). The problem is sufficiently large
that it has been used to argue for an additional long range
force in the dark sector (Farrar & Rosen 2007). Here we ex-
amine the possibility of such a large velocity in both CDM
and MOND.
One critical point that has only very recently been ad-
dressed is how the shock velocity relates to the collision ve-
locity of the clusters. Naively, one might expect the dissipa-
tional collision of the gas clouds to slow things down so that
the shock speed would provide a lower limit on the colli-
sion speed. Recent hydrodynamical simulations (Springel &
Farrar 2007; Milosavljevic et al. 2007) suggest the opposite.
A combination of effects in the two hydrodynamical sim-
ulations show that the shock velocity may be higher than
the impact velocity. The results of the two independent hy-
drodynamical simulations do not seem to be in perfect con-
cordance, and the precise result seems to be rather model
dependent. Nevertheless, it seems that the actual relative
velocity lies somewhere in the range 3500-4500 kms−1.
The difficulties posed by a high collision velocity for
CDM have been discussed previously by Hayashi & White
(2006) and Farrar & Rosen (2007). Whereas Springel & Far-
rar (2007) and Milosavljevic et al. (2007) consider the com-
plex hydrodynamic response of the two gas clouds during
Figure 1. Shows baryonic mass against circular velocity. Rotat-
ing galaxies (blue circles) are from McGaugh (2005) and clusters
(green triangles) are from Sanders (2003) using the measured tem-
perature to estimate the circular velocity assuming isothermality.
The solid orange line is the CDM M-V relation (Steinmetz &
Navarro 1999) assuming Mb = fbMvir with fb = 0.17 (Spergel et
al. 2006) and the dashed red line is the MOND prediction. The
spirals lie directly on the MOND prediction, but the clusters are
generally 2-3 times in mass below it. The CDM expectation is
nicely consistent with clusters, but implies many dark baryons in
spirals in addition to the non-baryonic dark matter.
the ongoing collision, here we investigate the ability of two
clusters like those comprising the bullet cluster to accelerate
to such a high relative velocity in the case of both CDM and
MOND prior to the merger. We compute a simple free fall
model for the two clusters in an expanding universe with
realistic mass models, and ask whether the observed colli-
sion velocity can be generated within the time available. We
take care to match the mass models to the specific observed
properties of the system appropriate to each flavor of grav-
ity in order to realistically evaluate the orbit of the clusters
prior to their collision.
2 MODELING THE FREEFALL
We wish to address a simple question. Given the observed
masses of the two clusters, is it possible to account for the
measured relative velocity from their gravitational freefall?
The expansion of the universe mitigates against large veloc-
ities, since the clusters must decouple from the Hubble flow
before falling together. Presumably it takes some time to
form such massive objects, though this is expected to occur
earlier in MOND than in ΛCDM (Sanders 1998, 2001; Mc-
Gaugh 1999, 2004; Nusser 2002; Stachniewicz & Kutschera
2002; Knebe & Gibson 2004; Dodelson & Liguori 2006). The
clusters are observed at z=0.3, giving at most 9Gyrs for
them to accelerate towards each other. This imposes an up-
per limit to the velocity that can be generated gravitation-
ally. Without doing the calculation, it is not obvious whether
the larger masses of the clusters in CDM or the stronger long
range force in MOND will induce larger relative velocities.
Since we know the state of the system directly prior
to collision, it makes sense to begin our simulations from
the final state and work backwards in time towards when
the relative velocity was zero. This point, where the clus-
c© 2007 RAS, MNRAS 000, 1–8
The collision velocity of the bullet cluster in conventional and modified dynamics 3
ters have zero relative velocity, is when they turned around
from the Hubble flow and began their long journey gravi-
tating towards each other. Working backwards in time leads
to potentially counter-intuitive discussions (such as Hubble
contraction), which we try to limit.
We must account for the Hubble expansion in a manner
representing the universe before z=0.3. The detailed form of
the expansion history of the universe a(t) is not known in
the case of MOND, so we take the scale factor of ΛCDM in
both models
da(t)
+ ΩΛa
. (1)
Where we take Ho = 72 km s
−1Mpc−1, Ωm=0.27 and ΩΛ =
0.73.
The important aspect is the basic fact that the uni-
verse is expanding and the mutual attraction of the clusters
must overcome this before they can plunge together at high
velocity.
We implement the scale factor in the simulations
through the equation of motion
[a(t)v] = g. (2)
Computing this numerically, from time step to time step we
calculate the ratio of the scale factor in the previous time
step to the current time step (i.e. ǫ = a(ti−1)/a(ti); we use
negative time steps to move backwards in time from the
presently known configuration, so ǫ > 1 (higher i means
earlier universe). We then have v(ti) = v(ti−1)ǫ+ g∆t.
The right hand side of Eq. 2 differs in MOND and CDM
not only because the law of gravity is altered, but also be-
cause the gravitating masses are higher in CDM.
The initial conditions are the crux of the problem, with
at least 4 unknowns. These include the masses of the two
clusters, the relative velocity of the clusters, and the dis-
tance of separation between the two when they had this
relative velocity. The separation is the same in MOND and
Newtonian gravity, but the Newtonian mass is higher.
The relative velocity of the two clusters can be mea-
sured because, in the last few 100Myrs, the less massive
sub cluster has passed through the centre of the more mas-
sive main cluster. The ram pressure has imposed a smooth
bow shock (Markevitch et al. 2004, Markevitch & Vikhlinin
2007) on the gas of the sub cluster. Since the relative veloc-
ity is the foundation of the problem we leave it free and try
to estimate it by fixing other variables. In our simulation,
we think it sensible to consider the separation of the two
clusters (i.e. of the two centres of mass) when they had the
calculated relative velocity to be when the leading edge of
the sub cluster’s gas cloud began to pass through the dense
region of gas belonging to the main cluster and separate
from the dark matter. It appears that the centre of the sub
cluster’s gas cloud (the location of the bullet) is preceeded
by the bow shock by around 200kpc further in the direc-
tion of travel. We take 200kpc to also be the radius within
which the gas of cluster 1 was dense enough to imprint the
bow shock. Indeed, the gas mass of the main cluster (sub
cluster) is only measured out to 180kpc (100kpc) and could
not be found further detailed in the literature. However,
the uncertainty is large enough that we wished to clarify
the impact of different initial separations by always using
a range of initial separations of between 350-500kpc. This
Figure 2. The total enclosed masses for the main cluster (black)
and sub cluster (red) for CDM (solid), MOND with standard µ
(dashed) and simple µ (dotted).
separation is defined as when the two pre-collision clusters
had the relative velocity of vrel related to the shock velocity
4740+710
−550 kms
−1. Now of course, they are on the opposite
sides on the sky after having passed through each other and
the gas has been offset from the DM.
3 THE COLLISION IN CDM
In the CDM framework, it is no problem to generate two
clusters in an N-body simulation and calculate all gravi-
tational accelerations exactly. However, in MOND we are
dealing with non linear gravity and the tools for such pur-
poses are only now being developed (Nipoti et al. 2007a).
Additionally, since we begin our simulations with the over-
lap of the two clusters, it is not guaranteed that the clusters
preserve their shapes as they separate. Furthermore, it was
not possible to simply include accretion history (Wechsler
et al. 2002) in the N-body simulations or easily vary the
truncation radius of the DM halo as was necessary.
Therefore, a better method was to semi-analytically ac-
count for these aspects in a simulation where gravity of one
cluster acting on the other is just the mass enclosed by a
sphere around the gravitating body’s centre of mass with
radius equal to the separation of the two cluster’s centres of
mass. For this procedure, the only two unknowns are the sep-
aration, which is initially known and computed each time-
step; and the mass enclosed. The enclosed mass depends on
the density profile of the two clusters and was fitted by Clowe
et al. (2006) using NFW profiles of the form (see Angus &
Zhao 2007)
ρ(r) =
m200r
r + r200c
= ln(1+c)− c
1 + c
where c is the concentration. The enclosed mass goes as
c© 2007 RAS, MNRAS 000, 1–8
4 G. W. Angus and S. S. McGaugh
Figure 3. The total enclosed masses for the main cluster as a
function of time, where 9.3Gyr marks the collision of the two
clusters and 0Gyr represents the Big Bang. The three lines corre-
spong to different mass assembly rates α=0.5 (black), 1.0 (blue)
and 1.5 (red). The mass loss is halted at m200/50.
m(r) = Am200
− 1 +
For the main cluster they give m200 = 1.5 × 1015M⊙,
r200 = 2100kpc and concentration c=1.94. For the sub clus-
ter, m200 = 1.5 × 1014M⊙, r200 = 1000kpc and c=7.12. We
augment the DM with a baryon fraction of 17% (Spergel et
al. 2006) which was part of the total mass during freefall.
We ran dynamical time steps (negative) such that
∆t = 10
d[pc]Myr (5)
Where d is the separation of the two centres of mass and
∆t has a maximum value of 1Myr. Since initially d∼ 400kpc,
the starting time steps are ∼0.06Myr. The simulations were
run until 9Gyr had elapsed.
The mass distributions as functions of radius for the
two clusters in CDM and the ones used in the MOND sim-
ulations are shown in Fig.2. A subtle point about the total
masses of the two clusters is that we do not expect the mass
to remain constant as we go back in time. Presumably they
grew from a seed of negligible mass at high redshift (see
discussion in Cameron & Driver 2007). This tends to im-
pede their freefall, reducing the maximum collision velocity
to ∼ 2900 kms−1 by the estimate of Farrar & Rosen (2007).
To include this, without the impedance, we use the proce-
dure of Wechsler et al. (2002) who used the relation
M(z) = M(z = 0.3)e
−α(z−0.3)
where α obviously encodes the speed of the accretion or
assembly of the halo. Typical values used in their work are
0.5< α <2.0. In Fig.3 we plot the mass enclosed within r200
for the two clusters as functions of reshift for α=0.5,1.0 and
1.5. Note we always keep a floor value of cluster mass of
m200/50 so the halo is never completely disassembled.
Another important point is whether mass integrated out
beyond r200 should be included, since the actual virial ra-
dius depends on both cosmology and redshift (Bullock et al.
2001). Indeed, the internal gravity of the main cluster has
not reached ao by r200 = 2100kpc meaning the MOND dy-
namical mass has not yet saturated. However, recall that it
takes ∼ (r200−d)/vrel = (2100kpc−400kpc)/3400 kms−1 ∼
500Myr for the clusters to separate enough for this extra
matter to even begin to manifest itself. The cluster is also
losing mass (backwards in time) due to accretion coupled
with the fact there must be overdensities on the opposite side
of the universe countering the influence of these overdensi-
ties. However, using Eq.4 it is straight-forward to include all
the enclosed mass out to any radius because the parameters
r200 and m200 do not explicitly force the enclosed mass to
truncate at r200, they simply define the shape of the profile.
4 THE COLLISION IN MOND
In MOND, the basic modification of purely Newtonian dy-
namics is
µ(x)g = gN, (7)
where gN is the Newtonian acceleration computed in the
usual way from the baryonic mass distribution, g is the ac-
tual acceleration (including the effective amplification due
to MOND conventionally ascribed to DM), ao is the char-
acteristic acceleration at which the modification becomes
effective (∼ 10−10 m s−2), x = g/ao, and µ(x) is an inter-
polation function smoothly connecting the Newtonian and
MOND regimes. In the limit of large accelerations, g ≫ ao,
µ → 1 and the Newtonian limit is obtained: everything be-
haves normally. The MOND limit occurs only for exceed-
ingly low accelerations, with µ → x for g ≪ ao. We im-
plement two possible versions of the interpolation function:
the ‘standard’ function traditionally used in fitting rotation
curves:
1 + x2
(e.g., Sanders & McGaugh 2002), and the ‘simple’ function
found by Famaey & Binney (2005) to provide a good fit the
terminal velocity curve of the Galaxy:
1 + x
. (9)
A well known problem with implementing the MOND
force law in numerical computations is that the original for-
mulation (Eq 7) does not conserve momentum (Felten 1984;
Bekenstein 2007). This was corrected with the introduction
of a Lagrangian formulation of MOND (Bekenstein & Mil-
grom 1984; Milgrom 1986) which has the modified Poisson
equation
∇ · [µ(|∇Φ|/ao)∇Φ] = 4πGρ. (10)
This formulation has been shown to obey the necessary
conservation laws (Bekenstein & Milgrom 1984; Bekenstein
2007). With some rearrangement, it leads to
µ(x)g = gN +∇× h, (11)
c© 2007 RAS, MNRAS 000, 1–8
The collision velocity of the bullet cluster in conventional and modified dynamics 5
which we recognize as Eq 7 with the addition of a curl field.
Unfortunately, implementing a numerical formulation of
the modified Poisson equation is not a simple one-line change
to typical N-body codes: this fails to obey the conservation
laws. Instead, one needs an entirely different numerical ap-
proach than is commonly employed. Progress has been made
along these lines (e.g., Brada & Milgrom 1995, 1999; Ciotti,
Londrillo, & Nipoti 2006; Nipoti, Londrillo & Ciotti 2007ab;
Tiret & Combes 2007; see also Nusser 2002; Knebe & Gib-
son 2004), but we do not seek here a full N-body treatment
of complex systems. Rather, we wish to develop and apply
a simple tool (Angus & McGaugh, in preparation) that can
provide some physical insight into basic problems. For the
specific case of the large collision velocity of the bullet clus-
ter, it suffices to treat the curl field as a small correction
to the center of mass motion (Milgrom 1986). The external
field effect (see Milgrom 1983a, Bekenstein 2007) is crudely
approximated as a constant of appropriate magnitude (Mc-
Gaugh 2004). We checked the effect of varying the external
field, which is modest. It is not possible to do better without
complete knowledge of the mass distribution in the environ-
ment of the clusters.
When modeling the bullet cluster in MOND, Angus et
al. (2007a) fitted the convergence map of Clowe et al. (2006)
using spherical potential models for the four mass compo-
nents. Their best fit gives masses for all four components in
MOND and standard gravity. Unfortunately, the map is only
sensitive out to 250kpc from the respective centres which ne-
glects an over large portion of the dynamical mass. So, in or-
der to remain consistent with the CDM simulations, we take
the NFW profile and calculate what the MOND dynamical
masses for the two commonly used interpolating functions
(Eq.8 & 9) are, as shown in Fig.2. The Newtonian mass for
the main cluster is twice that of the MOND dynamical mass
with the standard µ and three times when the simple µ is
used.
The mutual gravity imposed upon the sub cluster by
the main cluster is
|gsub + gex|
gsub = gn,sub = −GMmain(d)/d2 (12)
and we simply swap the subscripts around to find the mutual
gravity of the sub cluster upon the main cluster. Following
on from above, d is the distance between the two centres of
mass and Mmain(d) is the enclosed mass within a radius d
from the centre of mass of the main cluster. The gex is the
external field limiting the MOND correction which comes
from large scale structure and is always assumed orthogonal
to the direction of gsub, making the argument of the µ func-
tion more easily expressed as
(g2sub+g
. The direction
and amplitude of gex is unknown at all times. The MON-
Dian additional acceleration becomes minor when the accel-
eration drops below gex. We use gex = ao/30 (Aguirre et
al. 2001; McGaugh 2004) which is roughly the external field
imposed on the Milky Way by M31 and vice versa (Famaey
et al. 2007a, Wu et al. 2007).
5 N-BODY COLLISION
Our first attempt at simulating the collision in Newtonian
gravity was using a standard N-body tree code. The benefit
it gives is that in principal, we can more accurately compute
the mutual gravity at the beginning of the simulation when
the two clusters overlap. However, this is fraught with diffi-
culties and inconsistencies. The first being that tidal effects
undoubtedly stretch the two clusters and 2-body interac-
tions may eject particles from the two halos. Therefore, it
makes better sense to begin such a simulation from high
redshift where the clusters are greatly separated and tidal
effects are negligible and let them freefall in the expanding
universe and when they collide, the tidal effects will be well
accounted for. Of course, the problem is that it is not triv-
ial to then sample collision velocities because the separation
and time at which the two clusters began their freefall is not
simply related. Moreover, the truncation of the two halos
and different mass models are not easily varied. Neverthe-
less, we did attempt a CDM N-body model with truncation
at r200 for both halos. We found a similar result to that from
the semi-analytical models of 3800 kms−1.
6 RESULTS
The ability of the two clusters that comprise the bullet to
bring each other to a halt at a finite time in the past is sen-
sitive to both the flavour of gravity at work and the true
relative velocity. For velocities larger than the maximum,
the relative velocity never reaches zero and increase sharply
at early times (large z). The two clusters do not gravitate
strongly enough to generate such high velocities and would
have to have had a huge relative velocity towards each other
in the early universe in order to overcome the Hubble expan-
sion and fall together with such a high relative velocity at
z=0.3. Fig.4 shows how the relative velocity of the two clus-
ters varies with time for a large sample of initial (meaning
collisional) relative velocities for a CDM and MOND sample
simulation. A difference of just 100 km s−1 can have a signif-
icant impact on the time required to generate such a large
velocity and by the same token, the longer the two clusters
free-fall, the larger a velocity they can generate. Sadly, there
is only a finite time (∼ 9Gyr)) since the Big Bang for this
to happen.
In Table 1 we’ve put the key results of the simulations
so as to give the reader a feel for what the maximum rela-
tive velocity that can be achieved is. Each velocity is that
achieved with an initial separation of 425kpc, where taking
350 or 500kpc induces an increase or decrease of 100 kms−1
which we take as the minimum error. The most extreme
CDM model is to have no truncation of the DM halos, ex-
tending them out to r1. This absurd extreme allows a maxi-
mum relative velocity of 4500 kms−1. Then, if we still allow
the halos to extend to r1, but account for some assembly of
the halos with α = 1 then the relative velocity reduces to
4200 kms−1.
More realistically, if we truncate the halos at r200 and
try four different halo assembly rates such that α =0.0, 0.5,
1.0 & 1.5 we get respective maximum relative velocities of
4000, 3900, 3800 and 3800 km s−1. These numbers repre-
sent the plausible maximum relative velocities in the CDM
framework.
For the MOND case we ran simulations with both the
simple (Eq.9) and standard (Eq.9) µ functions. The stan-
dard function leads to higher dynamical masses from the
c© 2007 RAS, MNRAS 000, 1–8
6 G. W. Angus and S. S. McGaugh
Figure 4. Shows the relative velocity of the two clusters plotted against time (a) CDM and (b) MOND. Time=0Myr is the current (z=0.3)
relative velocity of the two clusters with larger times corresponding to higher redshifts. Black lines correspond to relative velocities that
are achievable, whereas red lines are not. In (a) we use the simulation (CDM2c) for which α=1.0, d=425kpc and we truncate the halos at
r200. The relative velocities used are vrel=3500-4200 km s
−1 in intervals of 100 km s−1. In (b) we use the simulation (MONDst2) which
uses the standard µ function and α=0.5, d=425kpc. The relative velocities used are vrel=4100-4800 km s
−1 in intervals of 100 km s−1. The
4 dashed lines are the predicted relative velocities according to the mean and 1σ error of the original relative velocity from Markevitch &
Vikhlinin (2007) in blue, the simulations of Milosavljevic (2007) in green and Springel & Farrar (2007) in turquoise. The high observed
collision velocity is more readily obtained in MOND than CDM.
Model Max Vrel [ km s
−1] Truncation Radius α Gravity
CDM1a 4500 r1 0.0 Newtonian
CDM1b 4200 r1 1.0 Newtonian
CDM2a 4000 r200 0.0 Newtonian
CDM2b 3900 r200 0.5 Newtonian
CDM2c 3800 r200 1.0 Newtonian
CDM2d 3800 r200 1.5 Newtonian
MONDst1 4800 r200 0.0 MOND-standard µ
MONDst2 4500 r200 0.5 MOND-standard µ
MONDsi1 4600 r200 0.0 MOND-simple µ
MONDsi2 4500 r200 0.5 MOND-simple µ
Table 1. Shows the parameters used in the different models and gives the maximum attainable relative velocity for each.
NFW profile, but lower 2-body gravity. The standard (sim-
ple) function with no accretion and with α = 0.5 gener-
ate 4800 (4600) and 4600 (4500) kms−1 respectively and for
comparison, the maximum CDM velocity with those reduced
masses is just 2700 (2300) kms−1. This is a clear demonstra-
tion of the expectation in MOND for larger peculiar veloc-
ities. We use the lower assembly parameter α = 0.5 be-
cause structure is expected to form more swiftly in MOND
(Sanders 1998, 2001).
An important factor is that of the fitted NFW density
profile to the convergence map, in which matter is extrapo-
lated to 2100kpc and 1000kpc for the main and sub cluster
respectively. Presumably the significance of the detection of
this mass is negligible and the NFW fit has been made as-
suming if we know the details in the central 250kpc, then we
know the density out to r200. The mass sheet degeneracy is
broken by constraining the mass at the edges of the fit based
on the slope of the profile in the inner regions - but if the
mass profile is wrong then it could lead to the completely
wrong measurement for the value of the mass sheet (Clowe,
de Lucia and King 2004).
All of this means that the density profiles of the two
clusters could be moderately different in reality. However,
the actual shape of any profile is less important to the rela-
tive velocity than simply the normalisation of the total mass.
To this end we have simulated the collision with 10% more
and 10% less mass for both clusters (with assembly param-
eter α = 1). The effect is to increase (10% more mass) or
decrease (10% less mass) the relative velocity by 200 kms−1
from 4800 kms−1 for model MONDst1.
Another concern is that the clusters are unlikely to be
spherically symmetric (Buote & Canizares 1996) and are
presumably elongated in the direction of motion. Again this
could lead to an incorrect density profile, whereas ellipticity
itself would have little effect on our results.
c© 2007 RAS, MNRAS 000, 1–8
The collision velocity of the bullet cluster in conventional and modified dynamics 7
7 SUMMARY
We have constructed specific mass models for the bullet
cluster in both CDM and MOND. We integrate backwards
from the observed conditions to check whether the large
(∼ 4700 km s−1) apparent transverse velocity can be at-
tained in either context. We find that it is difficult to
achieve vrel > 4500 kms
−1 under any conditions. Never-
theless, within the range of the uncertainties, the appropri-
ate velocity occurs fairly naturally in MOND. In contrast,
ΛCDM models can at most attain ∼ 3800 kms−1 and are
more comfortable with considerably smaller velocities.
Taken at face value, a collision velocity of 4700 km s−1
constitutes a direct contradiction to ΛCDM. Ironically, this
cluster, widely advertised as a fatal observation to MOND
because of the residual mass discrepancy it shows, seems to
pose a comparably serious problem for ΛCDM. It has often
been the case that observations which are claimed to falsify
MOND turn out to make no more sense in terms of dark
matter.
Two critical outstanding issues remain to be clarified.
The first is the exact density profiles and virial masses of the
two clusters and the second is how the observed shock ve-
locity relates to the actual collision velocity of the two grav-
itating masses. The recent simulations of Springel & Farrar
(2007) and Milosavljevic et al. (2007) seem to suggest that,
contrary to naive expectations, hydrodynamic effects reduce
the relative velocity of the mass with respect to the shock.
A combination of effects is responsible, being just barely
sufficient to reconcile the data with ΛCDM. Hydrodynam-
ical simulations are notoriously difficult, and indeed these
two recent ones do not agree in detail. It would be excellent
to see a fully self-consistent simulation including both hy-
drodynamical effects and a proper mass model and orbital
computation like that presented here.
There are a number of puzzling aspects to the hydrody-
namical simulations. First of all, Springel & Farrar (2007)
use Hernquist profiles for the DM distribution in the clus-
ters and not NFW halos. Furthermore, they find that the
morphology of the bullet is reproduced only for a remark-
ably dead head-on collision. If the impact parameter is even
12kpc — a target smaller than the diameter of the Milky
Way — quite noticeable morphological differences ensue.
This can be avoided if the separation of mass centres hap-
pens to be along our line of sight — quite a coincidence
in a system already remarkable for having the vector of its
collision velocity almost entirely in the plane of the sky. Fur-
thermore, the mass models require significant tweaking from
that infered from the convergence map and are unable to re-
produce the currently observed, post merger positions of the
gas and DM. It appears to us that only the first rather than
the last chapter has been written on this subject. Getting
this right is of the utmost importance, as the validity of both
paradigms rests on the edge of a knife, separated by just a
few hundred km s−1.
More generally, the frequency of bullet-like clusters may
provide an additional test. The probability of high collision
velocities drops with dramatic rapidity in ΛCDM (Hyashi &
White 2006). In contrast, somewhat higher velocities seem
natural to MOND. Naively it would seem that high impact
velocity systems like the bullet would be part and parcel
of what might be expected of a MOND universe. With this
in mind, it is quite intriguing that many bullet cluster like
systems have been detected (although none quite as unique).
The dark ring around Cl0024+17 tentatively observed by
Jee et al. (2007; see also Famaey et al. 2007c), the dark core
created by the “train wreck” in Abell 520 by Mahdavi et al.
(2007), Cl0152+1357 (Jee et al. 2005a), MS1054+0321 (Jee
et al. 2005b) and the line of sight merger with > 3000 kms−1
relative velocity observed by Dupke et al. (2007) for Abell
576 may all provide examples and potential tests.
ACKNOWLEDGEMENTS
We acknowledge discussions with Benoit Famaey, Tom Zlos-
nik, Douglas Clowe, HongSheng Zhao, Greg Bothun, Moti
Milgrom, Bob Sanders and Ewan Cameron. GWA thanks
Steve Vine for his N-body tree code. GWA is supported by
a PPARC scholarship. The work of SSM is supported in part
by NSF grant AST0505956.
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	Introduction
	Modeling the freefall
	The collision in CDM
	The collision in MOND
	N-body collision
	Results
	Summary
