arXiv:0704.0094v1  [astro-ph]  2 Apr 2007
Timing and Lensing of the Colliding Bullet Clusters: barely enough time and gravity
to accelerate the bullet
HongSheng Zhao
University of St Andrews, Scottish University Physics Alliances, KY16 9SS, UK
We present semi-analytical constraint on the amount of dark matter in the merging bullet galaxy
cluster using the classical Local Group timing arguments. We consider particle orbits in potential
models which fit the lensing data. Marginally consistent CDM models in Newtonian gravity are
found with a total mass MCDM = 1 × 10
M⊙ of Cold DM: the bullet subhalo can move with
VDM = 3000 kms
−1, and the ”bullet” X-ray gas can move with Vgas = 4200 kms
−1. These are
nearly the maximum speeds that are accelerable by the gravity of two truncated CDM halos in a
Hubble time even without the ram pressure. Consistency breaks down if one adopts higher end of
the error bars for the bullet gas speed (5000− 5400 kms−1), and the bullet gas would not be bound
by the sub-cluster halo for the Hubble time. Models with VDM ∼ 4500 kms
∼ Vgas would invoke
unrealistic large amount MCDM = 7× 10
M⊙ of CDM for a cluster containing only ∼ 10
M⊙ of
gas. Our results are generalisable beyond General Relativity, e.g., a speed of 4500 kms−1 is easily
obtained in the relativistic MONDian lensing model of Angus et al. (2007). However, MONDian
model with hot dark matter MHDM ≤ 0.6×10
M⊙ and CDM model with a halo mass ≤ 1×10
are barely consistent with lensing and velocity data.
PACS numbers: 98.10.+z, 98.62.Dm, 95.35.+d; submitted to Physical Review D, rapid publications
I. POTENTIAL FROM TIMING
Timing is a unique technique to establish the case for
dark matter halos, first and most throughly explored in
the context of the Local Group (Kahn & Woljter 1959,
Fich & Tremaine 1991, Peebles 1989, Inga & Saha 1998).
In its simplest version the Local Group consists of the
Milky Way and M31 as two isolated point masses, which
formed close to each other, moved apart due to the Hub-
ble expansion, and slowed down and moved towards each
other upto their present velocity ∼ 120 km s−1 and sepa-
ration (about 700 kpc) due to their mutual gravity. The
age of the universe sets the upper limit on the period of
this galaxy pair, hence the total mass of the pair through
Kepler’s 3rd law assuming Newtonian gravity.
Timing also finds a timely application in the pair of
merging galaxy clusters 1E0657-56 at redshift z = 0.3,
which is largely an extra-galactic grand analogy of the
M31-MW system. The sub-cluster, called the ”bullet”,
presently penetrates 400-700 kpc through the main clus-
ter with an apparent speed of ∼ 4750+710
−550 km s
−1 (Marke-
vitch 2006). The X-ray gas of the bullet (amounts to
2×1013M⊙) collides with the X-ray gas of the main clus-
ter (with the total gas up to 1014M⊙) and forms a Mach-3
cone in front of the ”bullet”. The two clusters have at
least four different centers, which are offset by 400 kpc
between the pair of X-ray gas centers and by 700 kpc
between the pair of star-light centers, which coincides
with the gravitational lensing centers and (dark matter)
potential centers (Clowe et al. 2006). The penetration
speed is unusually high, hard for standard cosmology to
explain statistically (Hayashi & White 2006), and modi-
∗Electronic address: hz4@st-andrews.ac.uk
fied force law has been suggested (Farrar & Rosen 2006,
Angus et al. 2007).
The timing method applies in in MONDian gravity
as well as Newtonian. Like lensing, timing is merely a
method about constraining potential distribution, and is
only indirectly related to the matter distribution. In this
Letter we model the bullet clusters as a pair of mass con-
centrations formed at high redshift, and set constraint on
their mutual force using the simple fact that their radial
oscillation period must be close to the age of universe
at z = 0.3. We check the consistency with the lensing
signal of the cluster and give interpretations in terms of
standard CDM and MOND.
First we can understand the speed of the bullet clus-
ter analytically in simplified scenarios. Approximate the
two clusters as points of fixed masses M1 and M2 on a
head-on orbit, we can apply the usual MW-M31 timing
argument. The total mass M0 = M1 + M2 is constant.
The radial orbital period is computed from
T = 2
∫ rmax
V (r)
, (1)
r3max
, Newtonian p = 2 (2)
2πrmax
, deep-MONDian, p = 1 (3)
∝ K−n/2r
max, for a K/r
p gravity, (4)
where rmax is the apocenter and is related to the present
relative velocity V (r) at separation r = 700 kpc by energy
http://arxiv.org/abs/0704.0094v1
conservation
V (r)2
= −GM0
Newtonian (5)
= V 2M (ln rmax − ln r) deep−MONDian (6)
r1−p − r1−pmax
K/(1− p) for a K/rp gravity,(7)
where VM =
ξ(GM0a0)
1/4 is the MOND cir-
cular velocity of two point masses, a0 equals one
Angstrom per square second and is the MOND
acceleration scale, and the dimensionless ξ ≡
3M1M2
∼ 0.81 ∼ 1 (cf. Mil-
grom 1994, Zhao 2007, in preparation) for a typical mass
ratio.
The predictions for simple Newtonian Keplerian grav-
ity are given in Fig. 1; the more subtle case for a MON-
Dian cluster is discussed in the final section. Setting the
orbital period T = 10Gyrs, the age of the universe at
the cluster redshift, yields presently V ∼ 3200 km s−1 in
Newtonian for a normal combined mass of M1 + M2 =
(0.7− 1)× 1015M⊙ for the clusters, which is about 7-10
times their baryonic gas content (∼ 1014M⊙) for Newto-
nian universe of Ω = 0.3 cold dark matter. In agreement
with Farrar & Rosen and Hayashi & White, the sim-
ple timing argument suggests that dark halo velocities
of 4750 km s−1, as high as the ”bullet” X-ray gas, would
require halos with unrealistically larger masses of dark
matter, ∼ 1016M⊙, an order of magnitude more than
what a universal baryon-dark ratio implies. As a sanity
check, assuming a conventional 3× 1012M⊙ Local Group
dark matter mass Fig.1 predicts the relative velocity of
∼ 100km/s for the M31-MW system at separation 700
kpc after 14 Gyrs, consistent with observation (Binney
& Tremaine 1987).
These analytical arguments, while straightforward, are
not precise given its simplifying assumptions. For one,
clusters do not form immediately at redshift infinity, and
the cluster mass and size might grow with time gradual-
lly. More important is that point mass Newtonian halo
models are far from fitting the weak lensing data of the
1E0657-56. A shallower Newtonian potential makes it
even more difficult to accelerate the bullet. On the other
hand, Angus, Shan, Zhao, Famaey (2007) show that there
are MOND-inspired potentials that fits lensing. As com-
mented in their conclusion, the same potential is deep
enough that a V = 4750 km s−1 ”bullet” is bound in an
orbit of apocenter rmax of a few Mpc, so the two clus-
ters could be accelerated by mutual gravity from a zero
velocity apocenter to 4750 km/s within the clusters’ life-
time. This line of thought was further explored by the
more systematic numerical study of Angus & McGaugh
(2007).
Our paper is a spin-off of these works and the works
of Hayashi & White and Farrar & Rosen. We emphasize
the unification of the semi-analytical timing perspective
and the lensing perspective, and aim to derive robust
constraints to the potential, without being limited to a
X-ray bullet speed
Baryon
12.5 13 13.5 14 14.5 15
Log (Combined Mass)
FIG. 1: Analytical timing-predicted dynamical mass vs. the
relative speed of two objects separated by 700 kpc after 10±
4 Gyrs (three lines in increasing order for increasing time)
assuming Keplerian potential of point masses. Three vertical
lines indicate typical Local Group Halo mass, Baryonic mass
in galaxy clusters, and most massive CDM halo masses. Three
horizontal lines indicate the error bar of the speed of the X-ray
”bullet” gas.
specific gravity theory or dark matter candidate.
Towards the completion of this work, we are made
aware by the preprint of Springel & Farrar (2007) that the
unobserved bullet DM halo could be moving slower than
its observed stripped X-ray gas. These authors, as well
as the preprint of Milosovic et al. (2007), emphasized the
effect of hydrodynamical pressure, which we will not be
able to model realisticly here. But to address the velocity
differences, instead we treat the X-ray gas as a ”bullestic
particle”. We argue that our hypothetical ballistic parti-
cle must move slow enough to be bound to vicinity of the
subhalo before the collision, but moves somewhat faster
than 4700+700
−550 km s
now, since it does not experience
ram pressure of the gas. This model follows the spirit
of classical timing models of the separation of the Large
and Small Magellanic Clouds and the Magellanic Stream
(Lin & Lynden-Bell 1982).
II. 3D POTENTIAL FROM LENSING
The weak lensing shear map of Clowe et al. (2006)
has been fitted by Angus et al. (2007) using a four-
component analytical potential each being spherical but
on different centres. For our purpose we redistribute the
minor components and simplify the potential into two
components centred on the moving centroid of galaxy
light of the main cluster with the present spatial coordi-
nates r1(t) = (−564,−176, 0) kpc and subcluster galaxy
centroid r2(t) = (145, 0, 0) kpc; the coordinate origin is
set at the present brightest point of the ”bullet” X-ray
gas; presently the cluster is at z = 0.3 or cosmic time
t = 10Gyrs. We also apply a Keplerian truncation to the
potential beyond the truncation radius rt. So the follow-
ing 3D potential is adopted for the cluster 1E0657-56 at
time t,
Φ(X,Y, Z, t) = (1800 km s−1)2φ (|r− r1|) (8)
+ (1270 km s−1)2φ (|r− r2|) ,
φ(|r − ri(t)|) = ln
|r − ri(t)|
180 kpc
+ cst, r < rt(9)
= − r̃t
|r− ri(t)|
, r ≥ rt(t) = C × t,(10)
where r̃t ≡ r
+1802
is to ensure a continuous and smooth
transition of the potential across the truncation radius rt.
The truncation rt evolves with time, since a pre-cluster
region collapses gradually after the big bang, and its
boundary and total mass grows with time till it reaches
the size of a cluster. In the interests of simplicity rather
than rigour, we use a linear model rt = C × t, where C
is a constant of the unit kpc/Gyr.
To check that the simplified potential is still consis-
tent with weak lensing data, we recompute the 3D weak
lensing convergence (Taylor et al. 2004) for sources at
distance D(0, zs) at the redshift zs,
κ(X,Y, zs) =
i=X,Y
∫ D(0,zs)
2D(z, zs)
(∂iΦ)dZ
where the integrations in square backets are the deflec-
tion angles for a source at zs, and the usrual lensing
effective distance is related to the comoving distances
by D(z, zs) = (1 + z)
−1D̃(z)
1− D̃(z)
D̃(zs)
= 587 Mpc is
for the bullet cluster z = 0.3 lensing sources at zs = 1;
the distance increases by a factor 1.3 to 1.6 for source
redshifts of 3 to infinity. Fig.2 shows the predicted κ
along the line joining the two dark centers; the result
is insensitive to the cluster truncation radius as long as
rt ≥ 1000kpc presently. The lensing model predicts a
signal in between that of the weak lensing data of Clowe
et al., and strong lensing data of Bradac et al. It is
known that these two data sets are somewhat discrepant
to each other. So the fit here is reasonable. The method
is deprojection is essentially similar to the decomoposi-
tion method of Bradac et al. whose explicit assumption
of Einsteinian gravity is however unnecessary.
The important thing here is that as far as deproject-
ing the above potential is concerned, no assumption is
needed on the gravity theory as long as light rays fol-
low geodesics, a feature built in most alternative grav-
ity theory. Similarly orbits of massive particles are also
(different) geodesics in these theories. The meaning of
potential in such theories is that the potential (scaled by
a factor 2/c2) represents metric perturbations to the flat
space-time, especially to the g00(cdt)
2 = −(1+ 2Φ
)(cdt)2
term, so the Christoffel Γi00 ∼ ∂∂XiΦ, it can be shown
–1200 –1000 –800 –600 –400 –200 0 200 400
X/kpc
FIG. 2: Predicted bullet cluster convergence (rescaled for
sources at infinity) along the line Y = 0.3X + cst connect-
ing our two potential centroids. The model predicts a lensing
signal in between that of observed weak lensing data from
sources at zs = 1 (Clowe et al, lower end of error bars) and
the united weak lensing and strong lensing (zs = 3) data
(Bradc et al. upper part of error bars); the mismatch of these
two datasets are presently unresolved.
that the geodesic equations have the same form as Ein-
steinian in the weak-field limit: d
R ≈ −(1 + v
)∇RΦ,
where R is the pair of spatial coordinates perpendicular
to the instantaneous velocity v; the pathes of light rays
are deflected twice as much by the metric perturbation
2Φ/c2 as those of low-speed particles.
III. ORBITS OF THE COLLIDING CLUSTERS
We now use this potential to predict the relative speed
of the two clusters. This is possible using the classical
timing argument, in the style of Kahn & Wolter (159),
Fich & Tremaine (1991) and Voltonen et al. (1998);
we postpone most rigourous least action models (Pee-
bles 1989, Schmoldt & Saha 1998) for later investiga-
tions since these require modeling a cosmological con-
stant and other mass concentrations along the orbital
path of the bullet clusters, which have technical issues in
non-Newtonian gravity. We trace the orbits of the two
centroids of the potentials according to the equation of
motion d
= −∇Φ(ri). We assign different relative ve-
locities presently (at z = 0.3), and integrate backward
in time and require the two centroids of the potential be
close together at a time 10 Gyrs ago. The motions are
primarily in the sky plane, but we allow for 600 km/s
relative velocity component in the line of sight. Clearly
at earlier times when t is small, the two centroids are
well-separated compared to their sizes, so they move in
the growing Keplerian potential of each other. At lat-
ter times the centroids came close and move in the cored
isothermal potential.
We shall consider models with a normal truncation
rt = C × t = 1000 kpc at time t = 10 Gyrs. We also
consider models with a very large truncation C × t =
10000 kpc. In the language of CDM, the truncation
means the virial radius of the halo. The present in-
stantaneous escape speed of the model can be com-
puted by Vesc =
−2Φ(X,Y, Z, t). We find Vesc ∼
4200− 4500 km s−1 in the central region of the shallower
potential model with a present truncation 1000 kpc. The
escape speed increases to Vesc ∼ 5700 km s−1 for models
with a present truncation 10000 kpc.
Fig. 3 shows the predicted orbits for different present
relative velocities VDM = |dr2dt −
|. Among models
with a normal truncation, we find VDM ∼ 2950 km s−1; a
model with relative velocity VDM < 2800 km s
−1 would
predict an unphysical orbital crossing at high redshift,
while models with VDM > 3000 km s
−1 would predict
that the two potential centroids were never close at high
redshift.
Larger halo velocities are only possible in models
with very large truncation. If the relative velocity is
4200 km s−1 < VDM < 4750 km s
−1 between two clus-
ter gravity centroids, then the truncation must be as big
as 10Mpc at z = 0.3.
We also track the orbit of the bullet X-ray gas cen-
troid as a tracer particle in the above bi-centric poten-
tial. We look for orbits where the bullet X-ray gas will
always be bound to one member of the binary system
since the ram pressure in a hydrodynamical collision is
unlikely to be so efficient to eject the X-ray gas out of
potential wells of both the main and sub-clusters. This
means that the bullet speed must not exceed greatly the
present instantaneous escape speed of the model, which is
∼ 4200− 4500 km s−1 in the central region of the shallow
potential of a model with a present truncation 1000 kpc.
The escape speed increases to ∼ 5700 km s−1 for models
with a present truncation 10000 kpc. The model with
normal truncation is marginally consistent with the ob-
served gas speed Vgas ∼ 4750+710−550 km s
−1. The problem
would become more severe if the potential were made
shallower by an even smaller truncation. The gas speed
is less an issue in models with larger truncation.
In short the present velocity and lensing data are eas-
ier explained with potential models of very large trun-
cation. Models with normal truncation have smaller
gravitational power, can only accelerate the subhalo to
3000 km s−1 in 10 Gyrs. Models with normal CDM trun-
cation can only accelerate the bullet X-ray gas cloud to
∼ 4200− 4400 km s−1, the escape speed, marginally con-
sistent with observations.
Above simulation results are sensitive to the present
cluster separation, but insensitive to the present direc-
tion of the velocity vector. Unmodeled effects such as
dynamical friction associated with a live halo will reduce
the predicted VDM for the same potential, but the effect
is mild since the actual collision is brief ∼ 0.1− 0.3Gyrs
and the factor exp(−M2/2) in Chandrasekhar’s formulae
Curve 1
Curve 2
Curve 3
Curve 4
Curve 9
Curve 10
Curve 11
V_DM=2850 
C=100
C=100
V_DM=2950 
C=1000
V_DM=4200 
C=1000kpc/Gyr
V_DM=4750 km/s
–2000
–4000 –2000 0 2000 4000
X kpc
FIG. 3: The orbit of the bullet subcluster X-ray gas (red,
with present Vgas = 5400 km s
−1 for the 10 Gyrs in the past,
and pink: for the future 4 Gyrs), and the orbits of the col-
liding main cluster halo (blue dashes) and subhalo (black
dashes) in the potential (eqs. 8-10) determined by lensing
data; dashes indicate length traveled in 0.5 Gyrs steps. No
explicit assumption of gravity is needed for these calcula-
tions. Orbits with different present halo relative velocity VDM
and halo growth rate C are shown after a vertical shift for
clarity. Timing requires the present cluster relative velocity
in between 2800 kms−1 < VDM < 3000 kms
−1 for poten-
tials of normal truncation (lowest panels where the cluster
truncation grows from zero to C × 10Gyr = 1000 kpc), and
4200 kms−1 < VDM < 4750 kms
−1 for potentials with large
truncation (two upper panels where the cluster truncation
grows from zero to C × 10 Gyr = 10000 kpc).
sharply reduces dynamical friction for a supersonic body,
where M ∼ 2− 3 is the Mach number for the bullet.
IV. NEWTONIAN AND MONDIAN MEANINGS
OF THE POTENTIAL MODEL
Assuming Newtonian gravity the models with normal
truncation rt = 1Mpc at t = 10Gyrs correspond to clus-
ter (dark) masses of M1 = 0.745 × 1015M⊙ and M2 =
0.345×1015M⊙; the larger truncation rt = 10Mpc corre-
sponds to M1 = 7.45×1015M⊙ and M2 = 3.45×1015M⊙
in Newtonian. All these models fit lensing.
Interpreted in the MONDian gravity, the truncation
is due to external field effect and cosmic background
so to make the MOND potential finite hence escapable
(Famaey, Bruneton, Zhao 2007). Beyond the trun-
cation radius, MOND potential becomes nearly Kep-
lerian. The MONDian models, insensitive to trunca-
tion, would have masses only M1 = 0.66 × 1015M⊙ and
M2 = 0.16 × 1015M⊙. These masses are still higher
than their baryonic content ∼ 1014M⊙, implying the
need for, e.g., massive neutrinos; the neutrino density
is too low in galaxies to affect normal MONDian fits to
galaxy rotation curves, but is high enough to bend light
and orbits significantly on 1Mpc scale. The neutrino-to-
baryon ration, approximately 7:1 in the bullet cluster,
would be a reasonable assumption for a MONDian uni-
verse with Ωb ∼ 0.04 plus 2eV neutrinos hot dark matter
ΩHDM ∼ 0.25 ∼ 7 × Ωb (Sanders 2003, Pointecoute &
Silk 2005, Skordis et al. 2006, Angus et al. 2007). The
amount of hot dark matter inferred here is the same as
Angus et al. (2007) since their potential parameters are
fixed by the same lensing data.
V. CONCLUSION
In short a consistent set of simple lensing and dynam-
ical model of the bullet cluster is found. The present
relative speeds between galaxies of the two clusters is pre-
dicted to be VDM ∼ 2900 km s−1 in CDM and VDM ∼
4500 km s−1 in µHDM (MOND + Hot Dark Matter) if
the two clusters were born close to each other 10 Gyrs
ago; both models assume close to universal gas-DM ratio
in clusters, i.e., about (0.6 − 1) × 1015M⊙ Hot or Cold
DM. Modeling the bullet X-ray gas as ballistic particle,
we find the gas particle with speed of Vgas = 4200km/s
(at the lower end of observed speed) is bound to the
potential of the subcluster for most part of the Hubble
time for both above models, insensitive to the preference
of the law of gravity. But if future relative proper mo-
tion measurements of the subcluster galaxy speed is as
high as VDM = 4500km/s, or the gas speed is as high as
Vgas ∼ 5400 km s−1, then Newtonian models would need
to invoke unlikely 7×1015M⊙ DM halos around 1014M⊙
[20] Angus, G.W. & McGaugh S.D. 2007, astro-ph/0703xxx
[20] Angus G.W., Shan H, Zhao H., Famaey B., 2007, ApJ,
654, L13
[3] Bekenstein J., 2004, Phys. Rev. D., 70, 3509
[4] Binney, J., & Tremaine, S. 1987, Galactic Dynamics,
Princeton University Press, Princeton, New Jersey, Ch.7
[5] Bradac M., Clowe D., Gonzalez A.H., et al., 2006,astro-
ph/0608408 (B06)
[6] Clowe D., Bradac M., Gonzalez A.H., et al., 2006,astro-
ph/0608407 (C06)
[20] Farrar G., & Rosen R.A., astro-ph/0610298
[20] Famaey B., Bruneton J.P., Zhao H.S. 2007, MNRAS, in
press (astro-ph/072275)
[20] Inga M.S. & Saha P. 1998, ApJ, 115, 2231
[20] Lin D.N.C. & Lynden-Bell D, 1982, MNRAS, 198, 707
[20] Markevitch M. 2006, in ESA SP-604: The X-ray Universe
2005, ed. A.Wlison 723
[20] Milgrom M. 1994, ApJ, 429, 540
[20] Kahn, F.D. & Woltjer L. 1959, ApJ, 130, 705
[20] Peebles P.J.E. 1989, ApJ, 344, L53
[20] Pointecoute E. & Silk J. 2005, MNRAS, 364, 654
[20] Skordis, C. et al. 2006, Phys. Rev. Lett, 96, 1301
[20] Sanders R. 2003, MNRAS, 343, 901
[20] Taylor A.,N., Bacon D.J., et al. 2004, MNRAS, 353, 1176
[20] Fich M. & Tremaine S. 1991,ARAA, 29, 409
[20] Voltonen M.J., Byrd G.G., McCall M., Innanen K.A.
1993, AJ 105, 886

  We present semi-analytical constraint on the amount of dark matter in the
merging bullet galaxy cluster using the classical Local Group timing arguments.
We consider particle orbits in potential models which fit the lensing data.
{\it Marginally consistent} CDM models in Newtonian gravity are found with a
total mass M_{CDM} = 1 x 10^{15}Msun of Cold DM: the bullet subhalo can move
with V_{DM}=3000km/s, and the "bullet" X-ray gas can move with
V_{gas}=4200km/s. These are nearly the {\it maximum speeds} that are
accelerable by the gravity of two truncated CDM halos in a Hubble time even
without the ram pressure. Consistency breaks down if one adopts higher end of
the error bars for the bullet gas speed (5000-5400km/s), and the bullet gas
would not be bound by the sub-cluster halo for the Hubble time. Models with
V_{DM}~ 4500km/s ~ V_{gas} would invoke unrealistic large amount M_{CDM}=7x
10^{15}Msun of CDM for a cluster containing only ~ 10^{14}Msun of gas. Our
results are generalisable beyond General Relativity, e.g., a speed of
$4500\kms$ is easily obtained in the relativistic MONDian lensing model of
Angus et al. (2007). However, MONDian model with little hot dark matter
$M_{HDM} \le 0.6\times 10^{15}\msun$ and CDM model with a small halo mass $\le
1\times 10^{15}\msun$ are barely consistent with lensing and velocity data.

arXiv:0704.0094v1  [astro-ph]  2 Apr 2007
Timing and Lensing of the Colliding Bullet Clusters: barely enough time and gravity
to accelerate the bullet
HongSheng Zhao
University of St Andrews, Scottish University Physics Alliances, KY16 9SS, UK
We present semi-analytical constraint on the amount of dark matter in the merging bullet galaxy
cluster using the classical Local Group timing arguments. We consider particle orbits in potential
models which fit the lensing data. Marginally consistent CDM models in Newtonian gravity are
found with a total mass MCDM = 1 × 10
M⊙ of Cold DM: the bullet subhalo can move with
VDM = 3000 kms
−1, and the ”bullet” X-ray gas can move with Vgas = 4200 kms
−1. These are
nearly the maximum speeds that are accelerable by the gravity of two truncated CDM halos in a
Hubble time even without the ram pressure. Consistency breaks down if one adopts higher end of
the error bars for the bullet gas speed (5000− 5400 kms−1), and the bullet gas would not be bound
by the sub-cluster halo for the Hubble time. Models with VDM ∼ 4500 kms
∼ Vgas would invoke
unrealistic large amount MCDM = 7× 10
M⊙ of CDM for a cluster containing only ∼ 10
M⊙ of
gas. Our results are generalisable beyond General Relativity, e.g., a speed of 4500 kms−1 is easily
obtained in the relativistic MONDian lensing model of Angus et al. (2007). However, MONDian
model with hot dark matter MHDM ≤ 0.6×10
M⊙ and CDM model with a halo mass ≤ 1×10
are barely consistent with lensing and velocity data.
PACS numbers: 98.10.+z, 98.62.Dm, 95.35.+d; submitted to Physical Review D, rapid publications
I. POTENTIAL FROM TIMING
Timing is a unique technique to establish the case for
dark matter halos, first and most throughly explored in
the context of the Local Group (Kahn & Woljter 1959,
Fich & Tremaine 1991, Peebles 1989, Inga & Saha 1998).
In its simplest version the Local Group consists of the
Milky Way and M31 as two isolated point masses, which
formed close to each other, moved apart due to the Hub-
ble expansion, and slowed down and moved towards each
other upto their present velocity ∼ 120 km s−1 and sepa-
ration (about 700 kpc) due to their mutual gravity. The
age of the universe sets the upper limit on the period of
this galaxy pair, hence the total mass of the pair through
Kepler’s 3rd law assuming Newtonian gravity.
Timing also finds a timely application in the pair of
merging galaxy clusters 1E0657-56 at redshift z = 0.3,
which is largely an extra-galactic grand analogy of the
M31-MW system. The sub-cluster, called the ”bullet”,
presently penetrates 400-700 kpc through the main clus-
ter with an apparent speed of ∼ 4750+710
−550 km s
−1 (Marke-
vitch 2006). The X-ray gas of the bullet (amounts to
2×1013M⊙) collides with the X-ray gas of the main clus-
ter (with the total gas up to 1014M⊙) and forms a Mach-3
cone in front of the ”bullet”. The two clusters have at
least four different centers, which are offset by 400 kpc
between the pair of X-ray gas centers and by 700 kpc
between the pair of star-light centers, which coincides
with the gravitational lensing centers and (dark matter)
potential centers (Clowe et al. 2006). The penetration
speed is unusually high, hard for standard cosmology to
explain statistically (Hayashi & White 2006), and modi-
∗Electronic address: hz4@st-andrews.ac.uk
fied force law has been suggested (Farrar & Rosen 2006,
Angus et al. 2007).
The timing method applies in in MONDian gravity
as well as Newtonian. Like lensing, timing is merely a
method about constraining potential distribution, and is
only indirectly related to the matter distribution. In this
Letter we model the bullet clusters as a pair of mass con-
centrations formed at high redshift, and set constraint on
their mutual force using the simple fact that their radial
oscillation period must be close to the age of universe
at z = 0.3. We check the consistency with the lensing
signal of the cluster and give interpretations in terms of
standard CDM and MOND.
First we can understand the speed of the bullet clus-
ter analytically in simplified scenarios. Approximate the
two clusters as points of fixed masses M1 and M2 on a
head-on orbit, we can apply the usual MW-M31 timing
argument. The total mass M0 = M1 + M2 is constant.
The radial orbital period is computed from
T = 2
∫ rmax
V (r)
, (1)
r3max
, Newtonian p = 2 (2)
2πrmax
, deep-MONDian, p = 1 (3)
∝ K−n/2r
max, for a K/r
p gravity, (4)
where rmax is the apocenter and is related to the present
relative velocity V (r) at separation r = 700 kpc by energy
http://arxiv.org/abs/0704.0094v1
conservation
V (r)2
= −GM0
Newtonian (5)
= V 2M (ln rmax − ln r) deep−MONDian (6)
r1−p − r1−pmax
K/(1− p) for a K/rp gravity,(7)
where VM =
ξ(GM0a0)
1/4 is the MOND cir-
cular velocity of two point masses, a0 equals one
Angstrom per square second and is the MOND
acceleration scale, and the dimensionless ξ ≡
3M1M2
∼ 0.81 ∼ 1 (cf. Mil-
grom 1994, Zhao 2007, in preparation) for a typical mass
ratio.
The predictions for simple Newtonian Keplerian grav-
ity are given in Fig. 1; the more subtle case for a MON-
Dian cluster is discussed in the final section. Setting the
orbital period T = 10Gyrs, the age of the universe at
the cluster redshift, yields presently V ∼ 3200 km s−1 in
Newtonian for a normal combined mass of M1 + M2 =
(0.7− 1)× 1015M⊙ for the clusters, which is about 7-10
times their baryonic gas content (∼ 1014M⊙) for Newto-
nian universe of Ω = 0.3 cold dark matter. In agreement
with Farrar & Rosen and Hayashi & White, the sim-
ple timing argument suggests that dark halo velocities
of 4750 km s−1, as high as the ”bullet” X-ray gas, would
require halos with unrealistically larger masses of dark
matter, ∼ 1016M⊙, an order of magnitude more than
what a universal baryon-dark ratio implies. As a sanity
check, assuming a conventional 3× 1012M⊙ Local Group
dark matter mass Fig.1 predicts the relative velocity of
∼ 100km/s for the M31-MW system at separation 700
kpc after 14 Gyrs, consistent with observation (Binney
& Tremaine 1987).
These analytical arguments, while straightforward, are
not precise given its simplifying assumptions. For one,
clusters do not form immediately at redshift infinity, and
the cluster mass and size might grow with time gradual-
lly. More important is that point mass Newtonian halo
models are far from fitting the weak lensing data of the
1E0657-56. A shallower Newtonian potential makes it
even more difficult to accelerate the bullet. On the other
hand, Angus, Shan, Zhao, Famaey (2007) show that there
are MOND-inspired potentials that fits lensing. As com-
mented in their conclusion, the same potential is deep
enough that a V = 4750 km s−1 ”bullet” is bound in an
orbit of apocenter rmax of a few Mpc, so the two clus-
ters could be accelerated by mutual gravity from a zero
velocity apocenter to 4750 km/s within the clusters’ life-
time. This line of thought was further explored by the
more systematic numerical study of Angus & McGaugh
(2007).
Our paper is a spin-off of these works and the works
of Hayashi & White and Farrar & Rosen. We emphasize
the unification of the semi-analytical timing perspective
and the lensing perspective, and aim to derive robust
constraints to the potential, without being limited to a
X-ray bullet speed
Baryon
12.5 13 13.5 14 14.5 15
Log (Combined Mass)
FIG. 1: Analytical timing-predicted dynamical mass vs. the
relative speed of two objects separated by 700 kpc after 10±
4 Gyrs (three lines in increasing order for increasing time)
assuming Keplerian potential of point masses. Three vertical
lines indicate typical Local Group Halo mass, Baryonic mass
in galaxy clusters, and most massive CDM halo masses. Three
horizontal lines indicate the error bar of the speed of the X-ray
”bullet” gas.
specific gravity theory or dark matter candidate.
Towards the completion of this work, we are made
aware by the preprint of Springel & Farrar (2007) that the
unobserved bullet DM halo could be moving slower than
its observed stripped X-ray gas. These authors, as well
as the preprint of Milosovic et al. (2007), emphasized the
effect of hydrodynamical pressure, which we will not be
able to model realisticly here. But to address the velocity
differences, instead we treat the X-ray gas as a ”bullestic
particle”. We argue that our hypothetical ballistic parti-
cle must move slow enough to be bound to vicinity of the
subhalo before the collision, but moves somewhat faster
than 4700+700
−550 km s
now, since it does not experience
ram pressure of the gas. This model follows the spirit
of classical timing models of the separation of the Large
and Small Magellanic Clouds and the Magellanic Stream
(Lin & Lynden-Bell 1982).
II. 3D POTENTIAL FROM LENSING
The weak lensing shear map of Clowe et al. (2006)
has been fitted by Angus et al. (2007) using a four-
component analytical potential each being spherical but
on different centres. For our purpose we redistribute the
minor components and simplify the potential into two
components centred on the moving centroid of galaxy
light of the main cluster with the present spatial coordi-
nates r1(t) = (−564,−176, 0) kpc and subcluster galaxy
centroid r2(t) = (145, 0, 0) kpc; the coordinate origin is
set at the present brightest point of the ”bullet” X-ray
gas; presently the cluster is at z = 0.3 or cosmic time
t = 10Gyrs. We also apply a Keplerian truncation to the
potential beyond the truncation radius rt. So the follow-
ing 3D potential is adopted for the cluster 1E0657-56 at
time t,
Φ(X,Y, Z, t) = (1800 km s−1)2φ (|r− r1|) (8)
+ (1270 km s−1)2φ (|r− r2|) ,
φ(|r − ri(t)|) = ln
|r − ri(t)|
180 kpc
+ cst, r < rt(9)
= − r̃t
|r− ri(t)|
, r ≥ rt(t) = C × t,(10)
where r̃t ≡ r
+1802
is to ensure a continuous and smooth
transition of the potential across the truncation radius rt.
The truncation rt evolves with time, since a pre-cluster
region collapses gradually after the big bang, and its
boundary and total mass grows with time till it reaches
the size of a cluster. In the interests of simplicity rather
than rigour, we use a linear model rt = C × t, where C
is a constant of the unit kpc/Gyr.
To check that the simplified potential is still consis-
tent with weak lensing data, we recompute the 3D weak
lensing convergence (Taylor et al. 2004) for sources at
distance D(0, zs) at the redshift zs,
κ(X,Y, zs) =
i=X,Y
∫ D(0,zs)
2D(z, zs)
(∂iΦ)dZ
where the integrations in square backets are the deflec-
tion angles for a source at zs, and the usrual lensing
effective distance is related to the comoving distances
by D(z, zs) = (1 + z)
−1D̃(z)
1− D̃(z)
D̃(zs)
= 587 Mpc is
for the bullet cluster z = 0.3 lensing sources at zs = 1;
the distance increases by a factor 1.3 to 1.6 for source
redshifts of 3 to infinity. Fig.2 shows the predicted κ
along the line joining the two dark centers; the result
is insensitive to the cluster truncation radius as long as
rt ≥ 1000kpc presently. The lensing model predicts a
signal in between that of the weak lensing data of Clowe
et al., and strong lensing data of Bradac et al. It is
known that these two data sets are somewhat discrepant
to each other. So the fit here is reasonable. The method
is deprojection is essentially similar to the decomoposi-
tion method of Bradac et al. whose explicit assumption
of Einsteinian gravity is however unnecessary.
The important thing here is that as far as deproject-
ing the above potential is concerned, no assumption is
needed on the gravity theory as long as light rays fol-
low geodesics, a feature built in most alternative grav-
ity theory. Similarly orbits of massive particles are also
(different) geodesics in these theories. The meaning of
potential in such theories is that the potential (scaled by
a factor 2/c2) represents metric perturbations to the flat
space-time, especially to the g00(cdt)
2 = −(1+ 2Φ
)(cdt)2
term, so the Christoffel Γi00 ∼ ∂∂XiΦ, it can be shown
–1200 –1000 –800 –600 –400 –200 0 200 400
X/kpc
FIG. 2: Predicted bullet cluster convergence (rescaled for
sources at infinity) along the line Y = 0.3X + cst connect-
ing our two potential centroids. The model predicts a lensing
signal in between that of observed weak lensing data from
sources at zs = 1 (Clowe et al, lower end of error bars) and
the united weak lensing and strong lensing (zs = 3) data
(Bradc et al. upper part of error bars); the mismatch of these
two datasets are presently unresolved.
that the geodesic equations have the same form as Ein-
steinian in the weak-field limit: d
R ≈ −(1 + v
)∇RΦ,
where R is the pair of spatial coordinates perpendicular
to the instantaneous velocity v; the pathes of light rays
are deflected twice as much by the metric perturbation
2Φ/c2 as those of low-speed particles.
III. ORBITS OF THE COLLIDING CLUSTERS
We now use this potential to predict the relative speed
of the two clusters. This is possible using the classical
timing argument, in the style of Kahn & Wolter (159),
Fich & Tremaine (1991) and Voltonen et al. (1998);
we postpone most rigourous least action models (Pee-
bles 1989, Schmoldt & Saha 1998) for later investiga-
tions since these require modeling a cosmological con-
stant and other mass concentrations along the orbital
path of the bullet clusters, which have technical issues in
non-Newtonian gravity. We trace the orbits of the two
centroids of the potentials according to the equation of
motion d
= −∇Φ(ri). We assign different relative ve-
locities presently (at z = 0.3), and integrate backward
in time and require the two centroids of the potential be
close together at a time 10 Gyrs ago. The motions are
primarily in the sky plane, but we allow for 600 km/s
relative velocity component in the line of sight. Clearly
at earlier times when t is small, the two centroids are
well-separated compared to their sizes, so they move in
the growing Keplerian potential of each other. At lat-
ter times the centroids came close and move in the cored
isothermal potential.
We shall consider models with a normal truncation
rt = C × t = 1000 kpc at time t = 10 Gyrs. We also
consider models with a very large truncation C × t =
10000 kpc. In the language of CDM, the truncation
means the virial radius of the halo. The present in-
stantaneous escape speed of the model can be com-
puted by Vesc =
−2Φ(X,Y, Z, t). We find Vesc ∼
4200− 4500 km s−1 in the central region of the shallower
potential model with a present truncation 1000 kpc. The
escape speed increases to Vesc ∼ 5700 km s−1 for models
with a present truncation 10000 kpc.
Fig. 3 shows the predicted orbits for different present
relative velocities VDM = |dr2dt −
|. Among models
with a normal truncation, we find VDM ∼ 2950 km s−1; a
model with relative velocity VDM < 2800 km s
−1 would
predict an unphysical orbital crossing at high redshift,
while models with VDM > 3000 km s
−1 would predict
that the two potential centroids were never close at high
redshift.
Larger halo velocities are only possible in models
with very large truncation. If the relative velocity is
4200 km s−1 < VDM < 4750 km s
−1 between two clus-
ter gravity centroids, then the truncation must be as big
as 10Mpc at z = 0.3.
We also track the orbit of the bullet X-ray gas cen-
troid as a tracer particle in the above bi-centric poten-
tial. We look for orbits where the bullet X-ray gas will
always be bound to one member of the binary system
since the ram pressure in a hydrodynamical collision is
unlikely to be so efficient to eject the X-ray gas out of
potential wells of both the main and sub-clusters. This
means that the bullet speed must not exceed greatly the
present instantaneous escape speed of the model, which is
∼ 4200− 4500 km s−1 in the central region of the shallow
potential of a model with a present truncation 1000 kpc.
The escape speed increases to ∼ 5700 km s−1 for models
with a present truncation 10000 kpc. The model with
normal truncation is marginally consistent with the ob-
served gas speed Vgas ∼ 4750+710−550 km s
−1. The problem
would become more severe if the potential were made
shallower by an even smaller truncation. The gas speed
is less an issue in models with larger truncation.
In short the present velocity and lensing data are eas-
ier explained with potential models of very large trun-
cation. Models with normal truncation have smaller
gravitational power, can only accelerate the subhalo to
3000 km s−1 in 10 Gyrs. Models with normal CDM trun-
cation can only accelerate the bullet X-ray gas cloud to
∼ 4200− 4400 km s−1, the escape speed, marginally con-
sistent with observations.
Above simulation results are sensitive to the present
cluster separation, but insensitive to the present direc-
tion of the velocity vector. Unmodeled effects such as
dynamical friction associated with a live halo will reduce
the predicted VDM for the same potential, but the effect
is mild since the actual collision is brief ∼ 0.1− 0.3Gyrs
and the factor exp(−M2/2) in Chandrasekhar’s formulae
Curve 1
Curve 2
Curve 3
Curve 4
Curve 9
Curve 10
Curve 11
V_DM=2850 
C=100
C=100
V_DM=2950 
C=1000
V_DM=4200 
C=1000kpc/Gyr
V_DM=4750 km/s
–2000
–4000 –2000 0 2000 4000
X kpc
FIG. 3: The orbit of the bullet subcluster X-ray gas (red,
with present Vgas = 5400 km s
−1 for the 10 Gyrs in the past,
and pink: for the future 4 Gyrs), and the orbits of the col-
liding main cluster halo (blue dashes) and subhalo (black
dashes) in the potential (eqs. 8-10) determined by lensing
data; dashes indicate length traveled in 0.5 Gyrs steps. No
explicit assumption of gravity is needed for these calcula-
tions. Orbits with different present halo relative velocity VDM
and halo growth rate C are shown after a vertical shift for
clarity. Timing requires the present cluster relative velocity
in between 2800 kms−1 < VDM < 3000 kms
−1 for poten-
tials of normal truncation (lowest panels where the cluster
truncation grows from zero to C × 10Gyr = 1000 kpc), and
4200 kms−1 < VDM < 4750 kms
−1 for potentials with large
truncation (two upper panels where the cluster truncation
grows from zero to C × 10 Gyr = 10000 kpc).
sharply reduces dynamical friction for a supersonic body,
where M ∼ 2− 3 is the Mach number for the bullet.
IV. NEWTONIAN AND MONDIAN MEANINGS
OF THE POTENTIAL MODEL
Assuming Newtonian gravity the models with normal
truncation rt = 1Mpc at t = 10Gyrs correspond to clus-
ter (dark) masses of M1 = 0.745 × 1015M⊙ and M2 =
0.345×1015M⊙; the larger truncation rt = 10Mpc corre-
sponds to M1 = 7.45×1015M⊙ and M2 = 3.45×1015M⊙
in Newtonian. All these models fit lensing.
Interpreted in the MONDian gravity, the truncation
is due to external field effect and cosmic background
so to make the MOND potential finite hence escapable
(Famaey, Bruneton, Zhao 2007). Beyond the trun-
cation radius, MOND potential becomes nearly Kep-
lerian. The MONDian models, insensitive to trunca-
tion, would have masses only M1 = 0.66 × 1015M⊙ and
M2 = 0.16 × 1015M⊙. These masses are still higher
than their baryonic content ∼ 1014M⊙, implying the
need for, e.g., massive neutrinos; the neutrino density
is too low in galaxies to affect normal MONDian fits to
galaxy rotation curves, but is high enough to bend light
and orbits significantly on 1Mpc scale. The neutrino-to-
baryon ration, approximately 7:1 in the bullet cluster,
would be a reasonable assumption for a MONDian uni-
verse with Ωb ∼ 0.04 plus 2eV neutrinos hot dark matter
ΩHDM ∼ 0.25 ∼ 7 × Ωb (Sanders 2003, Pointecoute &
Silk 2005, Skordis et al. 2006, Angus et al. 2007). The
amount of hot dark matter inferred here is the same as
Angus et al. (2007) since their potential parameters are
fixed by the same lensing data.
V. CONCLUSION
In short a consistent set of simple lensing and dynam-
ical model of the bullet cluster is found. The present
relative speeds between galaxies of the two clusters is pre-
dicted to be VDM ∼ 2900 km s−1 in CDM and VDM ∼
4500 km s−1 in µHDM (MOND + Hot Dark Matter) if
the two clusters were born close to each other 10 Gyrs
ago; both models assume close to universal gas-DM ratio
in clusters, i.e., about (0.6 − 1) × 1015M⊙ Hot or Cold
DM. Modeling the bullet X-ray gas as ballistic particle,
we find the gas particle with speed of Vgas = 4200km/s
(at the lower end of observed speed) is bound to the
potential of the subcluster for most part of the Hubble
time for both above models, insensitive to the preference
of the law of gravity. But if future relative proper mo-
tion measurements of the subcluster galaxy speed is as
high as VDM = 4500km/s, or the gas speed is as high as
Vgas ∼ 5400 km s−1, then Newtonian models would need
to invoke unlikely 7×1015M⊙ DM halos around 1014M⊙
[20] Angus, G.W. & McGaugh S.D. 2007, astro-ph/0703xxx
[20] Angus G.W., Shan H, Zhao H., Famaey B., 2007, ApJ,
654, L13
[3] Bekenstein J., 2004, Phys. Rev. D., 70, 3509
[4] Binney, J., & Tremaine, S. 1987, Galactic Dynamics,
Princeton University Press, Princeton, New Jersey, Ch.7
[5] Bradac M., Clowe D., Gonzalez A.H., et al., 2006,astro-
ph/0608408 (B06)
[6] Clowe D., Bradac M., Gonzalez A.H., et al., 2006,astro-
ph/0608407 (C06)
[20] Farrar G., & Rosen R.A., astro-ph/0610298
[20] Famaey B., Bruneton J.P., Zhao H.S. 2007, MNRAS, in
press (astro-ph/072275)
[20] Inga M.S. & Saha P. 1998, ApJ, 115, 2231
[20] Lin D.N.C. & Lynden-Bell D, 1982, MNRAS, 198, 707
[20] Markevitch M. 2006, in ESA SP-604: The X-ray Universe
2005, ed. A.Wlison 723
[20] Milgrom M. 1994, ApJ, 429, 540
[20] Kahn, F.D. & Woltjer L. 1959, ApJ, 130, 705
[20] Peebles P.J.E. 1989, ApJ, 344, L53
[20] Pointecoute E. & Silk J. 2005, MNRAS, 364, 654
[20] Skordis, C. et al. 2006, Phys. Rev. Lett, 96, 1301
[20] Sanders R. 2003, MNRAS, 343, 901
[20] Taylor A.,N., Bacon D.J., et al. 2004, MNRAS, 353, 1176
[20] Fich M. & Tremaine S. 1991,ARAA, 29, 409
[20] Voltonen M.J., Byrd G.G., McCall M., Innanen K.A.
1993, AJ 105, 886
