Reconstructing the Intrinsic Triaxial Shape of the Virgo Cluster
Bomee Lee and Jounghun Lee
Department of Physics and Astronomy, FPRD, Seoul National University, Seoul 151-747,
Korea
bmlee@astro.snu.ac.kr, jounghun@astro.snu.ac.kr
ABSTRACT
To use galaxy clusters as a cosmological probe, it is important to account for
their triaxiality. Assuming that the triaxial shapes of galaxy clusters are induced
by the tidal interaction with the surrounding matter, Lee and Kang recently
developed a reconstruction algorithm for the measurement of the axial ratio of
a triaxial cluster. We examine the validity of this reconstruction algorithm by
performing an observational test of it with the Virgo cluster as a target. We
first modify the LK06 algorithm by incorporating the two dimensional projec-
tion effect. Then, we analyze the 1275 member galaxies from the Virgo Cluster
Catalogue and find the projected direction of the Virgo cluster major axis by
measuring the anisotropy in the spatial distribution of the member galaxies in
the two dimensional projected plane. Applying the modified reconstruction al-
gorithm to the analyzed data, we find that the axial ratio of the triaxial Virgo
cluster is (1: 0.54 : 0.73). This result is consistent with the recent observational
report from the Virgo Cluster Survey, proving the robustness of the reconstruc-
tion algorithm. It is also found that at the inner radii the shape tends to be more
like prolate. We discuss the possible effect of the Virgo cluster triaxiality on the
mass estimation.
Subject headings: cosmology:theory — large-scale structure of universe
1. INTRODUCTION
Galaxy clusters provide one of the most powerful tools to constrain the key cosmological
parameters. In the era of precision cosmology, it is important to determine their mass as
accurately as possible before using them as a cosmological probe. Any kind of simplified
assumption about the properties of galaxy clusters could cause substantial systematics in
the mass estimation. The triaxial shapes of galaxy clusters are one of such properties.
http://arxiv.org/abs/0704.1400v1
– 2 –
It has been long known both observationally and numerically that the galaxy clusters
are noticeably triaxial (e.g., Frenk et al. 1988; West 1989; Plionis et al. 1991; Warren et al.
1992). While plenty of efforts have been already made to take into account the triaxial shapes
of galaxy clusters (Jing & Suto 2002; Fox & Pen 2001; Suwa et al. 2003; Lee & Suto 2004;
Kasun & Evrard 2005; Hopkins et al. 2005; Lee et al. 2005; Smith & Watts 2005; Hayashi et al.
2007), previous studies have been largely focused on the statistical treatment of cluster tri-
axiality. For the measurement of the gas mass fraction of galaxy clusters that can pro-
vide powerful constraints on the density parameter and the dark energy equation of state
(White et al. 1993; Lubin et al. 1996; Cen 1997; Evrard 1997; Cooray 1998; Laroque et al.
2006; Ferramacho & Blanchard 2007), however, it is necessary to deal with the individual
clusters and their triaxial shapes.
The standard picture based on the cosmic web theory (Bardeen et al. 1986; Bond et al.
1996) explains that galaxy clusters are rare events corresponding to the local maxima of
the initial density field and form at the dense nodes of the local filaments in the cosmic
web through tidal interactions with the surrounding matter. The tidal effect from the sur-
rounding matter results in the deviation of cluster shapes from spherical symmetry as well
as the preferential alignments of cluster galaxies (or halos) with the major axes of their host
clusters (Binggeli 1982; Struble & Peebles 1985; Hopkins et al. 2005; Kasun & Evrard 2005;
Lee et al. 2005; Algood et al. 2006; Altay et al. 2006; Paz et al. 2006).
In the frame of this standard scenario, Lee & Kang (2006, hereafter, LK06) have re-
cently developed an analytic algorithm to reconstruct the triaxial shapes of individual clus-
ters. The key concept of the LK06 algorithm is that the two axial ratios of a triaxial cluster
are related to the eigenvalues of the local tidal shear tensor. By measuring the spatial align-
ment of the cluster galaxies with the major axis of their host cluster, one can determine the
eigenvalues of the local tidal tensor, which will in turn yields the two axial ratios of a triaxial
cluster.
Testing their analytic model against high-resolution N-body simulations, LK06 have
shown that their algorithm works well within 20% errors. Now that the LK06 algorithm is
known to work in principle, it is time to test the algorithm against observations. Our goal
here is to apply the LK06 algorithm to real observational data and examine its validity in
practice. Here, we use the Virgo cluster as a target, whose triaxial shape has been very
recently measured observationally (Mei et al. 2007).
The organization of this paper is as follows. In §2, a brief overview of the LK06 algorithm
is provided and how to incorporate the two dimensional projection effect into the algorithm is
explained. In §3, the Virgo cluster data are analyzed and its triaxial shapes are reconstructed
using the LK06 algorithm. In §4, the results are summarized, and the advantages and the
– 3 –
caveats of our model are discussed.
2. THEORETICAL MODEL
2.1. Overview of the LK06 Algorithm
According to the LK06 algorithm, the cluster triaxial shape originates from its tidal
interaction with the surrounding matter distribution. This assumption has been verified from
high-resolution N-body simulation which demonstrated clearly that the tidal field elongates
the cluster shapes (e.g., Altay et al. 2006, and references therein). LK06 has shown that the
two axial ratios of a triaxial cluster are related to the three eigenvalues of the local tidal
tensor defined as the second derivative of the gravitational potential:
1− λ2
1− λ3
1− λ1
1− λ3
, (1)
where {a, b, c} (with a ≤ b ≤ c)are the three principal axis lengths of a cluster and {λ1, λ2, λ3}
(with λ1 ≥ λ2 ≥ λ3) are the three eigenvalues of the local tidal tensor, T. According to this
formula, one can estimate the cluster axial ratios if λ1, λ2, λ3 is fixed by fitting the probability
density distribution, p(cos θ) analytically to the observational data. LK06 suggested that
the preferential locations of the cluster galaxies near the cluster major axes given the local
tidal tensor be described as
〈x̂ix̂j |T̂ 〉 =
δij + sT̂ikT̂kj. (2)
where x̂ ≡ (x̂i) is the unit position vector of a cluster galaxy, T̂ ≡ T/|T| is the unit tidal
shear tensor, and s ∈ [−1, 1] is the correlation parameter that represents the correlation
strength between x̂ and T̂. Under the assumption that the minor axis of the tidal shear
tensor is in the direction of the cluster major axis, equation (2) basically describes the
alignment between the position of a cluster galaxy and the major axis of its host cluster. If
s = −1, there is a maximum alignment. If s = 1, there is a maximum anti-alignment. The
case of s = 0 corresponds to no alignment.
Let θ3d be the angle between the host cluster major axis and the galaxy position vector.
Under the assumption that the cluster major axis is in the direction of the minor principal
axis of the tidal shear tensor, The probability density distribution of cos θ3d was derived by
LK6 as
p(cos θ3d) =
(x̂i ·M
ij · x̂j)
2 , (3)
– 4 –
where φ is an azimuthal angle of x̂ measured in an arbitrary coordinate system. Here,
covariance matrix M is defined as M ≡ 〈x̂ix̂j |T̂ 〉. Note that equation (3) holds good for any
arbitrary coordinate system in which the tidal shear tensor is not necessarily diagonal.
In the principal axis frame of the tidal tensor, equation (4) can be expressed only in
terms of the three eigenvalues of the tidal shear tensor as
p(cos θ) =
(1− s+ 3sλ̂2i )
sin2 θ cos2 φ
1− s+ 3sλ̂2
sin2 θ sin2 φ
1− s+ 3sλ̂2
cos2 θ
1− s+ 3sλ̂2
dφ, (4)
where {λ̂i}
i=1 are the eigenvalues of T̂, related to {λi}
i=1 as
δcλ̂1
λ̂1 + λ̂2 + λ̂3
, λ2 =
δcλ̂2
λ̂1 + λ̂2 + λ̂3
, λ3 =
δcλ̂3
λ̂1 + λ̂2 + λ̂3
, (5)
where δc ≈ 1.68 is the linear density threshold for a dark halo (Eke et al. 1996) satisfying
the following constraint of δc =
i=1 λi.
The key concept of LK06 algorithm is that by fitting equation (3) to the observed
probability distribution, one can find the best-fit values of λ1, λ2 and s, and then determine
the cluster axial ratios using equation (1). Although LK06 algorithm allows us in principle
to reconstruct the three dimensional intrinsic triaxial shapes of individual clusters, it is
restricted to the cases where the informations on the three dimensional positions of the
cluster galaxies in the cluster principal axis frame are given. Unfortunately, for most clusters,
these informations are not available but only two dimensional projected images of clusters.
In §2.2, we attempt to modify the LK06 algorithm in order to incorporate the two
dimensional projection effect.
2.2. Projection Effect
Let us suppose that the position vectors of the cluster galaxies are all projected along
the line of sight direction onto the plane of a sky. Unless the major axis of the host cluster is
perfectly aligned with the line-of-sight direction to the cluster center, one would expect that
the projected position vectors of the cluster galaxies should show a tendency to be aligned
with the projected major principal axes.
Let θ2d be the angle between the projected cluster major axis and galaxy position vector
in the plane of sky. The probability distribution can be calculated by integrating equation
– 5 –
(3) along the line of sight as
p(cos θ2d) =
(x̂i ·M
ij · x̂j)
2dx̂3, (6)
where (x̂3) is now chosen to be in the direction of the line of sight. In other words, we
consider a certain Cartesian coordinate system in which the x̂3 direction is parallel to the
line of the sight to the cluster center of mass. Note that in this coordinate system the tidal
tensor is not necessarily diagonal.
Through the similarity transformation
T̂ = Rt · Λ̂T ·R, (7)
λ̂1 0 0
0 λ̂2 0
0 0 λ̂3
, (8)
one can express the nondiagonal unit tidal tensor T̂ in terms of its eigenvalues.Here, the
rotation matrix R has the form of (Binney 1985)
− sinψ − cosψ cos ξ cosψ sin ξ
cosψ − sinψ cos ξ sinψ sin ξ
0 sin ξ cos ξ
 , (9)
where (ξ, ψ) is the polar coordinate of the line-of-sight direction in the principal axis frame
of the cluster.
Through equations (6)-(9), we finally final an analytic expression for the probability
density distribution of cos θ2d in terms of {λ̂i}
i=1 and the correlation parameter s:
p(cos θ2d) =
(1 + s)2(1− 2s)
Q−3/2dx̂3, (10)
with the factor Q defined as
Q ≡ [(1 + s)2 − 3s(1 + s)](A1λ̂1 + A2λ̂2 + A3λ̂3)(1− x̂
+[(1 + s)2 − 3s(1 + s)](C1λ̂1 + C2λ̂2 + C3λ̂3)x̂
+6s(1 + s)(B1λ̂1 +B2λ̂2 +B3λ̂3)x̂3
1− x̂2
, (11)
where the coefficients {Ai}
i=1, {Bi}
i=1, {Ci}
i=1 are given as
A1 = cosψ cos θ2d(cosψ cos θ2d − 2 sinψ cos ξ sin θ2d) + sin
2 θ2d(sin
2 ψ + sin2 ξ cos2 ψ),
– 6 –
A2 = sinψ cos θ2d(sinψ cos θ2d + 2 cosψ cos ξ sin θ2d) + sin
2 θ2d(cos
2 ψ + sin2 ξ sin2 ψ),
A3 = cos
2 θ2d + cos
2 ξ sin2 θ2d,
B1 = − sin ξ cosψ(cos ξ sin θ2d cosψ + sinψ cos θ2d),
B2 = − sin ξ sinψ(cos ξ sin θ2d sinψ − cosψ cos θ2d),
B3 = cos ξ sin ξ sin θ2d,
C1 = sin
2 ψ + cos2 ξ cos2 ψ,
C2 = cos
2 ψ + cos2 ξ sin2 ψ,
C3 = sin
With this new modified algorithm in the two dimensional space, we can reconstruct the
intrinsic shape of a triaxial cluster halo from the observed two dimensional image.
3. APPLICATION TO THE VIRGO CLUSTER HALO
3.1. Observational Data and Analysis
The Virgo cluster is the nearest richly populated cluster of galaxies whose properties
has been studied fruitfully for long (e.g., Bohringer et al. 1994; West & Blakeslee 2000, and
references therein). It is known to have approximately 1275 member galaxies (Binggeli et al.
1985) and located at a distance of approximately 16.1 Mpc from us (Tonry et al. 2001) with
the major axis inclined at an angle of approximately 10o with respect to the line of sight
(West & Blakeslee 2000).
Near the center of the Virgo cluster is located the large ellipticity galaxy M87 (or Virgo
A). The major axis of the Virgo cluster is found to be inclined approximately 10◦ with respect
to the line of sight direction to M87 (West & Blakeslee 2000). We use data from the Virgo
Cluster Catalogue (Binggeli et al. 1985) which compiles the equatorial coordinates of total
1275 member galaxies.
3.2. Coordinate Transformation and Projection Effect
Let r and (α, δ) represent the three dimensional distance to a member galaxy and its
equatorial coordinates, respectively. Under the assumption that the position of M87 is the
center of mass of the Virgo cluster, the Cartesian coordinate of a member galaxy in the
center of mass frame can be written as
x1 = r cosα cos δ − x1va
x2 = r sinα cos δ − x2va
– 7 –
x3 = r sin δ − x3va (12)
where (x1va, x2va, x3va) represents the position of M87. The equatorial coordinates of M87
is measured to be αva = 187.71
◦ and δva = 12.39
◦. The distance to Virgo A, rva, from us is
also known to be approximately 16.1Mpc (Tonry et al. 2001; SBF survey). Thus,we have a
full information on (x1va, x2va, x3va).
Now let us consider a coordinate system where the third axis is in in the direction of the
line of sight to M87. Let (u, ϑ, ϕ) be the spherical polar coordinate of the member galaxy
in this coordinate system. It can be found through coordinate transformation as
sin ξva cosψva cos ξva cosψva − sinψva
sin ξva sinψva cos ξva sinψva cosψva
cos ξva − sin ξva 0
 (13)
where (ξva, ψva) is the polar coordinate of M87, so that ξva ≡ π/2− δva.
In the plane of sky projected along the line of sight direction, the spherical polar coordi-
nates of the member galaxy can be regarded as the Cartesian coordinates as r → 0, ϑ→ xl,
and φ → yl. Basically, it represents a two dimensional projected position of a Virgo cluster
member galaxy in the plane of sky with the position of M87 as a center.
For each member galaxy from the Virgo Cluster Catalogue, we have determined the two
dimensional projected position (xl, yl) using the given equatorial coordinates.
3.3. Virgo Cluster Reconstruction
To measure the alignments between the positions of the Virgo cluster galaxies and the
projected major axis of the Virgo cluster and compare the distribution of the alignment
angles with the analytic model (eq.[6]), it is necessary to find the direction of the projected
major axis in the coordinate system of (x1l, x2l). Equivalently, it is necessary to have an
information on the polar coordinates of the line of sight, (ξ, ψ) with respect to the three
dimensional principal axis of the Virgo cluster.
The seminal paper of West & Blakeslee (2000) provides us with the information on ξ
i.e., the angle between the three dimensional major axis of the Virgo cluster and the line of
sight direction as approximately 10◦. However, we still need the azimuthal angle ψ of the
major axis of the Virgo cluster.
To find the azimuthal angle ψ, we first let ξ = 10◦ and ψ = 0 in the analytic model
(eq.[6]). It implies that we choose a certain Cartesian coordinates (x1p, x2p) where the angle
– 8 –
ψ vanishes. Then, we transform the coordinate system of (x1l, x2l) into this new coordinate
system as
cosψ sinψ
− sinψ cosψ
. (14)
Here, note that (cosψ, sinψ) corresponds to the projected major axis of the Virgo cluster in
the (x1l, x2l) coordinate system. It is expected that in this new (x1p, x2p)-coordinate system
the the observationally measured distribution should fit the analytic model (eq.[6] best.
For a given ψ, we measure the alignment angle between the projected major axis and
position vector of each Virgo cluster galaxy as
cos θ2d = x̂1p cosψ + x̂2p sinψ, (15)
where x̂p ≡ xp/|xp|. Then, by counting the number of galaxies galaxy’s number density
as a function of cos θd, we can derive the probability distribution, p(cos θd). We fit this
observational distribution with the analytic model, adjusting the values of λ1, λ2 and s
through χ2-minimization. We repeart the whole process varying the value of ψ, and seek
for the value of ψ which yields the smallest χ2 value. As a final step, we determine the
corresponding best-fit values of λ1 and λ2.
Finally, we find the axial ratios of the Virgo cluster to be a/c = 0.54 and b/c = 0.73
by using equation (1) with the constraint of δc =
i=1 λi. The best-fit value of s is also
determined to be −0.25, indicating that the positions of the cluster galaxies are indeed
aligned with the major axes of the Virgo cluster. Figure [1] plots the two dimensional
projected positions of the Virgo cluster galaxies in the (x1p, x2p)-coordinate system. The
arrow represents the direction of the projected major axis determined from the chosen value
of ψ.
Figure 2 plots the probability distribution of the alignment angles for four different cases
of ψ. In each panel, the histogram with Poisson errors is the observational distribution while
the solid line represents the analytic fitting model. The dotted line stands for the case of no
alignment at all. The top left panel corresponds to the finally chosen value of ψ for which
the analytic fitting model and the observational result agree with each other best. The other
three panels show the three exemplary cases of ψ for which the analytic fitting model and
the observational result do not agree with each other well with relatively high χ2 value.
To investigate whether the reconstructed axial ratios of the Virgo cluster changes with
the radius from the center, we introduce a cut-off radius, Rcut, and remeasure the axial ratios
using only those galaxies located within Rcut from the center in the (x1p, x2p)- corrodinate
system. We repeat the same process but using four different values of Rcut. Table 1 lists the
values of the resulting best-fit axial ratios a/c and b/c, and the best-fit correlation parameter
– 9 –
for the four different cut-off radii of Rcut. As can be seen, at the inner radius smaller than
the maximum one, the shape tends to be more prolate-like, consistent with recent numerical
report (e.g., Hayashi et al. 2007). Note also that the value of s is consistently −0.25, which
implies the strength of the tidal interaction with surrounding matter is consistent.
Figure 3 plots the probability distributions for the four different cases of Rcut. The top
left panel shows the case of maximum cut-off radius. As can be seen, the agreements between
the analytic fitting model and the observational result are quite good even at inner radii.
4. SUMMARY AND DISCUSSION
We have modified the cluster reconstruction algorithm which was originally developed
by Lee & Kang (2006) to apply it to the two dimensional projected images of galaxy clusters
in practice. Assuming that unless the cluster major axes are in the line-of-sight direction,
we have found that the alignments between the galaxy positions and the projected major
axes can be used to reconstruct the two axial ratios of the triaxial clusters. We have applied
the modified algorithm to the observational data of the Virgo cluster and shown that the
reconstructed axial ratios of (1: 0.54 : 0.73) are in good agreement with the recent report
from the Virgo Cluster survey (Mei et al. 2007), which proves the validity and usefulness of
our method in practice.
Now that the Virgo cluster is found to be triaxial, let us discuss on the triaxiality
effect on the mass estimation. For simplification, let us assume that the Virgo cluster has a
uniform density. Then, the mass of the triaxial Virgo cluster with the axial ratios given as
(1 : 0.54 : 0.73) is estimated to be 2.6 times larger than the spherical case since the spherical
radius is close to the minor axis length of the Virgo cluster since the major axis of the Virgo
cluster is very well aligned with the line of sight direction. Therefore, it can cause maximum
∼ 50% errors to neglect the Virgo cluster triaxiality.
The most prominent merit of our method over the previous one is that it reconstructs
directly the intrinsic three dimensional structures of the underlying triaxial dark matter halo
using the fact that the cluster triaxiality originated from the tidal interaction. Convention-
ally, the triaxial shape of a cluster is found through calculating its inertia momentum tensor.
This conventional method, however, is unlikely to yield the intrinsic shape of the underlying
dark matter halo unless the target cluster is a well relaxed system. In addition, our method
does not resort to any simplified assumption like the axis-symmetry or the alignment with
the line-of-sight and etc.
Yet, it is worth mentioning here a couple of limitations of our method. First, it assumes
– 10 –
that the major axes of the clusters are not aligned with the line-of-sight direction so that
the alignments between the galaxy positions and the projected major axes can be measured.
Second, for the Virgo cluster, the crucial information on the angle ξ between the three
dimensional major axis and the line of sight has been already given West & Blakeslee (2000).
Which simplifies the whole of our method since it was only ψ that has to be determined.
For most clusters, however, this information is not given, so that both the values of ξ and ψ
have to be determined through fitting before finding the axial ratios.
Third, its success is subject to the validity of the LK06 algorithm. According to the
numerical test, the LK06 algorithm suffers approximately 20% errors for the case that the
number of the member galaxies is not high enough. It implies that the LK06 algorithm is
definitely restricted to the rich clusters with large numbers of galaxies. Therefore, it may be
necessary to refine and improve the LK06 algorithm itself for the application to poor cluster
samples. Our future work is in this direction.
This work is supported by the research grant No. R01-2005-000-10610-0 from the Basic
Research Program of the Korea Science and Engineering Foundation.
– 11 –
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This preprint was prepared with the AAS LATEX macros v5.2.
– 14 –
Fig. 1.— Positions of the Virgo member galaxies in the two dimensional projected space.
The arrow represents the direction of the projected major axes of the Virgo cluster.
– 15 –
Fig. 2.— Probability density distributions of the cosines of the angles between the position
vectors of the Virgo cluster galaxies and four different choices of the Virgo cluster major
axis in the two dimensional projected plane of sky. In each panel, the histogram with
Poisson errors represents the observational data points from the Virgo Catalog, the solid
line is the analytic fitting function based on the LK06 reconstruction algorithm, and the
dotted line corresponds to the case of no correlation. The top left panel corresponds to the
best-fit result according to which the major axes of the Virgo cluster in the two dimensional
projected plane is determined, while the other three panels show how the agreements between
the observational and the analytic results change if different directions other than the major
axes are used.
– 16 –
Fig. 3.— Comparison between the observational and the analytic results for the alignments
at different two dimensional cut-off radii (Rcut).
– 17 –
Table 1. The cut-off radius (Rcut) in the two dimensional projected space, the number of
the member galaxies enclosed within Rcut, the reconstructed two axial ratios, and the
best-fit value of the correlation parameter.
Rcut Ng a/c b/c s
(Mpc)
2.24 1275 0.53 0.73 −0.25
1.68 1221 0.64 0.69 −0.25
1.49 1154 0.64 0.69 −0.25
1.12 902 0.64 0.69 −0.25
	INTRODUCTION
	THEORETICAL MODEL
	Overview of the LK06 Algorithm
	Projection Effect
	APPLICATION TO THE VIRGO CLUSTER HALO
	Observational Data and Analysis
	Coordinate Transformation and Projection Effect
	Virgo Cluster Reconstruction
	SUMMARY AND DISCUSSION

  To use galaxy clusters as a cosmological probe, it is important to account
for their triaxiality. Assuming that the triaxial shapes of galaxy clusters are
induced by the tidal interaction with the surrounding matter, Lee and Kang
recently developed a reconstruction algorithm for the measurement of the axial
ratio of a triaxial cluster. We examine the validity of this reconstruction
algorithm by performing an observational test of it with the Virgo cluster as a
target. We first modify the LK06 algorithm by incorporating the two dimensional
projection effect. Then, we analyze the 1275 member galaxies from the Virgo
Cluster Catalogue and find the projected direction of the Virgo cluster major
axis by measuring the anisotropy in the spatial distribution of the member
galaxies in the two dimensional projected plane. Applying the modified
reconstruction algorithm to the analyzed data, we find that the axial ratio of
the triaxial Virgo cluster is (1: 0.54 : 0.73). This result is consistent with
the recent observational report from the Virgo Cluster Survey, proving the
robustness of the reconstruction algorithm. It is also found that at the inner
radii the shape tends to be more like prolate. We discuss the possible effect
of the Virgo cluster triaxiality on the mass estimation.

Reconstructing the Intrinsic Triaxial Shape of the Virgo Cluster
Bomee Lee and Jounghun Lee
Department of Physics and Astronomy, FPRD, Seoul National University, Seoul 151-747,
Korea
bmlee@astro.snu.ac.kr, jounghun@astro.snu.ac.kr
ABSTRACT
To use galaxy clusters as a cosmological probe, it is important to account for
their triaxiality. Assuming that the triaxial shapes of galaxy clusters are induced
by the tidal interaction with the surrounding matter, Lee and Kang recently
developed a reconstruction algorithm for the measurement of the axial ratio of
a triaxial cluster. We examine the validity of this reconstruction algorithm by
performing an observational test of it with the Virgo cluster as a target. We
first modify the LK06 algorithm by incorporating the two dimensional projec-
tion effect. Then, we analyze the 1275 member galaxies from the Virgo Cluster
Catalogue and find the projected direction of the Virgo cluster major axis by
measuring the anisotropy in the spatial distribution of the member galaxies in
the two dimensional projected plane. Applying the modified reconstruction al-
gorithm to the analyzed data, we find that the axial ratio of the triaxial Virgo
cluster is (1: 0.54 : 0.73). This result is consistent with the recent observational
report from the Virgo Cluster Survey, proving the robustness of the reconstruc-
tion algorithm. It is also found that at the inner radii the shape tends to be more
like prolate. We discuss the possible effect of the Virgo cluster triaxiality on the
mass estimation.
Subject headings: cosmology:theory — large-scale structure of universe
1. INTRODUCTION
Galaxy clusters provide one of the most powerful tools to constrain the key cosmological
parameters. In the era of precision cosmology, it is important to determine their mass as
accurately as possible before using them as a cosmological probe. Any kind of simplified
assumption about the properties of galaxy clusters could cause substantial systematics in
the mass estimation. The triaxial shapes of galaxy clusters are one of such properties.
http://arxiv.org/abs/0704.1400v1
– 2 –
It has been long known both observationally and numerically that the galaxy clusters
are noticeably triaxial (e.g., Frenk et al. 1988; West 1989; Plionis et al. 1991; Warren et al.
1992). While plenty of efforts have been already made to take into account the triaxial shapes
of galaxy clusters (Jing & Suto 2002; Fox & Pen 2001; Suwa et al. 2003; Lee & Suto 2004;
Kasun & Evrard 2005; Hopkins et al. 2005; Lee et al. 2005; Smith & Watts 2005; Hayashi et al.
2007), previous studies have been largely focused on the statistical treatment of cluster tri-
axiality. For the measurement of the gas mass fraction of galaxy clusters that can pro-
vide powerful constraints on the density parameter and the dark energy equation of state
(White et al. 1993; Lubin et al. 1996; Cen 1997; Evrard 1997; Cooray 1998; Laroque et al.
2006; Ferramacho & Blanchard 2007), however, it is necessary to deal with the individual
clusters and their triaxial shapes.
The standard picture based on the cosmic web theory (Bardeen et al. 1986; Bond et al.
1996) explains that galaxy clusters are rare events corresponding to the local maxima of
the initial density field and form at the dense nodes of the local filaments in the cosmic
web through tidal interactions with the surrounding matter. The tidal effect from the sur-
rounding matter results in the deviation of cluster shapes from spherical symmetry as well
as the preferential alignments of cluster galaxies (or halos) with the major axes of their host
clusters (Binggeli 1982; Struble & Peebles 1985; Hopkins et al. 2005; Kasun & Evrard 2005;
Lee et al. 2005; Algood et al. 2006; Altay et al. 2006; Paz et al. 2006).
In the frame of this standard scenario, Lee & Kang (2006, hereafter, LK06) have re-
cently developed an analytic algorithm to reconstruct the triaxial shapes of individual clus-
ters. The key concept of the LK06 algorithm is that the two axial ratios of a triaxial cluster
are related to the eigenvalues of the local tidal shear tensor. By measuring the spatial align-
ment of the cluster galaxies with the major axis of their host cluster, one can determine the
eigenvalues of the local tidal tensor, which will in turn yields the two axial ratios of a triaxial
cluster.
Testing their analytic model against high-resolution N-body simulations, LK06 have
shown that their algorithm works well within 20% errors. Now that the LK06 algorithm is
known to work in principle, it is time to test the algorithm against observations. Our goal
here is to apply the LK06 algorithm to real observational data and examine its validity in
practice. Here, we use the Virgo cluster as a target, whose triaxial shape has been very
recently measured observationally (Mei et al. 2007).
The organization of this paper is as follows. In §2, a brief overview of the LK06 algorithm
is provided and how to incorporate the two dimensional projection effect into the algorithm is
explained. In §3, the Virgo cluster data are analyzed and its triaxial shapes are reconstructed
using the LK06 algorithm. In §4, the results are summarized, and the advantages and the
– 3 –
caveats of our model are discussed.
2. THEORETICAL MODEL
2.1. Overview of the LK06 Algorithm
According to the LK06 algorithm, the cluster triaxial shape originates from its tidal
interaction with the surrounding matter distribution. This assumption has been verified from
high-resolution N-body simulation which demonstrated clearly that the tidal field elongates
the cluster shapes (e.g., Altay et al. 2006, and references therein). LK06 has shown that the
two axial ratios of a triaxial cluster are related to the three eigenvalues of the local tidal
tensor defined as the second derivative of the gravitational potential:
1− λ2
1− λ3
1− λ1
1− λ3
, (1)
where {a, b, c} (with a ≤ b ≤ c)are the three principal axis lengths of a cluster and {λ1, λ2, λ3}
(with λ1 ≥ λ2 ≥ λ3) are the three eigenvalues of the local tidal tensor, T. According to this
formula, one can estimate the cluster axial ratios if λ1, λ2, λ3 is fixed by fitting the probability
density distribution, p(cos θ) analytically to the observational data. LK06 suggested that
the preferential locations of the cluster galaxies near the cluster major axes given the local
tidal tensor be described as
〈x̂ix̂j |T̂ 〉 =
δij + sT̂ikT̂kj. (2)
where x̂ ≡ (x̂i) is the unit position vector of a cluster galaxy, T̂ ≡ T/|T| is the unit tidal
shear tensor, and s ∈ [−1, 1] is the correlation parameter that represents the correlation
strength between x̂ and T̂. Under the assumption that the minor axis of the tidal shear
tensor is in the direction of the cluster major axis, equation (2) basically describes the
alignment between the position of a cluster galaxy and the major axis of its host cluster. If
s = −1, there is a maximum alignment. If s = 1, there is a maximum anti-alignment. The
case of s = 0 corresponds to no alignment.
Let θ3d be the angle between the host cluster major axis and the galaxy position vector.
Under the assumption that the cluster major axis is in the direction of the minor principal
axis of the tidal shear tensor, The probability density distribution of cos θ3d was derived by
LK6 as
p(cos θ3d) =
(x̂i ·M
ij · x̂j)
2 , (3)
– 4 –
where φ is an azimuthal angle of x̂ measured in an arbitrary coordinate system. Here,
covariance matrix M is defined as M ≡ 〈x̂ix̂j |T̂ 〉. Note that equation (3) holds good for any
arbitrary coordinate system in which the tidal shear tensor is not necessarily diagonal.
In the principal axis frame of the tidal tensor, equation (4) can be expressed only in
terms of the three eigenvalues of the tidal shear tensor as
p(cos θ) =
(1− s+ 3sλ̂2i )
sin2 θ cos2 φ
1− s+ 3sλ̂2
sin2 θ sin2 φ
1− s+ 3sλ̂2
cos2 θ
1− s+ 3sλ̂2
dφ, (4)
where {λ̂i}
i=1 are the eigenvalues of T̂, related to {λi}
i=1 as
δcλ̂1
λ̂1 + λ̂2 + λ̂3
, λ2 =
δcλ̂2
λ̂1 + λ̂2 + λ̂3
, λ3 =
δcλ̂3
λ̂1 + λ̂2 + λ̂3
, (5)
where δc ≈ 1.68 is the linear density threshold for a dark halo (Eke et al. 1996) satisfying
the following constraint of δc =
i=1 λi.
The key concept of LK06 algorithm is that by fitting equation (3) to the observed
probability distribution, one can find the best-fit values of λ1, λ2 and s, and then determine
the cluster axial ratios using equation (1). Although LK06 algorithm allows us in principle
to reconstruct the three dimensional intrinsic triaxial shapes of individual clusters, it is
restricted to the cases where the informations on the three dimensional positions of the
cluster galaxies in the cluster principal axis frame are given. Unfortunately, for most clusters,
these informations are not available but only two dimensional projected images of clusters.
In §2.2, we attempt to modify the LK06 algorithm in order to incorporate the two
dimensional projection effect.
2.2. Projection Effect
Let us suppose that the position vectors of the cluster galaxies are all projected along
the line of sight direction onto the plane of a sky. Unless the major axis of the host cluster is
perfectly aligned with the line-of-sight direction to the cluster center, one would expect that
the projected position vectors of the cluster galaxies should show a tendency to be aligned
with the projected major principal axes.
Let θ2d be the angle between the projected cluster major axis and galaxy position vector
in the plane of sky. The probability distribution can be calculated by integrating equation
– 5 –
(3) along the line of sight as
p(cos θ2d) =
(x̂i ·M
ij · x̂j)
2dx̂3, (6)
where (x̂3) is now chosen to be in the direction of the line of sight. In other words, we
consider a certain Cartesian coordinate system in which the x̂3 direction is parallel to the
line of the sight to the cluster center of mass. Note that in this coordinate system the tidal
tensor is not necessarily diagonal.
Through the similarity transformation
T̂ = Rt · Λ̂T ·R, (7)
λ̂1 0 0
0 λ̂2 0
0 0 λ̂3
, (8)
one can express the nondiagonal unit tidal tensor T̂ in terms of its eigenvalues.Here, the
rotation matrix R has the form of (Binney 1985)
− sinψ − cosψ cos ξ cosψ sin ξ
cosψ − sinψ cos ξ sinψ sin ξ
0 sin ξ cos ξ
 , (9)
where (ξ, ψ) is the polar coordinate of the line-of-sight direction in the principal axis frame
of the cluster.
Through equations (6)-(9), we finally final an analytic expression for the probability
density distribution of cos θ2d in terms of {λ̂i}
i=1 and the correlation parameter s:
p(cos θ2d) =
(1 + s)2(1− 2s)
Q−3/2dx̂3, (10)
with the factor Q defined as
Q ≡ [(1 + s)2 − 3s(1 + s)](A1λ̂1 + A2λ̂2 + A3λ̂3)(1− x̂
+[(1 + s)2 − 3s(1 + s)](C1λ̂1 + C2λ̂2 + C3λ̂3)x̂
+6s(1 + s)(B1λ̂1 +B2λ̂2 +B3λ̂3)x̂3
1− x̂2
, (11)
where the coefficients {Ai}
i=1, {Bi}
i=1, {Ci}
i=1 are given as
A1 = cosψ cos θ2d(cosψ cos θ2d − 2 sinψ cos ξ sin θ2d) + sin
2 θ2d(sin
2 ψ + sin2 ξ cos2 ψ),
– 6 –
A2 = sinψ cos θ2d(sinψ cos θ2d + 2 cosψ cos ξ sin θ2d) + sin
2 θ2d(cos
2 ψ + sin2 ξ sin2 ψ),
A3 = cos
2 θ2d + cos
2 ξ sin2 θ2d,
B1 = − sin ξ cosψ(cos ξ sin θ2d cosψ + sinψ cos θ2d),
B2 = − sin ξ sinψ(cos ξ sin θ2d sinψ − cosψ cos θ2d),
B3 = cos ξ sin ξ sin θ2d,
C1 = sin
2 ψ + cos2 ξ cos2 ψ,
C2 = cos
2 ψ + cos2 ξ sin2 ψ,
C3 = sin
With this new modified algorithm in the two dimensional space, we can reconstruct the
intrinsic shape of a triaxial cluster halo from the observed two dimensional image.
3. APPLICATION TO THE VIRGO CLUSTER HALO
3.1. Observational Data and Analysis
The Virgo cluster is the nearest richly populated cluster of galaxies whose properties
has been studied fruitfully for long (e.g., Bohringer et al. 1994; West & Blakeslee 2000, and
references therein). It is known to have approximately 1275 member galaxies (Binggeli et al.
1985) and located at a distance of approximately 16.1 Mpc from us (Tonry et al. 2001) with
the major axis inclined at an angle of approximately 10o with respect to the line of sight
(West & Blakeslee 2000).
Near the center of the Virgo cluster is located the large ellipticity galaxy M87 (or Virgo
A). The major axis of the Virgo cluster is found to be inclined approximately 10◦ with respect
to the line of sight direction to M87 (West & Blakeslee 2000). We use data from the Virgo
Cluster Catalogue (Binggeli et al. 1985) which compiles the equatorial coordinates of total
1275 member galaxies.
3.2. Coordinate Transformation and Projection Effect
Let r and (α, δ) represent the three dimensional distance to a member galaxy and its
equatorial coordinates, respectively. Under the assumption that the position of M87 is the
center of mass of the Virgo cluster, the Cartesian coordinate of a member galaxy in the
center of mass frame can be written as
x1 = r cosα cos δ − x1va
x2 = r sinα cos δ − x2va
– 7 –
x3 = r sin δ − x3va (12)
where (x1va, x2va, x3va) represents the position of M87. The equatorial coordinates of M87
is measured to be αva = 187.71
◦ and δva = 12.39
◦. The distance to Virgo A, rva, from us is
also known to be approximately 16.1Mpc (Tonry et al. 2001; SBF survey). Thus,we have a
full information on (x1va, x2va, x3va).
Now let us consider a coordinate system where the third axis is in in the direction of the
line of sight to M87. Let (u, ϑ, ϕ) be the spherical polar coordinate of the member galaxy
in this coordinate system. It can be found through coordinate transformation as
sin ξva cosψva cos ξva cosψva − sinψva
sin ξva sinψva cos ξva sinψva cosψva
cos ξva − sin ξva 0
 (13)
where (ξva, ψva) is the polar coordinate of M87, so that ξva ≡ π/2− δva.
In the plane of sky projected along the line of sight direction, the spherical polar coordi-
nates of the member galaxy can be regarded as the Cartesian coordinates as r → 0, ϑ→ xl,
and φ → yl. Basically, it represents a two dimensional projected position of a Virgo cluster
member galaxy in the plane of sky with the position of M87 as a center.
For each member galaxy from the Virgo Cluster Catalogue, we have determined the two
dimensional projected position (xl, yl) using the given equatorial coordinates.
3.3. Virgo Cluster Reconstruction
To measure the alignments between the positions of the Virgo cluster galaxies and the
projected major axis of the Virgo cluster and compare the distribution of the alignment
angles with the analytic model (eq.[6]), it is necessary to find the direction of the projected
major axis in the coordinate system of (x1l, x2l). Equivalently, it is necessary to have an
information on the polar coordinates of the line of sight, (ξ, ψ) with respect to the three
dimensional principal axis of the Virgo cluster.
The seminal paper of West & Blakeslee (2000) provides us with the information on ξ
i.e., the angle between the three dimensional major axis of the Virgo cluster and the line of
sight direction as approximately 10◦. However, we still need the azimuthal angle ψ of the
major axis of the Virgo cluster.
To find the azimuthal angle ψ, we first let ξ = 10◦ and ψ = 0 in the analytic model
(eq.[6]). It implies that we choose a certain Cartesian coordinates (x1p, x2p) where the angle
– 8 –
ψ vanishes. Then, we transform the coordinate system of (x1l, x2l) into this new coordinate
system as
cosψ sinψ
− sinψ cosψ
. (14)
Here, note that (cosψ, sinψ) corresponds to the projected major axis of the Virgo cluster in
the (x1l, x2l) coordinate system. It is expected that in this new (x1p, x2p)-coordinate system
the the observationally measured distribution should fit the analytic model (eq.[6] best.
For a given ψ, we measure the alignment angle between the projected major axis and
position vector of each Virgo cluster galaxy as
cos θ2d = x̂1p cosψ + x̂2p sinψ, (15)
where x̂p ≡ xp/|xp|. Then, by counting the number of galaxies galaxy’s number density
as a function of cos θd, we can derive the probability distribution, p(cos θd). We fit this
observational distribution with the analytic model, adjusting the values of λ1, λ2 and s
through χ2-minimization. We repeart the whole process varying the value of ψ, and seek
for the value of ψ which yields the smallest χ2 value. As a final step, we determine the
corresponding best-fit values of λ1 and λ2.
Finally, we find the axial ratios of the Virgo cluster to be a/c = 0.54 and b/c = 0.73
by using equation (1) with the constraint of δc =
i=1 λi. The best-fit value of s is also
determined to be −0.25, indicating that the positions of the cluster galaxies are indeed
aligned with the major axes of the Virgo cluster. Figure [1] plots the two dimensional
projected positions of the Virgo cluster galaxies in the (x1p, x2p)-coordinate system. The
arrow represents the direction of the projected major axis determined from the chosen value
of ψ.
Figure 2 plots the probability distribution of the alignment angles for four different cases
of ψ. In each panel, the histogram with Poisson errors is the observational distribution while
the solid line represents the analytic fitting model. The dotted line stands for the case of no
alignment at all. The top left panel corresponds to the finally chosen value of ψ for which
the analytic fitting model and the observational result agree with each other best. The other
three panels show the three exemplary cases of ψ for which the analytic fitting model and
the observational result do not agree with each other well with relatively high χ2 value.
To investigate whether the reconstructed axial ratios of the Virgo cluster changes with
the radius from the center, we introduce a cut-off radius, Rcut, and remeasure the axial ratios
using only those galaxies located within Rcut from the center in the (x1p, x2p)- corrodinate
system. We repeat the same process but using four different values of Rcut. Table 1 lists the
values of the resulting best-fit axial ratios a/c and b/c, and the best-fit correlation parameter
– 9 –
for the four different cut-off radii of Rcut. As can be seen, at the inner radius smaller than
the maximum one, the shape tends to be more prolate-like, consistent with recent numerical
report (e.g., Hayashi et al. 2007). Note also that the value of s is consistently −0.25, which
implies the strength of the tidal interaction with surrounding matter is consistent.
Figure 3 plots the probability distributions for the four different cases of Rcut. The top
left panel shows the case of maximum cut-off radius. As can be seen, the agreements between
the analytic fitting model and the observational result are quite good even at inner radii.
4. SUMMARY AND DISCUSSION
We have modified the cluster reconstruction algorithm which was originally developed
by Lee & Kang (2006) to apply it to the two dimensional projected images of galaxy clusters
in practice. Assuming that unless the cluster major axes are in the line-of-sight direction,
we have found that the alignments between the galaxy positions and the projected major
axes can be used to reconstruct the two axial ratios of the triaxial clusters. We have applied
the modified algorithm to the observational data of the Virgo cluster and shown that the
reconstructed axial ratios of (1: 0.54 : 0.73) are in good agreement with the recent report
from the Virgo Cluster survey (Mei et al. 2007), which proves the validity and usefulness of
our method in practice.
Now that the Virgo cluster is found to be triaxial, let us discuss on the triaxiality
effect on the mass estimation. For simplification, let us assume that the Virgo cluster has a
uniform density. Then, the mass of the triaxial Virgo cluster with the axial ratios given as
(1 : 0.54 : 0.73) is estimated to be 2.6 times larger than the spherical case since the spherical
radius is close to the minor axis length of the Virgo cluster since the major axis of the Virgo
cluster is very well aligned with the line of sight direction. Therefore, it can cause maximum
∼ 50% errors to neglect the Virgo cluster triaxiality.
The most prominent merit of our method over the previous one is that it reconstructs
directly the intrinsic three dimensional structures of the underlying triaxial dark matter halo
using the fact that the cluster triaxiality originated from the tidal interaction. Convention-
ally, the triaxial shape of a cluster is found through calculating its inertia momentum tensor.
This conventional method, however, is unlikely to yield the intrinsic shape of the underlying
dark matter halo unless the target cluster is a well relaxed system. In addition, our method
does not resort to any simplified assumption like the axis-symmetry or the alignment with
the line-of-sight and etc.
Yet, it is worth mentioning here a couple of limitations of our method. First, it assumes
– 10 –
that the major axes of the clusters are not aligned with the line-of-sight direction so that
the alignments between the galaxy positions and the projected major axes can be measured.
Second, for the Virgo cluster, the crucial information on the angle ξ between the three
dimensional major axis and the line of sight has been already given West & Blakeslee (2000).
Which simplifies the whole of our method since it was only ψ that has to be determined.
For most clusters, however, this information is not given, so that both the values of ξ and ψ
have to be determined through fitting before finding the axial ratios.
Third, its success is subject to the validity of the LK06 algorithm. According to the
numerical test, the LK06 algorithm suffers approximately 20% errors for the case that the
number of the member galaxies is not high enough. It implies that the LK06 algorithm is
definitely restricted to the rich clusters with large numbers of galaxies. Therefore, it may be
necessary to refine and improve the LK06 algorithm itself for the application to poor cluster
samples. Our future work is in this direction.
This work is supported by the research grant No. R01-2005-000-10610-0 from the Basic
Research Program of the Korea Science and Engineering Foundation.
– 11 –
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This preprint was prepared with the AAS LATEX macros v5.2.
– 14 –
Fig. 1.— Positions of the Virgo member galaxies in the two dimensional projected space.
The arrow represents the direction of the projected major axes of the Virgo cluster.
– 15 –
Fig. 2.— Probability density distributions of the cosines of the angles between the position
vectors of the Virgo cluster galaxies and four different choices of the Virgo cluster major
axis in the two dimensional projected plane of sky. In each panel, the histogram with
Poisson errors represents the observational data points from the Virgo Catalog, the solid
line is the analytic fitting function based on the LK06 reconstruction algorithm, and the
dotted line corresponds to the case of no correlation. The top left panel corresponds to the
best-fit result according to which the major axes of the Virgo cluster in the two dimensional
projected plane is determined, while the other three panels show how the agreements between
the observational and the analytic results change if different directions other than the major
axes are used.
– 16 –
Fig. 3.— Comparison between the observational and the analytic results for the alignments
at different two dimensional cut-off radii (Rcut).
– 17 –
Table 1. The cut-off radius (Rcut) in the two dimensional projected space, the number of
the member galaxies enclosed within Rcut, the reconstructed two axial ratios, and the
best-fit value of the correlation parameter.
Rcut Ng a/c b/c s
(Mpc)
2.24 1275 0.53 0.73 −0.25
1.68 1221 0.64 0.69 −0.25
1.49 1154 0.64 0.69 −0.25
1.12 902 0.64 0.69 −0.25
	INTRODUCTION
	THEORETICAL MODEL
	Overview of the LK06 Algorithm
	Projection Effect
	APPLICATION TO THE VIRGO CLUSTER HALO
	Observational Data and Analysis
	Coordinate Transformation and Projection Effect
	Virgo Cluster Reconstruction
	SUMMARY AND DISCUSSION
