A&A 375, 338343 (2001)
DOI: 10.1051/00046361:20010847
The velocity field of collapsing spherical structures
Limitations of the spherical infall model in mass estimation
N. Hiotelis^{}
Lyceum of Naxos, Chora of Naxos, Naxos 84300, Greece
Received 15 January 2001 / Accepted 28 May 2001
Abstract
We assume that the amplitude of the caustics in
redshift space is a sum of two components: the first one can be
predicted by the spherical infall model with no random motion, and
the second is due to the random motion distribution. Smooth model
curves are used to estimate the maximum values of the first
component for the Coma cluster. Then, an approximation of the
radial component of the infall velocity  based on the above
curves  is derived and a mass profile of the cluster is
calculated. This mass profile, that is an upper limit for the
spherical infall model, combined with estimations given by other
authors provides an approximation of a lower limit for the mass of
the system.
Key words: galaxies: clusters: general 
galaxies: clusters: individual: coma cluster 
cosmology: largescale structure of Universe
The spherical infall model has been extensively discussed in the literature (Gunn &
Gott 1972; Silk 1974; Gunn 1977; Peebles 1976,
1980; Schechter 1980; Schectman 1982; Ostriker et al. 1988). This model assumes that galaxy clusters started as small
density perturbations in the early universe. These perturbations eventually deviate
from the general expansion and after reaching a maximum radius they start
collapsing. The most probable scenario is that galaxies formed during the expansion
phase of their cluster. Thus, the cluster consists of largely individual galaxies
during its collapse.
The observed velocities of these galaxies (along the
lineofsight) give important information about the dynamical
state of the cluster (Kaiser 1987). Regös & Geller
(1989) showed that plotting the observed velocities of
galaxies as function of their angular distances from the centre of
the cluster one obtains a velocity distribution that is bounded by
sharp, characteristic trumpetshaped curves. These curves, called
caustics, form an envelope containing the galaxies of the cluster.
The caustics are used to estimate the density parameter
of the universe (Regös & Geller 1989) and
the mass profile for the galaxy clusters (Geller et al.
1999; Reisenegger et al. 2000). In Sect. 2, we
show that the form of the caustics fully defines the profile of
the radial infall velocity in the case of a pure spherical
collapse. In Sect. 3, we describe the spherical model. This model
is applied in Sect. 4 to data from the Coma cluster. In Sect. 5,
the conclusions are given.
For the sake of simplicity, let us consider the xy plane of a
spherically symmetric cluster, undergoing a radial collapse
(Fig. 1), where
is the position vector of a
galaxy at K, relative to the centre C of the cluster. The velocity
of the galaxy at K relative to the observer is given by
the relation

(1) 
where
is the velocity of the centre C relative to
O. Defining
,
where r is the magnitude of
and multiplying both sides of Eq. (1) by the unit vector
along OK direction, we have

(2) 
where
is the observed velocity along the
lineofsight.
From Eq. (2), for r=R and
for
,
the observed velocity takes the same value
.
Thus,
takes extreme
values for r between R and .
We denote by
the radius of maximum expansion (turnaround radius)
where
.
For constant R, the quantity
does not depend on r and thus the
condition
for the extreme is written

(3) 
The solution r_{*} for r of Eq. (3)
is a function of R (
r_{*}=r_{*}(R)). This r_{*} is the
radial  relative to the centre of the cluster  distance of the
galaxy which shows the observer the maximum or minimum velocity at
a given R. Thus, the extreme values of
at Rare

(4) 
The curves described by Eq. (4) are known as caustics (Regös &
Geller 1989).
Following Diaferio et al. (1997), we define the amplitude
A of the caustics by the relation

(5) 
which is given by

(6) 
Note that by definition
is negative in the infall region of the
cluster. The amplitude contains only information about the dynamical situation of
the cluster, since effects due to the motion of the observer relative to the cluster
are excluded. Such effects are described by the term
in Eq. (4) and have been studied by Praton &
Schneider (1994).
Differentiating Eq. (6) with respect to R, we have

Figure 1:
Geometry of the problem. O: observer, K: galaxy, C: cluster's centre. 
Open with DEXTER 

(7) 
Because of Eq. (3), the term inside brackets
equals zero and this leads to

(8) 
Using Eqs. (6) and (8) we have

(9) 
Solving Eq. (9) for R and substituting in Eq. (8) the
radial velocity is fully defined. This is an interesting result, since it shows that
the exact data can give the exact profile of the infall velocity, in the case of a
completely radial collapse. It is interesting to note that the above relations lead
to Eq. (10), relating the logarithmic derivatives of the amplitude to the
infall velocity;

(10) 
The above equations refer to the ideal case of a pure spherical collapse
and completely accurate observations. In real systems, the velocity field is a
superposition of a radial systematic component and a component of a random nature.
The first one can be assumed as spherically symmetric while the second accounts for
the effects of smallscale substructure and observational errors. The effects of
smallscale substructure are clearly shown in the results of Nbody simulations
performed by van Haarlem & van de Weygaert (1993). They conclude that the
velocity profile, as predicted by the spherical infall model, compares badly with
the actual velocity field resulting from their simulations. In real systems, such as
the Coma cluster, the smearing of the form of the caustics is also clear (van Haarlem et al. 1993). However, if the systematic radial component of the
velocity field is known, a mass profile of the infall regions of the clusters can be
determined, applying the spherical infall model. This profile should give a lower
limit for the mass of the system since the effects of smallscale substructure
reported above increase the amplitude of the caustics. Thus, we apply the previous
equations assuming that
describes the systematic radial component of the velocity.
In Sect. 4 we use model functions for the amplitude A, calculating the profile of
by Eqs. (8) and (9), we apply the spherical infall model
and we derive a mass profile for the Coma cluster. These model functions for A are
smooth and decreasing, as it required by Eqs. (8) and (9), and
their approximation is based on the respective curves estimated by Geller et al.
(1999).
The equation of motion of a spherical shell of initial radius a is given by Newton's
law as

(11) 
where M(r) is the mass inside radius r and G is the gravitational
constant. If the mass inside the shell is conserved (no shells crossing), then M(r) is constant and equal to M(a). We assume that this condition holds in what
follows. Multiplying Eq. (11) by
and integrating, we have

(12) 
where
E=1/2 v_{i}^{2}(GM(a))/a is a constant with
dimensions of specific energy and v_{i} is the initial velocity.
The solution of the above equation, in the case of negative E, is
given in parametric form by the expressions

(13) 

(14) 
In order to synchronize all shells, we add the constant
given by
,
where t_{i} is the
age of the universe at the initial conditions and
is the initial phase of
the shell, given by Eq. (13) for r=a. A shell reaches the radius of
maximum expansion for
and completes its collapse for .
We
denote by
the fractional excess of mass of a sphere of radius a, relative
to the mass
of a sphere of the same radius that has a constant density
(equal to the mean density of the universe at the initial conditions). Using the
relation between the mean density of the universe
,
the Hubble constant Hand the density parameter ,
that is

(15) 
the mass inside radius a is given by

(16) 
where index i stands for the values at the initial
conditions. Assuming that the initial velocity of the shell is
that of Hubble's flow (
v_{i}=aH_{i}), the energy E is written

(17) 
Substituting Eq. (17) in Eqs. (13), (14) and
calculating the time derivative of the radius yields the relations

(18) 

(19) 

(20) 
where
.
Combining Eqs. (18)(20) results in

(21) 
where t is the present age of the shell. The time tis approximated by
,
where z is the
redshift of the cluster, H_{0} is the present value of the Hubble
constant and
is the value of the density parameter. The
function f for the FriedmannLemaitre cosmological model is
given in standard books of cosmology (see Zel'dovich & Novikov
1983) and its form depends on the value of relative to the critical value
.
Reasonable initial
conditions are those at the epoch of decoupling, when matter
starts playing a significant role. The value of
is very
small compared to t, so it can be omitted without significant
error in the calculation of the phase .
However,
Eq. (21) can also be solved in its full form based on
the following procedure. Solving Eq. (18) for
gives

(22) 
The substitution of the above expression in Eq. (20) leads to the
following cubic for a

(23) 
where
.
This has one real
and positive root, given by the relation

(24) 
where

(25) 
Thus,
can be expressed as a function of
and .
Then,
can also be expressed
in terms of the same variables, since it is given by the relation

(26) 
where
and t_{i} is
the age of the universe at the initial conditions (decoupling),
given in standard books of cosmology (see Zel'dovich & Novikov
1983).
The present method can be summarized as follows: A known profile of the amplitude of
the caustics, using Eqs. (8) and (9) leads to the profile of
radial velocity. Then, the solution of Eq. (21) defines the phase
of
any shell. Assuming no shell crossing during the evolution of the cluster, the
combination of Eqs. (16), (18) and (20) gives the mass
inside a spherical region with current radius r as

(27) 
The procedure described above is applied to estimate a lower limit
for the mass profile of the Coma cluster. For this purpose, we
employ data provided by Geller et al. (1999). The largest
sample presented by these authors contains 691 galaxies. The
cluster's velocity
(c is the speed of light and
is the redshift of the cluster centre) is
and for a Hubble constant
,
its distance is
70.9 h^{1} Mpc.

Figure 2:
Dashed line: profile of the amplitude of the caustics (Geller et al.
1999). Solid curve: approximation of this line using analytical smooth
functions (see Eq. (28) and text for more details.) 
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As noted in Sect. 2, our approach is based on the profile of the amplitude A of
the caustics. Departures from spherical symmetry, random motions due to the
development of substructure, the finite number of galaxies in the cluster and
observational errors are some of the reasons making the determination of caustics
nontrivial. A general approach, given by Diaferio (1999), is based on the
argument that caustics are high density regions (see Regös & Geller
1989) and is as follows. First, a smooth estimate for the density f of
observed galaxies on the
plane is calculated and then a cut is applied
at some density contour which is taken to correspond to the caustics. A similar
procedure is followed by Reisenegger et al. (2000) in their application to
the Sharpley Supercluster. Problems associated with the above procedure, such as the
choice of smoothing lengths and the density cutoff, are discussed in the above
papers. The dashed line  shown in Fig. 2  is Geller et al.'s
(1999) estimation of the amplitude
of the caustics derived by
the abovediscussed procedure. We assume that this can be written in the form
where A(R) stands for the component due to the radial
infall velocity and the positive
describes the

Figure 3:
Radial velocity profile corresponding to the
smooth curve plotted in Fig. 2. 
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"inflation'' of the caustics due to the development of
random motions. Unfortunately,
cannot be modelled properly
and an infinite number of curves lower than
could be
provided as models for A. For this reason we use the following
procedure. First, we use models for A(R) of the form

(28) 
Then, we estimate the values of the parameters
v_{0},r_{M} and k in order that these curves satisfy the condition
for
all values of R and minimize the sum
.
In this way, we derive a profile that could be
considered as an upper bound for the A component.
Such a curve is plotted in
Fig. 2. The parameters used in Eq. (28) are:
.
We note that
is not the
physical radius of the system, but a fitting parameter. The form of Eq. (28)
has mathematical advantages, since it allows an analytical evaluation of the profile
of the infall velocity, using Eqs. (8) and (9) reported in
Sect. 2. This is given by the relation

(29) 
with
.
In Fig. 3, the resulting profile of the radial infall
velocity is plotted.
Assuming that the above profile is a good approximation
of the radial systematic component of the infall velocity, we
apply the spherical infall model to calculate mass profiles, for various values of the density
parameter
of the universe, that are plotted
in Fig. 4. Starting from the bottom of the figure, the different curves correspond to
respectively. Larger values of
lead to steeper mass profiles at the outer regions. The
variation of
alters the profile at the outer regions of
the cluster and (as expected) larger values of the density
parameter of the universe give rise to larger amount of mass
inside a given radius. Notice that the mass scales as
,
roughly as expected (see Regös & Geller
1989). Thus, the matter inside 10 h^{1} Mpc
varies for various values of the density parameter of the universe
in the range 0.994 to
.
The
dashed curve, plotted also in this figure, is the cumulative mass of
an halo with a Navarro et al. (1995) profile. This is
given by the relation
,
where
and
are fitting parameters. The values of the
parameters used are:
,
Mpc. It is
clearly shown in Fig. 4 that there is a remarkable
agreement of the form of the cumulative mass profiles estimated by
the present approach (solid curves) with the Navarro et al.
(1995) mass profile (dashed curve). Note that the dashed
curve describes the mass profile in the virialized region of the
cluster.

Figure 4:
Mass profiles corresponding
to the radial velocity profile (Fig. 3)
for varying .
From the bottom of the figure, the curves
correspond to
respectively.
Dashed curve: Navarro et al. (1995)
mass profile. 
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Figure 5:
Mass profiles for various values of
as in Fig. 4. The bar at 0.5 shows the
limits given by Hughes (1989) while the bar at 2.5 shows the
limits given by Briel et al. (1992). The outer bar at
10 shows the estimation given by Geller et al. (1999).

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Several methods have been proposed in the literature regarding the
estimation of the mass of galaxy clusters.
A class of such methods, used to estimate the density profile of
these systems, is based on the use of Xray data (The & White
1988; Hughes 1989; Watt et al. 1992). It is
assumed that the gas is described by the equation of hydrostatic
equilibrium. The latter is solved using the density and
temperature profiles of the gas, that are measured using Xray
observations. This method is accurate in the region where the
hydrostatic equilibrium holds, that is the central region of
galaxy clusters. For example, Hughes (1989) determined
the minimum and maximum mass profiles consistent with all the data
for Coma  if the masstolight ratio of the data was not
constrained to be a constant  obtaining allowed mass ranges of
M_{0.5}=1.9
and
M_{2.5}=0.5
where M_{0.5} and M_{2.5} are the masses inside
0.5 h^{1} Mpc and
2.5 h^{1} Mpc respectively. Briel et al. (1992)
used their ROSAT survey image to measure the Xray surface
brightness out to 100' from the cluster center. They found that
the binding mass is more centrally concentrated than the Xray
gas, and obtain
M_{2.5}=0.6
.
Another class of methods employs the projected velocities
of the galaxiesmembers of the
cluster. The methods based on the form of the caustics in redshift
space belong among others in this class. In a series of papers
(Diaferio et al. 1997; Diaferio 1999; Geller et al. 1999), it has been proposed to estimate the mass
profile of a galaxy cluster using the equation

(30) 
Although there is no rigorous proof for this relation, its testing
against cosmological Nbody simulations shows that it approximates
well the mass profiles. The derivation of the above relation is
based on the argument that the amplitude of the velocity field of
galaxy clusters depends mainly on local dynamics, since the random
motion is significant. The mass profile is, in any case, roughly
proportional to the square of the amplitude of the caustics and
since the amplitude used by Geller et al. (1999) is
larger because it includes the effect of random motion, one
expects their estimation to result in a larger mass for the
system. They estimate a mass inside a radius of
10 h^{1} Mpc in the range
.
In Fig. 5, we compare the mass estimations provided by our
method for the Coma cluster to the mass estimations given by
Hughes (1989), Briel (1992) and Geller et al.
(1999). It is clear that the mass profile resulting from
our application shows the minimum estimated values of mass at
different radii and provides a reasonable estimation of the lower
limit of the mass profile of the system.
More detailed
observations could lead to a more accurate definition of the
caustics and improve the estimation of mass profiles .
Acknowledgements
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Copyright ESO 2001