Obscuration by Diffuse Cosmic Dust
Frank J. Masci, PASA, 15 (3), 299
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A number of studies
have attributed the existence of large deficits of
background sources behind nearby galaxy clusters as
due to extinction by dust.
Bogart & Wagner (1973) found that
distant rich Abell clusters were anticorrelated on the sky with nearby ones.
They argued for a mean extinction of 
mag
extending to 
times the optical radii of the nearby clusters.
Boyle et al. (1988)
however claimed a
deficit of background quasars within 4' of clusters
consisting of tens of galaxies.
These authors
attribute this to an extinction 
mag, and
deduce a dust mass of 
within 0.5Mpc
of the clusters.
Romani & Maoz (1992) found that optically-selected quasars
from the Véron-Cetty & Véron (1989) catalogue avoid
rich foreground Abell clusters.
They also found deficits of
out to radii
from the clusters, and postulate a
mean extinction, mag.
The numbers of background sources behind clusters is also expected to
be modified by
gravitational lensing (GL) by the cluster potential.
Depending on the intrinsic luminosity function of the background population,
and the limiting magnitude to which the sources are detected, GL
can cause either an enhancement or a deficit in the number of
background sources.
The GL effect has been used to explain various
reports of overdensities of both optically and radio-selected
quasars behind foreground clusters
(Bartelmann & Schneider 1993; Bartelmann et al. 1994;
Rodrigues-Williams & Hogan 1994;
Seitz & Schneider 1995).
The reported overdensities for optically-selected QSOs
are contrary to the studies above where
anticorrelations with foreground
clusters are found.
These overdensities however are claimed on angular scales
from the cluster centers, considerably
larger than scales on which most of the underdensities have been claimed,
which are of order a few arcminutes.
One interpretation is that
dust obscuration bias may be greater towards cluster centers due
to the presence of greater quantities of dust.
On the other hand, the reported anticorrelations on small
angular scales can perhaps be explained by the optically
crowded fields, where QSO identification may be difficult.
At present, the effects of clusters on background source counts still
remains controversial.
More direct evidence for the existence of intracluster dust
was provided by Hu et al. (1985) and
Hu (1992) who compared the Ly-
flux from
emission line systems in ``cooling flow'' clusters with
Balmer line fluxes at optical wavelengths.
Extinctions of
mag towards the cluster centers
were found, in good agreement with estimates
from the quasar deficits above.
Intracluster dust has been predicted to give rise to
detectable diffuse IR emission (eg. Dwek et al. 1990).
The most extensive search was conducted by Wise et al. (1993), who
claimed to have detected excess diffuse 60-100
m emission at the 2
level from a number of rich Abell clusters.
They derived dust temperatures in the range 24-34 K and
dust masses within radii of 
Mpc.
Recently,
Allen (1995) detected strong X-ray absorption and optical
reddening in ellipticals situated at the centers of rich cooling flow
clusters, providing strong evidence for dust.
These studies indicate that intracluster dust is
certainly present, however,
the magnitude of its effect on producing background source
deficits remains a controversial issue.
In this section, we
give some predictions that
may be used to further constrain cluster dust properties, or
help determine the dominant mechanism (ie. GL or extinction) by which clusters
affect background observations.
X-ray spectral measurements show the presence of hot, metal enriched
gas in rich galaxy clusters with 
solar metallicity.
This gas is believed to be both
of galactic and primordial
origin (ie. pre-existing IGM gas), with the bulk of metals being
ejected from cluster galaxies (see Sarazin 1986 for a review).
Ejection from galaxies may occur abruptly
through collisions between the cluster galaxies, `sudden' ram-pressure
ablation, or through continuous ram-pressure stripping by
intracluster gas (eg. Takeda et al. 1984).
The lack of significant amounts of dust (relative to what should have
been produced by stellar evolution) and interstellar gas in cluster
ellipticals provides evidence for a mass loss process.
On the other hand, in ellipticals that avoid dense cluster
environments, significant quantities
of neutral hydrogen, molecular gas and dust
have been detected (eg. Lees et al. 1991).
If the dust-to-gas ratio of intracluster gas
in rich clusters were similar to that of the Milky Way, then a radial
gas column density of typically 
with metallicity 
would produce an extinction
mag.
This however is much greater than the value observed.
The likely reason for the deficiency of dust in the intracluster medium is its
destruction by thermal sputtering in the hot gas, a process which operates on
timescales 
yr, where
is the gas density and
the grain radius (Draine & Salpeter 1979).
Dust injection timescales from galaxies
is typically of order a Hubble time
(eg. Takeda et al. 1984) and
hence, grains are effectively destroyed, with only the most recently
injected still surviving and providing possibly some measurable
extinction.
The spatial distribution in dust mass density remains a
major uncertainty. A number of authors have shown that
under a steady state of continuous injection from
cluster galaxies, destruction by thermal sputtering
at a constant rate, and assuming instantaneous mixing with the hot gas,
the resulting mass density in dust will be of order
(eg. Dwek et al. 1990)
where 
is the injected dust-to-gas mass ratio, assumed to be
equal to the mean value of the galactic ISM, 
(Pei 1992).
According to this simple model,
the dust mass density is independent of gas density and position in the
cluster.
If we relax the assumption of instantaneous mixing of dust with
the hot gas however, so that the spatial distribution of gas is
different from that of the injected dust, the radial distribution
of dust can significantly differ from uniformity throughout a cluster.
Such a non-uniform spatial dust distribution
may be found in clusters exhibiting cooling flows.
If, as suggested by Fabian et al. (1991), most of the cooled gas remains
cold and becomes molecular in cluster cores, then
a relatively large amount of dust may also form, resulting in a
dust distribution which peaks within the central regions.
We explore the radial dependence of extinction
optical depth through a cluster,
and the expected deficit in background sources
by assuming that dust is diffusely distributed and follows a
spatial density distribution:
where 
is a characteristic radius which we fix and
n is our free parameter.
Equation (9) with n=3/2 is the usual King profile which
with 
Mpc, represents a good approximation to
the galaxy distribution in clusters.
Thus for simplicity, we keep fixed at 
Mpc and vary n.
To bracket the range of possibilities in the distribution of
intracluster dust, we consider the range 
.
n=0 corresponds to the simple case where
constant, which may describe a situation
where injection of dust is balanced by its destruction by hot gas
as discussed above.
The value n=3/2 assumes that dust
follows the galaxy distribution. This profile may arise
if grain destruction by a similar distribution of
hot gas were entirely absent.
To model the spatial distribution of optical depth,
we assume that intracluster dust is distributed within a
sphere of radius 
.
The central dust mass density 
in equation (9)
is fixed by assuming
values for the total dust mass 
and such that
We assume a cluster dust radius of 
Mpc, which represents
a radius containing 
of the virial mass
of a typical dense cluster characterised by
galactic velocity dispersion 
(Sarazin 1986 and references therein).
We assume
that the total dust mass within Mpc
is 
. This value is
consistent with that derived from extinction measures by
Hu, Cowie & Wang (1985), IR emission detections by Wise et al. (1993) and
theoretical estimates of
the mean intracluster dust density as given by equation (8).
Using equation (3), the B-band optical depth through our
spherical intracluster dust distribution at some projected distance
r from its center can be written:
where 
is our assumed radial density distribution
(equation 9) and 
the mass density of an individual dust grain.
For a uniform dust density (
constant),
and our assumed values of and given above,
the radial dependence in dust optical depth can be written:
where 
is the optical depth through the center of our cluster,
which with
grain properties characteristic of the galactic ISM, will scale as
This value is about a factor three times lower than estimates of the
mean extinction derived from the deficit of QSOs
behind foreground clusters (eg. Boyle et al. 1988), and that implied by the
Balmer decrements of Hu et al. (1985).
For a fixed dust mass of 
however, we can achieve
larger values for the central optical depth by steepening the
radial dust-density distribution profile, determined by the slope n
in equation (9).
Figure 3a shows the optical depth
as a function of projected cluster radius
for the cases n=0, 0.5, 1, and 1.5.
The case n=0.5 approximately corresponds to
the model of Dwek et al. (1990), which included effects of mild
sputtering by hot gas in order to fit for the observed IR emission
from the Coma cluster.
As shown, the case n=0 (
constant) predicts that
the dust optical depth should be almost
independent of projected radius r.
Within all projected radii, the optical depths predicted by
our diffuse dust model lie in the range 
.
Turning back to the discussion of Section 2 where we show that
background obscuration by diffuse dust reaches its maximum
for 
, these optical depths satisfy this
condition for 
, typical of luminous
background galaxies and quasars.
Figure 2: (a) Optical depth as a function of projected cluster radius for
various dust density distributions as parameterised by equation (9).
(b) Differential number of background sources lost from a flux-limited
sample (arbitrary scale).
(c) Total fraction of background sources missing to some r
and (d) Cluster-QSO two-point angular correlation function,
filled circles: Boyle et al. (1988), squares: Romain & Maoz (1992),
open circles: Rodrigues-Williams & Hogan (1994), triangles:
Rodrigues-Williams & Hawkins (1995).
We now explore the effects of these models
on background source counts as a function of projected cluster radius.
We first give an estimate of the projected radius
at which the numbers of background sources lost from a flux-limited
sample is expected to be a maximum.
This is determined by investigating the dependence in the differential
number of
sources missing, 
, within an interval (r,r+dr) as a
function of projected radius r.
From equation (2), this
differential number will scale as
where 
is given by equation (11).
Figure 3b plots 
as a function of r for
our various models, where we have assumed 
.
Thus from observations,
an identification of the projected radius at which the
background source deficit peaks can be used to
constrain the spatial distribution of intracluster dust.
The cumulative fraction of background
sources missing within a projected cluster
radius is given by
This fraction is shown in Figure 3c.
As expected, the n=3/2 model which contains the largest amount of dust
within the inner few hundred kiloparsecs predicts the strongest
trend with r, while the opposite is predicted if the dust density
were completely uniform.
These predictions can be compared with a number of existing studies of the
observed two-point angular correlation function between clusters and
optically-selected QSOs.
This function is usually defined as
where 
is the average number
of observed cluster-QSO pairs within an angular radius 
and
is that expected in a random
distribution.
For our purposes, can be
replaced by the ``true'' number of cluster-QSO pairs expected in the
absence of dust,
and hence, we can re-write equation (16) as
We compare our models with a number of studies of 
for optically selected QSOs in Figure 3d.
These studies considerably differ from each other in the selection
of the QSO and cluster samples, and as seen, both
anticorrelations and correlations on different angular scales
are found.
The former have been interpreted in terms of extinction by intracluster
dust, while the latter with the GL phenomenon.
In most cases, the reported overdensities are too large to be
consistent with GL models given our current knowledge
of cluster masses and QSO distributions.
It is interesting to note that the studies which have reached the
smallest angular scales (
) are also those in which anticorrelations
between QSOs and foreground clusters have been reported.
This can be understood in terms of a larger dust concentration
and hence extinction towards cluster centers.
These studies however may not be free of selection effects,
such as in the detection of QSOs from the visual inspection of
objective prism plates.
From a cross-correlation analysis of galactic stars with their cluster
sample however,
Boyle et al. (1988) found that such selection effects are minimal.
The maximum dust radial extent assumed in our models,
Mpc, corresponds to angular scales at the mean
redshift of the clusters (
) used in
these studies.
Thus, as shown in Figure 3d,
our model predictions only extend to .
As shown in this figure, the n=3/2 model which corresponds to the
case where the dust density is assumed to follow the galaxy distribution,
provides the best fit to the Boyle et al. (1988) data.
We must note that this is the only existing study performed to
angular scales 
with which we can compare our models.
Further studies to such scales are necessary to
confirm the Boyle et al. result, and/or
provide a handle on any selection effects.
To summarise, we have shown that for a plausible value of the
dust mass in a typical rich galaxy cluster,
obscuration of background sources will be most
effective if dust is diffusely distributed on scales Mpc.
This conclusion is based on our predicted optical depths
(
) satisfying our condition for
`maximum' obscuration:
(see Section 2), where typically
for luminous background galaxies and QSOs.
We have explored the spatial distribution in dust optical depth
and background source deficits expected through a
typical rich cluster by assuming different
radial dust density profiles.
These predictions can be used to constrain cluster dust properties.
A dust density distribution with n=3/2 (equation 9) appears
to best satisfy the `small scale' cluster-QSO angular correlation study
of Boyle et al. (1988).
© Copyright Astronomical Society of Australia 1997