Diffuse Dust in Galaxy Clusters

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 60-100m 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.

Spatial Distribution of Cluster Dust

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.

Spatial Distribution of Dust Optical Depth and Background Source Deficits

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.

Summary

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