Compact versus Diffuse Dust Distributions

In this section, we explore the dependence of obscuration of background sources on the spatial distribution of a given mass of dust. For simplicity, we assume the dust is associated with a cylindrical face-on disk with uniform dust mass density. We quantify the amount of obscuration by investigating the number of background sources behind our absorber that are missed from an optical flux-limited sample.

The fraction of sources missing to some luminosity L relative to the case where there is no dust extinction is simply , where represents the observed number of sources in the presence of dust. For a uniform dust optical depth , . For simplicity, we assume that background sources are described by a cumulative luminosity function that follows a power-law: , where is the slope. This form for the luminosity function is often observed for `luminous' () galaxies and quasars, which dominate high redshift () populations in flux-limited samples. With this assumption, the fraction of background sources missing over a given area when viewed through our dusty absorber with uniform optical depth is given by
 
If the `true' number of background sources per unit solid angle is , then the total number of background sources lost from a flux limited sample within the projected radius R of our absorber can be written:
 
where D is the distance of the absorber from us.

To investigate the dependence of background source counts on the spatial dust distribution, we need to first determine the dependence of in equation (2) on the spatial extent R for a fixed mass of dust . This can be determined from the individual properties of grains as follows. The extinction optical depth at a wavelength through a slab of dust composed of grains with uniform radius a is defined as
 
where is the extinction efficiency which depends on the grain size and dielectric properties, is the number density of grains and is the length of the dust column along the line-of-sight. Assuming our cylindrical absorber (whose axis lies along the line-of-sight) has a uniform dust mass density: , where R is its cross-sectional radius, we can write, , where is the mass density of an individual grain. We use the extinction efficiency in the V-band as parameterised by Goudfrooij et al. (1994) for a graphite and silicate mixture (of equal abundances) with mean grain size m characteristic of the galactic ISM. The value used is . We use a galactic extinction curve to convert to a B-band extinction measure, where typically (eg. Pei 1992). Combining these quantities, we find that the B-band optical depth, , through our model absorber can be written in terms of its dust mass and cross-sectional radius as follows:
 
where we have scaled to a dust mass and radius typical of local massive spirals and ellipticals (eg. Zaritsky 1994). This measure is consistent with mean optical depths derived by other means (eg. Giovanelli et al. 1994 and references therein).

From equation (4), we see that the dust optical depth through our model absorber for a fixed dust mass varies in terms of its cross-sectional radius R as . For the nominal dust parameters in equation (4), the number of sources missed behind our model absorber (equation 2) can be written
 
where and
 
is the `true' number of background sources falling within the projected scale radius R=20kpc.

From the functional forms of equations (2) and (5), there are two limiting cases:

1. For optical depths , the factor in equation (2) is of order unity. This corresponds to values of R such that for the nominal parameters in equation (4). For values of R in this range, we have and the obscuration of background sources will depend most strongly on R.
2. For or equivalently , will approach a constant limiting value, independent of the dust extent R. From equation (5), this limiting value can be shown to be .

As a simple illustration, we show in Figure 2 the dependence of the number of background sources missing behind our model dust absorber on R, for a fixed dust mass of as defined by equation (5). We assumed a cumulative luminosity function slope of , typical of that for luminous galaxies and quasars. From the above discussion, we see that when , ie. (or when ), the obscuration will start to approach its maximum value and remain approximately constant as .

  
Figure 1: The number of sources missing from an optical flux-limited sample behind a face-on dusty disk with fixed mass as a function of its radial extent R. This number is normalised against the `true' number of background sources (in the absence of the absorber) that fall within the solid angle subtended by our nominal radius of 20kpc at distance D. See equation (5.)

We conclude that when dust becomes diffuse and extended on a scale such that the mean optical depth through the distribution satisfies , where is the cumulative luminosity function slope of background sources, obscuration will start to be important and is maximised for . The characteristic spatial scale at which this occurs will depend on the dust mass through equation (4). For the typical grain values in equation (4), this characteristic radius is given by
 
The simple model in Figure 2 shows that the obscuration of background sources due to a normal foreground galaxy will be most effective if dust is distributed over a region a few times its optical radius. This prediction may be difficult to confirm observationally due to possible contamination from light in the galactic absorber. In the following sections, we explore two examples of possible `large-scale' diffuse dust distributions that can be explored observationally.

© Copyright Astronomical Society of Australia 1997