Inhomogeneous Absorbers and Derived Column Densities
  We study the dependence of column densities derived from absorption lines on the spatial distribution of the ions in an absorber. In particular, we investigate four varieties of coverage by the absorber of the background source: the familiar homogeneous partial coverage (HPC), and three functional forms that parameterize inhomogeneous coverage: a powerlaw, an ellipse quandrant, and a Gaussian distribution. We calculate the residual line intensities obtained from our inhomogeneous coverage models and then use these intensities as ``observed'' quantities to compute the optical depth and covering factors assuming HPC. We find that the resulting spatially-averaged optical depths are comparable (within a factor of $\lesssim 1.5$) to the average optical depths of the input distributions, as long as the input distributions do not contain spatially narrow ``spikes.'' Such spikes (very large optical depths over a small coverage area) can profoundly affect the average optical depth in the absorber but have little impact on the observed intensities. We also study the converse approach: we start with HPC as the assumed physical model and then infer the parameters of the inhomogeneous coverage models through a doublet analysis. Again the resulting average optical depths are comparable to the corresponding quantity of the input distribution. Finally, we construct a more realistic two-dimensional optical depth distriubution based on a random distribution of absorbing clouds, and we use that to calculate observed intensities. A doublet analysis applied to those intensities shows all four of our simple analytic functions yield an accurate estimate of the true average optical depth, while the powerlaw yields the best approximation to the intial distribution. 
Bassem M. Sabra, Fred Hamann