Spectral line and continuum radiation propagation in a clumpy medium

We discuss the propagation of spectral line and continuum radiation in a clumpy medium and give general expressions for the observed absorption or emission from a cloud population. We show that the affect of the medium clumpiness can usually be characterised by a single number multiplying the mean column opacity. Our result provides a simpler proof and generalisation of the result of Martin et al (1984). The formalism provides a simple way to understand the effects of clumping on molecular line profiles and ratios, for example how clumping effects the interpretation of 13CO(1-0) to 12CO(1-0) line ratios. It can also be used as a propagation operator in physical models of clumpy media where the incident radiation effects the spectral line emissivity. We are working to extend the formalism to the propagation of masers in a clumpy medium, but in this case there are special difficulties because formal expectation values are not characteristic of observations because they are biased by rare events.