Free-Free Spectral Energy Distributions of Hierarchically Clumped HII   Regions
  In an effort to understand unusual power-law spectral slopes observed in some hypercompact HII regions, we consider the radio continuum energy distribution from an ensemble of spherical clumps. An analytic expression for the free-free emission from a single spherical clump is derived. The radio continuum slope (with F_\nu \nu^\alpha) is governed by the population of clump optical depths N(tau), such that (a) at frequencies where all clumps are thick, a continuum slope of +2 is found, (b) at frequencies where all clumps are optically thin, a flattened slope of -0.11 is found, and (c) at intermediate frequencies, a power-law segment of significant bandwidth with slopes between these two limiting values can result. For the ensemble distribution, we adopt a power-law distribution N(tau) tau^{-\gamma}, and find that significant power-law segments in the SED with slopes from +2 to -0.11 result only for a relatively restricted range of $\gamma$ values of 1 to 2. Further, a greater range of clump optical depths for this distribution leads to a wider bandwidth over which the intermediate power-law segment exists. The model is applied to the source W49N-B2 with an observed slope of \alphab +0.9, but that may be turning over to become optically thin around 2 mm. An adequate fit is found in which most clumps are optically thin and there is little shadowing of rearward clumps by foreground clumps (i.e., the geometrical covering factor C<<1). The primary insight gained from our study is that in the Rayleigh-Jeans limit for the Planck function that applies for the radio band, it is the distribution in optical depth of the clump population that is solely responsible for setting the continuum shape, with variations in the size and temperature of clumps serving to modulate the level of free-free emission. 
Richard Ignace Edward B Churchwell