Free‐Free Spectral Energy Distributions of Hierarchically Clumped HiiRegions

In an effort to understand unusual power-law spectral slopes observed in somehypercompact HII regions, we consider the radio continuum energy distributionfrom an ensemble of spherical clumps. An analytic expression for the free-freeemission from a single spherical clump is derived. The radio continuum slope(with F_\nu \nu^\alpha) is governed by the population of clump optical depthsN(tau), such that (a) at frequencies where all clumps are thick, a continuumslope of +2 is found, (b) at frequencies where all clumps are optically thin, aflattened slope of -0.11 is found, and (c) at intermediate frequencies, apower-law segment of significant bandwidth with slopes between these twolimiting values can result. For the ensemble distribution, we adopt a power-lawdistribution N(tau) tau^{-\gamma}, and find that significant power-law segmentsin the SED with slopes from +2 to -0.11 result only for a relatively restrictedrange of $\gamma$ values of 1 to 2. Further, a greater range of clump opticaldepths for this distribution leads to a wider bandwidth over which theintermediate power-law segment exists. The model is applied to the sourceW49N-B2 with an observed slope of \alphab +0.9, but that may be turning over tobecome optically thin around 2 mm. An adequate fit is found in which mostclumps are optically thin and there is little shadowing of rearward clumps byforeground clumps (i.e., the geometrical covering factor C<<1). The primaryinsight gained from our study is that in the Rayleigh-Jeans limit for thePlanck function that applies for the radio band, it is the distribution inoptical depth of the clump population that is solely responsible for settingthe continuum shape, with variations in the size and temperature of clumpsserving to modulate the level of free-free emission.Comment: Astrophysical Journal, in pres