Free-Free Radiation from Dense Interstellar Shock Waves

We present detailed calculations of the free-free emission behind shock waves in media where the postshock gas may be optically thick at radio frequencies. Using a numerical shock code, we solve the free-free radiative transfer equation at frequencies of 1.5, 5, and 15 GHz for a family of planar shock models that cover a wide range of densities (1000 cm^-3 ≤ n₀ ≤ 10⁹ cm^-3) and shock velocities (30 km s^-1 ≤ V₀ ≤ 300 km s^-1). These models also predict how different preshock magnetic fields and viewing angles affect the overall free-free emission. As the shock velocities and preshock densities increase, the free-free spectral indices generally rise from optically thin values (~-0.1) and approach optically thick ones (~2.0). We find that for V ≥ 120 km s^-1, the ionized precursor produced by the emergent postshock UV radiation can itself become optically thick to free-free radiation when the preshock density exceeds 10⁶ cm^-3. We also find that for some combinations of shock parameters, the superposition of radiation from the precursor onto that of the postshock region can produce spectral indices exceeding 2.0. In addition, we find that magnetic fields can suppress the free-free radiation only if they are strong enough to lower the magnetosonic Mach number below 4 for V ≥ 100 km s^-1. An observer who knows the resolved angular sizes, fluxes, and spectral indices of a thermal radio source can use our grids to determine the preshock density, shock velocity, and viewing angle to the source. Possible applications include HH objects, shocked cloudlets, and accretion shocks deep within molecular clouds. As an example, we use our results to predict shock parameters for HH 1-2 and several thermal radio sources in Cep A east. We find that the predicted shock parameters for HH 1 and 2 agree well with those inferred from optical line ratios and line profiles. Radio emission from sources in Cep A are explained reasonably well as arising from thermal radiation behind a dense, fast shock like one might expect from a stellar jet deeply embedded within the molecular cloud core.