A PDR model for the FIR mid-J CO ladder with universal rotational temperature in star forming region

A photon dominated region (PDR) is one of the leading candidate mechanisms for the origin of the warm CO gas with near universal $\sim$300~K rotational temperature inferred from the CO emission detected towards embedded protostars by Herschel/PACS. We have developed a PDR model in general coordinates, where we can use the most adequate coordinate system for an embedded protostar having outflow cavity walls, to solve chemistry and gas energetics self-consistently for given UV radiation fields with different spectral shapes. Simple 1D tests and applications show that FIR mid-$J$ ($14\leq J\leq 24$) CO lines are emitted from near the surface of a dense region exposed to high UV fluxes. We apply our model to HH46 and find the UV-heated outflow cavity wall can reproduce the mid-$J$, CO transitions observed by Herschel/PACS. A model with UV radiation corresponding to a blackbody of 10,000~K results in the rotational temperature lower than 300~K, while models with the Draine interstellar radiation field and the 15,000 K blackbody radiation field predict the rotational temperature similar to the observed one.