Multi-frequency, thermally coupled radiative transfer with TRAPHIC: Method and tests

We present an extension of TRAPHIC, the method for radiative transfer of ionising radiation in smoothed particle hydrodynamics simulations that we introduced in Pawlik & Schaye (2008). The new version keeps all advantages of the original implementation: photons are transported at the speed of light, in a photon-conserving manner, directly on the spatially adaptive, unstructured grid traced out by the particles, in a computation time that is independent of the number of radiation sources, and in parallel on distributed memory machines. We extend the method to include multiple frequencies, both hydrogen and helium, and to model the coupled evolution of the temperature and ionisation balance. We test our methods by performing a set of simulations of increasing complexity and including a small cosmological reionisation run. The results are in excellent agreement with exact solutions, where available, and also with results obtained with other codes if we make similar assumptions and account for differences in the atomic rates used. We use the new implementation to illustrate the differences between simulations that compute photoheating in the grey approximation and those that use multiple frequency bins. We show that close to ionising sources the grey approximation asymptotes to the multi-frequency result if photoheating rates are computed in the optically thin limit, but that the grey approximation breaks down everywhere if, as is often done, the optically thick limit is assumed.