## 5. ConclusionsThe aim of this paper has been to develop and test Monte Carlo techniques for deriving radiative equilibrium temperature distributions when the absorption and scattering coefficients depend on state variables and are thus coupled to the ambient radiation field. In solving a non-trivial test problem, we have demonstrated that the iterative approach effectively required for such problems is not precluded by Monte Carlo noise. Indeed, by constructing Monte Carlo radiation fields that are rigorously divergence-free, rapid convergence is achieved with a temperature-correction procedure that is formally identical to the notoriously unsuccessful -iteration method. This resurrection of the -iteration method is in fact of considerable importance since the resulting temperature-correction procedure - see Eq. (17) - is geometry-independent. Thus, unlike most standard temperature-correction schemes, which are constructed to achieve constant flux or constant luminosity variable, the procedure presented here immediately generalizes to 2- and 3-D problems. Moreover, this generalization is facilitated by the volume-based intensity estimators derived in Sect. 3.4. In conclusion, the successful reproduction of Castor's solution for an extended, non-grey atmosphere implies that realistic 2- and 3-D problems that similarly require solution by iteration can be tackled with some confidence using Monte Carlo techniques. Moreover, in sharp contrast with conventional methods, the Monte Carlo codes will remain modest in size and can therefore be quickly constructed and verified. © European Southern Observatory (ESO) 1999 Online publication: March 10, 1999 |