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In order to constrain the models describing circumstellar environments, it isnecessary to solve the radiative transfer equation in the presence ofabsorption and scattering, coupled with the equation for radiative equilibrium.However, solving this problem requires much CPU time, which makes the use ofautomatic minimisation procedures to characterise these environmentschallenging. The use of approximate methods is then of primary interest. Onepromising candidate method is the flux-limited diffusion (FLD), which recaststhe radiative transfer problem into a non-linear diffusion equation. Oneimportant aspect for the accuracy of the method lies in the implementation ofappropriate boundary conditions (BCs). We present new BCs for the FLDapproximation in circumstellar environments that we apply here to sphericallysymmetric envelopes. At the inner boundary, the entering flux (coming from thestar and from the envelope itself) may be written in the FLD formalism andprovides us with an adequate BC. At the free outer boundary, we used the FLDformalism to constrain the ratio of the mean radiation intensity over theemerging flux. In both cases we derived non-linear mixed BCs relating thesurface values of the mean specific intensity and its gradient. We implementedthese conditions and compared the results with previous benchmarks and theresults of a Monte Carlo radiative transfer code. A comparison with resultsderived from BCs that were previously proposed in other contexts is presentedas well. For all the tested cases, the average relative difference with thebenchmark results is below 2\% for the temperature profile and below 6\% forthe corresponding spectral energy distribution or the emerging flux. We pointout that the FLD method together with the new outer BC also allows us to derivean approximation for the emerging flux.

astronomy and astrophysics

New boundary conditions for the approximate flux-limited diffusion  radiative transfer in circumstellar environments: test case study for  spherically symmetric envelopes