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Astron. Astrophys. 344, 282-288 (1999)

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2. Test problems

In order to test Monte Carlo codes, it is necessary to apply them first to special cases where simplifying assumptions have allowed exact or highly accurate solutions to be derived with conventional analytic or numerical techniques. Here, where the aim is to derive the temperature distribution throughout a medium in radiative equilibrium, the simplest test cases are provided by the theory of grey stellar atmospheres. In this case, the exact solution is known for plane-parallel geometry (the Hopf function) and accurate numerical solutions are available for spherical, extended atmospheres.

Although results for grey atmospheres will be briefly reported, they are essentially trivial in the present context, in that they do not test our ability to improve solutions iteratively in the presence of Monte Carlo noise. Accordingly, a more meaningful and challenging test problem has been sought from the extensive literature on non -grey stellar atmospheres computed under the assumption of local thermodynamic equilibrium (LTE).

The chosen problem is that considered by Castor (1974). Motivated by evidence that the continuum-forming layers in the Of star [FORMULA] Puppis and in a number of Wolf-Rayet stars are extended, he created static model atmospheres having this characteristic by considering stars of exceptionally high luminosity-to-mass ratios. Using a method based on moment equations and requiring iteration on Eddington factors and on ratios of mean absorption coefficients, Castor computed both the structure of these extended atmospheres and their emergent spectra.

In view of its closely similar scientific motivation, Castor's work provided a natural test problem for testing Monte Carlo techniques that would allow the Abbott-Lucy code to include continuum formation in model winds for hot stars. However, because of this aim, an acceptable technique should not merely reproduce Castor's results, it must also permit the treatment of line transfer - fundamental to the dynamics of the winds - and allow radiative equilibrium to be imposed in the matter frame rather than the rest frame (Lucy & Abbott 1993). The technique described below meets these conditions as well as not being restricted to 1-D geometry.

In Castor's work, both the density and the temperature stratification were computed. However, given our present understanding of the dynamical causes of atmospheric extension in hot stars, solving for the complete structure of a static atmosphere that is extended because of high L/M now seems little more than an academic exercise. Accordingly, we focus here just on the problem of deriving the temperature distribution from the condition of radiative equilibrium. Thus, the principal test problem is the following: given the density stratification [FORMULA] predicted by Castor's code, derive the corresponding temperature stratification [FORMULA] as well as the resulting emergent luminosity density [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999