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

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1. Introduction

The theoretical investigation of many astrophysical phenomena requires the accurate determinations of ambient and emergent radiation fields. Nowadays, with the ready availability of powerful computers, this is usually accomplished by solving the equation of radiative transfer numerically. Thus, derivatives and integrals are approximated by differences and summations, and the resulting algebraic system solved by matrix inversion, with an outer iteration loop being required when the absorption and scattering coefficients are coupled to the radiation field due to their dependence on state variables. For problems with a high degree of symmetry, such as plane-parallel stellar atmospheres or spherically-symmetric stellar winds, this conventional numerical approach can hardly be faulted and has indeed led to powerful codes that accurately solve these 1-D problems, even with full non-LTE treatments of excitation and ionization of numerous elements. However, as astronomers' interests turn increasingly to 2- and 3-D problems, this approach becomes problematical, for the number of variables to be solved for then becomes huge, as do the codes themselves. Moreover, most of the iterative schemes fundamental to the success of these existing codes are specific to 1-D geometry.

When geometrical simplicity is lost, the Monte Carlo approach to transfer problems becomes attractive and may often be the only feasible technique. Indeed, inspection of the astrophysical literature reveals numerous cases where the Monte Carlo method has been applied specifically to treat complexities that would defeat or severely test the conventional approach. Some recent examples are: studying the penetration of UV radiation into the interiors of clumpy interstellar clouds (Boissé 1990), treating resonance-line scattering in accretion disk winds (Knigge et al. 1995), and computing polarization maps for the circumstellar envelopes of protostars (Fischer et al. 1994). It is noteworthy, however, that in these examples and indeed in most Monte Carlo transfer codes the absorption and scattering coefficients are not coupled to the radiation field. Evidently, such problems, which require solution by iteration, have for the most part been avoided by developers of Monte Carlo codes.

When a problem is solved iteratively, corrections are applied at each iteration, and these are derived from the residuals that express the previous solution's failure to satisfy the problem's basic equations. An obvious concern, therefore, when contemplating Monte Carlo techniques, is that these residuals will at some stage be dominated by sampling errors, thus possibly halting the convergence of the iterative sequence before a solution of the desired accuracy has been achieved. In fact, this is exactly the problem that thwarted Price (1969) in his ambitious attempt to calculate a non-LTE, plane-parallel, radiative equilibrium stellar atmosphere with Monte Carlo methods. At large optical depths, the flux residuals were dominated by sampling errors and so the temperature corrections were meaningless.

Nevertheless, with today's computer power and with appropriate Monte Carlo techniques, problems requiring solution by iteration are feasible. A recent example is the work of Och et al. (1998), who iteratively determined the temperature and ionization stratification for a photoionized nebula of uniform density using a Monte Carlo treatment of radiative transfer, obtaining good agreement with the predictions of conventional codes both for the nebula's structure and its emission line spectrum. Since their Monte Carlo code is not fundamentally restricted to spherical symmetry, it can readily be generalized to treat realistic 3-D models of inhomogeneous nebulae.

In this paper, another such 1-D test problem is treated, namely that of computing the temperature stratification and emergent spectrum of an extended spherical non-grey stellar atmosphere in LTE, a problem solved with conventional methods by Castor (1974). The particular investigation described here was initially carried out in 1988 with the aim of including continuum formation in a Monte Carlo code (Abbott & Lucy 1985) developed to investigate the dynamical consequences of multi-line transfer effects in hot star winds and was briefly alluded to by Schmutz et al. (1990). Although successful, this work was in the end omitted from the code developed for Wolf-Rayet winds (Lucy & Abbott 1993- see Sect. 2.3) in order to avoid two nested iteration loops. Nevertheless, the omitted technique is likely to be useful in other astrophysical contexts, as indicated by similar, but not identical work briefly described by Bjorkman & Wood (1997). As with the work of Och et al. (1998), the techniques described herein readily generalize to 2 and 3-D problems.

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999