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Astron. Astrophys. 335, 1003-1008 (1998)

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2. The standard model and the numerical techniques

The model calculations are of similar kind as described in previous papers (e.g. Hamann et al. 1994), and the reader is referred to these papers for more details and references. Only the basic assumptions and definitions are briefly repeated here.

A "standard" WR atmosphere is assumed to be expanding in a spherically-symmetric, homogeneous and stationary flow. With a given mass-loss rate [FORMULA], the density stratification [FORMULA] and the velocity field [FORMULA] are related via the equation of continuity.

The velocity field is pre-specified from plausible ad-hoc assumptions. For the supersonic part we adopt the usual [FORMULA]-law (cf., e.g., Hamann et al. 1993, Eq. 1) with the terminal velocity [FORMULA] being a free parameter. The exponent [FORMULA] is set to unity throughout this work. In the subsonic region the velocity field is defined such that a hydrostatic density stratification is approached.

The "stellar radius" [FORMULA], which is the inner boundary of our model atmosphere, corresponds per definition to a Rosseland optical depth of 20. The "stellar temperature" [FORMULA] is defined by the luminosity L and the stellar radius [FORMULA] via the Stefan-Boltzmann law, i.e. [FORMULA] denotes the effective temperature referring to the radius [FORMULA]. The problem of defining a reference radius and, correspondingly, of any "effective temperature" in spherically extended atmospheres has been addressed in previous papers (e.g. Hamann 1994).

Only Doppler broadening is accounted for in the profile function of the line absorption coefficient. The Doppler-velocity [FORMULA] reflects random motion on small scales and is generally set to 100 km s-1. This number has only marginal influence on the emergent line profiles; in the present context we tested that with 50 km s-1 the results are almost identical, but higher values have numerical advantages.

The line radiation transfer in the spherically expanding geometry is formulated in the comoving-frame of reference (CMF), treating correctly the overlap of blending lines. Continuum formation is treated with the moment equations and variable Eddington factors.

The equations of statistical equilibrium account for all relevant radiative and collisional transition rates. The temperature stratification is calculated from the assumption of radiative equilibrium. The consistent solution of both sets of equations, radiation transfer and statistical equilibrium, is achieved by "iteration with approximate lambda operators", taking advantage of "Broyden's method" in the solution algorithm (Koesterke et al. 1992). The obtained non-LTE population numbers enter the subsequent Formal Integration of the transfer equation in the observer's frame which is performed by straightforward integration along each ray.

Frequency redistribution of line photons by electron scattering is accounted by means of the appropriate angle-averaged redistribution function (Hummer 1962). The effect is only accounted for in the Formal Integration. From earlier test calculations we know that the influence of the redistribution mechanism on the "model" (i.e. on the stratification of temperature and population numbers) is negligible.

Adopting the described "standard model" assumptions, any particular WR atmosphere is specified by its basic parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA] and chemical composition (e.g. given in the form of mass fractions [FORMULA], [FORMULA] etc.).

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
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