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Astron. Astrophys. 350, 1007-1017 (1999)

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3. Stellar atmosphere codes

In this section we introduce and utilise the codes that are used to carry out spectral synthesis of WR124. CMFGEN (Hillier 1987, 1990; Hillier & Miller 1998, 1999) solves the transfer equation in the co-moving frame, subject to statistical and radiative equilibrium, assuming an expanding, spherically-symmetric, homogeneous or clumped, atmosphere. Populations and ionization structure are consistent with the radiation field.

To take into account the (microscopic and macroscopic) velocity structure expected in a real WR star, a uniform Doppler line width of [FORMULA]=50 km s-1 is assumed. Since this is substantially greater than the (species dependent) thermal Doppler width, we refer to it as a `broad-line' approximation. All lines within some multiple of [FORMULA] can interact within the Sobolev resonance zone. Because of the large line width used, the interaction between two lines may be overestimated in the broad-line approximation.

A simplifying `super level' approach is used for individual levels (Anderson 1989), particularly for iron-group elements. In this approach, several levels of similar energies and properties are treated as a single `super level', with only the populations of the super level included in the solution of the rate equations. The population of an individual atomic level in the full model atom is determined by assuming that it has the same departure coefficient as the corresponding super level to which it belongs.

Unfortunately, individual line blanketed co-moving frame calculations are generally computationally demanding, despite the use of super levels, so we first need to confirm that individual case studies, such as this, are in accord with observation. A computationally quick method is to (i) solve the transfer problem in the Sobolev approximation rather than the co-moving frame, which the code ISA -wind does (de Koter et al. 1993, 1997), and (ii) consider line blanketing via Monte Carlo sampling following Schmutz (1994, 1997) allowing the opacity of a huge number of lines to be considered. How consistent are results obtained with these different methods?


The model calculations are based on the iterative technique of Hillier (1987, 1990) named CMFGEN . Allowance is made for line blanketing and clumping following the formulation of Hillier & Miller (1998) to which the reader is referred for specific details. The model atom contains hydrogen, helium, carbon, nitrogen, silicon and iron as shown in Table 2. Details of each ion are included, such that for hydrogen, n=1 to 30 full levels are considered, which are grouped into 10 super levels. Test calculations indicated that [FORMULA]5000 iron transitions with [FORMULA] have negligible influence on the emergent spectrum, but required considerable computational effort, so these have been excluded. Within a super-level, representing a group of full levels, populations are assumed to be in LTE. Elemental abundances other than hydrogen and helium are fixed at cosmic (Si=0.1% by mass, Fe=0.2% by mass) or approximately CNO-equilibrium (C=0.03%, N=1 to 2% by mass) values for a solar metallicity environment.


Table 2. Summary of model atom used in CMFGEN and ISA -wind radiative transfer calculations, including full levels ([FORMULA]), super levels ([FORMULA]) and total number of transitions ([FORMULA]), following Hillier & Miller (1998) and de Koter et al. (1997). Note that transitions considered in CMFGEN include those between individual terms in multiplets. The Monte Carlo calculation used to derive line blanketing accounts for transitions of most metals, including Al, Si and Fe.

The stellar radius ([FORMULA]) is defined as the inner boundary of the model atmosphere and is located at Rosseland optical depth of [FORMULA]20 with the stellar temperature ([FORMULA]) defined by the usual Stefan-Boltzmann relation. Similarly, the effective temperature ([FORMULA]) relates to the radius ([FORMULA]) at which the Rosseland optical depth equals 2/3.

3.2. ISA-wind

The improved Sobolev approximation code (ISA -wind) is described in detail by de Koter et al. (1993, 1997). The principal differences with CMFGEN relate to: (a) the treatment of the line radiation transfer; (b) the wind electron temperature, which assumes a grey, LTE atmosphere rather than radiative equilibrium, where the temperature is fixed at a pre-assigned value in the outer regions, such that lines originating in the outer wind (e.g. He I [FORMULA]10830) may be incorrectly predicted; (c) the velocity structure deep in the atmosphere; (d) the specific atomic model treated, as listed in Table 2 ; (e) the neglect of clumping.

ISA -wind solves the line radiation transfer using an improved version of the Sobolev approximation. In co-moving frame codes, line photons with a finite (given) width interact with the continuum over a range of depth and frequency points, while in the Sobolev approximation the continuum opacity and source function are considered constant within the line resonance volume. In the improved version of the Sobolev code, ISA -wind, absorption of line radiation by the continuum is taken into account within the resonance volume. The Sobolev approximation should be ideal for the large velocity gradients of WR outflows. It introduces great simplification in the rate equations, and makes the overall iteration process about 10 times faster than a co-moving frame code, for the same number or ionic levels and grid points.

Turning to line blanketing, an iterative technique including the Monte Carlo method of Schmutz et al. (1991) and Schmutz (1994, 1997) is used. The method allows the computation of intensity-weighed effective opacity factors, which account for the presence of tens of thousands of spectral lines, dominated by Fe and Ni. Based on ISA -wind atmosphere calculations, the Monte Carlo (MC) code determines the line blanketing factors. An iterative procedure is used, such that blanketing factors are used by the non-LTE code to calculate a new atmosphere, which in turn is used to calculate new blanketing factors. A few iterations are generally sufficient. This is due to the fact that the scattering and absorption factors are not very sensitive to the specific model parameters.

The MC method deals with the radiative transfer in the correct manner, except that the ionization and excitation equilibrium of metal species is approximate. This dictates which lines are efficient at capturing photons for each point in the atmosphere. The ionization structure of H and He in the MC code is derived from ISA -wind. For other species, an representative ionization temperature is used, obtained from the principal ionization stages for each metal that is considered in ISA -wind. The MC code calculated ionization stratification can therefore be compared with that of ISA -wind for consistency.

Since line-line interactions are not accounted for in the MC calculation, explicit correction factors, named `photon loss' factors, need to be considered to allow for interaction between important resonance lines (He II [FORMULA]303, He I [FORMULA]584) and nearby metal lines (Schmutz 1997). We have calculated photon loss factors from the opacities of He II [FORMULA]303 and lines at nearby wavelengths, and find that line-line interactions are negligible for the stellar temperatures of WNL stars. (Pasquali et al. (1997) previously adopted a factor of 10-4 at He II [FORMULA]303 for WNL stars.) Additionally, we have determined that line-line interaction between He I [FORMULA]584 and adjacent metal lines would not affect the ionization balance or derived stellar parameters for WR124.

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© European Southern Observatory (ESO) 1999

Online publication: October 14, 1999