Astron. Astrophys. 323, 488-512 (1997)

Appendix A: atomic data

In this section, we describe our atomic models for hydrogen and helium and the data used for calculating the Stark-broadening in the formal integrals.

Our hydrogen model consists of 10 NLTE levels, defined by their principal quantum number. Since we are not including magnetic fields in our calculations, no splitting of the l values is required.

Radiative ionization cross-sections are calculated using the hydrogenic expression, which is proportional to . The appropriate Gaunt factors are taken from Gingerich (1964). For the collisional ionizations, a generic form is adopted (Mihalas 1978, p. 133), where the factor is fitted in powers of temperature T (all levels but ) or ( level). Free-free opacity is calculated using a hydrogenic form (Mihalas 1978) with Gaunt factors from Karzas & Latter (1961).

Our implementation of the treatment of lines requires only the oscillator strength as input for radiative processes; they are taken from Wiese et al. (1966). For collisional processes, we use the formula of Burke et al. (1967), scaled to the Sampson & Golden (1971) results.

The ionized helium model includes 14 levels, all of which are treated in full NLTE (with only one quantum number, n, to describe each level, and all possible transitions among them). Radiative bound-free and free-free cross-sections are calculated using the same formulae as for hydrogen, with the obvious changes. Collisional bound-free probabilities use the same two expressions as for hydrogen, where the four highest levels are treated in the manner described by Seaton (1962).

The oscillator strengths for radiative bound-bound transitions are also taken from Wiese et al. (1966). Collisional bound-bound cross-sections are those proposed by Hinnov (1966), scaled to fit the results from Sampson and Golden (1971), as in the hydrogen case.

The much more complicated model of neutral helium consists of 27 levels, 14 for the singlet and 13 for the triplet configuration, all of which are treated in NLTE. For , all possible L values are considered, while for we take two compressed levels per n, one for the singlets and the other for triplets, packing the different L states into one.

The number of possible transitions and formulae used is also increased compared to hydrogenic ions. Dielectronic recombination is not included at present; however, this should be an only minor contribution since such a process becomes dominant for HeI only at around  K (Burgess 1964). In any case, high temperatures are reached only in the deepest layers of our models, where the electron density is large enough to produce collisional ionization before the stabilizing transition can take place (Burgess and Summers 1969).

Allowed (radiative and collisional) and forbidden (only collisional) HeI transitions are considered and treated, up to a total number of around 250. For the different formulae employed, we refer the reader to the paper by Butler & Giddings (1985), where they describe the major features of DETAIL and all the cross-sections implemented in it.

We now turn to the data used for calculating the Stark-broadening of hydrogen and helium. For neutral hydrogen, Vidal, Cooper & Smith (1970, 1971, 1973) have developed an unified theory which has proven to be highly accurate ³. Schöning & Butler (1989a, 1989b) have extended this theory to HeII lines. Both groups have published tables of the corresponding line profiles under conditions of astrophysical interest. When the line we are considering is in the tables, we use their data; otherwise, we apply the theory of Griem (1960) as used by Auer & Mihalas (1972) with the improvements due to Simon (1979) to obtain Stark profiles for HeII lines. For neutral helium, we use tables computed with the theory of Barnard et al. (1969) for the lines at 4026 and 4388 Å, and that of Barnard et al. (1974) for those at 4922 and 4471 Å. For other HeI lines and in those cases where the tables for the upper four lines do not extend far enough in electron density, we use the "isolated line" approach described by Griem (1974).

Appendix B: the temperature stratification: NLTE Hopf function

The most tedious (and time-consuming) work in the construction of stellar atmospheres is the establishment of the temperature stratification from a rigorous treatment of radiative equilibrium. Considering the rather large number of present uncertainties, which at present influence the resulting structures (cf. Sect. 3.1), one may debate the extent to which this procedure is useful if we consider the expanding atmospheres of massive stars. Since it is of great importance, however, that the resulting atmospheres and line profiles are at least internally consistent - e.g, our results should reproduce the plane-parallel ones in cases of negligible winds - we have developed an approximate treatment of the problem which yields both the required consistency and costs almost no time and effort. This procedure consists of two steps, namely to adapt the results of available plane-parallel NLTE models (this section) and to relate them to the present situation of (spherically) atmospheres with a different density stratification (Appendix C).

The first step is based on a suggestion by R.-P. Kudritzki and exploits the fact that the decisive control parameter of the temperature run is the Rosseland opacity (and not, e.g., the column density). Thus, and at first neglecting the influence of sphericity, two models with different density structures should yield an analogous degree of flux conservation when both apply the same functional dependence . From this philosophy, it is straightforward to develop an adequate procedure. We take the output from a converged plane-parallel NLTE model available to our group -preferentially with the same elements as considered in the model to be constructed - and relate their apparent stratification via the well-known formula for the grey case, i.e., we introduce a "NLTE Hopf function" (Eq. 7). In this way, we account for NLTE effects leading to deviations from the grey case. Strictly speaking, this approach will only then result in the same degree of flux conservation if the -scale of the model to be constructed is also calculated on the basis of NLTE-opacities. However, due to the nature of Rosseland opacities which give the highest weight to the smallest opacities (which are then close to the pure Thomson background), the differences between (LTE) and (NLTE) are only small. Thus, significant differences of both scales, which would require an additional iteration cycle (not presently implemented in our code), are only to be expected if the ionization equilibrium of the dominant ions is drastically shifted due to NLTE effects.

Finally, and in order to obtain a practicable tool for establishing the new temperature structure, the NLTE Hopf function derived from the model output is parameterized in a way consistent with the (two-point) functional behaviour of the grey Hopf function (e.g., Mihalas 1978, p. 69), namely

where and are fit parameters to the run of . These parameters are chosen in such a way that the fitted stratification in the decisive part of the -scale - from outwards to the temperature minimum (in the typical NLTE case at ) - is the best compromise with the actual run. This approach implicitly assumes that for , if the temperature minimum is located at (cf. the discussion in Sect. 3.1). Fig. 25 and 26 give some examples of the precision obtained, and Table 2 presents the parameters of the NLTE Hopf function for a number of models discussed in this paper.

Fig. 25. NLTE Hopf function from plane-parallel NLTE H/He models OP2 (upper curves) and AP7 (lower curves, for parameters see Table 1). Dotted: approximation of Eq. B1 with parameters from Table A1. For convenience, both lower curves are shifted by -0.2.

Fig. 26. Temperature stratification of plane-parallel NLTE H/He models OP2 (upper curves) and AP7 (lower curves). Dotted: Temperature structure from approximate NLTE Hopf function as in Fig. 25. For convenience, both lower curves are shifted by -0.5.

Table 2. NLTE Hopf function parameterized as in Eq. B1 for a number of plane-parallel NLTE H/He models, all with .

Appendix C: the temperature stratification: inclusion of sphericity effects

The solution of the spherical grey problem has been extensively discussed by Hummer & Rybicki (1971 and references therein). However, instead of using their final (numerical) results, we require an approximate procedure which will allow us to generalize the approach of Appendix B for spherical geometries. In particular, we seek the appropriate transformation of the NLTE Hopf function derived from plane-parallel models to retain the flux-conserving properties (under NLTE conditions) of the scheme.

In spherical atmospheres, the asymptotic behaviour of the grey mean intensity is well known:

is the photospheric Eddington flux, is defined in Eq. 9 and C is an integration constant depending on the chosen boundary condition. Assuming now a power law stratification for the Rosseland opacity , the ratio of and is simply given by

In the outer atmospheres of the OBA-stars under consideration, Thomson-scattering dominates, so that with we can write

and thus, from Eq. C1

In the interior part of the atmosphere, we choose the integration constant in such a way that we can unify both regimes

and obtain (up to now from the LTE condition )

For power law opacities, this equation is equivalent to

Note that Eq. C7 agrees with the "conventional" spherical grey temperature law (Larson 1969), but has been derived here in a somewhat different spirit. This expression is now (almost) consistent with the analogous plane-parallel one. In this limit, and the offset in -scale relates well to the corresponding one in the Eddington-approximation, namely for and for , compared to the plane-parallel value of . Thus, we demand that the NLTE Hopf function derived from plane-parallel models (see above) shall correspond to this quantity and finally have

where the spherical analog of the NLTE Hopf function has been defined by

Appendix D: calculation of net continuum rates in the ALI formalism

In this appendix, we describe how we include the ionization/ recombination integrals into the rate equations. As pointed out in Sect. 2.3, it is especially this formulation where our approach differs mostly from other codes, except perhaps ISA (de Koter et al. 1993), which applies a similar philosophy. Since in our opinion the formulation of the net continuum rates is the very heart of any NLTE code and decisively controls the stability and convergence speed, we will give here our solution to the problem in some detail.

To begin with, the net continuum rate between transition is defined as

with lower occupation number , upper occupation number (in the continuum with respect to ), recombination rate and ionization rate . With ionization and recombination rates written explicitly, we obtain

The integration is understood to be performed always between threshold frequency and - formally - infinity. Rearranging with respect to the mean continuum intensity , we have

where an asterisk denotes the usual LTE ratio (given by the Saha-Boltzmann factor). Before we can further proceed, we have to establish the relation between mean intensity and source function with regard to the ALI-formalism.

The complete continuum source function can be written as:

where the superscript "t" denotes the thermal part of emissivity and opacity, is the Thomson opacity and the "quasi-thermal" (because of the Thomson contribution to the opacity) source function. Note that we have omitted the frequential dependence of and to avoid an accumulation of indices.

These thermal quantities include the contribution from all transitions involved, so that with

the quasi-thermal source function can be expressed as

and with Eq. (D4) the ALI formalism reads (e.g., Werner & Husfeld 1985)

The indices n and account for the iteration step in which the corresponding quantities have been calculated, is the usual Lambda-Operator acting on and the appropriate ALO. By assuming the Thomson coefficient, TH, to remain constant between two consecutive iterations, this equation can be put in a more compact form,

The result of this different formulation is a considerable acceleration of the convergence rate in cases of an optically thick scattering continuum (e.g., for frequencies far away from the edges), since the amplification "matrix" (Olson et al. 1986) obtains large values as .

Introducing this equation for J in Eq. (D3), we can in principle calculate the corresponding terms in the rate equations. The problem that arises now is the well known non-linearity related with this procedure: the current source function, , depends on the actual populations, which are still unknown; when the dependence is explicitly written, the resulting expression to be inserted in is non-linear. To avoid the problem of solving a set of N (number of levels) coupled non-linear equations, it is customary to manipulate the equations in order to retain linearity, typically by using some non-linear terms from the previous iteration. That is what Rybicki (1971) called preconditioning, and some authors (e.g., Herrero 1987, Voels et al. 1989, Rybicki & Hummer 1991) apply this idea in their codes. Preconditioning, in principle, decreases the power of the ALI cycle, as not all the information from the current source function, , is used, but the gain in computational time is large enough to prefer it. Other methods to solve this problem are the complete linearization, developed by Auer & Mihalas (1969), which is restricted to a rather limited number of levels, and the formulation by Pauldrach & Herrero (1988), which uses the results from the two previous iterations. (As an alternative to ALI and complete linearization, we mention here also the newly rediscovered method of successive overrelaxation (SOR, cf. Trujillo Bueno & Fabiani Bendicho 1995)).

We will follow the philosophy of preconditioning, however in our own formulation. The non-linearity appears because of the term in Eq. (D8), which is removed by an approximate treatment of Eq. (D6): for each transition , we suppose

which is equivalent to assuming that the transition under consideration is the dominant one, but weighting it by its real contribution to the total (quasi-thermal) source function.

Are we allowed to make such an approximation? Certainly, since by definition and thus , i.e., our modified ALO still underestimates the real operator, which ensures convergence (cf. Puls & Herrero 1988).

The second assumption of our approach is to require the ratio (which is much better than any absolute value) between transition opacity and total opacity to remain constant between iteration and n, so that Eq. (D8) finally obtains the form

Thus, we can remove the non-linear part in the second term of equation (D3) via the definition of the thermal source function and achieve linearity in both and :

In order to find a compact expression for , we use the definition of for calculating ,

and finally obtain the net continuum rate

with integrals to

In passing, we note that Eq. D13 results in the usual Lambda iteration for , whereas for (only one transition present) and (no electron scattering) this equation is consistent with the usual ALI formulation for local ALOs.

Appendix E: separation of line and continuum transfer in the formal integral

In the following section, we develop a formalism to separate the line and continuum transfer in the formal integral. This approach can save a large amount of computational time in those cases, where

i. the integral is solved on a radial micro-grid (cf. Sect. 2.4.1). ii. the resonance zones have an only small spatial extent compared to the rest of the ray. iii. the continuum source function is approximately frequency-independent over the line profile under consideration.

The maximum gain in computational speed by using the following approach is thus obtained if we consider pure Doppler profiles and a "simple" continuum source function. However, even for Stark-broadening a substantial acceleration is achieved, since this effect is present only in the lower atmosphere. In order to apply this approach universally, we have included the case of a frequency dependent source function, e.g., non-coherent scattering. If this option is chosen (which is necessary only for winds with a large or a significat photon injection rate from below the electron-scattering sphere), then the temporal advantage of the procedure is lost.

Under the assumption that the frequential variation of the continuum opacity over the line profile can be neglected, the transfer equation along a certain ray (usual geometry provided) can be written as

where x and are the observer's and CMF-frequency in units of the wind's maximum Doppler shift

( is the transition frequency). We now define a transformed intensity via

with

Defining also a transformed source function, ,

it is straightforward to derive a transformed equation of transfer

The reader should note that this equation corresponds to the transfer equation for the pure line case, i.e., it has to be solved only inside the resonance zone(s). (One zone per ray and line component, unless the components overlap within their intrinsic profiles). In between, the transformed intensity remains constant. Up to this point, we have strictly followed the procedure introduced by Hummer & Rybicki (1985) in order to derive a "Sobolev approximation with continuum". From now on, our approach is different, since we consider the "exact" case. With the obvious boundary conditions

the transformed transfer equation has the well-known solution

with an optical depth

In principle, we can calculate at every position along the ray, but in the following we are only interested in its value at to obtain emergent intensities and fluxes. Transforming back by means of Eq. (E3), the emergent intensity reads

Using the continuum optical depth along the ray,

we can write in a more compact way,

where

Note that and consist of pure continuum quantities and have to be calculated only once per line profile (unless becomes frequency dependent, see above). With these notations, the transformed line source function becomes

so that the integral for the emergent intensity is finally given by

The advantages of this formula compared to the conventional solution are obvious. After having established our micro-grid (typcially with five points per thermal Doppler velocity, see Sect. 2.4.1) we can calculate all continuum quantities ( and in advance and find, per ray and frequency, the boundaries of the resonance zone(s). This can be done without much computational effort, if we organize the profile calculation in such a way that the outer loop comprises the angular integration ( rays) and the inner loop is the frequential one. In this case then and when we proceed from one observer's frequency to the next, the boundaries of the resonance zones are shifted by only a few number of depth points which can be found almost instantaneously due to our micro-grid concept.

Inside the resonance zones, we calculate the modified line source function (bracketed term above) and calculate the integral. Outside the resonance zones, nothing has to be done. Having integrated throughout a given ray, the emergent intensity is obtained by adding the first two terms in Eq. E14.

Hence, in the most favourable case of pure Doppler profiles, we have to perform typically 30 integration steps per ray, frequency and component. (assuming a width of thermal Doppler widths).

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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