Astron. Astrophys. 358, 956-992 (2000)

6. The line force parameterization

Based on the properties of the force-multiplier M, we will develop now the parameterization of the line acceleration.

6.1. Minor simplifications

To this end and in order to reduce the computational effort, we adopt the simplifications described in the following.

In our approach, we neglect any diffuse radiation, i.e., we account only for the direct stellar radiation originating from the stellar surface. In a spherically-symmetric and radially expanding wind, the material around a specific location expands locally into all directions. If we superimpose a differential rotation law to the radial expansion, we cannot exclude that the matter experiences a compression along certain directions , provided that the additional velocity components are large enough compared to the radial one. In such a case, a purely local formulation of the line acceleration becomes questionable if there are non-local resonance points in the wind with zero projected velocity relative to (C ommon P oint resonance zones) and the purely local expression for the line force (Eq. (22)) has to be replaced by a formulation of radiative transfer that accounts for the influence of non-local processes (see Rybicki & Hummer 1978, Mazzali 1990).

However, a parameter study performed by Petrenz (1994) has clarified that such CP resonance zones may arise in rotating winds only for very shallow radial expansion laws combined with extreme rotation rates (). Even then, the CP zones subtend only a minor part of . Therefore, the local description of the radiative transfer is a fairly good approximation and we assume the incident intensity () to depend only on the physical conditions on the stellar surface. For our test calculations, the radiation field at location on the surface follows either a black-body law (with , see Sect. 3.1) or is taken from a model flux distribution (corresponding to the local values of and ).

Furthermore, we also neglect the rotational Doppler shift of the photospheric spectrum when calculating as well as the directional dependence of resulting from limb darkening, since both effects are of only secondary importance.

In our time-dependent 2-D simulations, the local force-multiplier parameters , , are repeatedly updated in the course of the hydrodynamic evolution to guarantee their convergence consistently with the flow. At every co-latitude of the hydrodynamic grid, this procedure requires the force-multipliers M to be calculated at a certain number () of radial grid points (SG = "subgrid", see below). These values have to be determined before every update of , , . Despite our simplified NLTE formalism, the computation of the sum in Eq. (24) for such a large number of grid points would be very cumbersome. Therefore, we determine the force-multipliers by interpolation on tables which cover the parameter space of maximum width (with effective temperatures as defined further below). Once computed, these force-multiplier tables can be employed for any other or additional time-dependent hydrodynamic simulation.

6.2. Quantities of the radiation field

According to Eq. (24), the force-multiplier depends on several quantities related to the radiation field: Firstly, on the ionizing flux and secondly on the line strengths . is a direction-weighted vector quantity, , whereas the line strengths depend implicitly - via occupation numbers - on the mean intensity , which is an angle-weighted scalar. Both and depend on radiation temperatures () at the stellar surface. With present computational capacities, a simultaneous exact integration of and over solid angle and frequency is not possible. Since we have seen that it is the frequency dependence of which is crucial for the resulting force-multipliers (cf. Sect. 5), we have to account for this dependence at least in an appproximate way. To this end, we will define below angle-averaged, frequency-independent effective temperatures, at every location .

For these averaged effective temperatures (which can be regarded to represent different atmospheric models), we take the according flux distributions from a grid of plane-parallel Kurucz fluxes (as function of ), which, in course, finally provide the frequency-dependent radiation temperatures required to establish the local ionization structure. ⁶

6.2.1. Radiation temperatures for black-body irradiation

Before we define the mean effective temperatures mentioned before, we investigate how strongly the radiation temperature varies throughout the wind if the stellar surface emits as a black-body. Neglecting limb darkening, we can simultaneously account for the dependence of on and : In this case, the mean intensity

provides the radiation temperature averaged over :

can be computed numerically via Eq. (30), with , .

For the model of a rapidly rotating B2 dwarf (corresponding to the "S-350" model discussed by OCB), Fig. 8 displays the frequency dependence of for two representative radii r at different co-latitudes . At the stellar surface , the mean intensity is determined entirely by local conditions with radiation temperature independent of frequency, given by . For larger distances, different surface elements covering a wider temperature range contribute to . In consequence, decreases over the poles because cooler regions near the equator contribute to the radiation field, and, vice versa, increases over the equator. Simultaneously, the difference of for different becomes smaller.

Fig. 8. Averaged radiation temperatures for a rapidly rotating B2 main-sequence star with 20000 K, 350 and black-body surface at two radii (, ). Stellar oblateness and gravity darkening based on the von Zeipel law have been considered. Other relevant parameters are = 486 , , effective temperature at the pole  K and at the equator  K, polar gravity and normal gravity at the equator .

varies only weakly with frequency, even in the short wavelength range where the sum of the different local Wien law emissions causes a slight increase of . For large , follows the Rayleigh-Jeans law, and does not longer vary with wavelength.

6.2.2. Mean effective temperatures

In the following, we will account for the simultaneous dependence of on physical properties of the stellar surface (i.e. at ) and on frequency in an approximate way.

Our method uses quantities depending either on the angle-weighted mean intensity (for calculating ionization structure, occupation numbers) or on the direction-weighted radiative flux (for calculating force-multipliers). To describe the frequency dependence of and , respectively, we define two different effective temperatures being independent of frequency, in the spirit of our introductory remarks above.

1. According to Eq. (24), the force-multiplier M is a flux-weighted quantity with factor , which is finally absorbed in the force-multiplier parameter (see Eq. (44)).

If the radiative flux is taken from a model atmosphere flux distribution, the latter should correspond to a local "continuum effective temperature" which ensures that the total flux given by the flux distribution, , equals the stellar flux given by integration over . In other words, the definition of should follow from radiative flux conservation in . Thus we have on the one side

(Note the vector character of = , which guarantees that is oriented perpendicularly to the stellar surface, thus having both a radial and a non-vanishing polar component.)

On the other side, frequency integration of the local photospheric flux distribution on the stellar surface, , yields the local radiative flux . Comparing both sides, we obtain the desired effective temperature

which can be easily calculated in dependence on the surface deformation. For constant surface temperature, , of course.

2. In a slightly different way, we address the question for the flux distribution providing the radiation temperatures which determine ionization structure (Eq. (28)) and occupation numbers (line strengths ). The crucial quantity here is the angle-weighted mean intensity , which is a frequency-dependent quantity. Note that the local radiation temperature should depend differently on frequency at various co-latitudes , in particular, if the stellar surface does not emit a black-body spectrum.

With , we can at first define an average temperature ,

with from Eq. (27). Requiring the conservation of radiative flux yields the local effective temperature :

If we insert now the expression for (Eq. (34)) into Eq. (35) and change the order of integration over and , we find

The integration over on the RHS of Eq. (36) yields . Thus, the effective temperature reads

In contrast to the direction-weighted temperature , we have to integrate over rather than over , due do the different angular weighting process inherent to mean intensity and flux, respectively. Fig. 9 displays both effective temperatures, and , and their difference for the above B2 dwarf model. Primarily, both quantities are a function of co-latitude . For , they are identical since they depend on purely local conditions, with . For larger radii r, the polar wind is illuminated by additional cooler surface elements and both and drop (and vice versa for the equator). As for the black-body case (Sect. 6.2.1), even for large distances from the surface the mean radiation field remains strongly affected by the physical conditions at the foot point : An easy geometrical consideration shows that no radiation exactly from the pole contributes at locations in the equatorial plane and that the polar wind remains unaffected from equatorial radiation.

Fig. 9. Mean effective temperatures and for the same model as in Fig. 8. Top panel: averaged continuum effective temperature ; middle panel: averaged effective temperature ; bottom panel: difference .

Compared to their variation with co-latitude, the difference between and is surprisingly small. Its order of magnitude can be easily checked, if one assumes a spherical stellar surface and a linear dependence on µ,

which is justified by Fig. 9 (top and middle panel).

E.g., over the pole at we obtain = 42450 K and = 42560 K. Thus, the difference between both temperatures has the same order of magnitude as the exact numerical result, and slightly exceeds . The larger r, of course, the more the radiation field becomes radial, and the difference between both temperatures vanishes at all.

The fact that both effective temperatures vary much stronger with co-latitude than with distance r justifies also our concept of a representative illuminating flux distribution taken for a mean effective temperature: In , this flux distribution is in accordance with a spectrum at least similar to the emission at .

The flux distribution is a function not only of effective temperature , but also of gravity . Since the Kurucz fluxes depend only weakly on the variations of we are interested in, we take, for all in the wind, the surface-averaged value (PP96, Eq. (30)).

6.2.3. The dilution factor

Owing to stellar oblateness, the dilution factor W (Eq. (27)) becomes a function not only of r but also of . Fig. 10 displays the ratio ( from Eq. (20)) for our B2-dwarf model. is normalized to the polar radius .

Fig. 10. The ratio for the B2-star from Fig. 8 with 20000 K, 350 , at different radii r and co-latitudes .

For all , is greater or equal , and W always exceeds . Due to oblateness, this discrepancy between W and is largest close to the star with the maximum effect in the equatorial plane. The radial variation of for constant is caused both by curvature and asphericity of the stellar surface.

Over the poles, the surface curvature is weaker than in the spherical case (see Collins 1963, Fig. 1). Therefore, the solid angle always exceeds the value for a spherical star with (except for , ). For larger r, grows compared to the spherical value, due to the enlarged surface.

At the equator, the surface curvature is stronger than in the spherical case, and the significant discrepancy between W and for decreases for larger r.

Far away from the star, the solid angle subtended by the stellar disk observed equator-on is smaller than viewed pole-on. In the latter case, one looks upon a rotationally-symmetric object with the same maximum extent for all azimuthal angles (cf. Fig. 1).

As Fig. 10 clarifies, the difference between W and only marginally exceeds a factor of 2. With respect to the weak dependence of the force-multiplier M on W (see Sect. 5), this difference is not relevant for qualitative studies.

6.3. Depth-dependent force-multiplier parameterization

Figs. 3 and 4 showed a clearly non-linear dependence of the force-multiplier on and : On the one hand, the approximation of the line-strength distribution function by a power law with constant exponent is not entirely sufficient. As shown by Puls et al. (2000), varies systematically as a function of line strength , thus implying a curvature of as a function of . On the other hand, the exponent depends also on the ionization structure, and, as a consequence, becomes a non-linear function of .

Because of this non-linear behaviour, the classical fit with global , , may lead to severe discrepancies between the fit values and the actual force-multipliers M (see Kudritzki et al. 1998, hereafter Ku98, Fig. 5).

For this reason, Ku98 proposed an improved parameterization of the force-multiplier, using the most simple fit formula of higher order for which allows for a linear dependence of and on and ,

with and corresponding to the CAK parameter . The fit is performed via the least-square solution of the over-determined system of linear equations for the unknown , , , , , :

with , the number of input points of and , respectively.

As shown by Ku98, this parameterization provides fits of the force-multipliers with errors smaller than dex in , where those fits cover completely the ranges in and relevant for the corresponding spectral type of the star.

Since we will determine the parameters , , in our hydrodynamic simulations by means of Eqs. (39-41), we have to check how well these values agree with the actual ones, i.e., those obtained by a piecewise differentiation of with respect to and , respectively. For some typical cases, Fig. 11 demonstrates that the new parameterization reproduces and in the correct order of magnitude and also succeeds in a qualitative reproduction of as a function of .

Fig. 11. Force-multiplier parameters and as a result of piecewise differentiation (non-linear curves ), compared to a determination via Eqs. (40/41) (linear functions, being degenerate for ). Top panel: = 20000 K, irradiation by Kurucz flux distribution ( = 3.0); bottom panel: = 40000 K, Planck fluxes.

At this point of reasoning, the obvious question arises why we employ this particular method to determine , , instead of deriving them directly by piecewise differentiation of M, moreover since the latter method yields results with higher precision. Briefly, there are two reasons:

(1) During the hydrodynamic evolution of the flow, one does not know a priori whether the momentary local values of , , will guarantee hydrodynamic stability for subsequent time steps. Since local variations of by two orders of magnitudes are quite possible (e.g., if a disturbance propagates through the wind), one needs a description for , , which covers the entire range of possible values for and .

(2) Secondly, the line acceleration can vary drastically on relatively small spatial scales due to strong variations of and the corresponding force-multiplier parameters and if a piecewise differentiation of M is employed. This lack of a "smooth" acceleration may cause severe disturbances of the flow and prevent the flow from converging to its stationary solution. Test calculations, both for 1-D and 2-D winds, have confirmed this behaviour.

If, on the other hand, we formulate by means of parameters fitted as described above, we avoid these problems, of course at the expense of variations in and not reproduced very precisely (cf. Fig. 11). This lack of precision results from our approximation of using only linear functions in and for and , respectively, and from the notion that the fit procedure has to cover a relatively broad range in and .

As pointed out already above, however (see Ku98), the multiplication of the three factors , and finally yields a good reproduction of the force-multiplier itself: On the average, local discrepancies in and are compensated by the fit value of . Fig. 13 clearly demonstrates this behaviour.

The actual procedure to determine , and in our hydrodynamical simulations is described in the next section. By means of these quantities then, the line acceleration

is calculated. Inaccurately determined values of may lead to a biased direction-weighting of the contributions , if varies strongly as function of . In our 2-D approach, however, we will restrict ourselves to this conceptionally simple representation of the force-multiplier parameters, especially in view of problem (2 ) discussed above.

As an obvious advantage, our parameterization is consistent for all spectral types and requires minimum computational effort during the hydrodynamic calculation. In particular - and contrasted to previous 2-D approaches (models by BC and Owocki and collaborators, restricted to constant line-force parameters) - it allows for a quantitatively reasonable description of the line force that accounts both for local physical conditions and the non-local stellar radiation field.

6.4. Local line force parameterization in 2-D winds

In the following, we generalize the parameterization introduced in Sect. 6.3 to 2-D winds from rotating stars, in particular accounting for the polar gradient of the surface temperature . Considering the line acceleration (Eq. (22)), the generalized expression obviously would read (cf. Sect. 3.2)

with direction-dependent CAK parameter

Since we neglect the dependence of the ratio / on direction, we can replace this expression by the average flux weighting factor and extract from the integral. Also and are replaced by average values , , so that the line force is finally given by

This approximation - compared to the "exact" expression (Eq. (43)) - can by justified as follows:

With respect to employing an average value for , we avoid otherwise severe problems with normalization, since the flux distribution F will be taken at an effective temperature (Eq. (33)) which ensures flux conservation in .

For , the formal dependence on direction is actually redundant, since parameterizes the force-multiplier as a function of ionization structure. The latter depends only on isotropic quantities, and , and we can extract the factor from the angular integral. Thus, becomes a function of only.

describes the reaction of the force-multiplier on changes in . In the expression for (Eq. (45)), we integrate over different values of as function of direction. The force-multiplier M is an explicit function of , and does not depend explicitly on any particular direction . Therefore, the integral

evaluated with a value of that has been determined from representative 1-D force-multipliers (see below) constitutes a plausible approximation of the "exact" expression

with as an average value of the directional dependent quantities .

The actual (average) values of , , are now calculated in the following way:

At first, we derive local force-multipliers

(with radial depth parameter ) at typically 10... 12 radial grid points and all co-latitudes , which are a subset of the numerical 2-D hydro grid , with between and a maximum radius (see also Sect. 7.1). This calculation bases on our approximate NLTE solution described in Sect. 4.

The radiation field for the flux weighting factor is given by the flux distribution at (direction averaged) effective temperature (Sect. 6.2.2), thus ensuring flux conservation in . The ionizing radiation field (decisive for ionization structure and line strengths ) is taken for the angle-averaged effective temperature (Sect. 6.2.2), thus accounting for its crucial frequency-dependence. Since and are frequency-independent effective temperatures locally averaged over , our model also accounts for the dependence of the radiation field on location. ⁷

Therefore, the force-multipliers determined in this way may be regarded as representative quantities (indirectly) averaged over . Consequently, the according parameters (as function of only) derived from these force-multipliers represent average values of the corresponding direction-dependent quantities etc.

To save computational time, the are interpolated from a pre-calculated table as function of , , and , with time t (cf. Sect. 6.1).

As shown in Fig. 9, and vary much weaker with radius r (for constant ) than with co-latitude (for constant r). Thus, the anyhow weak temperature dependence of M (cf. Sect. 5) for constant is negligible compared to its dependence on and , respectively. In so far, it is sufficient to parameterize as a function of and along a radial ray at constant co-latitude .

For all then, we determine the force multipliers at the radial subgrid described above, and additionally the values + , + , with , and . These additional force-multipliers allow us to include the possible range of the non-radial depth parameters .

By means of these M-values, we determine the six parameters , , , , , separately for every co-latitude via Eq. (42). Inserting these parameters into Eqs. (40) and (41) provides and , which depend on time via and , and allows to calculate finally the line acceleration via Eq. (45).

During the temporal evolution of our simulations, the above procedure is repeated typically each 100 time steps , for each co-latitude .

With the local values of as central values of the squares in the --subspace defined above, we force the fit range to be concentrated about the actual values of and , which are the only input quantities of our procedure that evolve with time (in contrast to the fixed quantities and ). Thus, the adaptation of the force-multiplier parameters to the flow becomes more consistent as if one determines a set of "global" values , etc. for the entire range of possible and before the simulation.

As extensive numerical tests have shown, satisfying fits of the force multipliers require a certain minimum width of the intervals and . In particular, if the interval is too narrow, the errors in , etc. may become much larger than the resulting parameters, and the resulting , and may obtain unphysical values, preventing a stable convergence of the flow. Empirically, we found a width of 1.5... 2.0 (1.0... 1.5) for the intervals in and , respectively.

6.5. Synopsis: calculation of a consistent 2-D line force by means of the force-multiplier concept

At every co-latitude , our procedure provides six constant parameters , , , , , by means of local force-multipliers and the values on neighbouring coordinates in the - subspace. With these parameters, which are repeatedly updated in the course of the hydrodynamic evolution, directional independent values of , and are determined. These latter values parameterize the line acceleration and depend on the local, time-dependent quantities and .

Since the force-multipliers are computed for flux distributions evaluated at local effective temperatures and averaged over , the resulting force-multiplier parameters can be regarded as representative values averaged over .

The primarily polar variation of and suggests a fit for , etc. along radial rays for fixed co-latitudes . It is obvious that the directional independent values of and may differ from the "exact" ones and for distinct directions . However, this problem may occur only in intermediate wind regions, i.e., at locations where the incident radiation field originates from surface elements with different temperatures. For larger distances, the irradiation becomes radial and its representation by an adequate flux distribution is certainly reasonable. Close to the star, finally, the incident intensity is characterized by purely local conditions.

© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000
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