8. Summary, conclusions and future work
In this paper, we have extended previous 2-D models of line-driven winds from rotating hot stars by accounting for the dependence of ionization structure and occupation numbers on local physical properties (density, velocity field) and the non-local stellar radiation field (variation with temperature and frequency).
To this end, we have formulated for the first time an approximate non-LTE description of 2-D winds. Since the exact dependence of the incident radiation field on direction and frequency (if gravity darkening is considered) cannot be accounted for simultaneously because of limited computational capacities and time, we have proposed the concept of a "mean irradiating atmosphere". This Ansatz allows, in an approximate way, to consider the frequency dependence of the incident photospheric radiation field, which most importantly determines the local ionization equilibrium, at least if realistic flux distributions are used. (For stellar surfaces emitting locally, i.e., as function of , a black body spectrum, the mean radiation temperature in the wind varies only weakly with frequency.)
In order to describe this frequency dependence of for all values of in a satisfactory way, we have defined local, frequency-independent mean effective temperatures , which, in connection with the appropriate gravitational acceleration, just define the above mean irradiating atmospheres and thus the frequency-dependent flux distributions in . It has been shown that varies much stronger with latitude than with distance from the star and decreases, for all iso-contours of r, monotonically from pole to equator, since the influence of the radiation field from the corresponding foot-point is significant at all radii.
By means of the resulting flux-distributions and local conditions, we have calculated 2-D NLTE occupation numbers, force-multipliers and according force-multiplier parameters, as function of (). The hydrodynamic models constructed in this way are entirely self-consistent, and converge to a stationary solution in the same way as models for globally constant , , , with the only difference that the physical convergence time can be larger by a factor of 5... 10.
For mass-loss rates , the procedure as outlined in this paper is not suitable for quantitative calculations (due to saturation effects arising from all lines becoming optically thin), although a generalization would be easily possible. Since, however, it is rather questionable whether a one-component plasma model is valid for such extremely thin winds at all, we have restricted our investigation to winds with mass-loss rates . For these winds, our algorithm to parameterize the force multiplier yields a very satisfactory accuracy of % (considering the complexity of the problem), if compared with the "exact" values found from summing up the individual line accelerations.
Our study was aimed at an estimate of maximum effects arising from rotation, in particular with respect to the differential behaviour as function of latitude, if compared with previous models allowing only for predefined, global force-multipliers. Thus, we have concentrated on rapidly rotating B-star winds, since in this spectral range the ionization structure is much more sensitive to local conditions and the variation of the radiation field than at other temperatures, e.g., for O-stars. For all models of our grid, which has been constructed to allow for a variety of luminosities, we have checked whether the Lyman continuum is optically thin, a prerequisite to definitely inhibit any bi-stability effect (to be investigated in a forthcoming paper, see below).
In all considered cases, we found a prolate wind structure, if gravity darkening is accounted for, with maximum ratios of polar to equatorial mass-loss rate up to a factor of 30. Non-radial components of the line (and continuum) acceleration inhibit any wind compression and cause negative polar velocities of order . Even if the non-radial line force components are neglected, a globally prolate wind is created, however with a moderately compressed disk in the equatorial plane.
Thus, the "-effect" suggested by Maeder (1999, cf. Sect. 1) is actually not occuring, at least for winds with an optically thin continuum. Since we have considered a temperature regime with maximum effects of ionization and have also understood the physical origin why a consistent NLTE approach gives rise to an even enhanced prolate wind morphology (Sect. 7.2.1 and below), we are convinced that this statement does not only apply to the limited spectral range considered here, but also at least for the complete OB-star range. (A-type winds might deserve a special investigation.)
For B-star winds with significant mass-loss (), the density contrast between the flow over the poles and in the equatorial plane reaches values of order . This ratio grows with rotation rate and decreases from thin winds (, ) to denser ones (B-supergiants, , ).
For our reference model B30-30, we have also calculated the wind structure which would arise if we neglect any 2-D effects with respect to occupation numbers and force-multipliers. For this purpose, we estimated global averages for , and from our self-consistent solution. The according model shows a density contrast of only half of the value in the original one. Compared to this "zero order" approximation of 2-D winds, our self-consistent parameterization leads to a moderately enhanced concentration of wind material over the poles and a significant reduction in the equatorial plane, as a consequence of the ionization effects summarized in the following.
Inspecting the contribution from various elements to the line force, the dominant rôle of iron in winds with considerable densities is obvious. The CNO group is most important (but still much less effective than iron) in the equatorial wind, since the flow evaporates for larger co-latitudes (at constant r), and the meta-stable iron lines do not accelerate the matter as effectively as in the denser polar wind. For an illumination by Kurucz fluxes, Feiii provides the major part of acceleration. Only for extremely rapid rotators ), Feii becomes most decisive in the coolest regions close to the equator, due to the significant reduction of the mean radiation temperature induced by gravity darkening ( K).
The radial variation of the ionization fractions over the pole is determined by the decrease of both effective wind density and with increasing distance from the star. Both effects act on the ionization equilibrium in the opposite way and thus have a stabilizing effect, with the consequence of rather frozen-in (however sometimes non-monotonic) ionization fractions. In contrast, the radially decreasing density and the increasing mean radiation temperature near the equator act in parallel and lead to a monotonic increase of the ionization equilibrium towards higher stages, which have, on the average, fewer lines and thus provide less acceleration. This different behaviour over the poles and in the equatorial plane is the final reason for the enhanced polar density contrast in self-consistent wind solutions, and should be valid in the complete OB-star range (though with a different degree of significance), as pointed out above.
We have also investigated briefly the importance of using realistic flux distributions by comparing our self-consistent models to winds illuminated by Planck fluxes. Such winds would be much thinner since the ionization equilibrium of the most contributing elements is shifted, on the average, to one stage higher than for Kurucz fluxes.
Our most important finding with regard to the influence of rotation on global wind properties is that the total mass-loss rate deviates from its 1-D value (for ) by at most 10... 20 %. This turned out to be true even for the highest rotation rates considered here (), with the only exception of winds from supergiants close to the Eddington-limit, where differences up to a factor of 2 become possible.
The enhanced mass-loss over the pole is compensated for by the reduced density in the equatorial plane, both effects resulting from gravity darkening. Since, under the assumption of an optically thin continuum, the ionization effects for a large part of the stellar surface are moderate (for equatorial regions, they are decisive, of course), and the total luminosity remains conserved under rotation, these small changes of the total 2-D mass-loss rate, compared to the non-rotating wind, are plausible, and have been explained by using some relevant scaling relations.
In conclusion, a quantitatively correct description of line-driven winds from rapidly rotating hot stars requires a self-consistent parameterization of the line force in all those cases, where the variation of at the stellar surface can induce a (significantly) stratified ionization equilibrium. For the hotter O-stars, this is possible only for large rotation rates, whereas in the B-star domain also moderate rates have to be included, especially for lower luminosities with thinner winds.
The resulting ionization pattern varying as function of latitude, if significant, implies a dependence of the spectral signature of corresponding lines on stellar inclination. Future effort on decent spectrum synthesis calculations is required to prove or disprove the theoretical predictions given in this paper.
In Sect. 7.2.5, we have investigated also the "worst-case" scenario with respect to the application of the WLR, if the winds are rapidly rotating, but mass-loss rates are derived by a 1-D analysis. Although a maximum scatter of 1.5 dex seems to be apparent, this scatter is reduced significantly if a mass-loss indicator is used which depends "only" on the lower wind conditions (e.g., ). Since the density contrast is smaller in those regions than in the outer wind (by roughly a factor of four, with a minimum value of order , cf. Eq. (49)), and since, as already discussed, the total line profile is influenced both from polar and equatorial regions, the actual contamination by rotational effects is definitely smaller.
PP96 have estimated the maximum error in the derivation of introduced by the standard 1-D analysis of to %. In their analysis, they used the simple analytic WCZ model with compression factors 4. Thus, this estimate does not necessarily correspond to all other wind models one might imagine. However, since the density contrast in our self-consistent simulation for a rotation speed is of the same order of magnitude as found in the WCZ model (although reversed!), the same degree of contamination might be expected.
Therefore, the influence of stellar rotation on mass-loss rates derived from is, in most cases, a second-order effect, regarding the fact that any application of the WLR uses the values of a large sample of stars, and, moreover, rapid rotators can be excluded from those samples easily unless they are observed pole-on. Note, however, that erroneously determined mass-loss rates for individual objects may have severe consequences with respect to their evolutionary status: An error in of only a factor of two is able to alter the evolution history drastically, as shown by Maeder (1991).
Future work. We have to admit that a number of aspects have been simplified in this investigation. In a "perfect" description, line-overlap and line-shadowing neglected here might introduce additional complications, as well as the impact of X-rays on the ionization equilibrium in the outer wind. In particular, the consequences of an optically thick Lyman continuum and the bi-stability effect on the wind dynamics have to be considered and will be discussed in a forthcoming paper.
A further phenomenon which is important both for the diagnostics of stellar winds and their dynamics is the presence or absence of clumps in the wind. If they actually exist and their spatial scale is small, the line acceleration could be drastically reduced and the inertial terms might become decisive again. Thus, the WCD/WCZ model might be resurrected. Whether final results of a first study on the influence of the line instability in rotating winds (Owocki 1999) support this scenario, has still to be awaited for. Finally, magnetic fields, if present with sufficient strength, could again alter our conclusions given above.
Of course, any improvement of the theory (especially regarding the considerable effort required) makes only sense if the models can be discriminated by a comparison with observations.
Thus, a re-investigation of previously analyzed winds from rapidly rotating early-type stars by means of systematically refined hydrodynamic wind models seems to be desirable. In particular, the determination of inclination, i.e., of absolute rotational velocities , remains the major problem and is presently possible only for few objects (e.g., the rapid rotator HD 93521, which exhibits spectral signatures that indicate a wind-compressed equatorial disk - despite all theoretical arguments against this scenario, cf. Bjorkman et al. 1994 and Massa 1995).
Fortunately, the different theoretical ideas and models yield also different results regarding the density structure. For a final clarification, an analysis of several spectral ranges is desirable to gain further and complementary insight (e.g., by infrared lines or IR/radio continua). In these spectral ranges, the line/continuum formation process takes place at much larger distances from the star as for the emission, and the density contrast, at least present in the theoretical simulations, is largest. Finally, the direct observation of the wind morphology by means of (optical) long-baseline interferometry is supposed to make excellent progress within the next years.
© European Southern Observatory (ESO) 2000
Online publication: June 20, 2000