## 8. Summary, conclusions and future workIn 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 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 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 In all considered cases, we found a Thus, the "-effect" suggested by
Maeder (1999, cf. Sect. 1) is actually 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 The 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 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 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 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).
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, 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 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 © European Southern Observatory (ESO) 2000 Online publication: June 20, 2000 |