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

7. Self-consistent models

In this section, we present our numerical results for self-consistent wind models accounting for both the local physical conditions and the non-local ionizing radiation field, as described in Sect. 6.4. To clarify to what extent this approach influences the physical signatures/quantities in the wind, we will differentially compare the properties of these new models with those following from more simple 1-D/2-D simulations using global values for , , .

7.1. Numerical aspects and tests for 1-D models

While the models evolve to their stationary state, the force-multiplier parameters consistently adapt to the flow, implying an additional time-dependence of , which is anyway a function of the hydrodynamic variables , and . Since , , are determined separately for every co-latitude , they are coupled with all force-multipliers within the radial fit range. To facilitate flow convergence, the points of the radial mesh are distributed over a range . ⁸ After initial disturbances originating at the wind base have been advected to radii , the force-multiplier parameters and thus the total force-multiplier relax to their stationary value.

The effective temperatures (Eq. (33)) and the line force components (Eq. (22)) are determined (the latter ones for every time step) by a standard quadrature over 9 5 equidistant integration points in . To avoid numerical inaccuracies in the outer wind region, it turned out that has to be fixed very accurately ().

For s physical evolution time (roughly 50 days), the computation of a self-consistent 2-D model typically required CPU hours on a 400 Mflops machine.

Model atmospheres. The flux distributions are taken from plane-parallel LTE Kurucz model atmospheres (Kurucz 1992), computed by N. Przybilla (Univ. Observatory, Munich). For our preferentially considered temperature range ( K), these fluxes provide the presently best approximation (compromise flux-blocking/blanketing vs. NLTE), although for low gravities the grid becomes incomplete due to the problem of too large line accelerations, being inconsistent with the hydrostatic assumption inherent to those models. In such cases, we used the fluxes from a model atmosphere with the lowest value of possible for given . Note, however, that this problem occurs only for very rapidly rotating supergiants (low and high temperatures over the poles).

1-D models. In a first step and as a test of our method, we have computed a number of non-rotating 1-D models in the considered parameter range. The convergence time of self-consistent models is larger by a factor 5... 10, compared to models with constant , , , owing to their additional "time-dependence". A comparison with the results by Kudritzki et al. (1998, based on the alternative stationary approach) showed an almost perfect agreement.

For very thin winds (), however, our method as proposed in this paper does not work: Due to the maximum value of line strength present in any wind plasma, the force-multipliers M saturate for ,

In consequence, the slope of (i.e., ) approaches zero for , and shows a significant curvature in this transition region, in contrast to our assumption of being a linear function of . Thus, the values for arising from our procedure become inaccurate in this regime, and are mostly drastically overestimated for the lowest values of . A solution of this problem is to restrict the fit range to some minimum value of and to continue with an analytical solution for in the outer wind, where .

In view of the additional uncertainties present for thin winds (e.g., the supposed decoupling of the accelerated metal ions from the rest material of the wind, see Springmann & Pauldrach 1992, Porter & Skouza 1999) and the notion that winds with such small mass-loss rates are too thin to have any observable effect on optical/IR emission lines and IR/radio continua, we did not follow this possibility, however restricted our further studies to winds with minimum mass-loss rates .

7.2. 2-D models

As outlined in Sect. 3.3, we have concentrated our investigations on the consequences of rotation for the winds of B-(super)giants. A thorough understanding of these physical effects will allow a future application to the analysis of and IR emission lines as well as the investigation of additional problems discussed there.

In a first step, we will analyze some basic effects by means of a self-consistent reference model. One of the most important issues concerns the question whether our description accounting for 2-D processes still predicts a concentration of wind material above the poles (as is true when using global line-force parameters). With regard to a subsequent application to line diagnostics, we will also quantify the density contrast between equatorial and polar wind.

In a second step, then, we will present the results for a model grid comprising very rapidly rotating B-stars (mostly ). The model parameters have been chosen in such a way that a broad range of luminosities () is covered and the (anticipated) mass-loss rates are well above the minimum value discussed in the last section. This investigation will facilitate an estimate of maximum effects of stellar rotation on the resulting wind structure and the corresponding deviations from the mean 1-D wind-momentum luminosity relation, both in a global sense and for the polar and equatorial wind. Table 1 introduces this model grid, and Table 2 tabulates the according parameters related to rotation. Note, that different rotation rates have been adopted for model B30-30, which will serve as our reference model described in the following.

Table 2. Model grid: Parameters related to rotation. Equatorial radius in units of , polar/equatorial effective temperature in K, in . and denote the equatorial normal gravity and escape velocity, respectively (for 1-D Thomson acceleration). The corresponding polar values and are listed in Table 1. Bold: Reference Model B30-30 with .

7.2.1. 2-D reference model for a B-supergiant wind

In this section, we investigate basic effects of rotation on the wind structure for a distinct B-star wind model in the framework of our self-consistent theory. From our grid, we have chosen the model "B30-30" with = 20000 K, , and a polar radius = 30 (all other parameters can be found in Tables 1 and 2) as a typical representative with moderate mass-loss.

Since we concentrate on winds with an optically thin continuum in the present investigation, we have to check whether the (final) mass-loss rate lies well below the critical value for which the wind becomes optically thick in the Lyman continuum. Otherwise, an abruptly enhanced mass-loss at co-latitudes with at the wind base (owing to the bi-stability effect, see Sect. 2.2) is to be expected. For our reference model with = , we have (), and the assumption of an optically thin continuum is justified. (Eq. (7) from Lamers & Pauldrach 1991yields at the pole and at the equator, cf. Table 3). Fig. 12 displays the resulting density structure and the radial velocity field for our model with Kurucz fluxes, after a flow time of s. Characteristic numerical results of this and further simulations are summarized in Table 4.

Fig. 12. Density and radial velocity component for the wind model B30-30 (KU) and = , with consistent force-multiplier parameters. The arrows indicate the polar velocities, with a maximum (absolute) value .

Table 3. Optical depth in the Lyman continuum , according to Lamers & Pauldrach (1991, Eq. (7)) for the different B-star wind models at different co-latitudes .
The assumption of an optically thin continuum is invalid for the models B45-30 and B60-30, since is larger than unity for all co-latitudes. P: Planck irradiation. Bold: Reference Model B30-30 with .

Table 4. Numerical results for 2-D wind models. : total (i.e., surface-integrated) mass-loss rate in ; : polar mass-loss rate at the wind base; : polar/equatorial mass-loss rate for ; rotational velocities in ; taken at . P/KU: irradiation with Planck/Kurucz fluxes, GL: Model with global f.-m. parameters. Bold: Reference Model B30-30(KU) with .

The wind structure is clearly prolate , and the polar density contrast significantly increases towards larger radii, with maximum values . (Note that from here on we use polar density contrasts, i.e., the inverse of the equatorial density contrast introduced in Sect. 3.3.) The wind is fast over the poles, with , and slowest in the equatorial plane, with . The matter is deflected towards the polar regions, and the negative polar velocity reaches a maximum (absolute) value of . Responsible for this deflection is, as mentioned in Sect. 3.3, the negative polar component arising from local asymmetric line resonances: For this model, is negative in the major part, and consequently also .

To illustrate both the order of magnitude of the force-multiplier parameters and the quality of our fit procedure, Fig. 13 (top panel) displays these quantities as function of and r. The lower panel shows the total force-multipliers , either taken from our pre-calculated table as function of local variables , , W and (i.e., the "actual" force-multiplier ; left) and the corresponding values resulting from the fit along radial rays, (right). Interestingly, the force-multiplier M is a non-monotonic function of co-latitude (for fixed r). We will explain this behaviour in Sect. 15.

Fig. 13. Top panel: force multiplier parameters for the model B30-30 () with Kurucz fluxes, taken from the fit along radial rays, cf. Eq. (42). Bottom panel: "actual" force-multipliers interpolated from pre-calculated table (left) and values reproduced by the fit (right).

The resulting parameters , , correspond, to order of magnitude, to the values one would expect from detailed 1-D non-LTE calculations for the considered temperature range (e.g., decreases for lower temperatures, as discussed in Puls et al. 2000). Considering the complexity of the covered parameter space, our fit procedure yields a very good reproduction of the actual values M by , with maximum errors of 5... 10%.

Physical properties. Fig. 14 displays the contribution from various elements to the line force as function of radius over the pole and in the equatorial plane, respectively, for our self-consistent 2-D model. As in Sect. 5, we have plotted (roughly the number of optically thick driving lines, see Eq. (29)) over atomic number Z and radius.

Fig. 14. Model B30-30 (KU): Contribution of various elements to the line acceleration over the pole (left) and in the equatorial plane (right), as function of .

At first, let us discuss some general features. Over the pole, the contribution from the iron group (in particular, iron itself) clearly dominates for all radii. In the outer wind and with increasing co-latitude , however, the relative acceleration from iron (although always dominating) significantly decreases, compared to the one from other species. This global behaviour results mainly from the decreasing (effective) wind density for larger r and (cf. Fig. 12, left). The contribution from iron is mainly due to several hundred thousand unsaturated meta-stable lines, whereas a few saturated resonance lines are decisive for the CNO group (in particular, in the outer wind), as discussed in Sect. 5. Consequently, iron is more important in the dense wind over the poles than in the equatorial plane, where the wind is thinner and the unsaturated meta-stable lines are less effective than the strong resonance lines (see also Puls et al. 2000).

For all elements, is maximum close to the star, where the density is highest, and decreases with radius (with the exception of carbon in the equatorial plane, see below), however to a much larger extent at the equator than over the pole. This effect is readily understood if one accounts for the increasing radiation temperature as function of radius in equatorial regions (cf. Fig. 9): the larger the distance to the star, the more also hot surface elements near the pole contribute to the mean radiation field there. In combination with the decreasing density, this effect supports higher ionization stages with a smaller line number: Thus, decreases drastically with radius. In polar regions, vice versa, the decreasing density is partly compensated by a decreasing radiation temperature, and the ionization remains much more constant, with the result that also remains much more constant as function of radius.

For the remaining discussion, we will restrict ourselves to iron and carbon which are the two elements contributing most to the line force.

Concerning these elements, two aspects are striking in the equatorial plane (see Fig. 14, right panel): Firstly, close to the star, the contribution from iron is significantly larger than over the pole (note the logarithmic scale!), and secondly, increases for carbon for larger r, as mentioned already above.

To understand the first point, we have plotted the corresponding polar ionization structure at the wind base in Fig. 16 (top panel). Shown are the ionization fractions

with the number density of element k and the one of ionization stage j.

The polar variation of these fractions is essentially controlled by the variation of the mean radiation field, which decreases from pole to equator (with for ). Throughout the entire wind, Feiii is the major ionization stage. For and close to the star, however, the fraction of Feii significantly increases, providing much more driving lines than Feiii ⁹. In consequence, Feiii dominates over the hot pole (all radii), whereas Feii is decisive for the line force at the equator (lower wind, compare the left and right bottom panels of Fig. 15).

Fig. 15. As Fig. 14, however contribution of various ionization stages of carbon (top) and iron (bottom).

Fig. 16. Model B30-30(KU): Ionization fractions of carbon (left) and iron (right) at distinct radial points , as function of co-latitude.

For minor ionization stages, strongly varies with co-latitude only close to the star, as is obvious from the sequence in Fig. 16: The larger r, the more becomes constant, since varies only mildly with in the outer wind (cf. Fig. 9).

To illuminate the behaviour of carbon in the equatorial plane ( increases with r in the outer wind), let us concentrate on Fig. 15, upper panel. Over the pole (left), the contribution from Cii exceeds the one from Ciii, since the spectrum of Cii has numerous unsaturated lines which are most effective in the dense part of the wind. Because of the counteracting rôle of decreasing density and decreasing radiation temperature, the ionization structure remains roughly constant, and is only mildly varying. In the equatorial plane, however, the ionization fraction of Ciii is continuously increasing (decreasing density and increasing temperature working in the same direction), and the strong resonance lines of Ciii become crucial for the line driving in the thinner wind (cf. AB82, Table 5). Thus, increases with r for carbon.

Table 5. Characteristic results for stellar winds utilizing different approximations, for stellar model B30-30. GLOBAL: 2-D model with global force-multiplier parameters. No NRF: self-consistent force-multiplier, however purely radial radiation force. r in , mass-loss rates over the pole and in the equatorial plane in , velocities in .

To underline and summarize our findings, Fig. 17 shows the radial ionization structure of carbon and iron, over the pole and in the equatorial plane. Very close to the star, the effective wind density is largest and favours low ionization stages. In polar regions, however, the local effective temperature is at maximum and supports higher ionization stages.

Fig. 17. Model B30-30(KU): Ionization fractions of carbon (right) and iron (left) over the pole (top) and in the equatorial plane (bottom), as function of radius.

Over the pole and for larger r, both and decrease (with less variable than at the wind base), and the balance of their opposite effect determines the ionization equilibrium, keeping it roughly constant in the "high" state. Trace ions show rather constant ionization fractions, too.

In the equatorial plane, the density decreases for larger r, while grows. Both effects support higher ionization stages, and the degree of ionization is monotonically increasing with r.

Since close to the star the ionization is lower for equatorial regions, also the number of driving lines () is larger than over the poles, due to the larger number of lines being present. Nevertheless, as we have seen, the wind is slow and thin in the equatorial plane and fast and dense over the poles.

This morphology is a consequence of the dependence of the total illuminating flux on (which over-compensates the larger number of driving lines, !) and the lack of an abrupt transition to lower ionization stages, when going from pole to equator. Such an effect (as supposed to be present in the bi-stability scenario) might induce a significantly enhanced equatorial mass-loss, if the lower ionization stages begin to dominate from a certain co-latitude on, with increasing much more than in the models we have considered here.

Specific effects due to 2-D NLTE description: Comparison with models based on global force-multiplier parameters. So far, we have discussed some major results of our self-consistent 2-D NLTE description. In the following, we will investigate the extent to which these models differ from those utilizing global force-multiplier parameters (including, of course, the same assumptions, i.e., allowing for gravity darkening and non-radial radiative forces). This comparison will allow to estimate the importance of a self-consistent approach.

Such global parameters represent estimated averages of the corresponding parameters for our self-consistent model, and have been derived from the results of our previous section. For the model described above, Fig. 13 (top panel) shows that , and can be considered as appropriate average values.

Using these global parameters at first in a 1-D simulation at  K, the differences between the self-consistent and the global approach turned out to be of only minor nature, related to the same unique radiation temperature in both models which causes an almost identical ionization structure and effective line-number . The only differences we have found concern the density stratification and are marginal.

On the other side, the differences for a 2-D model are much more severe, thus immediately pointing to the dominant rôle of the radiation temperature (varying as function of ). The results for the corresponding 2-D simulation with global force-multiplier parameters (as given above) are listed in Tables 4 and 5, and Figs. 22, 23, 24 compare, among others, the density and velocity stratification of our self-consistent and the "global" model. The total mass-loss rate of the latter agrees at least qualitatively with the value found for the self-consistent model (). ¹⁰ Compared to the model with global mean ¹¹ force-multiplier parameters, however, our self-consistent model reveals a moderately enhanced concentration of wind material over the poles and a significant evaporation of the wind in the equatorial plane; the corresponding density contrast is amplified by roughly a factor of three.

This difference in density structure is related to the fact that any model using global line-force parameters cannot react on ionization gradients, which are mostly a function of co-latitude, as discussed above.

For the converged global model, we have also calculated the radial ionization structure and the effective number of optically thick driving lines as posteriori, which are then, of course, in no way self-consistent. A comparison with the corresponding values from our self-consistent approach reveals that the differences over the pole are almost negligible. In the equatorial plane, however, the self-consistent model shows stronger radial ionization gradients than present in the global model, and is much smaller, especially for iron. Again, these differences arise from the additive effect of increasing radiation temperature and decreasing density, where the latter is significantly lower in the self-consistent model (cf. Table 4).

Thus, we see that our self-consistent treatment is especially important in equatorial regions , since any disturbance of the hydrodynamical stratification is effectively amplified via the back-reaction of the induced disturbance of the ionization structure, whereas the counteracting effects of density and radiation field near the pole lead to a much smoother behaviour.

In conclusion, it is the specific radial dependence of density and mean radiation field (acting in opposite or parallel) which leads to a density contrast between polar and equatorial wind which is even stronger than for any global model: For larger distances from the star, higher ionization stages with fewer driving lines are favoured near equatorial regions, whereas over the poles the line acceleration remains efficient over a larger radius interval, as a consequence of a much more constant ionization structure.

Note, that these results should apply under fairly general conditions, with the only exception of iron, if a transition from Feiii to Feiv (which has an extremely rich spectrum, cf. Springmann & Puls 1998) would occur in the lower wind region where the mass-loss rate is adjusted. In the temperature region discussed here, however, Feiv is completely unimportant.

Thus, our new and quantitative approach of the wind dynamics does not weaken the prolate wind structure arising from non-radial line forces and gravity darkening or even induce the formation of an oblate structure, but rather suggests the contrary effect, i.e., increases the degree of "prolateness"! Of course, this conclusion concerns only those cases with an optically thin Lyman continuum everywhere.

7.2.2. Gravity darkening alone

As we have mentioned a number of times, two effects control the wind morphology and the actual degree of asphericity: The polar component of the line acceleration, , causes a polewards redistribution of wind matter. Gravity darkening increases the polar flux and amplifies the polar density contrast. Additionally, it controls the ionization structure in such a way that this contrast becomes even stronger than in models with constant radiation temperature (as function of ()).

It is interesting to isolate the pure effect of gravity darkening from the impact of the non-radial line-forces and to investigate the resulting wind- and especially ionization structure. The according model (with , stellar parameters as for B30-30(KU), consistent force-multipliers ) is shown in Fig. 18, and characteristical results are summarized in Table 5.

Fig. 18. Density and radial velocity field for the wind of model B30-30, with a purely radial radiative acceleration. The arrows indicate the polar velocity component (directed equatorwards) with a maximum value .

The density structure of this model again is globally prolate , however with an equatorwards directed polar velocity of maximum value (supersonic) near the equatorial plane. Thus, a wind compressed disk is formed. The radial outflow of the disk matter is markedly slower than the outflow of polar material (). ¹² The polar density contrast never exceeds a factor of roughly three, to be compared with the much higher values suggested for the original WCD model for the winds of main sequence stars, cf. Sect. 3.3. This rather small value is the consequence of the increased polar mass-loss due to gravity darkening, the reduced values of arising in parallel and the lower values of valid for our supergiant model, resulting in a faster acceleration of the lower wind material.

Additionally, this density contrast behaves non-monotonically as function of radius, since the disk evaporates relatively faster than the polar wind for larger radii.

The ionization structure close to the stellar surface (Fig. 19, right) and the contribution from the various ionization stages in the equatorial plane (Fig. 19, left) show an interesting behaviour: Compared to the original model B30-30 including non-radial line/continuum accelerations, we find a significantly enhanced contribution from the iron group in the equatorial disk. (The small disturbances in are due to the fact that this model has been calculated for only  s of physical time, and has not entirely converged yet.) In accordance with the higher density in the disk, Feii (Feiv) is slightly more (less) abundant. However, Feiii behaves almost identical and is the major ionization stage in in both models. This result illustrates very clearly that the ionization equilibrium is mainly determined by the non-local radiation field rather than by the local density (provided that the local densities of the individual models do not differ by several orders of magnitude).

Fig. 19. Model B30-30, with purely radial radiative acceleration. (left) and ionization fractions of iron (right) in the equatorial plane.

With respect to these findings, the influence of X-rays generated in the shock zones confining the equatorial disk (neglected in our approach) might be of importance. In particular for (very) high ionization stages, the ionization equilibrium will be changed considerably (cf. MacFarlane et al. 1993, Pauldrach et al. 1994), and we have to admit that our modeling is not entirely self-consistent yet concerning this point.

In any case, this model impressively demonstrates the importance of non-radial line-forces for the wind morphology. Only if these forces are included, an unambiguously prolate wind structure is formed, whereas their neglect gives rise to an equatorial disk even for models with a self-consistent line force parameterization.

7.2.3. Irradiation by Planck flux distributions

Employing a realistic illuminating energy distribution is essential for the wind dynamics, as demonstrated in the following by means of a wind model similar to model B30-30, however irradiated by Planck fluxes. The numerical results of this simulation are listed in Table 4.

A comparison of these data (B30-30, PL) with those obtained for an irradiation by Kurucz fluxes (B30-30, KU) yields the following differences: In the Planck case, the mass-loss rate is smaller by roughly a factor 30, whereas the velocities are significantly higher, especially at the equator ( vs. for the KU-model). Note also that for Planck-fluxes the equatorial wind is faster than the polar one, in contrast to our reference model (see below).

The major ionization stages are, for Planck irradiation and on the average, one stage higher than in the Kurucz case, in agreement with our findings from Sect. 5 (esp. Fig. 6). In consequence, the number of accelerating lines is smaller, implying a significantly reduced mass-loss rate. Fig. 20 displays for different elements over the pole and in the equatorial plane. Now, the major contribution is due to the CNO group and silicon, sulfur and argon (Z=14, 16, 18) rather than to iron, in agreement with the findings by Puls et al. (2000) that these light ions are the effective ones in such thin winds. Only close to the stellar surface, iron dominates because of the higher density.

Fig. 20. As Fig. 14, however for model B30-30 illuminated by Planck fluxes. Note the different scaling for .

Almost throughout the entire wind, Ciii is the dominant ion (again one stage higher than for Kurucz irradiation) and one of the major contributors to the line force. The behaviour of Civ, however, is more interesting. Note at first that at the wind base a strong polar gradient in the ionization fraction of this ion is present (Fig. 21), leading to a significantly smaller contribution from Civ in the equatorial plane than over the poles. Due to the increasing radiation temperature and decreasing density, the contribution of Civ increases with radius in equatorial regions (as was true for Ciii in the Kurucz case). For the thin wind studied here, however, carbon is essential for the acceleration, and the increasing contribution by Civ acts as an "after-burner" for the equatorial flow. This explains the higher terminal velocities found above.

Fig. 21. Model B30-30 for Planck irradiation. Ionization fractions for carbon at the pole and the equator.

7.2.4. Dependence of wind properties on rotation rate

So far, we have investigated the physical properties for our reference model and an extremely high rotation rate, , to estimate maximum rotational effects on and differences between the polar and the equatorial wind, with special emphasis on the NLTE aspect. In the following, we will study the wind morphology as function of rotation rate, again for model B30-30 (KU). Additionally, we will check basic predictions from simple scaling arguments.

Fig. 22 displays the density contrast for three different values of at , where the flow has become more or less purely radial and, therefore, has already converged to its maximum value. For (), the data indicate only a minor deviation from unity, , and mass-loss rates varying by the same factor, as shown in Fig. 24. For this low rotation rate, the latitudinal dependence of is marginal (; see Fig. 23).

Fig. 22. Ratio of local density to the corresponding equatorial value, , for model B30-30 (KU) and different rotational velocities. Additionally, the results for the 2-D model with global force-multiplier parameters (B30-30 GL) are displayed (long dashes).

Fig. 23. As above, however for the terminal velocity, .

Fig. 24. As above, however ratio of local mass-loss rates to the value resulting from 1-D self-consistent calculations, with .

The concentration of wind material toward the poles becomes more significant for (), with of the same order of magnitude as for models with global force-multiplier parameters and comparable rotation rates (cf. Petrenz 1999, Table 7.6). For extreme rotation rates (), the density contrast becomes very large (!). Note, that for all rotation rates the density contrast increases monotonically towards the pole, as expected for models with gravity darkening and negative values of throughout the entire wind.

For comparison, Fig. 22 shows also the density contrast for our model with global force-multiplier parameters, B30-30 (GL). As discussed already in Sect. 9, this quantity is significantly smaller than for the self-consistent simulation, related mostly to processes near the equator. The evident difference points again to the importance of a self-consistent treatment, at least in the case of extreme rotation and if equatorial regions are of interest.

For the same model sequence, Fig. 23 displays the run of terminal velocity, . For all rotation rates, it decreases from pole to equator. ¹³ The difference between its polar and equatorial value, however, is only small for all considered cases. (). This is also true for the GL-model, where the terminal velocity changes by from pole to equator.

The most interesting quantity to be analyzed, of course, is the mass-loss rate. Before going in further details, we like to stress one of the major results of our present investigation, following already from the data listed in Table 4:

For any rotation rate, the surface-integrated mass-loss rate does almost not differ from the corresponding value for a non-rotating wind , .

Due to gravity darkening, the equatorial mass-loss is diminished and the polar one enhanced, compared to the non-rotating case. Both effects, however, almost entirely compensate each other!

In Fig. 24, we have displayed the ratio at for model B30-30 (KU) and different . For , this ratio varies by a factor of 4 or larger, with an according value for the contrast between polar and equatorial mass-loss rate. Such variations will have considerable impact on the formation of optical (or IR) recombination lines, and have to be accounted for in any analysis of observed line profiles, in particular for the determination of local and global mass-loss rates. Note, e.g., that for such large latitudinal variations () of density or mass-loss rate, derived by means of a conventional 1-D -analysis might overestimate the actual value by %, as shown by PP96.

Before proceeding further, we will try to understand our finding by means of simple scaling arguments, which also show the validity and limitations of such relations.

If we define a "local" critical velocity

and consider at first models with constant force-multiplier parameters, the usual scaling relations for radiation driven winds can be applied by assuming that the mass-loss is created close to the stellar surface. In this case, the combined effect of centrifugal accelerations reducing the effective gravity on the one side and the dependence of flux on latitude due to gravity darkening on the other leads to a scaling of

This equation has been discussed by Owocki et al. 1997 and Puls et al. 1999, who have neglected the stellar oblateness which is additionally considered here. We have used the approximation (F the flux, the von Zeipel constant as function of angular velocity , the normal and the radial component of the effective gravitational acceleration including centrifugal terms), which should be valid unless the star is rotating very close to break-up. Note, that the final scaling-law normalized to the polar mass-loss rate (for given as indicated above) is independent of the von Zeipel constant.

To a lesser degree of precision, the terminal velocity should scale as

A more accurate expression (again neglecting the stellar oblateness) has been given by Puls et al. 1999, Eq. (4), which is, however, of no interest for the following discussion. Fig. 25 compares now these predictions with the results from our numerical simulation for model B30-30 (GL), which, because we have used global line-force parameters, just fulfills the above requirements. (In this figure, we have neglected the term . Thus, the displayed limiting values at are correct, whereas in between the actual curve should lie slightly higher.)

Fig. 25. Model B30-30 (GL). Left: Comparison of local mass-loss rates at the wind base, , normalized to the polar value, with analytical prediction Eq. (49), neglecting the term (see text). Right: The same for , compared with Eq. (50).

For the comparison with our numerical mass-loss rates as function of , we have used the values in the sonic region (with ), to avoid any disturbance by the latitudinal redistribution of matter. In particular, the polar mass-loss rate increases with r if gravity darkening is accounted for. For all models we have calculated, however, the radial variation of has never exceeded a factor of two (cf. Table 4, column 4 vs. 5).

The comparison clearly shows that our simulation is in close agreement with the analytical predictions, except for equatorial regions (), where the actual mass-loss rate is lower, with a maximum deviation by a factor of two. Since the considered model has been entirely converged in the inner wind regions, we are convinced that this difference is real. We attribute it to the increasing influence of the radial dependence of the centrifugal acceleration, which is maximum in equatorial regions and has been neglected in the analytical approach leading to Eq. (49). ¹⁴ In agreement with the lower mass-loss rate in equatorial regions, the terminal velocities of our simulations are higher there.

Since Eq. (49) is valid at least for the major part of the lower wind and constant , and , we can integrate over to obtain the total mass-loss rate (again assuming )

which is independent of radius due to its global conservation. is usual ratio of angular velocity to the critical one (for oblate stars)

related to the various critical velocities in the following way:

The quantity in Eq. (51) accounts mainly for the deformation of the stellar surface (e.g., CO)

and can be approximated (with a precision better than 1%) by a polynom in ,

Thus, for rapidly rotating stars (and constant force-multiplier parameters), the total mass-loss rate should be of order 60% of the polar one. By comparing with our model B30-30(GL) (), we find a slightly smaller value of 57%, since the actual equatorial mass-loss rate is lower than predicted, as discussed above.

We have to relate now with the appropriate 1-D value. The polar radius of our 2-D models corresponds to the 1-D stellar radius, and the local gravities are equal by definition. Thus, we have to account only for the different illuminating fluxes, by means of the well-known scaling relations for radiatively driven winds (, e.g., Puls et al. (1996)) and find

with . At this stage, the von Zeipel constant

with Boltzmann constant and surface integrated normal gravitational acceleration becomes important. Utilizing the polynomial fit provided by CO (their Eq. (32))

and the equality of gravity and radius for the 2-D and the 1-D model at the pole, we obtain

where the latter approximation holds for . For typical values of in the OB-star domain, the maximum value () predicted by the exact version of this equation is of order 1.4, and for it is 1.2.

Thus, finally accounting for the fact that the total mass-loss rate is somewhat lower than predicted, our finding can be understood at least for models with constant , and (assumed to be equal in the 1-D and 2-D case, of course). With respect to our self-consistent simulations, the above analysis is hampered by the assumption of constant force-multiplier parameters. Anyhow, for almost all calculated models the ratio of to follows the analytical prediction (51), with a value of roughly 0.5 for (cf. Table 4, column 4 vs. 7). This on a first glance astonishing result is primarily related to the fact that, close to the surface, the major impact of our self-consistent approach concerns the equatorial regions, whereas the differences in polar regions are minor (cf. Sect. 9 and Fig. 27). Since the strongly reduced mass-loss in equatorial regions has an only weak influence on the total one, the validity of Eq. (51) and consequently Eq. (60) is explained also for the self-consistent models. In the outer regions, of course, the differences become larger, again mostly near the equator (Fig. 24). However, these differences do not play any rôle for the total mass-loss: Once created at the wind base, it remains constant throughout the wind due to its global conservation.

As a consequence of the above arguments, the relation should be valid as long as there is no dramatic difference between polar and intermediate latitude ionization structure. For models with gravity darkening, this seems to be almost impossible (provided that the continuum is optically thin), and we actually have found the above "identity" not only for models at 20,000 K, but also for models of different spectral type, e.g., for O-stars (cf. Petrenz 1999). ¹⁵

One might argue that our finding is self-evident, since the decisive factor controlling the total mass-loss is the total luminosity, being conserved under rotation. However, the additional dependence of on effective mass modified by centrifugal acceleration is non-negligible, and is compensated for only if gravity darkening is taken into account. Otherwise, the scaling for as function of behaves differently (actually, directed in the opposite sense, with the bracket in Eq. (49) modified by an exponent of ), and it is easy to show that in this case becomes considerably larger than for corresponding non-rotating winds.

We can continue now with our comparison of latitudinal wind structure as function of rotation rate. One of the most striking features is the significant enhanced density contrast when increasing from to , cf. Fig. 22.

This different morphology of intermediate and rapidly rotating winds is, of course, the consequence of the drastically diminished temperature contrast between equator and pole for the model with . In particular, the equatorial temperature ( K) is much closer to the nominal value than for the rapid rotator ( K).

Thus, Feii with its many lines is much less important at the equator (see Fig. 26 for the ionization fractions of iron), and the latitudinal variations of at the wind base have almost vanished, as displayed in Fig. 27. As well, also the radial variation of is less significant than for : becomes a mildly and monotonically decreasing function of radius at all latitudes, also for carbon in the equatorial plane.

Fig. 26. Model B30-30 for . Ionization fractions of iron at the wind base (left); force multiplier (right).

Fig. 27. Model B30-30 (KU): at the wind-base Left: ; right: .

Since is now a well-behaved function and also the illuminating flux () varies to a much lesser extent, the wind is by far not as asymmetric as for .

Also the force multiplier M displays a monotonic behaviour, and has a much lower maximum ( for , Fig. 26) than for the rapid rotator (, Fig. 13). This brings us finally back to the question concerning the non-monotonic polar variation of M for the latter model, as shown in Fig. 13: Starting at the poles, M initially increases as function of due to the increasing evaporation of the wind (). Near the equatorial plane, however, both the stellar oblateness and the slower radial expansion imply a larger value of , and decreases again, although grows in accordance with the increasing contribution from Feii. Since for lower rotation rates the polar variation of both the flux and the effective line number is much lower, the "density effect" remains the stronger one, and does not change its behaviour.

7.2.5. Dependence of wind properties on stellar luminosity

In order to estimate possible maximum effects of rotation on the application of radiation driven wind theory to stellar evolution calculations and the wind-momentum luminosity relation, we will finally discuss the dependence of wind properties on luminosity, by means of rapidly rotating models ().

To this end, we have calculated a sequence of stellar winds from models at  K and different luminosities (cf. Table 1). The most important numerical results have been listed in Table 4.

Fig. 28 displays the ratio of polar to equatorial mass-loss as function of luminosity. In agreement with our previous results, we find also here a clearly prolate wind structure in all considered cases, with a maximum ratio for main sequence models with lowest mass-loss rates. This contrast decreases towards highest luminosities, since the latitudinal ionization gradients of the CNO group are most effective for the thinner winds (see Sect. 7.2.3, Fig. 21). For larger wind densities, the mass-loss from more or less all latitudes is controlled by iron, displaying a less significant ionization shift and thus favourizing a reduced density contrast. (To clarify this trend unambiguously, we have included also models for B-supergiant winds which actually should become optically thick in the Lyman continuum (, cf. Table 3). For these models, our simplified non-LTE description is certainly no longer correct, and the bi-stability effect might begin to operate.)

Fig. 28. Ratio of polar to equatorial mass-loss rate as function of stellar luminosity, for rapidly rotating B-star models with  K (, ratio evaluated at 4 and 4 , respectively). The non-monotonic behaviour at is due to the difference in stellar mass for the two models B15-10 and B15-15.

If we compare the ratio of surface-integrated 2-D mass-loss rate to the corresponding value for the non-rotating star, we find, in agreement with the results from our reference model and the reasoning given in the previous section, for almost all models a close agreement, except for the one with highest luminosity. The borderline (at  K) appears to be situated at , from where on the discrepancy between and becomes larger, owing to the strongly enhanced Thomson acceleration over the poles. (The term is no longer negligible for such models, becoming a strongly increasing function of latitude because of the polewards increase of , cf. also Sect. 3.3)

In consequence, the total mass-loss rate might become larger than at highest luminosities, by roughly a factor of two. This result, however, has to be taken with caution, since we have neglected effects due to the optically thick Lyman continuum.

Consequences for the wind-momentum luminosity relation.

The mutual compensation of polar and equatorial mass-loss in comparison with the 1-D value even for rapid rotators implies some interesting consequences for the application of the wind-momentum luminosity relation, as displayed in Fig. 29.

Fig. 29. Wind-momentum luminosity relation for the same models as in Fig 28. Compared are the modified wind-momenta resulting from our 2-D simulations with mass-loss rate , respectively, and from corresponding (self-consistent) 1-D models (). See text.

This figure shows the modified wind-momentum as defined by Kudritzki et al. (1995), plotted against stellar luminosity, where the values for and have been taken for the following cases (for simplicity, we have always used ): For the polar and equatorial wind, for the global wind with total mass-loss rate and some average value (remember that in most cases varies only marginally), and for the according self-consistent 1-D model.

In all four cases, the mass-loss rate has been evaluated at , where the polar density contrast has already saturated to its maximum value. By means of this diagram, we can estimate the maximum contamination of empirically derived WLRs due to rotation: If the wind is observed under arbitrary inclination, the polar and equatorial value of may differ by up to 1.5 dex, thus implying a considerable scatter in the observed WLR. This difference becomes smaller for higher stellar luminosities, because of the smaller mass-loss/density contrast for denser winds, as discussed above.

Note, however, that this maximum deviation is certainly overestimated if the mass-loss rate is derived from . Firstly, most of this emission originates from the inner wind (), where the density contrast is smaller than in the outer wind, even for extreme rotation rates. A typical number for this reduced density contrast in the lower wind is a factor of four (cf. Petrenz 1999, Table 8.1), so that the above maximum scatter is reduced to a value of 0.9 dex. Secondly, the observed emission consists of a superposition of polar and equatorial emission. Even if one observes the star directly pole-on, there will be always a lower emission from equatorial regions (and vice versa for equator-on observations), which partly compensates for the deviation from the emission predicted by simple 1-D wind models. Only detailed line synthesis calculations (e.g., as performed by PP96 for the case of wind-compressed zone models) will finally allow to check the actual scatter arising in the WLR, if the winds were observed either pole-on or equator-on.

A very promising result, however, is the very good agreement of derived for the global wind ( and the according self-consistent 1-D model. Note especially that even for the model of highest luminosity (with ) both quantities agree, due to the compensation by a lower terminal velocity. In so far, it is evident that the most important scaling relation relying on the principal mechanism, radiative driving, is not affected if the global wind is considered: The primary effect of rotation is "only" a redistribution of matter . Thus, the assumption of non-rotating, spherical winds is a reasonable simplification if a description of global quantities is aimed at, e.g., in the context of (1-D) stellar evolutionary models, and the models are situated not too close to the Eddington or limit.

© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000
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