## 7. Self-consistent models
In this section, we present our numerical results for self-consistent
wind models accounting for both the ## 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
. The effective temperatures
(Eq. (33)) and the line force components
(Eq. (22)) are determined (the
latter ones for 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.
For In consequence, the slope of
(i.e., ) approaches zero for
, and shows a significant
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 modelsAs 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
## 7.2.1. 2-D reference model for a B-supergiant windIn 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
The wind structure is clearly 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
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
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
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 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
To understand the first point, we have plotted the corresponding
with the number density of
element 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
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 To illuminate the behaviour of carbon in the equatorial plane
( increases with
To underline and summarize our findings, Fig. 17 shows the
Over the pole and for larger In the equatorial plane, the density decreases for larger 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
Such global parameters represent estimated Using these global parameters at first in a On the other side, the differences for a This difference in density structure is related to the fact that
any model using global line-force parameters 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 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 aloneAs 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),
The 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
With respect to these findings, the influence of 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 ## 7.2.3. Irradiation by Planck flux distributionsEmploying 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 (
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.
## 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 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).
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. 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:
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 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
( 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.)
For the comparison with our numerical mass-loss rates as function
of , we have used the values in the
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). 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). 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.
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 ## 7.2.5. Dependence of wind properties on stellar luminosity
In order to estimate possible 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.)
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.
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.
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 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 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
© European Southern Observatory (ESO) 2000 Online publication: June 20, 2000 |