## 6. The line force parameterization
Based on the properties of the force-multiplier ## 6.1. Minor simplificationsTo this end and in order to reduce the computational effort, we adopt the simplifications described in the following. In our approach, we neglect any diffuse radiation, i.e., we account
only for the However, a parameter study performed by Petrenz (1994) has clarified that such CP resonance zones may arise in rotating winds only for very shallow radial expansion laws combined with extreme rotation rates (). Even then, the CP zones subtend only a minor part of . Therefore, the local description of the radiative transfer is a fairly good approximation and we assume the incident intensity () to depend only on the physical conditions on the stellar surface. For our test calculations, the radiation field at location on the surface follows either a black-body law (with , see Sect. 3.1) or is taken from a model flux distribution (corresponding to the local values of and ). Furthermore, we also neglect the rotational Doppler shift of the photospheric spectrum when calculating as well as the directional dependence of resulting from limb darkening, since both effects are of only secondary importance.
In our time-dependent 2-D simulations, the local force-multiplier
parameters ,
,
are repeatedly updated in the course of the hydrodynamic evolution to
guarantee their convergence consistently with the flow. At every
co-latitude of the hydrodynamic
grid, this procedure requires the force-multipliers ## 6.2. Quantities of the radiation fieldAccording to Eq. (24), the force-multiplier
depends on several quantities
related to the radiation field: Firstly, on the ionizing flux
and secondly on the line strengths
.
is a For these averaged effective temperatures (which can be regarded to
represent different atmospheric models), we take the according flux
distributions from a grid of plane-parallel Kurucz fluxes (as function
of ), which, in course, finally
provide the ## 6.2.1. Radiation temperatures for black-body irradiation
Before we define the mean effective temperatures mentioned before, we
investigate how strongly the radiation temperature varies throughout
the wind if the stellar surface emits as a black-body. Neglecting limb
darkening, we can simultaneously account for the dependence of
on
provides the radiation temperature averaged over : can be computed numerically via Eq. (30), with , . For the model of a rapidly rotating B2 dwarf (corresponding to the
"S-350" model discussed by OCB), Fig. 8 displays the frequency
dependence of for two
representative radii
varies only weakly with frequency, even in the short wavelength range where the sum of the different local Wien law emissions causes a slight increase of . For large , follows the Rayleigh-Jeans law, and does not longer vary with wavelength. ## 6.2.2. Mean effective temperaturesIn the following, we will account for the simultaneous dependence of on physical properties of the stellar surface (i.e. at ) and on frequency in an approximate way. Our method uses quantities depending either on the angle-weighted mean intensity (for calculating ionization structure, occupation numbers) or on the direction-weighted radiative flux (for calculating force-multipliers). To describe the frequency dependence of and , respectively, we define two different effective temperatures being independent of frequency, in the spirit of our introductory remarks above.
If the radiative flux
is taken from a model atmosphere flux distribution, the latter should
correspond to a local "continuum effective temperature"
which ensures that the total flux
given by the flux distribution, ,
equals the stellar flux given by integration over
. In other words, the definition of
should follow from (Note the On the other side, frequency integration of the local photospheric flux distribution on the stellar surface, , yields the local radiative flux . Comparing both sides, we obtain the desired effective temperature which can be easily calculated in dependence on the surface deformation. For constant surface temperature, , of course.
With , we can at first define an average temperature , with from Eq. (27). Requiring the conservation of radiative flux yields the local effective temperature : If we insert now the expression for (Eq. (34)) into Eq. (35) and change the order of integration over and , we find The integration over on the RHS of Eq. (36) yields . Thus, the effective temperature reads In contrast to the direction-weighted temperature
, we have to integrate over
rather than over
, due do the different angular
weighting process inherent to mean intensity and flux, respectively.
Fig. 9 displays both effective temperatures,
and
, and their difference for the above
B2 dwarf model.
Primarily, both quantities are a function of co-latitude
. For
, they are identical since they
depend on purely local conditions, with
. For larger radii
Compared to their variation with co-latitude, the difference
between and
is surprisingly small. Its order of
magnitude can be easily checked, if one assumes a spherical stellar
surface and a linear dependence on which is justified by Fig. 9 (top and middle panel). E.g., over the pole at
we obtain
= 42450 K and
= 42560 K. Thus, the difference
between both temperatures has the same order of magnitude as the exact
numerical result, and slightly
exceeds . The larger The fact that both effective temperatures vary much stronger with
co-latitude than with distance
The flux distribution is a function not only of effective temperature , but also of gravity . Since the Kurucz fluxes depend only weakly on the variations of we are interested in, we take, for all in the wind, the surface-averaged value (PP96, Eq. (30)). ## 6.2.3. The dilution factorOwing to stellar oblateness, the dilution factor
For all ,
is greater or equal
, and Over the poles, the surface curvature is weaker than in the
spherical case (see Collins 1963, Fig. 1). Therefore, the solid angle
always exceeds the value for a
spherical star with (except for
,
). For larger At the equator, the surface curvature is stronger than in the
spherical case, and the significant discrepancy between Far away from the star, the solid angle subtended by the stellar disk observed equator-on is smaller than viewed pole-on. In the latter case, one looks upon a rotationally-symmetric object with the same maximum extent for all azimuthal angles (cf. Fig. 1). As Fig. 10 clarifies, the difference between ## 6.3. Depth-dependent force-multiplier parameterizationFigs. 3 and 4 showed a clearly non-linear dependence of the force-multiplier on and : On the one hand, the approximation of the line-strength distribution function by a power law with constant exponent is not entirely sufficient. As shown by Puls et al. (2000), varies systematically as a function of line strength , thus implying a curvature of as a function of . On the other hand, the exponent depends also on the ionization structure, and, as a consequence, becomes a non-linear function of . Because of this non-linear behaviour, the classical fit with global
, ,
may lead to severe discrepancies
between the fit values and the
actual force-multipliers For this reason, Ku98 proposed an improved parameterization of the force-multiplier, using the most simple fit formula of higher order for which allows for a linear dependence of and on and , with and corresponding to the CAK parameter . The fit is performed via the least-square solution of the over-determined system of linear equations for the unknown , , , , , : with , the number of input points of and , respectively. As shown by Ku98, this parameterization provides fits of the force-multipliers with errors smaller than dex in , where those fits cover completely the ranges in and relevant for the corresponding spectral type of the star. Since we will determine the parameters , , in our hydrodynamic simulations by means of Eqs. (39-41), we have to check how well these values agree with the actual ones, i.e., those obtained by a piecewise differentiation of with respect to and , respectively. For some typical cases, Fig. 11 demonstrates that the new parameterization reproduces and in the correct order of magnitude and also succeeds in a qualitative reproduction of as a function of .
At this point of reasoning, the obvious question arises why we
employ this particular method to determine
, ,
instead of deriving them directly by
piecewise differentiation of
If, on the other hand, we formulate by means of parameters fitted as described above, we avoid these problems, of course at the expense of variations in and not reproduced very precisely (cf. Fig. 11). This lack of precision results from our approximation of using only linear functions in and for and , respectively, and from the notion that the fit procedure has to cover a relatively broad range in and . As pointed out already above, however (see Ku98), the multiplication of the three factors , and finally yields a good reproduction of the force-multiplier itself: On the average, local discrepancies in and are compensated by the fit value of . Fig. 13 clearly demonstrates this behaviour. The actual procedure to determine , and in our hydrodynamical simulations is described in the next section. By means of these quantities then, the line acceleration is calculated. Inaccurately determined values of
may lead to a biased
direction-weighting of the contributions
, if
varies strongly as function of
. In our 2-D approach, however, we
will restrict ourselves to this conceptionally simple representation
of the force-multiplier parameters, especially in view of problem
( As an obvious advantage, our parameterization is ## 6.4. Local line force parameterization in 2-D winds
In the following, we generalize the parameterization introduced in
Sect. 6.3 to 2-D winds from rotating stars, in particular accounting
for the with direction-dependent CAK parameter Since we neglect the dependence of the ratio
/
on direction, we can replace this expression by the This approximation - compared to the "exact" expression (Eq. (43)) - can by justified as follows: With respect to employing an average value for
, we avoid otherwise severe problems
with normalization, since the flux distribution
For , the formal dependence on
direction is actually redundant, since
parameterizes the force-multiplier
as a function of ionization structure. The latter depends only
on describes the reaction of the
force-multiplier on changes in . In
the expression for (Eq. (45)), we
integrate over different values of
as function of direction. The force-multiplier evaluated with a value of that
has been determined from with as an average value of the directional dependent quantities . The actual (average) values of , , are now calculated in the following way: At first, we derive (with The radiation field for the flux weighting factor
is given by the flux distribution
at (direction averaged) effective temperature
(Sect. 6.2.2), thus ensuring flux
conservation in . The Therefore, the force-multipliers
determined in this way may be regarded as To save computational time, the
are interpolated from a As shown in Fig. 9, and
vary much weaker with radius
For all then, we determine the
force multipliers at the radial
subgrid described above, and By means of these During the temporal evolution of our simulations, the above procedure is repeated typically each 100 time steps , for each co-latitude . With the local values of as
central values of the squares in the
--subspace
defined above, we force the fit range to be concentrated about the
As extensive numerical tests have shown, satisfying fits of the force multipliers require a certain minimum width of the intervals and . In particular, if the interval is too narrow, the errors in , etc. may become much larger than the resulting parameters, and the resulting , and may obtain unphysical values, preventing a stable convergence of the flow. Empirically, we found a width of 1.5... 2.0 (1.0... 1.5) for the intervals in and , respectively. ## 6.5. Synopsis: calculation of a consistent 2-D line force by means of the force-multiplier conceptAt every co-latitude , our procedure provides six constant parameters , , , , , by means of local force-multipliers and the values on neighbouring coordinates in the - subspace. With these parameters, which are repeatedly updated in the course of the hydrodynamic evolution, directional independent values of , and are determined. These latter values parameterize the line acceleration and depend on the local, time-dependent quantities and . Since the force-multipliers are computed for flux distributions evaluated at local effective temperatures and averaged over , the resulting force-multiplier parameters can be regarded as representative values averaged over . The primarily © European Southern Observatory (ESO) 2000 Online publication: June 20, 2000 |