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

3. The radiative acceleration

In this section, we introduce our notation and give a brief summary of basic expressions for the radiative acceleration. For detailed derivations, we refer the reader to the references cited in the following.

The general expression for radiative acceleration reads

with the specific intensity and the mass-absorption coefficient for line (continuum) radiation. denotes the direction vector which points from the considered location into the direction of radiation and, thus, is anti-parallel to the direction vector introduced in Fig. 1, i.e. .

3.1. The continuum acceleration

As discussed above, we assume an optically thin continuum in the wind and neglect any diffuse radiation, i.e., the irradiation origins at the stellar surface and the corresponding intensity experiences no attenuation between the star and . Thus, the intensity is determined solely by the physical conditions at the location where the ray pointing from to the stellar surface intercepts the latter for the first time.

In winds from early-type hot stars, the continuum radiative acceleration essentially arises from Thomson acceleration and is a vector quantity given by (cf. C095, Eq. (43))

with the stellar luminosity, the electron scattering opacity, the Thomson cross-section and the electron number density. denotes the solid angle subtended by the stellar disk, and is the local polar angle confining (cf. Fig. 1). The electron density follows from

with the helium abundance, the proton rest mass and the number of free electrons per He atom (O-stars: , B-stars: , A-stars: ).

Note that the Thomson acceleration is a vector quantity and may have a non-radial component in polar direction (if the stellar surface deviates from spherical symmetry or/and varies with co-latitude, see below), whereas the azimuthal component vanishes due to symmetry about the rotational axis. In this case, both the polar and the radial component of the Thomson acceleration become a function of co-latitude . Especially the latter implies some consequences for rapidly rotating supergiants with gravity darkening, as discussed in Sect. 3.3).

denotes the effective temperature at the stellar surface defined via the emerging frequency-integrated flux (see Sect. 3.3),

3.2. The line acceleration

By means of the frequency-integrated line opacity

with the gf-value, the occupation numbers and statistical weights of the lower (l) and upper (u) levels, e the elementary charge and the electron rest mass, we define the line-strength

with the line transition frequency and the Doppler-width. denotes the thermal velocity.

As the continuum acceleration, the line acceleration is a vector quantity (cf. C095, Eq. (44)), defined in the Sobolev approximation as the sum over an ensemble of lines:

with the optical depth in Sobolev approximation ¹,

denotes the generalized depth parameter depending on the absolute value of the directional velocity derivative , which, for spherical symmetry, collapses to the depth parameter introduced by CAK, .

According to the findings by CAK, Abbott (1982) and Puls et al. (2000), the number distribution of spectral lines with frequency and line-strength can be approximated by a power-law line-strength distribution function

with and a frequency distribution independent of line strength. The quantity normally varies with location .

The additional exponential factor with maximum line-strength has been introduced by Owocki et al. (1988) in order to prevent the number of (strong) lines from becoming smaller than unity. Note that this parameterization does not consider line-overlaps and multi-line effects, i.e., we restrict ourselves to the single-line scattering case where the photons interact with the ions in such a way as if each line would be well separated from its neighbour, independently of line density.

Employing this line distribution function makes the transition possible from the sum in Eq. (15) to an integral. After some algebra, we obtain with Eq. (16) for (cf. C095) and assuming

In this expression, is still considered as a global parameter constant throughout the wind , i.e., independent of the actual local physical conditions. We will drop this assumption below (see Sect. 6.3).

is the frequency-integrated stellar intensity, and we have neglected the dependence of on direction . In principle, the polar surface temperature gradient owing to gravity darkening may have an actual impact, if the intensities at the poles, respectively at the equator, have their maxima in different frequency ranges with a different number of contributing lines. However, we accept this approximation, in particular with respect to the line force parameterization proposed in Sect. 6.

To account for the dependence of the ionization structure on particle density and dilution of the stellar radiation field, Abbott (1982) introduced an additional factor which represents the largest part of variation of ,

with the spherical dilution factor (the generalization for arbitrary stellar surfaces is given by Eq. (27)),

Introducing now the force-multiplier parameter (cf. CAK, Puls et al. 2000)

the line force finally reads

Like , also (essentially ) and are considered as global parameters so far. We will drop this approximation in Sect. 6.3.

Finally, we need the expression for the directional derivative . Due to symmetry about the polar axis, this quantity simplifies to the expression given by C095 (Eqs. (41/42)). ²

For a spherically-symmetric wind from a uniformly bright stellar surface, the line acceleration given by Eq. (22) reads (cf. CAK, PPK, Friend & Abbott 1986)

denotes the force-multiplier , given by the ratio of radial line to Thomson acceleration

This quantity is usually parameterized by means of the force-multiplier parameters , , (for an alternative parameterization, see Gayley 1995):

With the knowledge of , , as a function of effective temperature (and metallicity), the stellar mass-loss rate and the terminal velocity can be easily calculated for every evolutionary state of the star, e.g., by means of the analytical formulae provided by Kudritzki et al. (1989).

3.3. Impact of non-radial line forces and gravity darkening on the wind dynamics

The influence of non-radial line force components and gravity darkening on the radiative line and the wind dynamics has been investigated in detail by C095, Owocki et al. (1996, 1997) and, with some refinements, by Petrenz (1999). In all these investigations, the line force has been parameterized by global parameters , , adopted from 1-D non-LTE calculations. In the following, we only summarize the main results.

As a consequence of asymmetries in the local directional derivative of the velocity field, the line acceleration has non-radial components along the polar and along the azimuthal direction, which imply following effects: Firstly, the polar component scales with the polar gradient of the radial velocity field . The latter is negative close to the star, where is of the same order of magnitude as the inertial (centrifugal and coriolis) accelerations () and becomes decisive for the wind dynamics. As a result of the balance of these accelerations, the disk formation is inhibited and the wind material is redistributed towards the polar regions with maximum (absolute) polar velocities , whereas the radial component stabilizes the flow against gravity (with ).

The density contrast between the equatorial and the polar wind ranges in between 2... 8, and typical terminal velocities are and for B-(O-)star winds.

The azimuthal component scales with the radial gradient of the azimuthal velocity field . If the latter is smaller than (i.e., for any rotation law with increasing with r smaller than for rigid body rotation), becomes negative and spins down the wind . In the equatorial plane, where the maximum effects occur, is diminished by up to , in comparison with an angular momentum conservation law (for an analytical discussion of the spin-down effect, see Gayley & Owocki 2000).

In comparison to a non-rotating star with an uniformly bright surface, a star with gravity darkening is expected to show a higher (lower) mass-loss over the poles (at the equator) owing to the enhanced (diminished) radiative flux. Typical quantitative effects on the surface properties of rotating stars owing to gravity darkening become clear from the data listed in Table 1 /2.

Table 1. Model grid: Stellar and wind parameters. : effective temperature for = 0; : Eddington-; : normal gravity at the pole; : escape velocity for spherical stellar surface; : critical velocity for spherical stellar surface; : critical velocity for a-spherical stellar surface (given by PP96, Eq. (20)). Velocities in , and solar helium abundance assumed for all models. Bold: Reference model B30-30.

Isolating the pure effect of gravity darkening (still accounting for the inertial acceleration terms, however assuming only a radial line force) yields, again for constant force-multiplier parameters, a slightly oblate wind structure with a moderately compressed disk in the equatorial plane. The density contrast 3 is much smaller than in the pure WCD model ( ), and terminal velocities of order , and are found for typical B-star parameters.

If one additionally accounts for the non-radial line force components, finally a prolate wind morphology results, with density contrasts of order 0.2 ... 0.5. For supergiants with considerable Eddington-luminosity, gravity darkening has an additional impact on the wind dynamics via Thomson acceleration , which approximately scales with the fourth power of the mean radiation field temperature and, consequently, is more (less) effective over the poles (in the equatorial plane) than in the wind of a uniformly bright star. This behaviour induces an even stronger concentration of wind material over the poles, with , where for a wind with purely radial 1-D Thomson acceleration, this ratio would be only .

In view of the studies performed so far and their specific shortcomings (especially the use of globally defined force-multiplier parameters neglecting the 2-D situation, in particular if gravity darkening is accounted for), we will concentrate on the B-star domain ( K) in our following investigations, and comment on hotter winds only when necessary. Briefly, the reasons for this strategy are:

• In the considered temperature range, the reaction of the ionization equilibrium on small variations of (as function of , if gravity darkening is accounted for) is at maximum. Thus, also the effects resulting from a consistent 2-D NLTE treatment should be at maximum in this domain. Note that for O-star winds the variation of ionization structure with temperature is much weaker, and the assumption of constant force-multiplier parameters may be much better justified there.

• The generalization of our results to cases of higher mass-loss will allow for a future discussion concerning the formation of B[e] star disks, one of the most prominent manifestations of the inter-relation between wind, rotation and (2-D) radiative driving, at least if one follows the presently favoured scenario (cf. Lamers & Pauldrach 1991, Lamers et al. 1999, however also Owocki et al. 1997for the counter-acting rôle of gravity darkening).

• Finally, all prior investigations resulting in detailed (numerical) models have concentrated on the B-star regime, as a consequence of the work by BC, who suggested the wind-compression scenario as one possible explanation for the Be-star phenomenon. In order to compare our findings with those more simpler, however well-documented approaches, and to investigate the importance of a consistent description, an inspection of this parameter range is inevitable anyway.

© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000
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