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

1. Introduction

During the last three decades, the theory of line-driving has been successfully applied to the winds from early-type hot stars, which exhibit mass-loss rates and terminal velocities . Based on the pioneering work by Lucy & Solomon (1970), Castor (1974) and Castor et al. (1975, hereafter CAK), Pauldrach et al. (1986, hereafter PPK) and Friend & Abbott (1986) established the so-called standard model as the quantitative basis for numerous successful applications of this theory (for a review, see Kudritzki 1997).

The most promising application, which has been discovered originally on a completely empirical basis, is the wind-momentum luminosity relation (WLR) between the modified wind-momentum rate and the stellar luminosity (Kudritzki et al. 1995). This relation (with an additional dependence on stellar metallicity) has also been understood from a theoretical point of view (Puls et al. 1996, 1998), and has been successfully used in several investigations (McCarthy et al. 1995, 1997, Kudritzki et al. 1999, Venn et al. 1999). In order to use this relation, a reliable value for the mass-loss derived from the observation must be available. In general, a successful determination of requires a detailed knowledge of the conditions in the stellar wind (see, e.g., Petrenz & Puls 1996, hereafter PP96, and the references given therein). All investigations carried out so far may suffer from the assumption of the standard model for the stellar wind, which comprises the most simple flow geometry possible, namely a radially expanding spherically-symmetric wind. Thus, the influence of stellar rotation, which breaks the radial symmetry, is neglected in the hydrodynamic treatment. However, a significant fraction of early-type stars shows projected rotational velocities up to 50... 80% of the critical value (cf. Conti & Ebbetts 1977). Additionally, their spectra often show time-dependent features clearly correlated with the estimated rotational period (for most recent advances, see Wolf et al. 1999).

Of course, also stellar evolution is affected by rotational effects. Herrero et al. (1992) detected a systematic "mass discrepancy" in their quantitative spectroscopic study of a large sample of galactic O-stars. They found that for evolved O-stars approaching the Eddington-limit the stellar masses derived from evolutionary tracks and stellar luminosities are significantly larger than the masses obtained from spectroscopic gravities. Additionally, all rapid rotators of their sample showed a severe surface He-enrichment, the so-called "He discrepancy". Although an analysis by means of unified model atmospheres accounting for the presence of winds (Herrero et al. 2000) has mitigated the mass discrepancy (however not ruled it out), the He discrepancy could be confirmed. Recent evolutionary calculations by Langer and Heger (1998), Maeder (1999), Meynet (1998) have indicated that stellar evolution including rotation and rotationally induced interior mixing could enhance the luminosity significantly for given masses. In addition, enhanced mass-loss along the evolutionary track would further reduce the mass and bring the evolutionary masses into agreement with the spectroscopic (and wind) masses (Langer et al. 1994).

From all of these perspectives, the neglect of rotation in the description of hot star atmospheres (photosphere + wind!) seems questionable. In recent years, this problem has been addressed by several theoretical studies. PPK and Friend & Abbott (1986) estimated the influence of stellar rotation on the wind dynamics in a global sense by solving the 1-D equations of motions in the equatorial plane only, considering the additional centrifugal acceleration term there. In result, they obtained an upper limit for the actual mass-loss rate, which should range in between the polar () and the equatorial value derived by this method. In general, they expected the effects of rotation to become decisive for .

The assumption of spherical symmetry has been dropped the first time by Bjorkman & Cassinelli (1993, hereafter BC). Employing the supersonic approximation, they solved for the particle trajectories in the polar plane, thus adopting azimuthal symmetry about the rotational axis. Additionally to the inertial accelerations, they considered a purely radial line force in the equations of motion. In this way, they provided simple expressions for density structure and the radial, polar and azimuthal components of the velocity field . In their model, they calculated the line force within the force-multiplier concept, where the corresponding force-multiplier parameters , , (cf. CAK, Abbott 1982) were assumed to be constant throughout the wind , i.e., independent both of r and . In the 1-D case, these force-multiplier parameters allow a very simple and accurate parameterization of the line force which depends on the contribution of several hundred thousand lines of metal ions (for a thorough physical discussion based on a detailed study of the underlying line statistics, see Puls et al. 2000).

The BC model predicts a polar deflection of the wind material towards the equatorial plane , owing to the combined effect of centrifugal, gravitational and radiative acceleration. Thus, the spherical symmetry is broken, and the density contrast between equatorial and polar wind increases with distance r from the star. For extreme rotation rates ( for O(B)-stars), one expects the formation of a wind-compressed disk (WCD) in the equatorial plane, caused by the collision of wind material deflected from both hemispheres with polar velocities , with the (isothermal) speed of sound. The matter is decelerated in a shock front about the equatorial plane and feeds a slowly outward propagating stationary disk. Owocki et al. (1994, "OCB") confirmed the principal validity of this model by detailed radiation hydrodynamic simulations, and recovered some interesting additional effects (e.g., an accretion of the innermost disk material onto the stellar surface). For an independent confirmation of their results (with some refinements), see Petrenz (1999).

So far, the analytical BC model has been applied, e.g., for simulating the formation of wind-compressed zones (WCZ) in the winds from Wolf-Rayet stars (Cassinelli et al. 1995), as a possible explanation for the shape of the Homonculus nebula around Carinae (MacLow et al. 1996), as underlying hydrodynamic wind model for line synthesis (PP96) and for the investigation of wind compression effects in different areas of the HRD (Ignace et al. 1996).

Owocki et al. (1997, 1998) extended the BC model and studied the effects of non-radial line force components (acting in the polar and azimuthal direction, respectively), where the azimuthal components had been discussed the first time by Grinin (1978), and gravity darkening (i.e., the polar gradient of the stellar surface temperature from hot poles to cooler equatorial regions) on the wind dynamics.

The non-radial components of the line force lead to an inhibition of disk formation and a redistribution of the wind material towards the poles . Additionally, the gravity darkening implies an enhanced (reduced) photon flux over the poles (in the equatorial plane) generating a prolate density structure of the wind with a denser and faster outflow over the rotational poles and a slower and thinner wind about the equatorial plane.

These results are in entire contradiction to the predictions by the simple BC model and, meanwhile, have been independently confirmed by Petrenz (1999); for a recent review, see Puls et al. (1999). Although severely challenging the validity of the wind compression mechanism, the results by Owocki et al. and Petrenz do not exclude the possibility that this mechanism might be decisive for outflows driven by other physical processes than line-driving.

One of the major problems of the described investigations is their assumption of force-multiplier parameters , , being constant throughout the wind. Moreover, these parameters have been taken from 1-D calculations, where, in the actual 2-D situation, they should be affected by at least two processes. Firstly, both density and ionizing radiation field are a function of co-latitude , and secondly, the assumption of constant force-multiplier parameters (with respect to radius) is rather questionable even for a uniformly bright stellar surface (cf. Kudritzki et al. 1998).

Thus, global 1-D parameters do not correctly account for the influence of local physical conditions (density, velocity field) and the non-local stellar radiation field on ionization stratification and occupation numbers (line opacities) in the wind.

Maeder (1999) has suggested the interesting possibility that due to the specific dependence of the force-multiplier parameters on temperature, the enhanced polar mass ejection due to gravity darkening described above may be progressively compensated by the effects of larger bound-free and line opacities, which then might favour a larger mass-flux in the cooler equatorial regions. Whether this suggestion ("-effect") actually works can be answered only by a more realistic 2-D non-LTE model which accounts for the complicated physical conditions in winds from rapidly rotating early-type stars.

In this paper, we will introduce a procedure providing a 2-D parameterization of the line acceleration by means of force-multiplier parameters that are consistent both with the local hydrodynamic properties of the flow and the non-local latitude-dependent radiation field.

With this improved model we will try to give an estimate both for the consequences of stellar rotation on the wind dynamics and on maximum errors introduced by the assumption of a spherically-symmetric wind when, e.g., using mean relationships as the WLR.

The paper is organized as follows: In Sect. 2 we introduce the wind model and discuss the basic simplifications and approximations. In Sect. 3, we provide the basic formulae for the vector line and continuum acceleration. Sect. 4 describes how we determine the ionization equilibrium and the occupation numbers in the wind. Sect. 5 gives an overview over the basic properties of the line force-multiplier M with respect to different physical quantities. On the basis of this investigation, in Sect. 6 we introduce an approximate method for the parameterization of the line force in 2-D winds. In Sect. 7, we present the results for winds from rapidly rotating hot stars and estimate the differences to corresponding models with global force-multiplier parameters. In Sect. 8, we finally discuss the results and give future perspectives.

© European Southern Observatory (ESO) 2000

Online publication: June 20, 2000
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