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

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2. The model

2.1. Physical processes

Our 2-D wind model (with photospheric boundary) comprises the following physical effects:

  • We consider the inertial (pseudo) accelerations in the equations of motion, i.e., the centrifugal and coriolis terms.

  • As a consequence of rotation, the stellar surface is oblated, and the latitudinal variation of stellar radius is usually calculated on basis of a Roche model (e.g., Collins 1963, 1965). In accordance with previous investigations by OCB and Cranmer & Owocki (1995, "CO95") and for the sake of simplicity, we have also used this approach. Note, however, that the Roche model applies in principle only to conservative angular velocity distributions (i.e., those with a cyclindrical symmetry, in particular rigid body rotation). Concerning the more appropriate case of "shellular" rotation (constant rotation rate on isobars, cf. Zahn 1992), see Meynet & Maeder (1997, Appendix A).

  • As shown by Owocki et al. (1997, 1998), Petrenz (1999) and Gayley & Owocki (2000), the non-radial components of both the continuum and line acceleration have a significant impact on the wind dynamics and have to be taken into account.

  • The gravity darkening effect (see, e.g., Kippenhahn & Weigert 1991) leads to an enhanced (diminished) radiative acceleration over the poles and in the equatorial plane, respectively (cf. C095, Owocki et al. 1997, 1998 and Petrenz 1999).

  • The most important point in our work concerns the ionization structure and occupation numbers in the wind, which will be determined consistently with both the local physical properties of the wind material and the non-local stellar radiation field.

2.2. Physical approximations

To limit the complexity of our model and to facilitate the required computational effort, we will adopt following simplifications:

  • We neglect the instability of the radiative line force (e.g., Owocki 1991, Feldmeier 1995), i.e., we assume a "smooth" wind without clumps and shocks which may arise from this instability. For studies of the large-scale wind morphology, this assumption is reasonable since the values of both density and velocity field averaged over small spatial length scales correspond to a macroscopic description by a smooth wind. Moreover, the correct numerical treatment of line force instabilities would require at least 2-D simulations that exceed available computational capacities (see, however, Owocki 1999).

  • For the calculation of the stellar surface shape, we neglect the variation of the Thomson acceleration with stellar co-latitude owing to the polar temperature gradient. This effect may be important at most for extremely rapidly rotating OB-supergiants and has been controversely discussed by Langer (1998) and Glatzel (1998). From the viewpoint of stellar evolution theory, however, there is a still ongoing, controversial debate about the stellar surface properties of early-type massive stars. In particular, the competing processes of shear turbulence and meridional circulation counteract on the transport of angular momentum inside the star, thus being crucial for chemical mixing and stellar surface properties (see Meynet 1998, Maeder 1999).

    Since our present study concentrates on B-stars (cf. Sect. 3.3) with [FORMULA], we consider the Thomson acceleration only in an averaged way for calculating the shape of the stellar surface and replace, in the gravitational acceleration, [FORMULA] by the corresponding effective mass [FORMULA] (cf. OCB, C095 and PP96). In this way, at least part of the radiative continuum acceleration is accounted for its influence on the scale height. (The influence of the line-acceleration in the quasi-hydrostatic part of the envelope which forms the stellar surface is only small, because of the vanishing velocity fields in this region.)

  • In analogy to prior investigations (e.g., C095), gravity darkening is considered in the framework of the von Zeipel (1924) theorem. Detailed studies recently performed by Maeder (1999) have revealed that the accuracy of this approximation is much better than expected, of order 10%.

  • Since there is no observational evidence for a lower limit of the (global) magnetic field strength at the surface of early-type stars, we do not consider magnetic fields in our hydrodynamic simulations (for a review, see Mathys 1999).

  • In this first paper, we consider only (O)B-star winds with an optically thin continuum in the line-driving spectral range. Rapidly rotating B-supergiants with larger mass-loss rates ([FORMULA] [FORMULA]) and increasing optical depth in the Lyman continuum, which are thought to be candidates for the B[e] phenomenon via the bi-stability effect (Lamers & Pauldrach 1991, Lamers et al. 1999) will be covered in a forthcoming paper. In any case, the restriction to winds with an optically thin continuum is not a severe one and actually covers the largest part of radiatively driven winds: Excluding Wolf-Rayets and the most luminous "normal" objects ([FORMULA], [FORMULA] [FORMULA]) and excluding B-Supergiants close to the bi-stability jump around [FORMULA] K (cf. Vink et al. 1999and Sect. 7.2), all OB-star winds are actually optically thin in the line-driving continuum, i.e. in the range 229 Å [FORMULA] 10000 Å (see, e.g., Herrero et al. 2000, their Sect. 6.1).

  • Since we are primarily interested in the consequences of stellar rotation for the large scale wind morphology , we will not consider explicitly time-dependent processes (e.g., non-radial pulsations, rotational modulation). Observations of non-stationary spectral features have indicated dynamic variability on only rather small spatial scales.

2.3. Geometry and co-ordinate systems

Adopting rotational symmetry about the polar axis, we can formulate the hydrodynamic equations in spherical polar co-ordinates [FORMULA] (see Fig. 1). r denotes the distance from the stellar center, [FORMULA] the co-latitude ([FORMULA] at the pole) and [FORMULA] the stellar azimuthal angle. At every location [FORMULA], a local right-handed orthonormal system [FORMULA] is introduced, and the local vector [FORMULA] (which is needed, e.g., to calculate the cone angle subtended by the stellar disk) is given by (cf. Fig 1)

[EQUATION]

with directional vector [FORMULA]. Finally, the stellar surface is described by [FORMULA].

[FIGURE] Fig. 1. Co-ordinate systems, see text.

2.4. The hydrodynamic equations

We solve the time-dependent hydrodynamic equations for the wind. Their stationary solution is found if the numerical simulation has relaxed to a converged state. This procedure has the additional advantage that we may also perform simulations of non-stationary phenomena which can be described by time-dependent boundary conditions.

Due to azimuthal symmetry about the rotational Z-axis, the hydrodynamic equations reduce to the polar plane [FORMULA]. Their solution depends only on the polar co-ordinates [FORMULA], and all derivatives with respect to [FORMULA] vanish (for a derivation, see, e.g., Batchelor 1967):

[EQUATION]

with time t, velocity field [FORMULA], density [FORMULA] and pressure p. The external acceleration [FORMULA] is given by the sum of gravitational, line and Thomson acceleration

[EQUATION]

In Eq. (6), we have assumed an isothermal equation of state, which is justified at least for a first qualitative study, since we do not account for line-driven instabilities that might generate strong shocks, and since the time scales for other heating- and cooling processes are significantly smaller than typical flow times. Note also that the only true body forces in the equations of motion are pressure, gravitation and line force.

2.5. Numerical specifications

To solve the hydrodynamic equations Eqs. (2-6), we have adapted the time-dependent finite-difference 2-D Eulerian code ZEUS-2D (Stone & Norman 1992) to our specific problem. This well-tested and frequently used code allowed an independent test of the results published by OCB and Owocki et al. (1996, 1997) who employed the hydrodynamic code VH-1 (developed by J. Blondin and collaborators) for their simulations.

ZEUS-2D solves the hydrodynamic equations for three velocity components [FORMULA] in the 2-D polar plane [FORMULA], i.e., for the 2.5-D geometry used to formulate our problem.

The flow variables are specified on a 2-D spatial grid [FORMULA] in radius [FORMULA] and co-latitude [FORMULA]. The mesh in radius is defined from an initial zone ([FORMULA]) out to a maximum zone at [FORMULA]. These zones are concentrated near the stellar surface where the flow gradients are steepest. The initial spacing ([FORMULA]) is constant at [FORMULA] [FORMULA][FORMULA]. It then increases by 6% per zone out to the maximum radius [FORMULA]. The equidistant grid in co-latitude [FORMULA] is defined from an initial zone [FORMULA] adjacent to the pole ([FORMULA]) to a maximum zone [FORMULA] adjacent to the equatorial plane ([FORMULA]).

As extensive test calculations have shown, this resolution of the computational domain is entirely sufficient for an investigation of the large-scale wind morphology. (Doubling the resolution both in radius and co-latitude only showed some correspondingly greater detail in small-scale flow structure.)

The most important numerical issue is to define an appropriate lower boundary condition, since the rotationally induced oblate stellar surface [FORMULA] does not fit the curvi-linear orthogonal co-ordinate lines [FORMULA].

This issue has been addressed the first time by OCB who proposed to specify the lower boundary condition along a "staircase" that straddles the equipotential surface [FORMULA] of the rigidly rotating star. We have chosen the same strategy, in accordance with the fact that this is the only "natural" way to treat the lower boundary condition for the geometry used to formulate our problem.

To guarantee a mildly subsonic inflow of wind material at the lower wind base which turned out to be essential for the stability of the flow and its convergence to a stationary state, we adjust the base density to a fixed value (if necessary, varying with co-latitude) and set the radial base velocity by constant slope extrapolation from the interior of the calculational domain.

To avoid meridional flows directly at the stellar surface, we assume a no-slip condition for the polar velocity component and rigid body rotation for the azimuthal velocity. Due to symmetry reasons, we choose reflecting boundary conditions along the polar axis and symmetric ones about the equatorial plane.

For a further discussion of specific technical details and problems, we refer the reader to OCB and Petrenz (1999, Sect. 6.).

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Online publication: June 20, 2000
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