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Astron. Astrophys. 343, 251-260 (1999)

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4. 1D polytropic winds

4.1. Parker winds

Our starting point is the well-known analytic Parker (1958) solution for a spherically symmetric, isothermal ([FORMULA]) outflow from a star of mass [FORMULA] and radius [FORMULA]. Given the magnitude of the escape speed [FORMULA], one can construct a unique `wind' solution which starts subsonically at the stellar surface and accelerates monotonically to supersonic speeds. This solution is transonic at the critical position [FORMULA], where [FORMULA] is the constant isothermal sound speed. Since we know the position of the critical point [FORMULA], we can easily determine the flow profile [FORMULA] and the corresponding density profile [FORMULA]. The radial velocity is obtained from the iterative solution of the transcendental equation

[EQUATION]

The density profile results from the integrated mass conservation equation. Since the radial velocity reaches a constant supersonic value asymptotically, the corresponding density vanishes at infinity as [FORMULA].

Choosing units such that [FORMULA], [FORMULA] with [FORMULA], we initialize a 1D spherically symmetric, polytropic (with [FORMULA]) hydrodynamic outflow with this analytic isothermal Parker wind solution on a non-uniform mesh ranging through [FORMULA]. We use 1000 grid points and exploit a grid accumulation at the stellar surface, where the acceleration due to the pressure gradient is largest. In the ghost cells used to impose boundary conditions at the stellar surface, we fix the value of the base density to unity, and extrapolate the radial momentum continuously from its calculated value in the first grid cell. At [FORMULA], we extrapolate both density and radial momentum continuously into the ghost cells. We then use a fully implicit time integration to arrive at the corresponding steady-state, spherically symmetric, polytropic Parker wind solution. The obtained solution [FORMULA], [FORMULA] can be verified to have a constant mass flux [FORMULA] as a function of radius and energy integral

[EQUATION]

where [FORMULA]. Also, determining the sonic point [FORMULA] where [FORMULA] and the base radial velocity [FORMULA], the solution can be checked to satisfy

[EQUATION]

Note how the isothermal [FORMULA] case is the only polytropic wind solution where the position of the critical point can be determined a priori .

In practice, we increased the polytropic index gradually from [FORMULA] through 1.05, 1.1, 1.12, 1.125 to [FORMULA], each time relaxing the obtained steady-state solution for one polytropic index to the unique transonic wind solution for the next value. In the top panel of Fig. 1, we plot the radial variation of the Mach number [FORMULA] for the isothermal [FORMULA] Parker wind with [FORMULA], and for similar polytropic winds with [FORMULA] and 1.13. The vertical dashed lines indicate the agreement of the positions of the sonic points where [FORMULA] with the calculated right hand side of Eq. (6). Note the outward shift of the sonic point with increasing polytropic index and the corresponding decrease of the asymptotic radial velocity.

[FIGURE] Fig. 1a and b. Polytropic Parker winds with [FORMULA]. Top panel: Mach number [FORMULA] as a function of radial coordinate for spherically symmetric Parker winds for polytropic index [FORMULA] (isothermal), [FORMULA] and [FORMULA]. Bottom panel: equatorial solutions for rotating [FORMULA] polytropic Parker winds for various rotation parameters [FORMULA]. Critical points are indicated.

When we relax the restriction of spherical symmetry by allowing a rigid stellar rotation rate [FORMULA], we can easily construct a solution for the equatorial plane only. Indeed, ignoring variations perpendicular to this plane, one simply adds a toroidal velocity profile where [FORMULA] and then solves for [FORMULA], [FORMULA], and [FORMULA]. The boundary conditions on the toroidal momentum keep [FORMULA] fixed at the base and extrapolate it continuously at the end of the computational domain. A polytropic, rotating Parker solution for the equatorial regions is found by relaxation from a non-rotating wind with the same polytropic index [FORMULA]. In the bottom panel of Fig. 1, we show the Mach number [FORMULA] for [FORMULA] and [FORMULA] Parker winds where [FORMULA] equals [FORMULA], [FORMULA] and [FORMULA]. The solution with [FORMULA] hardly differs from its non-rotating thermally driven counterpart shown in Fig. 1, as expected. The additional centrifugal acceleration causes an increase in the base velocity [FORMULA] and in the asymptotic radial velocity. Again, the solution can be verified to have a constant radial mass flux [FORMULA], Bernoulli function

[EQUATION]

and constant specific angular momentum [FORMULA]. The positions of the critical point(s) [FORMULA] are now obtained from a generalization of Eq. (6), namely from the solutions of

[EQUATION]

This equation reduces to a second degree polynomial for a [FORMULA] isothermal, rotating, Parker wind so it is evident that rotation rates exist that introduce a second critical point. In Fig. 1, only the [FORMULA] solution exhibits two critical points, shown as vertical dotted lines, within the domain. We determined the critical point(s) by solving Eq. (7) using the calculated base speed [FORMULA]. The close-up of the radial variation of [FORMULA] for [FORMULA] at the base reveals that this thermo-centrifugally driven wind passes the first critical point while being decelerated, then starts to accelerate and finally becomes supersonic at the second critical point. To correctly capture the dynamics close to the stellar surface it is clear that we need a high grid resolution, especially at the stellar surface. Indeed, the first critical point for the [FORMULA] solution is situated at [FORMULA].

4.2. Weber-Davis winds

The magnetized Weber-Davis (WD) solution (Weber & Davis 1967) represents a valuable extension to the rotating, polytropic Parker wind solution for the equatorial plane. Again assuming that there is no variation perpendicular to this plane, one now needs to solve for an additional two magnetic field components [FORMULA] and [FORMULA]. One is trivially obtained from the [FORMULA] equation, namely [FORMULA]. The analytic treatment reveals that the magnetized polytropic wind solution has a total of two critical points, namely the slow [FORMULA] and the fast [FORMULA] critical point. These are determined by the zeros of [FORMULA]. In between lies the Alfvén point [FORMULA], defined as the radius at which the radial velocity [FORMULA] equals the radial Alfvén speed [FORMULA]. Since the equatorial fieldline is prescribed to be radial in the poloidal plane and the transfield force balance is not taken into account, this Alfvénic transition is not a critical point in this model.

In the fully implicit time stepping towards a steady-state WD wind for specific values of [FORMULA], [FORMULA], [FORMULA], and for the base radial Alfvén speed [FORMULA], we initialize [FORMULA], [FORMULA], and [FORMULA] with the corresponding non-magnetic, polytropic rotating Parker solution. We fix [FORMULA] to its known [FORMULA] dependence throughout the time evolution, and initialize [FORMULA] to zero. The boundary conditions at [FORMULA] extrapolate all quantities we solve for continuously into the ghost cells. At the base, we keep the density fixed, the radial momentum and toroidal field component are extrapolated linearly from the first two calculated mesh points, while the toroidal momentum [FORMULA] is coupled to the magnetic field ensuring

[EQUATION]

This expresses the parallelism of the velocity and the magnetic field in the frame rotating with the stellar angular velocity [FORMULA]. Using these initial and boundary conditions, we arrive at the unique WD wind solution for the given parameters. This magnetized polytropic wind solution for the equatorial plane is shown in Fig. 2. The solution agrees exactly with the analytic WD wind: we obtain five constants of motion, namely the mass flux [FORMULA], the magnetic flux [FORMULA] which is constant by construction, the validity of Eq. (8) over the whole domain, the Bernoulli integral

[EQUATION]

and the constant total specific angular momentum [FORMULA]. The positions of the critical points are [FORMULA] and [FORMULA], while the Alfvén point is at [FORMULA], as indicated in Fig. 2. This agrees with the values given in the Appendix to Keppens et al. (1995), where the same WD solution was calculated in a completely different fashion. Indeed, the WD solution for given values of [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA], can alternatively be calculated as a minimization problem in a six-dimensional space (see Belcher & MacGregor 1976) where one solves for the six unknowns [[FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA]]. This can be done using standard Newton-Raphson iteration provided one has an educated initial guess, but it can even be obtained by the use of a genetic algorithm , as first demonstrated by Charbonneau (1995). The fact that the, for our method initially unknown, base velocities appear in the determining set of variables for these minimization methods again indicates that a high base resolution is absolutely essential. Values for these six unknowns found from the solution in Fig. 2 are [FORMULA] and these agree with the Newton-Raphson solution.

[FIGURE] Fig. 2a and b. Weber-Davis wind solution for the equatorial plane. Top panel: radial variation of the radial Alfvén speed [FORMULA], sound speed [FORMULA], and velocities [FORMULA] and [FORMULA], all normalized to the base sound speed [FORMULA]. Bottom panel: the corresponding poloidal Alfvén Mach number [FORMULA], and poloidal slow and fast Mach numbers, determining the positions of the critical points and the Alfvén point. See text for parameters.

The calculation of the WD polytropic wind by the stepwise relaxation from an isothermal Parker wind is thus an excellent test for the numerics, as every step (from isothermal to polytropic, from non-rotating to rotating, from Parker to WD) can be verified precisely to agree with the known solutions. It should be clear that we can construct WD wind solutions where the acceleration results from the combined action of thermal, centrifugal, and magnetic forces. However, our interest is in the generalization of these 1D models out of the equatorial plane. We will again proceed in logical steps towards this goal.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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