5. Axisymmetric 2D polytropic HD winds
To arrive at a crude model for the coronal expansion of a rigidly rotating star, we set forth to construct an axisymmetric, steady-state, polytropic wind solution valid throughout a poloidal cross-section. With the polar axis as rotation and symmetry axis, we need to generalize the rotating, polytropic Parker wind which succesfully modeled the equatorial regions. Whereas the Parker solution had at least one critical point, its 2D extension is expected to give rise to critical curves in the poloidal plane. The degree of rotation determines the deviation from perfect circles arising in the non-rotating, spherically symmetric case.
To initialize a 2D fully implicit time-stepping procedure to arrive at a steady-state wind, we use the 1D Parker solution with identical escape speed , polytropic index , and rotational parameter . We use a spherical grid in the poloidal plane, where the grid spacing is equidistant in , but is accumulated at the base in the radial direction. We take a grid and only model a quarter of a full poloidal cross-section. The density is initialized such that for all angles , the radial variation equals the 1D Parker wind appropriate for the equator. Writing the Parker solution as , , , we set , and similarly, we set and so that it vanishes at the pole , while everywhere. Since we now use a coarser radial resolution, we interpolate the Parker solution linearly onto the new radial grid. Boundary conditions then impose symmetry conditions at the pole () and the equator (). The radial coordinate still covers , as in the 1D calculations. Since the solutions are supersonic at , the boundary conditions there merely extrapolate the density and all three momentum components linearly in the ghost cells. The stellar rotation enters as a boundary condition in the toroidal momentum component, which enforces , where are the cartesian coordinates in the poloidal plane. Note that the toroidal momentum may still change in the process, since we can no longer fix the density at the stellar surface to a -independent constant value. This is because in steady-state, the density profile should establish a gradient in the direction to balance the component of the centrifugal force in that direction. In the purely radial direction, the inwards pointing gravity must be balanced by the combination of the pressure gradient and the radial component of the centrifugal force. We therefore extrapolate the density linearly at the base. To enforce the total mass flux as in the equatorial Parker solution, we determine the constant from the 1D calculation, and fix and at the stellar surface for its 2D extension.
An elementary analytic treatment for a 2D polytropic steady-state wind solution proceeds by noting that mass conservation is ensured when the poloidal momentum is derived from an arbitrary stream function such that . It is then easily shown that the toroidal momentum equation is equivalent with the existence of a second arbitrary function , corresponding to the conservation of specific angular momentum along a poloidal streamline. Similarly, energy conservation along a streamline introduces
Across the poloidal streamlines all forces must balance out.
We show streamlines and the contours of constant poloidal Mach number for two hydrodynamic wind solutions for , , and with (top panel) and (bottom panel) in Fig. 3. We restricted the plotting region to about . For the imposed mass flux parameter, we used the values for the slow rotator and for the faster rotator, as found from the equatorial Parker solution for the same and . Note how for low rotation rates, the wind solution is almost spherically symmetric with nearly radial streamlines and circular Mach curves. For higher rotation rates, the critical Mach curve where moves inwards at the equator and outwards at the pole when compared to a non-rotating case. The streamlines show the equatorward deflection when material is released from the stellar surface due to the centrifugal force. Such equatorward streamline bending due to rotation is discussed in detail in the analytical study by Tsinganos & Sauty (1992). For these solutions, we can then verify that the specific angular momentum L, as well as the total energy E, are conserved along the streamlines.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999