Magnetic Braking by a Stellar Wind--V APPROXIMATE DETERMINATION OF THE POLOIDAL FIELD

Abstract

The exact equation which determines the steady, axisymmetric structure of the poloidal magnetic field in the stellar wind problem is derived from the transfield component of the equation of motion in the case of an isothermal corona. It has a critical point at the Alfvènic point and the critical condition there yields the boundary condition for the solution inside the Alfvènic point. Thus the result obtained by Weber & Davis for their assumed radial field can be generalized as follows: conditions downstream from the Alfvènic point influence neither the flow structure nor the field structure upstream. Then the field structure that deviates least from the equilibrium in the transfield direction is selected inside the Alfvènic point from a variety of the poloidal fields assumed in Paper IV for the three non-dimensional parameters |$l,\,\zeta \,\text{and}\,\kappa $| measuring the coronal temperature, the field strength and the centrifugal force, respectively, at the coronal base. For the thermal– centrifugal wind |$(l\sim 3\,\text{and}\,\kappa \gtrsim\,1)$| the field structure is nearly radial near the rotation axis and nearly dipolar in other regions of the wind zone. For the thermal wind (l, κ small) the structure may be intermediate between dipolar and radial, depending upon ζ, and for the centrifugal wind (l, κ large) the structure should be nearly dipolar. In the vicinity of the boundary between the dead and wind zones the assumed poloidal fields as well as the method of deciding the boundary field line in Paper IV break down. The field structure in the dead zone is well approximated by the dipole field in the framework of the assumed fields.