Magnetic Rotator Winds and Keplerian Disks of Hot Stars

We set up equations and boundary conditions for magnetic rotator winds and disks in axially symmetric hot stars in a steady state. We establish a theorem stating that if a disk region has no meridional motion but its magnetic field has a normal component at a point Q on its shock boundary, the angular velocity of the disk region at Q is the same as the angular velocity of the star at the point P_* at which the magnetic field line through Q is anchored. When there is equatorial symmetry, all points of the disk along the field line through Q will have the same angular velocity as P_*. Also, we show that for a given value of the magnetic field strength, if the rotation rate is too high or the flow velocity into the shock boundary is too low, a Keplerian disk region will not be formed. We consider the formation of disks in magnetic rotators through the processes of fill-up and diffusion into Keplerian orbits. At the end of the fill-up stage the density of the disk increases significantly and the magnetic force in the disk becomes negligible. If the meridional component B_m of the field at the surface is larger than a minimum value B_m,min, a Keplerian disk can form. The radial extent of the Keplerian region is larger for larger values of B_m and is largest when B_m equals an optimal value B_m,opt. The extent does not increase when B_m is larger than B_m,opt. If α is the ratio of rotational speed to the critical rotation speed at the photosphere, the inner and outer radii of the maximal quasi-steady Keplerian disk region are given by α^-2/3R and 2^4/3α^-2/3R, respectively, where R is the stellar radius. For models with dipole-type fields, the values of B_m,min in B-type stars are of the order of 1-10 G, and in O-type stars they are about 500 G. Because the values of B_m required for disk formation in B-type stars are relatively small and the fill-up time is short, we suggest that meridional circulation may play a role in some of the time variation observed in disks of Be stars through its effect on the magnetic field near the photosphere.