Magnetospheres

An axisymmetric force-free magnetic field in spherical coordinates has a relationship between its azimuthal component to its poloidal flux-function. A power law dependence for the connection admits separable field solutions but poses a nonlinear eigenvalue boundary-value problem for the separation parameter (Low and Lou, Astrophys. J. 352, 343 (1990)).When the atmosphere of a star is rotating the problem complexity increases. These Notes consider the nonlinear eigenvalue spectrum, providing an understanding of the eigen functions and relationship between the field's degree of multi-polarity, the rotation and rate of radial decay as illustrated through a polytropic equation of state. The Notes are restricted to uniform rotation and to axisymmetric fields. Dominant effects are presented of rotation in changing the spatial patterns of the magnetic field from those without rotation. For differential rotation and non-axisymmetric force-free fields there may be field solutions of even richer topological structure but the governing equations have remained intractable to date. Perhaps the methods and discussion given here for the uniformly rotating situation indicate a possible procedure for such problems that need to be solved urgently for a more complete understanding of force-free magnetic fields in stellar atmospheres.