Do magnetized winds self-collimate?

We discuss the asymptotic behaviour of steady, axisymmetric, non-relativistic, polytropic, perfect magnetohydrodynamic (MHD) winds from rotating magnetized bodies, which have passed smoothly through three critical surfaces. A quantity defined by ζ≡αρϖ2/η plays a crucial role in the asymptotic domain far from the Alfvénic surface. Here α is the angular velocity of field lines, η is the mass flux per unit flux tube and ρ is the mass density. In general, ζ is a function of the magnetic stream function P and the distance s measured along each field line. In the asymptotic domain, ϖBt≈-ζ and the electric current parallel to the poloidal field is j∥∝-(∂ζ/∂P)sBp, while the perpendicular one is j⊥∝-∂ζ/∂s)P. Also ζ(P,s) as a function of s measures the change of the Poynting energy flow along a given field line and as a function of P measures the total electric current passing within the surface of revolution defined by P and with s fixed. The asymptotic transfield equation reduces to ρvp2/R≈j∥Bt/c, where R is the curvature radius of poloidal field lines. This equation indicates that when viewed from the rotation-axis side, the field geometry is convex (R>0) in the range of field lines with ingoing current (j∥0, i.e. (∂ζ/∂s)P