Outflows from magnetic rotators I. Inner structure

A simplified model for the stationary, axisymmetric structure of magnetized winds with a polytropic equation of state is presented. The shape of the magnetic surfaces is assumed to be known (conical in this paper) within the fast magnetosonic surface. The model is non-self-similar. The regularity of the magnetic surfaces at critical surfaces is ensured by the Alfv\`en regularity condition and criticality conditions at the slow and fast magnetosonic critical points. These conditions are used to evaluate three constants of motion, the total energy, angular momentum, and the ratio of mass to magnetic flux $\alpha$, as well as the shape of the critical surfaces. The rotation rate $\Omega$ and the polytropic constant $Q$ as a function of the magnetic surfaces, together with the mass-to-magnetic flux ratio on the axis $\alpha_0$ entirely specify the model. Analytic results are given for limiting cases, and parameter studies are performed by numerical means. The model accepts any boundary conditions. Given the properties of astrophysical objects with outflows, the model allows their classification in terms of a rotation parameter. Critical surfaces are nearly spherical for slow rotators, but become strongly distorted for rapid rotators. The fast point remains at a finite distance for finite entropy flows, in contrast to cold flows.It is found that for a given mass loss rate, the rotation rate is limited.