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. Rather than solving the equilibrium perpendicular to the flux surfaces everywhere, solutions are found at the Alfvèn surface where it takes the form of the Alfvèn regularity condition and at the base of the flow. This constrains the Transfield equilibrium in that the Alfvèn regularity condition is imposed and the regularity of the magnetic surfaces at the Alfvèn critical surface is ensured. The model imposes criticality conditions at the slow and fast magnetosonic critical points using the Bernoulli equation. These Alfvèn regularity and criticality conditions are used to evaluate three constants of motion, the total energy, angular momentum, and the ratio of mass to magnetic flux α, as well as the shape of the critical surfaces. The rotation rate Ω and the polytropic constant Q as a function of the magnetic surfaces, together with the mass-to-magnetic flux ratio on the axis α₀ 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. Numerical calculations yield the value of the rotation parameter ω. Rotators can be defined as slow, intermediate or fast according to whether ω is much less or close to unity or near its maximum value for fast rotators, (3/2)^3/2. Given the properties of astrophysical objects with outflows, the model allows their classification in terms of the 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.