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The stationary axisymmetric outflow from a rotating sphere with a (split) monopole magnetic field is considered. The stream equation describing the outflow is linearized in terms of the Michel magnetization parameter σ−1 ≪ 1, which allows a self‐consistent analysis of the direct problem. It is shown that for a finite σ the fast magnetosonic surface is located at a finite distance ∼ σ1/3RL (RL = c/ΩF is the light cylinder). We have also found that the particle energy at the fast surface is just equal to the Michel value γ ∼ 1/3σ. The particle acceleration and magnetic field collimation are shown to become ineffective outside the fast magnetosonic surface.

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The problem of the efficiency of particle acceleration for a paraboloidal poloidal magnetic field is considered within the approach of steady axisymmetric magnetohydrodynamic (MHD) flow. For the large Michel magnetization parameter σ it is possible to linearize the stream equation near the force‐free solution and to solve the problem self‐consistently as was done by Beskin, Kuznetsova & Rafikov for a monopole magnetic field. It is shown that, on the fast magnetosonic surface (FMS), the particle Lorentz factor γ does not exceed the standard value σ1/3. On the other hand, in the supersonic region, the Lorentz factor grows with the distance z from the equatorial plane as γ≈ (z/RL)1/2 up to the distance z≈σ2RL, where RL=c/ΩF is the radius of the light cylinder. Thus, the maximal Lorentz factor is γmax≈σ, which corresponds to almost the full conversion of the Poynting energy flux into the particle kinetic one.