Relativistic centrifugal winds

Centrifugal winds in the gravity- and pressure-free limit are investigated in the magnetohydrodynamic approximation. The critical condition at the magnetosonic point at infinity determines the angular momentum flow per unit flux tube β in terms of $$\Phi_\infty = (B_\text p \varpi^2)_\infty$$ and the three integrals; α, the angular velocity of field line, $$\eta = \rho\kappa = \rho\upsilon_\text p/B_\text p$$ the mass flux per unit flux tube and µ, the non-dimensional energy related to the flow, where B_p is the poloidal magnetic field. The flow variables such as the Lorentz factor γ are determined in terms of α, η and µ if $$\phi = \Phi_\infty/\Phi = (B_\text p\varpi^2)_\infty/(B_\text p\varpi^2)$$ is specified as a function of axial distance ϖ together with Φ_∞ and $$\sigma = (\gamma^2_\infty-1)^{3/2}=\alpha^2\Phi_\infty/4\pi\eta c^3$$ decides the degree of relativity of the critical flow solution along each field line. It is argued that the extreme-relativistic limit of $$\sigma \simeq \gamma^3_\infty\rightarrow \infty$$ is a result of the massless, inertia-free or force-free limit of $$\eta \rightarrow 0$$, in which the (pure-) Alfvénic point is at infinity and the critical condition there yields the same expression for β as derived from the massless limit of the critical condition at the magnetosonic point. There is a value of σ (say σ₀) at which the pure-Alfvénic cylinder defined by $$4\pi\rho\gamma\kappa^2 = 1$$ coincides with the light cylinder and for $$\sigma \gt \sigma_0$$ the pure-Alfvénic cylinder is outside the light cylinder. The light cylinder is situated between the pure-Alfvénic point and the magnetosonic point at infinity for $$\sigma \gt \sigma_0$$ and reduces to infinity as $$\sigma \rightarrow 0$$ in the non-relativistic limit. The Alfvénic point defined by $$\alpha^2\varpi^2/c^3+4\pi\rho\gamma\kappa^2 = 1$$ always lies inside the light cylinder and the pure-Alfvénic point, which reduces to the former in the massless limit and to the latter in the non-relativistic limit.