The efficiency of the magnetic acceleration in relativistic jets

Using steady, axisymmetric, ideal magnetohydrodynamics (MHD) we analyze relativistic outflows by means of examining the momentum equation along the flow and in the transfield direction. We argue that the asymptotic Lorentz factor is ~ mu-sigma_M, and the asymptotic value of the Poynting-to-matter energy flux ratio - the so-called sigma function - is given by sigma/(1+sigma) \~ sigma_M / mu, where sigma_M is the Michel's magnetization parameter and mu c^2 the total energy-to-mass flux ratio. We discuss how these values depend on the conditions near the origin of the flow. By employing self-similar solutions we verify the above result, and show that a Poynting-dominated flow near the source reaches equipartition between Poynting and matter energy fluxes, or even becomes matter-dominated, depending on the value of sigma_M / mu.