Relativistic Jet Flow from a One Dimensional Magnetic Nozzle - Analytic Solutions

Magnetohydrodynamic devices that can accelerate plasmas to speeds of the order of hundreds of kilometres per second have been designed and built for nearly forty years. Up to the time of writing, however, the theory for such devices has been exclusively non-relativistic. In this paper we derive the special relativistic magnetohydrodynamic (SRMHD) equations and use them to obtain the relativistic, magnetic nozzle equation which describes the production of jet flows with speeds approaching the speed of light. We obtain analytic solutions to this equation and show that, in principle, magnetic field gradients can accelerate a plasma to highly relativistic speeds. We also show that the exit kinetic energy, E_K, of a particle is given by the equation E_K= m₀C²_FR, where m₀ is the rest mass of the particle and C_FR is the fast magnetosonic speed at the start of the flow. The relativistic nozzle differs in a number of ways from the non-relativistic case. A non-relativistic nozzle has a relatively symmetric converging/diverging shape, while a highly relativistic nozzle converges in the usual manner, but diverges, in an abrupt fashion, at the very end of the nozzle. The gentle divergence of non-relativistic nozzles causes the exit plasma densities and magnetic fields of the flow to have values that are small relative to their values at the start of the nozzle. The abrupt divergence of a highly relativistic nozzle implies that, for a less than perfect nozzle, the exit values of the mass density and the magnetic field strength are comparable to their initial values. This unexpected dichotomy in behaviour may have future application in understanding the `radio-loud' and `radio-quiet' relativistic jets that are produced from astrophysical sources.