Analytical Solution of Magnetically Dominated Astrophysical Jets and Winds: Jet Launching, Acceleration, and Collimation

We present an analytical solution of a highly magnetized jet/wind flow. The left side of the general force-free jet/wind equation (the "pulsar" equation) is separated into a rotating and a nonrotating term. The two equations with either term can be solved analytically, and the two solutions match each other very well. Therefore, we obtain a general approximate solution of a magnetically dominated jet/wind, which covers from the nonrelativistic to relativistic regimes, with the drift velocity well matching the cold plasma velocity. The acceleration of a jet includes three stages. (1) The jet flow is located within the Alfv\'{e}n critical surface (i.e. the light cylinder), has a nonrelativistic speed, and is dominated by toroidal motion. (2) The jet is beyond the Alfv\'{e}n critical surface where the flow is dominated by poloidal motion and becomes relativistic. The total velocity in these two stages follows the same law $v\Gamma=\Omega R$. (3) The evolution law is replaced by $v\Gamma\approx1/\left(\theta\sqrt{2-\nu}\right)$, where $\theta$ is the half-opening angle of the jet and $0\leq\nu\leq2$ is a free parameter determined by the magnetic field configuration. This is because the earlier efficient acceleration finally breaks the causality connection between different parts in the jet, preventing a global solution. The jet has to carry local charges and currents to support an electromagnetic balance. This approximate solution is consistent with known theoretical results and numerical simulations, and it is more convenient to directly compare with observations. This theory may be used to constrain the spin of black holes in astrophysical jets.