Simulations of ultrarelativistic magnetodynamic jets from gamma-ray burst engines

Long-duration gamma-ray bursts (GRBs) require an engine capable of driving a jet of plasma to ultrarelativistic bulk Lorentz factors of up to several hundred and into narrow opening angles of a few degrees. We use global axisymmetric stationary solutions of magnetically dominated (force-free) ultrarelativistic jets to test whether the popular magnetic-driving paradigm can generate the required Lorentz factors and opening angles. Our global solutions are obtained via time-dependent relativistic ideal magnetodynamical numerical simulations which follow the jet from the central engine to beyond six orders of magnitude in radius. Our model is primarily motivated by the collapsar model, in which a jet is produced by a spinning black hole or neutron star and then propagates through a massive stellar envelope. We find that the size of the pre-supernova progenitor star and the radial profile of pressure inside the star determine the terminal Lorentz factor and opening angle of the jet. At the radius where the jet breaks out of the star, our well-motivated fiducial model generates a Lorentz factor γ ∼ 400 and a half-opening angle θj ∼ 2◦, consistent with observations of many longduration GRBs. Other models with slightly different parameters give γ in the range 100–5000 and θj from 0. ◦1 to 10◦, thus reproducing the range of properties inferred for GRB jets. A potentially observable feature of some of our solutions is that the maximum Poynting flux in the jet is found at θ ∼ θj with the jet power concentrated in a hollow cone, while the maximum in the Lorentz factor occurs at an angle θ substantially smaller than θj also in a hollow cone. We derive approximate analytical formulae for the radial and angular distribution of γ and the radial dependence of θj . These formulae reproduce the simulation results and allow us to predict the outcome of models beyond those simulated. We also briefly discuss applications to active galactic nuclei, X-ray binaries and short-duration GRBs.