Three-dimensional magnetohydrodynamic simulations of the Crab Nebula

In this paper, we give a detailed account of the first three-dimensional (3D) relativistic magnetohydrodynamic simulations of pulsar wind nebulae, with parameters most suitable for the Crab nebula. In contrast to the previous 2D simulations, we also consider pulsar winds with much stronger magnetization, up to σ ≃ few. The 3D models preserve the separation of the post-termination shock flow into the equatorial and polar components, but the polar jets are disrupted by the kink mode of the current driven instability and 'dissolve' into the main body of the nebula after propagation of several shock radii. With the exception of the region near the termination shock, the 3D models do not exhibit the strong z-pinch configuration characteristic of the 1D and 2D models. Contrary to the expectations based on 1D analytical and semi-analytical models, the 3D solutions with highly magnetized pulsar winds still produce termination shocks with radii comparable to those deduced from the observations. The reason for this is not only the randomization of magnetic field observed in the 3D solutions, but also the magnetic dissipation inside the nebula. Assuming that the particle acceleration occurs only at the termination shock, we produced synthetic maps of the Crab nebula synchrotron emission. These maps retain most of the features revealed in the previous 2D simulations, including thin wisps and the inner knot. The polarization and variability of the inner knot is in a particularly good agreement with the observations of the Crab nebula and the overall polarization of the inner nebula is also reproduced quite well. However, the polar jet is not as bright as observed, suggesting that an additional particle acceleration, presumably related to the magnetic dissipation, has to be invoked.