A (semi)-analytic view of the inner structure of Pulsar Wind Nebulae

When the wind of an active pulsar impacts on the surrounding medium, it forms a termination shock (TS) that feeds a relativistic and magnetized bubble, known as "Pulsar Wind Nebula". About thirty years ago, Kennel Coroniti investigated this scenario, but unfortunately their results failed to match the observed morphologies. That model was in principle correct, but its main drawback was the assumption of a spherical symmetry. More recently, numerical codes have been used to simulate in detail the dynamical structure of PWNe: they have shown complex morphologies, with a closer resemblance with observations. We show how Kennel Coroniti model can be generalized to two dimensions, by solving the jump equations for an oblique TS, and then the relativistic MHD equations in the downstream regions closest to the TS. In this way we can obtain two dimensional, steady state solutions, which in the inner regions agree quite well with the numerical ones. This method is semi-analytic and computationally rather light: given the shape of the TS (in an analytic form), the spatial behaviour of the physical quantities (like velocity, pressure, magnetic field) is derived. Maps of the synchrotron emission are also obtained. A final goal is to use semi-analytic modelling, together with numerical simulations, to improve inversion techniques, aimed at deriving the pulsar-wind parameters from observations