The pulsar force-free magnetosphere linked to its striped wind: time-dependent pseudo-spectral simulations

(abridged) Pulsar activity and its related radiation mechanism are usually explained by invoking some plasma processes occurring inside the magnetosphere. Despite many detailed local investigations, the global electrodynamics around those neutron stars remains poorly described. Better understanding of these compact objects requires a deep and accurate knowledge of their immediate electromagnetic surrounding within the magnetosphere and its link to the relativistic pulsar wind. The aim of this work is to present accurate solutions to the nearly stationary force-free pulsar magnetosphere and its link to the striped wind, for various spin periods and arbitrary inclination. To this end, the time-dependent Maxwell equations are solved in spherical geometry in the force-free approximation using a vector spherical harmonic expansion of the electromagnetic field. An exact analytical enforcement of the divergenceless of the magnetic part is obtained by a projection method. Special care has been given to design an algorithm able to look deeply into the magnetosphere with physically realistic ratios of stellar $R_*$ to light-cylinder $\rlight$ radius. We checked our code against several analytical solutions, like the Deutsch vacuum rotator solution and the Michel monopole field. We also retrieve energy losses comparable to the magneto-dipole radiation formula and consistent with previous similar works. Finally, for arbitrary obliquity, we give an expression for the total electric charge of the system. It does not vanish except for the perpendicular rotator. This is due to the often ignored point charge located at the centre of the neutron star. It is questionable if such solutions with huge electric charges could exist in reality except for configurations close to an orthogonal rotator. The charge spread over the stellar crust is not a tunable parameter as is often hypothesized.