Pulsars are rapidly spinning highly magnetised neutron stars. Their spin period is observed to decrease with time. An early analytical model for this process was the vacuum retarded dipole (VRD) by Deutsch (1955, AnAp, 18). This model assumes an idealised star and it finds that the rotational energy is radiated away by the electromagnetic fields. This model has been superseded by more realistic numerical simulations that account for the non-vacuum like surroundings of the neutron star. However, the VRD still provides a reasonable approximation and is a useful limiting case that can provide some qualitative understanding. We provide detailed derivations of the spin down and related electromagnetic field equations of the VRD solution. We also correct typographical errors in the general field equations and boundary conditions used by Deutsch (1955, AnAp, 18).

While predominantly dipolar, large-scale magnetic fields are usually assumed in most studies involving neutron stars, there are multiple observational, theoretical hints and numerical simulations highlightening the importance of non-dipolar components. I review here the most important observational facts and numerical studies pointing towards the existence of magnetospheric currents and internal small-scale structures, arising from multipoles of poloidal and toroidal fields. This holds for all neutron star stages: at birth, during their lifetime and after a merger.

Massive stars (M ≥ 10M⊙) end their lives with spectacular explosions due to gravitational collapse. The collapse turns the stars into compact objects such as neutron stars and black holes with the ejection of cosmic rays and heavy elements. Despite the importance of these astrophysical events, the mechanism of supernova explosions has been an unsolved issue in astrophysics. This is because clarification of the supernova dynamics requires the full knowledge of nuclear and neutrino physics at extreme conditions, and large-scale numerical simulations of neutrino radiation hydrodynamics in multi-dimensions. This article is a brief overview of the understanding (with difficulty) of the supernova mechanism through the recent advance of numerical modeling at supercomputing facilities. Numerical studies with the progress of nuclear physics are applied to follow the evolution of compact objects with neutrino emissions in order to reveal the birth of pulsars/black holes from the massive stars.