Charges and Magnetic Fields of the Slowly-rotating Neutron Stars

In association with the Goldreich-Julian's magnetic braking mechanism to explain the luminosity of radio pulsars as a result of their spin-down, it is of some interest to explore how much of magnetic flux can actually penetrate the surface of the rotating neutron stars at least in idealized situations. In order to address this issue, one needs to figure out the amount of charge on and the structure of magnetic field around a rotating neutron star. In the present work, based on the solution-generating method given by Wald, the magnetic fields around both the uncharged and (slightly) charged neutron star have been obtained. Particularly for the charged neutron star, it has been demonstrated following again the argument by Wald that the neutron star will gradually accrete the charge until it reaches the equilibrium value $\tilde{Q}=2B_{0}J$. Then next, the magnetic flux through one half of the surface of the rotating neutron star has been computed as well. With the nonvanishing accretion charge having value in the range $0 < \tilde{Q} \leq 2B_{0}J$, the total magnetic flux through the neutron star has been shown to be greater than that without the accretion charge.