Modelling of the surface magnetic field in neutron stars: Application to radio pulsars

We propose vacuum gap (VG) model which can be applied uniformly for normal and high-magnetic-field pulsars. The model requires a strong and non-dipolar surface magnetic field near the pulsar polar cap. We assume that the actual surface magnetic field B-s in pulsars results from the superposition of a global dipole field B-d and a crust-anchored small scale magnetic anomaly B- m. We provide a numerical formalism for modelling such structures of the surface magnetic field and explore it within the framework of the VG model, which requires strong surface fields B-s greater than or similar to 10(13) G. Thus, in order to increase the resultant surface field to values exceeding 10(13) G, in low magnetic field pulsars with B-d << 10(13) G it is required that B-m >> B-d, with the same polarities (orientations) of B-d and B-m. However, if the polarities are opposite, the resultant surface field can be lower than the dipolar surface component inferred from the pulsar spin-down. We propose that high-magnetic-field pulsars (HBPs) with the inferred global dipole field B-d exceeding the so-called photon splitting threshold B-cr similar to 4 x 10(13) G, can generate observable radio emission "against the odds", provided that the surface dipolar magnetic field B-d is reduced below Bcr by the magnetic anomaly Bm of the right strength and polarity. We find that effective reduction is possible if the values of Bd and Bm are of the same order of magnitude, which would be expected in HBPs with B-d > B-cr. The proposed VG model of radio emission from HBPs, in which pair production occurs directly above the polar cap, is an alternative to the recently proposed lengthened space-charge-limited-flow (SCLF) model, in which the pair formation front is located at relatively high altitudes, where the dipole field is degraded below Bcr. Our model allows high B-d radio-loud pulsars not only just above B-cr but even above 2 x 10(14) G, which is the upper limit for HBPs within the lengthened SCLF model