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

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