Multi-wavelength emission modelling of PSR J0437–4715

Context. The diversity of pulsar light curves and radio polarisation properties originates in the structure of the magnetic field close to the stellar surface. For millisecond pulsars, this complexity is particularly puzzling. Fortunately, some means exist to uncover the magnetic field topology that indeed impacts the emission within the magnetosphere but also on the surface through its hot spot thermal radiation. Aims. We aim at deducing a plausible magnetic field geometry for the millisecond pulsar J0437−4715 by using combined information from the soft X-ray hot spot geometry deduced from NICER observations by pulse profile modelling and from radio and γ -ray pulse profile fitting. We also check the consistency between the obtained geometry and the radio polarisation data. Methods. Our γ -ray light-curve shapes rely on the striped wind model, whereas the radio polarisation fits rely on the rotating vector model. The magnetosphere structure is obtained from dipolar force-free magnetosphere simulations. Results. We demonstrate that a slightly off-centred dipole augmented by a small-scale dipole located on one polar cap explains simultaneously the shape of the hot spot and the radio and γ -ray data with a magnetic obliquity of α ≈ (42 ± 5)° and a line-of-sight inclination angle of ζ ≈ (136 ± 5)°. Conclusions. Our simple dipole model reproduces all the radio and γ -ray characteristics of PSR J0437−4715, including its radio polarisation data. It shows that the radio emission could be produced in regions where the magnetic field is mainly of dipolar nature.