Multi-wavelength emission modelling of PSR~J0437$-$4715

Abstract

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 which indeed impacts the emission within the magnetosphere but also on the surface through its hot spot thermal radiation. 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 geometry obtained and the radio polarisation data. 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. 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 $α\approx (42\pm5) \degr$ and a line-of-sight inclination angle of $ζ\approx (136 \pm5) \degr$. 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.

Figures (9)

• Figure 1: Multi-wavelength pulse profile of PSR J0437$-$4715 as observed in radio by Parkes at $1.4$ GHz (red line), by NICER in soft X-rays (0.3-3.0 keV, green line) and in $\gamma$-rays by Fermi/LAT (black line). Note that the zero line of the NICER data is arbitrary, since technically, the pulsed emission lies on top of an unpulsed pulsar emission + background emission (all origins combined). Note that this is also true for the radio data, for which the profile baseline has been subtracted to lie at zero.
• Figure 2: Example of a good fit of the $\gamma$-ray pulse profile ($\geq$ 0.1 GeV). The radio profile is shown in red, our model is displayed in orange, the $\gamma$-ray light-curve in black and its fit in blue.
• Figure 3: Color map showing the isocontours of the reduced ${\raisebox{\depth}{$\chi$}}^2$ fit for the $\gamma$-ray light-curve for the angles $\alpha$ and $\zeta$. The $1\sigma$, $2\sigma$ and $3\sigma$ confidence intervals are also shown. The minimum is located at ${(\alpha,\zeta)=(138^\circ,136^\circ)}$ (corresponding to the mirror angle ${\alpha=42^\circ}$) and depicted by a red circle.
• Figure 4: Example of a dipole producing an annular hot spot on one pole. Magnetic field lines are shown in blue and the neutron star surface as a solid black circle of normalised radius. The stellar interior is shown in light gray.
• Figure 5: Projection of the emission cone rims onto the sky for different emission lengths $s$ in units of $r_\textrm{L}$. The phase location of the magnetic axis at $0.5$ is shown as a vertical dashed line, whereas the line of sight at $\zeta=136^\circ$ is shown as a horizontal dashed line.