Physics motivated models of pulsar X-ray hotspots: off-center dipole configurations

TL;DR

This work develops a physics-based, non-axisymmetric model of pulsar X-ray hotspots by computing magnetospheric currents for an off-center dipole within a force-free framework, then mapping these currents to surface heating and a temperature distribution. The approach leverages Euler potentials and a phenomenological heating law, with automatic differentiation to evaluate derivatives efficiently, and uses X-PSI to generate light curves for comparison with NICER data. Bayesian MCMC inference shows that including volumetric return current improves parameter constraints and can reproduce the observed pulse profiles of PSR J0030+0451 and PSR J0437--4715, though limitations in the current prescription and fixed NS structure remain. The results suggest that off-center dipole configurations, augmented by return-current physics, can capture major features of the X-ray emission and motivate further exploration of multipole magnetic structures and more detailed microphysical heating models.

Abstract

Recently, it was proposed that an off-center dipole magnetic configuration, together with a non-trivial temperature profile, may be the best model to explain the X-ray light curve of PSR J0030+0451 observed by the Neutron Star Interior Composition Explorer (\emph{NICER}). Using a theoretical model for the electric current density in a force-free pulsar magnetosphere, we compute from first principles the distribution of electric current over the polar cap associated with an off-center magnetic dipole. We then use a simple prescription to compute the resulting temperature distribution, which allows us to derive the observed X-ray light curve. We investigate the role of the volumetric return current region in the polar cap and find that although it does not make a big difference in an aligned dipole case, the difference can be bigger in the case of an off-center dipole. Finally, we apply Markov Chain Monte Carlo (MCMC) fitting to the X-ray light curves of pulsars PSR J0030+0451 and PSR J0437--4715 with and without the volumetric return current, and find that our model can reasonably recover the observed X-ray light curves.

Figures (8)

• Figure 1: Temperature distribution of different hotspost configurations (magnetic inclination $\iota = 45^\circ$.):(a) Uniform centered dipole circular hotspots with $T = 10^6\,K$.(b) The centered dipole with computed temperature distribution without considering return current. (c) The centered dipole with computed temperature distribution with considering return current region. All the temperature profile normalized by the maximum temperature of this profile. The cyan boundary shared by a,b,c represent the polar cap boundary delineated by $\alpha = \alpha_0$ from Equation \ref{['eq:alpha0']}.
• Figure 2: Pulse profile of bolometric flux computed from the centered dipole field in three cases: Uniform hotspot temperature (red), Without return current (blue), With return current (green). All the other parameters to generate this plot are listed in Table \ref{['tab1:xpsi_para']}. The magnetic inclination angle $\iota=45^{\circ}$, observer inclination angle $I = 90^{\circ}$.
• Figure 3: 4-current density squared $J^{2}$ and temperature $T$ distribution of shifted hotspot projected on the star surface, with magnetic inclination $\iota = 45^{\circ}$ and dipole shift $\delta = R_{\star}/2$ along different directions. Panels (a) and (b): $x$--direction shift; panels (c) and (d): $y$--direction; panels (e) and (f): $z$--direction shift. Each color map is normalized by the maximum value of $J^{2}$ and $T$.
• Figure 4: Pulse profile computed from the shifted dipole field in three cases: $x$--direction shift (blue), $y$--direction shift (red) and $z$--direction shift (green), comparing with centered dipole field computation without including any shift (black dashed). All the other parameters to generate this plot are listed in Table \ref{['tab1:xpsi_para']} with magnetic inclination $\iota = 45^{\circ}$, observer inclination $I = 90^{\circ}$
• Figure 5: The posterior of the observer inclination $I$, magnetic inclination angle $\iota$ (both in radians), $x$--$y$--$z$ direction shifts and $\log_{10}T_{s}$ after applying constraints to the observed pulse profile of J0030. Blue is the posterior with Case 1: without return current. Red posterior is result in Case 2: with return current. The contour levels in the corner plot, going from deep to light colours, correspond to the 68%, 84% and 98.9% levels. The title of this plot indicates the median of the distribution as well as the range of the 68% credible interval. Here, $x$--$y$--$z$ shifts are measured in $\mathrm{km}$, while $T_s$ is in numerical units.