The impact of superconductivity and the Hall effect in models of magnetised neutron stars

Abstract

Equilibrium configurations of the internal magnetic field of a pulsar play a key role in modelling astrophysical phenomena from glitches to gravitational wave emission. In this paper, we present a numerical scheme for solving the Grad–Shafranov equation and calculating equilibrium configurations of pulsars, accounting for superconductivity in the core of the neutron star, and for the Hall effect in the crust of the star. Our numerical code uses a finite difference method in which the source term appearing in the Grad–Shafranov equation, which is used to model the magnetic equilibrium is non-linear. We obtain solutions by linearising the source and applying an under-relaxation scheme at each step of computation to improve the solver’s convergence. We have developed our code in both C++ and Python, and our numerical algorithm can further be adapted to solve any non-linear PDEs appearing in other areas of computational astrophysics. We produce mixed toroidal–poloidal field configurations, and extend the portion of parameter space that can be investigated with respect to previous studies. We find that in even in the more extreme cases, the magnetic energy in the toroidal component does not exceed approximately 5% of the total. We also find that if the core of the star is superconducting, the toroidal component is entirely confined to the crust of the star, which has important implications for pulsar glitch models which rely on the presence of a strong toroidal field region in the core of the star, where superfluid vortices pin to superconducting fluxtubes.

Pulsar glitches: observations and physical interpretation

“…This region has a spatially confined extent in the core that varies with the total (interior) magnetic field strength. Early results suggest that the mixed toroidal-poloidal field region might reside only in the crust for internal field intensities below ∼10 13 G [183,184]. Even if pinning to fluxtubes does not result in the core region to act as angular momentum reservoir, it can still raise the coupling timescales locally, reducing I α and contributing to the observed recoveries [54].…”

“…In particular, many models rely on toroidal components of the magnetic field in the interior to provide the pinning region and the angular momentum reservoir [54]. Realistic equilibrium models for magnetic fields in superconducting neutron stars, however, suggest that the toroidal field is entirely confined to the crust of the star for magnetic field intensities like the ones expected in regular pulsars [183,184]. Additional theoretical work is required to understand if this scenario is viable.…”

Magnetically confined mountains on accreting neutron stars in general relativity

“…Magnetically confined mountains are formed by the accretion of conducting plasma onto the magnetic polar cap of a neutron star in systems like low-mass X-ray binaries (LMXBs). Several features have been incorporated previously into the theoretical model, including mountain stability (Vigelius & Melatos 2008;Mukherjee & Bhattacharya 2012;Mukherjee et al 2013), Ohmic and thermal relaxation (Vigelius & Melatos 2009;Suvorov & Melatos 2019), core superconductivity (Passamonti & Lander 2014;Sur & Haskell 2021), mountain sinking (Choudhuri & Konar 2002;Wette et al 2010), various equations of state (Priymak et al 2011;Mukherjee 2017), triaxial configurations (Singh et al 2020), higher magnetic multipole moments and toroidal fields (Suvorov & Melatos 2020;Fujisawa et al 2022). Recently, Rossetto et al (2023) formulated the problem of magnetically confined mountains on neutron stars in general relativity.…”

Continuous Gravitational Wave Emissions from Neutron Stars with Pinned Superfluids in the Core

“…In particular the presence of a stronger toroidal field in the interior of the star, which leads to regions of strong pinning of vortices to superconducting flux-tubes (see [22,43] and references therein), could explain some glitch features [44,45] like glitch overshoots [42,46]. From the theoretical point of view the situation is unclear, as models have been constructed with toroidal components of the field that can be up to two orders of magnitude stronger than the inferred exterior dipole [47], but the inclusion of superconductivity in the models generally leads to the expulsion of the toroidal flux to the crust of the star [48,49], restricting the amount of angular momentum that could be stored in the outer core to power a glitch.…”

“…The equilibrium structure of the magnetic field in the interior of a NS is still an open problem[63], nevertheless several models have been developed, and there is a general consensus that to be stable, the field must have a twisted-torus configuration, in which a strong interior toroidal component stabilises the poloidal component of the field which stretches outside the star[47,64,65]. The relative strength of the toroidal field is particularly interesting, as while many models require this component to be somewhat weaker (roughly an order of magnitude) than the poloidal component[48,49], there are several models that predict stronger toroidal components in the interior[52,66], which would lead to regions of strong vortex-flux-tube pinning[22]. From the observational side, there is some indication that a pinned superfluid must exist in the core of at least the Vela pulsar, to explain the large fraction of spin-down that is inverted by its glitches over time (the activity, see the discussion in[39]), but also to explain the difference in post-glitch relaxation with respect to the Crab pulsar[41], and the behaviour of moderately active pulsars displaying glitches of large size like PSR J1341-6220[40].In order to calculate the strongest GW signal we consider the case of an orthogonal rotator, in which the magnetic field axis is perpendicular to the rotation axis.…”