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

Abstract

Equilibrium configurations of the internal magnetic field of a pulsar play a key role in modeling 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, used to model the magnetic equilibrium is nonlinear. We obtain solutions by linearizing 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 nonlinear 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 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.

Parameter estimation of a two-component neutron star model with spin wandering

“…Such forces are typically important dynamically with respect to the small lag |Ω s − Ω c | Ω c , Ω s . The steady-state coexistence of differential rotation and internal magnetic fields with open and closed topologies is a subtle problem which has been studied analytically (Easson 1979;Melatos 2012;Glampedakis & Lasky 2015) and numerically (Anzuini & Melatos 2020;Sur et al 2020;Sur & Haskell 2021). The time-dependent interaction between differential rotation and the Lorentz force has also been studied in the context of magnetized Couette flows and instabilities (Mamatsashvili et al 2019;Rüdiger et al 2020).…”

Long-term GRMHD simulation of magnetic field in isolated neutron stars

“…The evolution shows that the toroidal field attains a quasi-stable equilibrium with energies similar to the ones obtained from solving the Grad-Shafranov equation (see e.g. Lander & Jones (2009); Armaza et al (2015); Sur & Haskell (2021)) which gives equilibrium solutions but doesn't say anything about the stability of these equilibrium fields. The star continues to lose energy till the end of our simulation but the ratio of poloidal and toroidal energies to the total magnetic energy is seen to settle at a stationary value for pS64 but not in pS128 or pS256 (Figure 3 left panel).…”

“…Understanding equilibria requires us to solve the so-called Grad-Shafranov equation which yields various magnetic field configurations with varying poloidal and toroidal field energies (Lander & Jones 2009;Ciolfi & Rezzolla 2012Gourgouliatos et al 2013;Armaza et al 2015;Sur & Haskell 2021). However, these solutions do not tell us anything about the stability of the magnetic field with time.…”

Heating in Magnetar Crusts from Electron Captures

“…Recalling that the thickness δR and the (gravitational or baryonic) mass δM of the outer crust are very small compared to the neutron star radius R and mass M, respectively, typically, δR ∼ 10 −2 R and δM ∼ 10 −5 M (see, e.g., [60]), we adopted the plane parallel approximation. Since the toroidal component of the magnetic field is expected to be confined in the crust of thickness ∆R ∼ 0.1R (see, e.g., [61]), the last term in the right-hand side of Equation ( 43) of order B 2 /(4πR) is much smaller than the second term in the left-hand side of order B 2 /(8π∆R). We thus dropped the last term in Equation (43) as in [9].…”