Poloidal magnetic fields in superconducting neutron stars

Abstract

We develop the formalism for computing the magnetic field within an axisymmetric neutron star with a strong type II superconductor core surrounded by a normal conductor. The formalism takes full account of the constraints imposed by hydrostatic equilibrium with a barotropic equation of state. A characteristic of this problem is that the currents and fields need to be determined simultaneously and self-consistently. Within the core, the strong type II limit B ≪ H allows us to compute the shapes of individual field lines. We specialize to purely poloidal magnetic fields that are perpendicular to the equator, and develop the ‘most dipolar case’ in which field lines are vertical at the outer radius of the core, which leads to a magnetic field at the stellar surface that is as close to a dipole as possible. We demonstrate that although field lines from the core may only penetrate a short distance into the normal shell, boundary conditions at the inner radius of the normal shell control the field strength on the surface. Remarkably, we find that for a Newtonian N = 1 polytrope, the surface dipole field strength is B_surf ≃ H_bϵ_b/3, where H_b is the magnetic field strength at the outer boundary of the type II core and ϵ_bR is the thickness of the normal shell. For reasonable models, H_b ≈ 10¹⁴ G and ϵ_b ≈ 0.1 so the surface field strength is B_surf ≃ 3 × 10¹² G, comparable to the field strengths of many radio pulsars. In general, H_b and ϵ_b are both determined by the equation of state of nuclear matter and by the mass of the neutron star, but B_surf ∼ 10¹² G is probably a robust result for the ‘most dipolar’ case. We speculate on how the wide range of neutron star surface fields might arise in situations with less restrictions on the internal field configuration. We show that quadrupolar distortions are ∼−10⁻⁹(H_b/10¹⁴ G)² and arise primarily in the normal shell for B ≪ H_b.