Magnetic Field Evolution in Neutron Star Crusts Due to the Hall Effect and Ohmic Decay

Abstract

We present calculations of magnetic field evolution by the Hall effect and Ohmic decay in the crust of neutron stars (NSs). In accreting NSs, Ohmic decay is always the dominant effect due to the large resistivity. In isolated NSs with relatively pure crusts, the Hall effect dominates Ohmic decay after a time t switch ≃ 10 4 yr B −3 12 , where B 12 is the magnetic field strength in units of 10 12 G. We compute the evolution of an initial field distribution by Ohmic decay, and give approximate analytic formulas for both the surface and interior fields as a function of time. Due to the strong dependence of t switch on B 12 , early Ohmic decay can alter the currents down to the base of the crust for B ∼ 10 11 G, neutron drip for B ∼ 10 12 G, and near the top of the crust for B 10 13 G, respectively. We then discuss magnetic field evolution by the Hall effect. Several examples are given to illustrate how an initial field configuration evolves. Hall wave eigenfunctions are computed, including the effect of the large density change across the crust. We estimate the response of the crust to the magnetic stresses induced by Hall waves, and give a detailed discussion of the boundary conditions at the solid-liquid interface. Finally, we discuss the implications for the Hall cascade proposed by Goldreich & Reisenegger.

Magnetic fields in neutron stars: A theoretical perspective

“…A complementary approach is to study analytic solutions of the Hall equation (with or without the resistive term), which has been done by Vainshtein et al [40], Cumming et al [10], and by our group [1,25]. In the latter, which extends and generalizes the work of Ref.…”

“…where the two terms on the right-hand side correspond to the Hall drift and resistive diffusion, respectively, whose relative importance in the neutron star crust is still a matter of controversy [10,16]. Here, we discuss some physical issues that are likely to determine the evolution of magnetic field under the Hall effect, without attempting to cover the many simulations recently performed to address how specific magnetic field configurations might decay in real neutron stars [32,14].…”

Hall cascades versus instabilities in neutron star magnetic fields

“…None of the fully nonlinear calculations, of ourselves or numerous other authors (Biskamp et al 1996(Biskamp et al , 1999Dastgeer et al 2000;Dastgeer & Zank 2003;Cho & Lazarian 2004;Shaikh & Zank 2005;Cho & Lazarian 2009), which have a broad variety of different initial conditions including sufficiently strong magnetic fields, have ever found anything other than a standard Hall cascade. Cumming et al (2004) also consider the possibility of a Hall instability, and its relevance for the evolution of neutron star magnetic fields. They clarify the nature of the instability: a background shear in the electron velocity drives growth of long-wavelength, i.e., large-scale, perturbations.…”

Magnetic Axis Drift and Magnetic Spot Formation in Neutron Stars with Toroidal Fields

“…The crust of a neutron star can be approximated to good accuracy by an ion Coulomb lattice, where only electrons have the freedom to move (Cumming et al 2004). Therefore, the electric current will be carried solely by electrons so that = -j v en e e , where j is the electric current density, e the elementary electron charge, n e the electron number density, and v e the electron velocity.…”