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

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 because of the large resistivity. In isolated NSs with relatively pure crusts, the Hall effect dominates ohmic decay after a time t_switch ≃ 10⁴ yr B, where B₁₂ is the magnetic field strength in units of 10¹² G. We compute the evolution of an initial field distribution by ohmic decay and give approximate analytic formulae for both the surface and interior fields as a function of time. Because of the strong dependence of t_switch on B₁₂, early ohmic decay can alter the currents down to the base of the crust for B ~ 10¹¹ G, neutron drip for B ~ 10¹² G, and near the top of the crust for B ≳ 10¹³ G. 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.