Hall drift in the crust of neutron stars - necessary for radio pulsar activity?

The radio pulsar models based on the existence of an inner accelerating gap located above the polar cap rely on the existence of a small scale, strong surface magnetic field $B_s$. This field exceeds the dipolar field $B_d$, responsible for the braking of the pulsar rotation, by at least one order of magnitude. Neither magnetospheric currents nor small scale field components generated during neutron star’s birth can provide such field structures in old pulsars. While the former are too weak to create $B_s \gtrsim 5\times 10^{13}$G$;\gg B_d$, the ohmic decay time of the latter is much shorter than $10^6$ years. We suggest that a large amount of magnetic energy is stored in a toroidal field component that is confined in deeper layers of the crust, where the ohmic decay time exceeds $10^7$ years. This toroidal field may be created by various processes acting early in a neutron star’s life. The Hall drift is a non-linear mechanism that, due to the coupling between different components and scales, may be able to create the demanded strong, small scale, magnetic spots. Taking into account both realistic crustal microphysics and a minimal cooling scenario, we show that, in axial symmetry, these field structures are created on a Hall time scale of $10^3$-$10^4$ years. These magnetic spots can be long-lived, thereby fulfilling the pre-conditions for the appearance of the radio pulsar activity. Such magnetic structures created by the Hall drift are not static, and dynamical variations on the Hall time scale are expected in the polar cap region.

💡 Research Summary

The paper addresses a long‑standing problem in radio pulsar theory: the inner accelerating gap above the polar cap, which is required for pair creation and coherent radio emission, demands a surface magnetic field component $B_{\rm s}$ that is at least an order of magnitude stronger than the large‑scale dipole field $B_{\rm d}$ inferred from spin‑down. Observations and gap models suggest $B_{\rm s}\gtrsim5\times10^{13}$ G, whereas $B_{\rm d}$ is typically $10^{12}$–$10^{13}$ G. Conventional sources of such a strong, small‑scale field are inadequate. Magnetospheric return currents can only generate fields of order $10^{12}$ G, far below the required strength. Small‑scale fields that might be imprinted at birth decay on an Ohmic timescale $\tau_{\rm ohm}\sim L^{2}/\eta$ that is much shorter than $10^{6}$ yr for the crustal depths where they would reside, so they cannot survive in old pulsars.

The authors propose that a substantial amount of magnetic energy is stored in a toroidal (azimuthal) component located deep in the neutron‑star crust, where the electrical conductivity is high and the Ohmic decay time exceeds $10^{7}$ yr. This toroidal field could be generated early in the star’s life by processes such as differential rotation, convective dynamos, or magneto‑thermal instabilities. The key mechanism that can bring this deep‑seated energy to the surface is the Hall drift, a non‑linear term in the induction equation arising from the motion of electrons in a strong magnetic field. The Hall term, $\mathbf{J}\times\mathbf{B}/(en_{e})$, couples different spatial scales and magnetic components, effectively transferring energy from large‑scale toroidal fields to small‑scale poloidal structures.

To test this idea, the authors perform axisymmetric (2‑D) magneto‑thermal simulations that incorporate realistic crustal microphysics (temperature‑dependent electrical and thermal conductivities, electron‑ion scattering) and a minimal cooling scenario (neutrino cooling plus photon surface emission). The initial configuration consists of a strong toroidal field confined to depths of a few hundred meters to a kilometer, superimposed on a weaker dipolar field. The simulations follow the evolution of the magnetic field over $10^{5}$ yr.

The results show that Hall drift operates on a characteristic Hall timescale $t_{\rm Hall}\sim 4\pi n_{e}eL^{2}/(cB)\approx10^{3}$–$10^{4}$ yr for the chosen parameters. Within this period, the toroidal energy is redistributed, producing localized magnetic “spots” on the surface with lateral sizes of a few hundred meters and field strengths $B_{\rm s}\sim10^{14}$ G—well above the threshold required for gap formation. Because the underlying toroidal reservoir decays only on the much longer Ohmic timescale, these spots can persist for $>10^{6}$ yr, i.e., throughout the observable lifetime of a typical radio pulsar. The strong field gradients associated with the spots provide the necessary electric potential to accelerate particles to energies that trigger curvature radiation and subsequent pair creation.

An important implication of the Hall‑drift scenario is that the surface magnetic configuration is not static. The non‑linear Hall evolution leads to quasi‑periodic rearrangements of the spot pattern on the Hall timescale, implying that the polar‑cap magnetic field—and therefore the gap properties—may vary over thousands of years. Such long‑term variability could manifest as secular changes in pulse profiles, drifting sub‑pulses, or mode switching observed in some pulsars.

The paper concludes that Hall drift offers a natural, physically robust pathway to generate and maintain the strong, small‑scale surface magnetic fields required for radio pulsar activity. It bridges the gap between deep‑crust toroidal reservoirs (which are long‑lived) and the polar‑cap conditions needed for pair creation, while also predicting observable long‑term magnetic variability. This mechanism thus enriches the theoretical framework for pulsar magnetospheres and suggests new avenues for interpreting timing and emission irregularities in the pulsar population.