Revisiting thermoelectric effects in the crust of neutron stars

Abstract

Context. Large thermal variations have been observed in neutron stars that typically are not aligned with density gradients. Such terms may activate the Biermann battery effect, leading to thermoelectric interactions and to the generation of electromotive force. Aims. We aim to identify the possible impact of a temperature anisotropy on the crust of a neutron star can have in the evolution of its magnetic field, through the thermoelectric terms. Methods. We consider a neutron star crust with large temperature gradients, associated with long-lived hot spots, described by a localized Gaussian-type function. We simulate the interplay between the battery term and the Hall and Ohmic evolution numerically for axisymmetric systems. Results. The results indicate that for crust temperatures of $\sim$$10^9$ K the toroidal field can be amplified up to $\sim$$10^{14}-10^{15}$ G near the points of maximum temperature gradients, and it locally changes the architecture of the poloidal field lines. For internal crustal temperatures of $\sim$$10^8$ K, the temperature gradient generates fields that are lower by about two orders of magnitude. In these cases, saturation is achieved after some hundred thousand years, after which the battery and Ohmic dissipation balance each other, whereas the Hall drift contributes comparatively little to the final field strength, but it can affect its structure. Conclusions. We conclude that the thermoelectric effect can impact the overall magnetic field evolution, provided that the thermal gradient is maintained for a sufficiently long time. Neutron stars endowned with moderate-strength magnetic fields may be affected by the thermoelectric effect if the hotspots survive for timescales of a few kiloyears.