Hall Attractor in Axially Symmetric Magnetic Fields in Neutron Star Crusts (1311.7345v2)

Abstract: We have found an attractor for an axially symmetric magnetic field evolving under the Hall effect and subdominant ohmic dissipation, resolving the question of the long term fate of the magnetic field in neutron star crusts. The electron fluid is in isorotation, analogous to Ferraro's law, with its angular velocity being approximately proportional to the poloidal magnetic flux, $\Omega \propto \Psi$. This equilibrium is the long term configuration of a magnetic field evolving because of the Hall effect and ohmic dissipation. For an initial dipole dominated field the attractor consists mainly of a dipole and an octupole component accompanied by an energetically negligible quadrupole toroidal field. The field dissipates in a self-similar way: although higher multipoles should have been decaying faster, the toroidal field mediates transfer of energy into them from the lower ones, leading to an advection diffusion equilibrium and keeping the ratio of the poloidal multipoles almost constant. This has implications for the structure of the intermediate age neutron stars, suggesting that their poloidal field should consist of a dipole and a octupole component accompanied by a very weak toroidal quadrupole. For initial conditions that have a higher multipole $\ell$ structure the attractor consists mainly of $\ell$ and $\ell+2$ poloidal components.

• The paper demonstrates that the Hall effect directs the evolution of neutron star crust magnetic fields toward a stable attractor state combining dipole and octupole configurations.
• The authors use time-evolution simulations in spherical coordinates to show that electron angular velocity aligns with poloidal magnetic flux in an isorotational equilibrium.
• These findings challenge traditional views on neutron star magnetism and offer new insights into spin-down rates and thermal evolution in middle-aged stars.

Hall Attractor in Axially Symmetric Magnetic Fields in Neutron Star Crusts

The paper, authored by Konstantinos N. Gourgouliatos and Andrew Cumming, addresses the evolution of magnetic fields within neutron star crusts under the influence of the Hall effect and ohmic dissipation. The study uncovers an attractor state for axially symmetric magnetic fields, resolving the longstanding question concerning the long-term fate of such fields in neutron star crusts. The implications span both the theoretical understanding of magnetohydrodynamics (MHD) in such unique environments and the practical interpretation of magnetic field structures in neutron stars.

Theoretical Insights and Methodology

The study begins by explaining the dynamics governing magnetic fields in neutron star crusts processed through the Hall effect, which results from the interplay of electromagnetic forces in contexts where high electrical conductivity and a single species of charge carriers (electrons) are present. A central aspect of the research is whether the Hall effect induces a turbulent magnetic cascade or stabilizes into a distinct attractor state. The authors employ simulations to demonstrate that rather than leading to complete dissipation via turbulent cascades, the Hall effect drives the system towards an isorotational equilibrium. This equilibrium is characterized by the angular velocity of the electron fluid being approximately proportional to the poloidal magnetic flux, symbolized as $\Omega \propto \Psi$.

In terms of methodology, the paper utilizes time-evolution simulations of axially symmetric magnetic fields in spherical coordinates to investigate their long-term behavior under varying conditions, including initial multipole structures beyond dipoles. These simulations reveal that after initial Hall-driven dynamism, the fields settle into a predictable structural state dictated by a constant electron angular velocity along field lines, reminiscent of Ferraro’s law.

Numerical Findings and Implications

One of the pivotal findings is that neutron star magnetic fields, initially configured with a dominant dipole or higher-order multipoles, stabilize into a state composed mainly of a dipole and octupole configuration, consistent with the proposed attractor state. Furthermore, the study suggests a self-similar energy dissipation process, wherein energy from the lower multipoles is redirected to higher ones, maintaining a consistent poloidal multipole ratio. Notably, the decay of multipoles does not follow the intuitive $\ell^{-2}$ decay due to this continual energy transfer mediated by the toroidal field, yielding a near-stable ratio between the dipole and octupole components.

Theoretical implications extend to a deeper understanding of magnetic field evolution under non-ideal MHD conditions, particularly in crusts of neutron stars where the Hall drift assumes significant dynamical importance. Practically, the results challenge traditional assumptions about neutron star magnetism, particularly in middle-aged stars, impacting interpretations of spin-down rates and surface magnetic field configurations. Predicting neutron star field configurations with more complexity than a classic dipole has profound implications for the understanding of neutron star thermal evolution and emission profiles.

Future Directions

The paper hints at several avenues for future research, including the exploration of similar attractor states in non-axially symmetric fields. Extending simulations to three dimensions or incorporating rotational dynamics more intricately could reveal further intricacies or affirm the robustness of the discovered attractor state. Moreover, understanding how these findings translate to observable electromagnetic spectra or influence neutron star collisions remains an open area of exploration.

In summary, this paper provides significant insight into the magnetic field dynamics of neutron star crusts, offering a new lens through which to understand the complex interactions governed by the Hall effect and ohmic dissipation. By discovering a magnetic field attractor state, the study not only addresses theoretical gaps but also contributes to the broader astrophysical discourse on neutron star magnetism.