Interlinking internal and external magnetic fields of relativistically rotating neutron stars

Laboratory of Universe Sciences, Department of Physics, University of Patras, Patras, Rio 26504, Greece

^⋆ Corresponding authors: d.ntotsikas@upatras.gr; kngourg@upatras.gr

Received: 7 March 2025
Accepted: 31 May 2025

Abstract

Context. Both the internal and external magnetic field of neutron stars have been studied thoroughly via sophisticated methods that incorporate relativistic effects at the exterior and a magneto-hydrostatic equilibrium at the interior, or even more complex regimes. However, the fundamental issue of linking the internal to the external field in a self-consistent way remains unresolved. To achieve a realistic depiction of the magnetic field, both the internal and external configurations need to be addressed in a single, simultaneous calculation.

Aims. Our aim is to solve for the structure of the magnetic field of a neutron star within the stellar interior and the relativistically rotating magnetosphere, adhering to barotropic equilibria in the interior and the relativistic force-free condition at the exterior.

Methods. We solved the axisymmetric pulsar equation for the magnetosphere and the associated equilibrium equations for the neutron star’s interior by employing an elliptic solver and using the method of simultaneous relaxation for the magnetic field inside and outside the star. Appropriate boundary conditions were implemented at the interior of the star, the light cylinder, and the external boundary of the integration domain.

Results. We have found self-consistent solutions corresponding to a variety of combinations of internal and external fields. In all cases, the external field satisfies the force-free axisymmetric pulsar equation. The internal field satisfies a barotropic equilibrium and extends to the centre of the star. If a toroidal field is included at the interior of the star, it either has the form of a twisted torus confined within the flux surfaces that close inside the star, or it extends to the magnetosphere but is contained to the field lines that close within the light cylinder.

Conlcusions. This work presents a global solution for the internal and external field of an axisymmetric rotating neutron star. It is shown that the twist of the internal field affects the external field by increasing the number of open field lines and eventually the spin-down rate of the star. This effect is far more drastic if the toroidal field, and consequently the poloidal current flowing within the star, is allowed to populate the closed field lines of the magnetosphere instead of remaining confined in the star. We further remark that the internal field structure depends on the presence of a twisted magnetosphere: if the twist current is not allowed to flow into the magnetosphere, it only occupies a narrow toroid at the interior of the star, whereas if the twist currents are allowed to flow into the magnetosphere, the internal toroidal field can occupy a significant volume of the stellar interior. Strong magnetospheric currents may also impact the emission mechanisms and lead to fluctuations in magnetar spin-down rates, the moding and nulling of pulsars, a correlation between angular shear and twist, and the general morphology of the pulsar magnetic field resulting in various observational manifestations. The magnetospheric toroidal fields may dissipate, and thus the system may switch from a global twist to an internal twist and consequently exhibit transient behavior.