The force-free twisted magnetosphere of a neutron star - II: Degeneracies of the Grad-Shafranov equation

TL;DR

This paper investigates the equilibrium solutions of force-free neutron star magnetospheres, revealing degeneracies and energy differences, and discusses the stability and physical plausibility of various configurations.

Contribution

It extends previous work by identifying multiple degenerate solutions, including higher energy states, and analyzes their properties and potential stability implications.

Findings

Multiple solutions exist for the same parameters, indicating degeneracy.

Lower energy solutions have up to 25% more energy than vacuum case.

Higher energy solutions can have up to 80% more energy, larger helicity, and stronger dipole fields.

Abstract

We extend our previous study of equilibrium solutions of non-rotating force-free magnetospheres of neutron stars. We show that multiple solutions exist for the same sets of parameters, implying that the solutions are degenerate. We are able to obtain configurations with disconnected field lines, however, in nearly all cases these correspond to degenerate higher energy solutions. We carry out a wide parametric search in order to understand the properties of the solutions. We confirm our previous results that the lower energy solutions have up to $\sim 25\%$ more energy than the vacuum case, helicity of the order of $\sim 5$ (in some defined units), maximum twist of $\sim 1.5$ rad, and a dipole strength that is up to $\sim 40\%$ larger than the vacuum dipole. Including the degenerate higher energy solutions allows for larger theoretical limits of up to $\sim 80\%$ more energy with respect to the vacuum case, helicity of the order of $\sim 8$, and a dipole strength that can now be up to four times that of the vacuum dipole, while the twist can be significantly larger and even diverge for configurations with disconnected domains. The higher energy solutions are probably unstable, therefore, it is unlikely that such magnetospheres exist under normal conditions in magnetars and high magnetic field pulsars.