Magnetic equilibria of relativistic axisymmetric stars: The impact of flow constants

TL;DR

This paper explores the role of flow constants in the magnetic equilibria of relativistic axisymmetric stars, revealing constraints on magnetic field configurations and providing new theoretical and numerical insights into neutron star models.

Contribution

It revisits the flow constant formalism to derive new results on magnetic equilibria and presents numerical solutions for multipolar magnetic fields in relativistic stars.

Findings

Mixed poloidal-toroidal fields are incompatible with no meridional flows, except in singular cases.

Flow constants can be used to identify features of neutron star magnetic equilibria.

Numerical solutions demonstrate the application of the formalism to realistic star models.

Abstract

Symmetries and conservation laws associated with the ideal Einstein-Euler system, for stationary and axisymmetric stars, can be utilized to define a set of flow constants. These quantities are conserved along flow lines in the sense that their gradients are orthogonal to the four-velocity. They are also conserved along surfaces of constant magnetic flux, making them powerful tools to identify general features of neutron star equilibria. One important corollary of their existence is that mixed poloidal-toroidal fields are inconsistent with the absence of meridional flows except in some singular sense, a surprising but powerful result first proven by Bekenstein and Oron. In this work, we revisit the flow constant formalism to rederive this result together with several new ones concerning both nonlinear and perturbative magnetic equilibria. Our investigation is supplemented by some numerical solutions for multipolar magnetic fields on top of a Tolman-VII background, where strict power-counting of the flow constants is used to ensure a self-consistent treatment.