Equilibrium models of relativistic stars with a toroidal magnetic field

TL;DR

This paper models rotating relativistic stars with toroidal magnetic fields, analyzing their shape and internal distortions, revealing new limits and formulas relevant for understanding magnetars and gravitational wave emissions.

Contribution

It introduces a comprehensive relativistic model for magnetized, rotating stars and derives a simple formula for their deformation coefficients, improving upon Newtonian estimates.

Findings

Non-rotating models can have unlimited magnetization and deformation.

Rotating models face a mass-shedding limit at lower frequencies than unmagnetized stars.

The derived formula accurately predicts deformations for highly magnetized, rotating neutron stars.

Abstract

We have computed models of rotating relativistic stars with a toroidal magnetic field and investigated the combined effects of magnetic field and rotation on the apparent shape (i.e. the surface deformation), which could be relevant for the electromagnetic emission, and on the internal matter distribution (i.e. the quadrupole distortion), which could be relevant for the emission of gravitational waves. Using a sample of eight different cold nuclear physics equations of state, we have computed models of maximum field strength, as well as the distortion coefficients for the surface and the quadrupolar deformations. Surprisingly, we find that non-rotating models admit arbitrary levels of magnetization, accompanied by a growth of size and quadrupole distortion to which we could not find a limit. Rotating models, on the other hand, are subject to a mass-shedding limit at frequencies well below the corresponding ones for unmagnetized stars. Overall, the space of solutions can be split into three distinct classes for which the surface deformation and the quadrupole distortion are either prolate and prolate, oblate and prolate, or oblate and oblate, respectively. We also derive a simple formula expressing the relativistic distortion coefficients, which allows one to compute the surface deformation and the quadrupole distortion up to significant levels of rotation and magnetization, essentially covering all known magnetars. Such a formula replaces Newtonian equivalent expressions that overestimate the magnetic quadrupole distortion by about a factor of 6 and are inadequate for strongly relativistic objects like neutron stars.