Equilibrium models of relativistic stars with a toroidal magnetic field

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 magnetisation, 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 unmagnetised 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 and that allows one to compute the surface deformation and the quadrupole distortion up to significant levels of rotation and magnetisation, essentially covering all known magnetars. Such formula replaces Newtonian equivalent expressions that overestimate the quadrupole distortion by about a factor of five and are inadequate for strongly-relativistic objects like neutron stars