We have computed models of rotating relativistic stars with a toroidalmagnetic field and investigated the combined effects of magnetic field androtation on the apparent shape (i.e. the surface deformation), which could berelevant for the electromagnetic emission, and on the internal matterdistribution (i.e. the quadrupole distortion), which could be relevant for theemission of gravitational waves. Using a sample of eight different cold nuclearphysics equations of state, we have computed models of maximum field strength,as well as the distortion coefficients for the surface and the quadrupolardeformations. Surprisingly, we find that non-rotating models admit arbitrarylevels of magnetization, accompanied by a growth of size and quadrupoledistortion to which we could not find a limit. Rotating models, on the otherhand, are subject to a mass-shedding limit at frequencies well below thecorresponding ones for unmagnetized stars. Overall, the space of solutions canbe split into three distinct classes for which the surface deformation and thequadrupole distortion are either prolate and prolate, oblate and prolate, oroblate and oblate, respectively. We also derive a simple formula expressing therelativistic distortion coefficients, which allows one to compute the surfacedeformation and the quadrupole distortion up to significant levels of rotationand magnetization, essentially covering all known magnetars. Such a formulareplaces Newtonian equivalent expressions that overestimate the magneticquadrupole distortion by about a factor of 6 and are inadequate for stronglyrelativistic objects like neutron stars.
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