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Distortion of neutron stars with a toroidal magnetic field

Models of rotating relativistic stars with a toroidal magnetic field have been computed for a sample of eight equations of state of cold dense matter. Non-rotating models admit important levels of magnetization and quadrupole distortion accompanied by a seemingly unlimited growth in size. Rotating models reach the mass-shedding limit at smaller angular velocities than in the non-magnetized case according to the larger circumferential equatorial radius induced by the magnetic field. Moreover, they can be classified as prolate-prolate, oblate-prolate, or oblate-oblate with respect to surface deformation and quadrupole distortion. Simple expressions for surface and quadrupole deformation are provided that are valid up to magnetar field strengths and rapid rotation.