We investigate the effects of the purely toroidal magnetic field on the
equilibrium structures of the relativistic stars. The master equations for
obtaining equilibrium solutions of relativistic rotating stars containing
purely toroidal magnetic fields are derived for the first time. To solve these
master equations numerically, we extend the Cook-Shapiro-Teukolsky scheme for
calculating relativistic rotating stars containing no magnetic field to
incorporate the effects of the purely toroidal magnetic fields. By using the
numerical scheme, we then calculate a large number of the equilibrium
configurations for a particular distribution of the magnetic field in order to
explore the equilibrium properties. We also construct the equilibrium sequences
of the constant baryon mass and/or the constant magnetic flux, which model the
evolution of an isolated neutron star as it loses angular momentum via the
gravitational waves. Important properties of the equilibrium configurations of
the magnetized stars obtained in this study are summarized as follows ; (1) For
the non-rotating stars, the matter distribution of the stars is prolately
distorted due to the toroidal magnetic fields. (2) For the rapidly rotating
stars, the shape of the stellar surface becomes oblate because of the
centrifugal force. But, the matter distribution deep inside the star is
sufficiently prolate for the mean matter distribution of the star to be
prolate. (3) The stronger toroidal magnetic fields lead to the mass-shedding of
the stars at the lower angular velocity. (4) For some equilibrium sequences of
the constant baryon mass and magnetic flux, the stars can spin up as they lose
angular momentum.Comment: 13 figures, 7 tables, submitted to PR
