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Relativistic stars with purely toroidal magnetic fields with realistic equations of state

We investigate equilibrium sequences of relativistic stars containing purely toroidal magnetic fields with four kinds of realistic equations of state (EOSs) of SLy (Douchin et al.), FPS (Pandharipande et al.), Shen (Shen et al.), and LS (Lattimer & Swesty). We numerically construct thousands of equilibrium configurations. Particularly we pay attention to the equilibrium sequences of constant baryon mass and/or constant magnetic flux, which model evolutions of an isolated neutron star. Important properties obtained in this study are summarized as follows ; (1) The dependence of the mass-shedding angular velocity on the EOSs is determined from that of the non-magnetized case. The stars with Shen(FPS) EOS reach the mass-shedding limit at the smallest(largest) angular velocity, while the stars with SLy or Lattimer-Swesty EOSs take the moderate values. (2) For the supramassive sequences, the equilibrium configurations are found to be generally oblate for the realistic EOSs in sharp contrast to the polytropic stars. For FPS(LS) EOS, the parameter region which permits the prolately deformed stars is widest(narrowest). For SLy and Shen EOS, it is in medium. Furthermore, the angular velocities $Ω_{\rm up}$, above which the stars start to spin up as they lose angular momentum, are found to depend sharply on the realistic EOSs. Our analysis indicates that the hierarchy of this spin up angular velocity is $Ω_{\rm up,SLy} > Ω_{\rm up,FPS} > Ω_{\rm up,LS}>Ω_{\rm up,Shen}$ and this relation holds even if the sequences have strong magnetic fields. Our results suggest the EOSs within the relativistic stars containing purely toroidal magnetic fields can be constrained by observing the angular velocity, the gravitational wave, and the signature of the spin up.