General relativistic compact stars with exotic matter

For such studies on compact stars, general relativistic effects are fundamentally important, since baryon density is comparable to pressure, ρ 0 c 2 ∼ P, hence the gravitational forth is strong. Moreover, strong magnetic field may change hydrostatic equilibriums of compact stars.

In this paper, we study the structures of general relativistic hybrid stars with purely toroidal magnetic field. To calculate magnetized compact stars, we adopt the scheme based on axisymetric and stationary formalism including purely toroidal magnetic field [16]. A star with pure toroidal magnetic field is unstable, however it becomes another stable star in which toroidal magnetic field is dominant in the dynamical simulation [17]. Moreover, Heger et al. have suggest that the toroidal magnetic field may dominate 10 5 times lager than the poloidal magnetic field at the last stage of the main sequence [18].

The organization of this paper is as follows. Adopted EOSs are briefly discussed in Sec. 2 with equilibriums of magnetized rotating compact stars. In Sec. 3, we show our nuemirical results. In Sec. 4, we discuss the consequence of our calculations.

The hardness of EOSs is an important ingredient for determining the equilibrium configurations as mentioned in Sec. 1. Though proto-neutron stars left after supernova explosions are very hot, T ∼ 50 MeV, they cool down to cold neutron stars (T ∼ 1MeV) in some tens of seconds [19]. Therefore we assume that the temperature for hydrostatic compact stars is zero, since EOSs at such low temperature (T ∼ 1MeV) show almost same stiffness as at zero-temperature. It means the temperature of compact stars is always much smaller than the typical chemical potentials. Moreover, the loss of neutrinos means that the chemical potentials of the neutrinos may be set to zero. Thus, we impose the barotropic condition of the EOSs (P = P(ε)) by assuming zerotemperature, zero-neutrino fraction, and beta-equilibrium.

Our theoretical framework for the hadronic phase of matter is the nonrelativistic Brueckner-Hartree-Fock approach including hyperons such as Σ -and Λ [20].

For quark phase, we adopt the MIT bag model for quark phase of u, d ,s-quarks, because this is the first step for the study on strucures of hybrid stars with rotation and magnetic field in the general relativistic formulations, though it is a toy model. We use massless u and d quarks, and s-quark with a current mass of m s = 180 MeV. We set that the bag constants, B, is 100 MeV fm -3 in this paper. Values of B > 150 MeV fm -3 can also be excluded within our model, because we do not obtain any more a phase transition in beta-stable matter in combination with our hadronic EOS.

For mixed phase, we must take into account the Gibbs condition, which require the pressure balance and the equality of the chemical potentials between the two phases besides the thermal equilibrium [21]. We use the Thomas-Fermi-approximation for the density plofiles of hadrons and quarks. In each cell, we must calculate the balance of the colomb interaction and the surface tension. However, there are a wide range of uncertainties about the surface tension, σ ∼ 10-100 MeV fm -2 , which are suggested by some theoretical estimates based on the MIT bag model for strangelets [22] and lattice guage simulations at finite temperature [23,24].

As for this uncertainity, Maruyama et al. have showed that the surface tensions of σ > 40 MeV fm -2 do not change the hardness of EOSs [25]. Then, we adapt two values of surface tension as σ = 10 MeV fm -2 for the minimum case, and σ = 40 MeV fm -2 for the maximum case.

At the maximum densities higher than two times of saturation density, muons may appear [26,27]. However, we neglect it, since the muon contribution to pressure at the higher density has been pointed to be very small [28].

Finally, we show our EOSs on Fig. 1. The left two panels show the pressure versus the baryon density for the quark-hadron mixed phase. For weak surface tension (σ = 10 MeV fm -2 ), the droplet structure does not appear, wheres, for strong surface tension (σ = 40 MeV fm -2 ), the rod structure does not appear. The number of hyperons are supressed because of the appearnce of quarks for both surface tensions. The right panel is for EOSs of nucleis with/without hyperons.

Master equations for the rotating relativistic stars containing purely toroidal magnetic fields are based on the assumptions summarized as follows [16]; (1) Equilibrium models are stationary and axisymmet

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