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Astron. Astrophys. 357, 968-976 (2000)

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3. Model EOS with deconfinement phase transition

For the investigation of the deconfinement phase transition expected to occur in neutron star matter at densities above the nuclear saturation density [FORMULA] several approaches to quark confinement dynamics have been discussed, see e.g. Blaschke et al. (1990) , Blaschke et al. (1999) , Drago et al. (1996)which lead to interesting conclusions for the properties of quark matter at high densities. Most of the approaches to quark deconfinement in neutron star matter, however, use a thermodynamical bag-model for the quark matter and employ a standard two-phase description of the equation of state (EOS) where the hadronic phase and the quark matter phase are modeled separately and the resulting EOS is obtained by imposing Gibbs' conditions for phase equilibrium with the constraint that baryon number as well as electric charge of the system are conserved (Glendenning 1992, 1997). Since the focus of our work is the elucidation of qualitative features of signals for a possible deconfinement transition in the pulsar timing, we will consider here such a rather standard, phenomenological model for an EOS with deconfinement transition.

The total pressure [FORMULA] as a thermodynamical potential is given by

[EQUATION]

where

[EQUATION]

is the EOS of the relativistic [FORMULA] mean-field model (Walecka model) for nuclear matter (Walecka 1974; Kapusta 1989), where the masses and chemical potentials have to be renormalized by the mean-values of the [FORMULA] and [FORMULA] fields [FORMULA], [FORMULA]. The pressure for two-flavor quark matter within a bag model EOS is given by

[EQUATION]

where B denotes the phenomenological bag pressure that enforces quark confinement and the transition to nuclear matter at low densities. For our numerical analyses in the present work, we assume here a value of [FORMULA] which allows us, e.g., to discuss a neutron star of 1.4 solar masses with an extended quark matter core.

In a neutron star, these phases of strongly interacting matter are in [FORMULA] equilibrium with electrons and muons which contribute to the pressure balance with

[EQUATION]

In the above expressions [FORMULA] is the partial pressure of the Fermion species i as a sum of particle and antiparticle contributions defined by

[EQUATION]

where [FORMULA]. All the other thermodynamic quantities of interest can be derived from the pressure (32) as, e.g., the partial densities of the species

[EQUATION]

The chemical equilibrium due to the direct and inverse [FORMULA]- decay processes imposes additional constraints on the values of the chemical potentials of leptonic and baryonic species (Glendenning 1997; Sahakian 1995). Only two independent chemical potentials remain according to the corresponding two conserved charges of the system, the total baryon number [FORMULA] as well as electrical charge Q

[EQUATION]

The deconfinement transition is obtained following the construction which obeys the global conservation laws and allows one to find the volume fraction of the quark matter phase [FORMULA] in the mixed phase where [FORMULA], so that at given [FORMULA] and T the total pressure under the conditions (38), (39) is a maximum, see Glendenning (1992, 1997).

In Fig. 1 we show the model EoS with deconfinement transition as described above. Note that in the density region of the phase transition there is a monotonous increase of the pressure which gives rise to an extended mixed phase region in the compact star after solution of the equations of hydrodynamic stability (13). For comparison, the relativistic mean-field EoS of Glendenning (1989)including [FORMULA] mesons, hyperons and muons (incompressibility [FORMULA] MeV, dotted line) and that of Glendenning (1992)including a deconfinement transition to three-flavor quark matter in a bag model with [FORMULA] MeV4 are shown, see also the monographs by Glendenning (1997)and Weber (1999).

[FIGURE] Fig. 1. Model equation of state for the pressure of hybrid star matter in [FORMULA]-equilibrium as a function of the baryon number density. The hadronic equation of state is a relativistic mean-field model ([FORMULA]), the quark matter one is a two-flavor bag model with [FORMULA] MeV fm-3. For comparison the relativistic mean-field EoS of Glendenning (1989)including [FORMULA] mesons, hyperons and muons (incompressibility [FORMULA] MeV, dotted line) and that of Glendenning (1992)including a deconfinement transition to three-flavor quark matter in a bag model with [FORMULA] MeV4 (dashed line) are shown.

In Fig. 2 we show the composition of the hybrid star matter as a function of the total baryon density at [FORMULA]. Solving the Tolman-Oppenheimer-Volkoff equations (13)- (15) for the hydrodynamical equilibrium of static spherically symmetric relativistic stars with the above defined EOS, we find that a configuration at the stability limit could have a quark matter core with a radius as large as [FORMULA] of the stars radius.

[FIGURE] Fig. 2. Composition of hybrid star matter in [FORMULA]-equilibrium as a function of baryon number density for the model EoS used in this work.

What implications this phase transition for rotating star configurations might have will be investigated in the next section by applying the method developed in Sect. 2 for the above EOS.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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