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

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4. Results and discussion

The results for the stability of rotating neutron star configurations with possible deconfinement phase transition according to the EOS described in the previous section are shown in Fig. 3, where the total baryon number, the total mass, and the moment of inertia are given as functions of the equatorial radius (left panels) and in dependence on the central baryon number density (right panels) for static stars (solid lines) as well as for stars rotating with the maximum angular velocity [FORMULA] (dashed lines).

[FIGURE] Fig. 3. Baryon number N, mass M, and moment of inertia I as a function of the equatorial radius (left panels) and the central density (right panels) for neutron star configurations with a deconfinement phase transition. The solid curves correspond to static configurations, the dashed ones to those with maximum angular velocity [FORMULA]. The lines between both extremal cases connect configurations with the same total baryon number [FORMULA].

The dotted lines connect configurations with fixed total baryon numbers [FORMULA] and it becomes apparent that the rotating configurations are less compact than the static ones. They have larger masses, radii and momenta of inertia at less central density such that for suitably chosen configurations a deconfinement transition in the interior can occur upon spin-down.

In order to demonstrate the consistency of our perturbative approach, we show in Fig. 4 the values of the expansion parameter for the maximally attainable rotation frequencies of stationary rotating objects ([FORMULA]) as a function of the central density characterizing the configuration.

[FIGURE] Fig. 4. Expansion parameter for the maximally attainable rotation frequencies of stationary rotating objects [FORMULA] as a function of the central density characterizing the configuration. Vertical dashed lines indicate the density band for which a mixed phase occurs, see Fig. 2.

In Fig. 5 we show the critical regions of the phase transition in the inner structure of the star configuration as well as the equatorial and polar radii in the plane of angular velocity [FORMULA] versus distance from the center of the star. It is obvious that with the increase of the angular velocity the star is deforming its shape. The maximal eccentricities of the configurations with [FORMULA], [FORMULA] and [FORMULA] are [FORMULA], [FORMULA] and [FORMULA], respectively. Due to the changes of the central density the quark core could disappear above a critical angular velocity.

[FIGURE] Fig. 5. Phase structure of rotating hybrid stars in equatorial direction in dependence of the angular velocity [FORMULA] for stars with different total baryon number: [FORMULA]. Dahsed vertical lines indicate the maximum frequency [FORMULA] for stationary rotation.

In Fig. 6 we display the dependence of the moment of inertia on the angular velocity for configurations with the same total baryon number [FORMULA] together with the different contributions to the total change of the moment of inertia. As it is shown the most important contributions come from the mass redistribution and the shape deformation. The relativistic contributions due to field and rotational energy are less important. In the same Fig. 6 we show the decrease of the spherical moment of inertia due to the decrease of the central density for high angular velocities which tends to partially compensate the further increase of the total moment of inertia for large [FORMULA]. There is no dramatic change in the slope of [FORMULA] at [FORMULA] kHz.

[FIGURE] Fig. 6. Contributions to the dependence of the moment of inertia on the angular velocity.

Fig. 7 shows the dependence of the moment of inertia as a function of the angular velocity. It is demonstrated that the behavior of [FORMULA] for a given total number of baryons [FORMULA] strongly depends on the presence of a pure quark matter core in the center of the star. If the core does already exist or it does not appear when the angular velocity increases up to the maximum value [FORMULA] then the second order derivative of the moment of inertia [FORMULA] does not change its sign. For the configuration with [FORMULA] the critical value for the occurrence of the sign change is [FORMULA] kHz while for [FORMULA] it is close to [FORMULA] kHz.

[FIGURE] Fig. 7. Moment of inertia as a function of angular velocity with (solid lines) and without (dashed lines) deconfinement phase transition for fixed total baryon number [FORMULA] (left panel), [FORMULA] (middle panel), [FORMULA] (right panel).

In order to point out possible observable consequences of such a characteristic behaviour of [FORMULA] we consider two possible scenarios for changes in the pulsar timing: (A) dipole radiation and the resulting dependence of the braking index on the angular velocity as suggested by Glendenning et al. (1997)and (B) mass accretion onto rapidly rotating neutron stars.

4.1. Magnetic dipole radiation

Due to the energy loss by magnetic dipole radiation plus emission of electron-positron wind the star has to spin-down and the resulting change of the angular velocity can be parametrized by a power law


where K is a constant and [FORMULA] is the braking index


where we used the notation [FORMULA], with the corresponding definition of [FORMULA] (see also Glendenning et al. 1997; Grigorian et al. 1999).

In Fig. 8 we display the result for the braking index [FORMULA] for a set of configurations with fixed total baryon numbers ranging from [FORMULA] up to [FORMULA], the region where during the spin-down evolution a quark matter core could occur for our model EOS. We observe that only for configurations within the interval of total baryon numbers [FORMULA] a quark matter core occurs during the spin-down as a consequence of the increasing central density (see also Fig. 5) and the braking index shows variations. The critical angular velocity [FORMULA] for the appearance of a quark matter core can be found from the minimum of the braking index Eq. (41). As can be seen from Fig. 8, all configurations with a quark matter core have braking indices [FORMULA] and braking indices significantly larger than 3 can be considered as precursors of the deconfinement transition. The magnitude of the jump in [FORMULA] during the transition to the quark core regime is a measure for the size of the quark core. It would even be sufficient to observe the maximum of the braking index [FORMULA] in order to infer not only the onset of deconfinement ([FORMULA]) but also the size of the quark core to be developed during further spin-down from the maximum deviation [FORMULA] of the braking index. For the model EOS we used a significant enhancement of the braking index which does only occur for pulsars with periods [FORMULA] ms (corresponding to [FORMULA] kHz), which have not yet been observed in nature. Thus the signal seems to be a weak one for most of the possible candidate pulsars. However, this statement is model dependent since, e.g., for the model EOS used by Glendenning et al. (1997), which includes the strangeness degree of freedom, a more dramatic signal at lower spin frequencies has been reported. Therefore, a more complete investigation of the braking index for a set of realistic EOS should be performed.

[FIGURE] Fig. 8. Braking index due to dipole radiation from fastly rotating isolated pulsars as a function of the angular velocity. The minima of [FORMULA] indicate the appearance/ disappearance of quark matter cores.

4.2. Mass accretion

A higher spin-down rate than in isolated pulsars might be possible for rotating neutron stars with mass accretion. In that case, at high rotation frequency the angular momentum transfer from accreting matter and the influence of magnetic fields can then be neglected (Shapiro & Teukolsky 1983) so that the evolution of the angular velocity is determined by the dependence of the moment of inertia on the total mass, i.e. baryon number,


where [FORMULA] has been assumed.

In Fig. 9 we consider the change of the pulsar timing due to mass accretion with a constant accretion rate [FORMULA] for fixed total angular momentum as a function of the total baryon number. Here the change from spin-down to spin-up behaviour during the pulsar evolution signals the deconfinement transition. When the pulsar has developed a quark matter core then the change of the moment of inertia due to further mass accretion is negligible and has no longer influence on the pulsar timing. However, in real systems the transfer of angular momentum from the accreting matter can lead to a spin-up already. Then the transition to the quark matter core regime should be observable as an increase in the spin-up rate.

[FIGURE] Fig. 9. Total baryon number dependence of the spin-down rate [FORMULA] in units of the baryon number accretion rate ([FORMULA]) and the corresponding angular velocity (lower panel) for different (conserved) total angular momenta [FORMULA]. The suggested signal for a deconfinement transition in rapidly rotating neutron stars with baryon number accretion is a transition from a spin-down to a spin-up regime, i.e. a zero in the spin-down rate.

It will be interesting to investigate in the future whether e.g. low-mass X-ray binaries (LMXBs) with mass accretion might be discussed as possible candidates for rapidly rotating neutron stars for which consequences of the transition to the quark core regime due to mass accretion might be detected. Recently, quasi-periodic brightness oscillations (QPO's) with frequences up to [FORMULA] Hz have been observed (Lamb et al. 1998) which entail new mass and radius constraints for compact objects. Note in this context that the assumption of a deconfined matter interior of the compact star in some LMXBs as e.g. SAX J1808.4-3658 (Li et al. 1999) seems to be more consistent than that of an ordinary hadronic matter interior.

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Online publication: June 5, 2000