Astron. Astrophys. 363, 1005-1012 (2000)

3. Rotating star configurations

We have calculated important properties of the uniformly rotating strange stars described by the SS1 and SS2 equations of state using the multi-domain spectral methods developed by Bonazzola et al. 1998. This method has been used previously for calculating rapidly rotating strange stars described by the MIT bag model (Gourgoulhon et al. 1999, Zdunik et al., 2000b). The multi-domain technique allows one to address the density discontinuity at the surface of self-bound stars even with a very high .

We construct equilibrium sequences of rotating compact strange stars with constant baryon mass, i.e. the so-called evolutionary sequences (for example a pulsar keeps its baryon mass constant while slowing down; neglecting accretion a compact star keeps its rest mass constant during evolution). We identify normal and supramassive stars as done for neutron stars. A sequence is called normal if it terminates at the zero angular momentum limit with a static, spherically symmetric solution, and it is called a supramassive sequence if it does not. The boundary between these two sequences is the sequence with the maximum baryon mass of a static configuration. The angular momentum (and the central density) of a star changes monotonically along each sequence. Note that the rotational frequency f does not necessarily change monotonically along a sequence, since a star changes its shape and, consequently, moment of inertia with increasing angular momentum.

3.1. Equilibrium sequences

Stable solutions for rotating neutron stars have to satisfy four different constraints (Cook et al. 1994): the static constraint - for the normal evolutionary sequence of rotating stars, when angular momentum goes to zero the configuration should be identical to the one described by OV equations for the same baryon mass; the low mass constraint below which a neutron star cannot form, the mass-shed (Keplerian) constraint, and the constraint of stability to quasi-radial perturbations.

The first three constraints provide bounds on normal sequences stars (those are always stable to quasi-radial perturbations), while the two last limits provide bounds on the supramassive stars.

The mass-shed limit is reached when the velocity at the equator of a rotating star is equal to the velocity of an orbiting particle (a star becomes unstable when gravitational attraction is not sufficient to hold matter bound to the surface). For normal sequences of neutron stars and low mass supramassive neutron stars, Keplerian configurations are always these with the maximal rotational frequency. For high mass supramassive neutron stars the Keplerian configuration is the one with lowest rotational frequency in the sequence.

The fourth constraint is the requirement of stability to axisymmetric perturbations. For an evolutionary sequence parameterized by the central density, the model is (secularly) stable if (or ) and unstable otherwise (Friedman et al. 1986). The stability constraint imposes a limit which begins at the maximum mass static configuration and terminates at the Keplerian limit sequence near the maximum mass rotating configuration (see for example gravitational mass versus central density dependence at Fig. 1 in Cook et al. (1994). Normal sequences begin at the static limit and terminate at the mass-shed limit. Along such sequences angular momentum increases while central density decreases. Supramassive star sequences begin at the stability limit with a minimal angular momentum. As the angular momentum increases, the central density decreases until the configuration terminates at the Keplerian limit. The intersection of the mass-shed and the stability limits on the gravitational mass - central density plane give us the locations of the configuration rotating with maximum frequency. For neutron stars the maximum mass Keplerian configuration is not necessarily stable against axisymmetric perturbations. For some equations of state of neutron stars the maximum mass configuration is the same as the configuration with the maximum frequency.

3.2. Limits on gravitational mass

To find equilibrium sequences of compact strange stars we take only three of the above constraints into account: the low mass constraint is not relevant for self-bound mat- ter. Other instabilities (for example to nonaxisymmetric perturbation) are not considered here since we cannot study them with our numerical code. Taking into account all con- straints described above we found limits on masses and rotation frequencies for the SS2 model, shown as thick lines in Fig. 2. The mass shed limit is shown as a thick short-dashed line, the stability limit as long-dashed thick line, the fastest normal and low mass supramassive configurations as a dot-dashed line, and the "low rotational frequency" maximum mass configurations as a thick solid line. The thin solid lines in Fig. 2 correspond to normal and massive supramassive equilibrium evolutionary sequences and are labelled with their baryon mass. In Fig. 3 we show an evolutionary sequence with a baryon mass of as an example of the low mass supramassive stars (with baryon mass lower than 1.9 ). The marginally stable configurations with respect to quasi-radial perturbation is marked with an open circle. The angular momen- tum increases along each curve from for static configurations ( for supramassive stars) to for the Keplerian ones represented by filled circles for normal and low mass supramassive sequences. The evolutionary sequence with separates normal stars from the supramassive ones.

Fig. 2. Gravitational mass as a function of the rotation frequency for the SS2 model. Thin solid lines correspond to evolutionary sequences with fixed baryon mass labelled close to each line. The angular momentum increases along each curve from for static configurations ( for supramassive stars) to for the Keplerian ones represented by filled circles for normal and low massive supramassive stars. The thick lines correspond to the upper limits on gravitational mass and rotational frequency; the dashed line corresponds to the mass-shed limit, the long-dashed line is the quasi-radial stability limit and the dashed-dotted line shows the fastest rotating configurations.

Fig. 3. Limits on gravitational mass and on the rotation frequency and evolutionary supramassive sequences with rest mass 1.7 and 1.8 . All lines are as indicated in Fig. 2. The marginally stable configuration with respect to quasi-radial oscillation is shown as an open circle.

The maximum mass configuration is found by considering a sequence of stars with a constant rotational frequency and parameterized by the central density. The maximum mass models for each sequence are represented by thick lines in Fig. 4. Chosen sequences are shown as thin solid lines. The Keplerian configurations are shown as a dashed line, while marginally stable configurations with respect to quasi- radial oscillation as long-dashed line. On one end of each sequence there is a Keplerian configuration (with the lowest central density in the sequence), and on the other end the last stable configuration with respect to axisymmetric perturbations (the densest object in the sequence). For kHz the maximum mass configuration (shown as thick solid line) for each sequence is close to the marginally stable one. For fast rotating configurations, kHz the maximum mass configurations are the Keplerian ones (for comparison see Fig. 1b in Gondek-Rosinska et al. 2000 for strange stars described by the MIT bag model).

Fig. 4. Gravitational mass vs. radius for stars described by the SS2 model. The thin solid lines correspond to sequences of stars with constant rotational frequency . The rotational frequency is labelled close to each line. The thick lines (solid and short-dashed) correspond to configurations with the maximal mass in each sequence. The dashed line corresponds to the mass-shed limit and the long dashed line is the quasi-radial stability limit. The intersection of the mass-shed and the stability limits gives the location of the configuration rotating with maximum frequency.

The thick solid line and thick short dashed line in Fig. 2 correspond to the maximal gravitational mass of a rotating strange star described by the SS2 equation of state as a function of the rotational frequency. There are two cases as the rotational frequency increases, first for rotational frequencies lower than (in the case of the SS2 model) and second for . In the first case the maximal mass limit line goes very close to the stability limit (shown as thin long dashed line). For rotation frequencies lower than kHz the stellar configuration is only slightly affected by rotation - the increase in the maximal mass is only a second order effect (where both the maximum mass line and the normal evolutionary sequence with maximum baryon mass are very close to each other) ie. , where is the static mass configuration. For higher frequencies increases faster. The maximum mass of rotating compact strange star in this case is approximately greater than the maximum mass of a static stellar configuration. For kHz we see a strong increase in the maximum mass (thick dashed line). In this case the configurations with maximum mass are Keplerian. The maximum mass of a rotating compact strange star in this regime (the maximum allowed mass of rotating configurations) is greater than the maximum mass of a static stellar configuration.

Note that the maximal mass configurations for a given rotational frequency are always supramassive i.e. no static configuration with such a mass can exist. It is worth noting that the central density of the rotating configurations is always lower than the central density of the maximum mass static star. This makes us confident that the rotating configurations found with the use of theapproximation of Eq. 1 are at least as accurate as the static configuration calculations.

3.3. Limits on rotational frequency

Let us now consider upper limits on rotational frequency of compact strange stars. In the case of high mass supramassive stars the maximum rotational frequency is determined by the condition of stability to axisymmetric perturbations, shown as a thick long dashed line in Fig. 2. Similarly to the supramassive neutron stars, the configurations to the left of this line are stable to radial collapse - they spin up loosing their angular momentum as discussed by Cook et al. (1994) in the case of supramassive neutron stars and by Gourgoulhon et al. (1999) in the case of supramassive MIT bag model strange stars.

In the case of normal and low mass supramassive stars, the maximum rotational frequency is slightly above the rotation frequency of the Keplerian configuration. The fastest configurations are shown as a thick dash-dotted line in Fig. 2 while the Keplerian one by filled circles. In Fig. 3 we present the details of the intermediate region of Fig. 2 for to show low mass supramassive star sequences.

The rotational frequency decreases for large values of the angular momentum and the sequences turn back in Fig. 2 and Fig. 3 before reaching the Keplerian configuration. At this small part of an evolutionary sequence, configurations spin up by loosing the angular momentum (or slow down when obtaining the angular momentum). The mass-shed configurations are reached due to the increase of the equatorial radius relative to the deformation of the rotating star. The difference between the Keplerian frequency and the maximal rotation frequency for these evolutionary sequences SS2 model is of the order of . Such a phenomenon was discussed by Zdunik et al. (2000b) in the case of normal sequences of MIT bag model strange stars. It is interesting to note that no such behavior was noticed for neutron stars. This feature is characteristic of stars described by a self-bound linear EOS.

The mass shed limit and the stability limit lines intersect twice: at frequency kHz and at the frequency of kHz. The intersection of the mass-shed limit (solid thick line in Fig. 2) and stability limit (solid dashed line in Fig. 2) determines the configuration rotating with maximum allowed frequency. Note that the maximum frequency occurs for only one extreme supramassive model and that this model is at the mass-shed limit and stability limit.

The configuration with the absolute maximum mass lies on the Keplerian limit line. Note that the maximum mass configuration is not rotating with the maximum allowed rotational frequency. For both SS1 an SS2 models the maximum mass rotating configuration is on the stable side of the mass-shed limit line.

In Fig. 5 we present the regions of the parameter space given by mass and rotation frequency where strange stars described by the SS1 (dashed line) and SS2 (solid line) equations of state can exist. The maximal mass and maximal frequency configurations lie on the Keplerian limit line. Details of the Keplerian configurations with the maximum mass are given in the right columns of Table 1. The maximum mass of rotating strange star given by SS1 and SS2 model is larger ( the equatorial radius) than in the case of a non-rotating star with maximum mass. For strange stars with massless and not interacting quarks these values are and respectively. The ratios of masses and equatorial radii do not depend on the parameter , since static and rotating maximum mass configurations scale identically, and are functions only of a (sound velocity) in the case of stars described by Eq. (1).

Fig. 5. Limits on gravitational mass and the rotation frequency for SS1 (dashed lines) and SS2 (solid lines).

For neutron stars the maximum mass which can be supported when the star is rotating uniformly increases by to depending on the equation of state while the radius increases by to (e.g. Cook et al. 1994; Datta et al. 1998) We note that a large increase (higher than in the case of neutron stars) of the maximum mass and corresponding equatorial radius in the case of strange stars of different EOS due to rotation is related to the fact that these EOS are self-bound with a very high density at the surface. They are much more compact than neutron stars and much higher rotation frequencies are required to deform them to reach Keplerian configurations.

3.4. ratio for rotating strange stars

It has been shown by Gourgoulhon et al. (1999), Stergioulas et al. (1999) and Gondek-Rosinska et al. (2000) (where in addition to the treatment of first two papers, mass and interaction between quarks were taken into account) that can be very high for strange stars described by the MIT bag model. In Fig. 6 we present the ratio of the rotational kinetic energy to the absolute value of the gravitational potential energy for evolutionary sequences for SS2 model. Each sequence is labeled with its baryon mass. The thin dashed line corresponds to the sequence of configurations rotating with the same rotational frequency. Note that for a rotating strange star the value of is significantly higher than that for an ordinary NS (eg. Cook et al. 1994). The large value of results from a flat density profile combined with strong equatorial flattening of rapidly rotating strange stars. It increases as mass decreases for Keplerian configurations. This property is universal for all self-bound linear EOS. does not depend on and its dependence on a is very weak. We can easily obtain a similar diagram for the simplest MIT model EOS of SS using the scaling relations , (Stergioulas et al 1999, for Keplerian configurations), where the subscript D denotes the EOS considered here. We check that the above scaling relations hold for any model in the evolutionary sequences. The difference in mass and rotational frequency between numerical results and rescaled ones are of the order of and . The value of is quite large for all configurations close to Keplerian, so it is possible that the point of onset of secular instability to non-axisymmetric normal modes has already been passed. For mass-shed configurations varies from 0.21 for maximum mass configuration to 0.27 for the low mass one (0.25 and 0.26 for gravitational mass 1.4 for SS1 and SS2 Keplerian models). This could be an indicator that rapidly rotating SS may constitute strong sources of gravitational waves (Gourgoulhon et al. 1999; Gondek-Rosinska et al. 2000; Gondek-Rosinska & Gourgoulhon 2000).

Fig. 6. The ratio of the rotational kinetic energy to the absolute value of the gravitational potential energy for SS2 evolutionary sequences with fixed baryon mass labelled above each line in sollar mass units. The solid thick line corresponds to Keplerian configurations. The thin dashed line correspond to sequences of configuration rotating with the same rotational frequency. Models located to left and below of this line rotate with lower frequency.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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