A&A 382, 939-946 (2002)
DOI: 10.1051/0004-6361:20011701
R. Ouyed
Nordic Institute for Theoretical Physics, Blegdamsvej 17, 2100 Copenhagen, Denmark
Received 5 July 2001 / Accepted 15 November 2001
Abstract
In a previous paper, using an equation of state of
dense matter representing a fluid of Skyrmions we constructed
the corresponding non-rotating compact-star models
in hydrostatic equilibrium; these are mostly
fluid stars (the Skyrmion fluid) thus naming them Skyrmion Stars.
Here we generalize our previous calculations by
constructing equilibrium sequences of rotating Skyrmion stars
in general relativity using
the computer code
developed by Stergioulas.
We calculated their masses and radii to be
,
and
,
respectively (R being
the circumferential radius of the star). The period of the maximally rotating
Skyrmion stars is calculated to be
.
We find that a gap (the height between the star surface
and the inner stable circular orbit) starts to appear for
.
Specifically, the Skyrmion star mass range with an existing gap
is calculated to be
with the corresponding orbital frequency
.
We apply our model
to the 4U 1820-30 low mass X-ray binary
and suggest a plausible
Skyrmion star candidate in the 4U 1636-53 system. We
discuss the difficulties encountered by our model in the 4U
0614+09 case with the highest known Quasi-Periodic Oscillation frequency
of 1329 Hz. A comparative study of Skyrmion stars and models of neutron
stars based on recent/modern equations of state is also presented.
Key words: dense matter - equation of state - stars: rotation
In this paper, we compute models of rotating Skyrmion stars using the RNS code written and made publicly available by N. Stergioulas (Stergioulas & Friedman 1995). RNS constructs models of rapidly rotating, relativistic, compact stars and assumes uniform rotation. The computation solves for the hydrostatic and Einstein field equations for uniformly rotating mass distributions, under the assumptions of stationarity, axial symmetry about the rotation axis, and reflection symmetry about the equatorial plane. The paper is presented as follows: in Sect. 2, we describe the equations solved by the RNS code and then construct the corresponding rotating Skyrmion stars. The results are analyzed and discussed in Sect. 3. A comparative study of Skyrmion stars and neutron stars is done in Sect. 4 before concluding in Sect. 5.
Here we present a brief outline of the equations solved for by
the RNS code. We start by describing the
space-time around a rotating compact star in quasi-isotropic coordinates,
as a generalization of Bardeen's metric (Bardeen 1970):
From the relativistic equations of motion, the equation of
hydrostatic equilibrium for a barytropic fluid (the field
equations are integrated by assuming various zero-temperature, barotropic
EOS of the form
)
may be obtained
as:
The geodesic motion for circular orbits around rotating relativistic stars is also presented in Bardeen (1970) and correspond to the motion of ZAMO yielding two possible values for the velocity corresponding to corotating and counter-rotating orbits. The innermost stable circular orbits are given by solving for V,rr = 0 where V is the effective potential as written in Bardeen (1972); the comma followed by rrrepresent a second order partial derivative with respect to radius r.
The highest energy density used in all the computed rotating models is the value which gives the maximum mass non-rotating star as calculated in OB. We specify the equation of state, and the central energy density and the code computes models with increasing angular velocity until the star is spinning with the same angular velocity as a particle orbiting the star at its equator^{}.
Figures 1a and 1b show the resulting stellar masses and radii as a function of the central density, respectively. The mass range is while the corresponding radii (equatorial circumferential radius; (proper equatorial circumference)/2) are calculated to be . In Fig. 1c, we show the resulting Mass-Radius plane. In general, for a given central density, rotation allows the masses and radii to increase by 30% and 40%, respectively, when compared with the non-rotating cases. The amount of baryonic mass in the outer region of the star (the crust region where is constructed using the EOS of Baym et al. 1971; is the nuclear saturation density) decreases drastically with rotation. Rotating Skyrmion stars crust constitute less than 5% of the total baryonic mass while it averages 20% for the static configurations. The nuclear crust is further addressed in OB to which we refer the interested reader.
The minimum spin period as a function of mass is shown in Fig. 2a and is calculated to be . The inertial frame dragging at the center of the star to the rotation rate is plotted in Fig. 2b which shows a significant value throughout and reaches a maximum value of 0.8 for the most massive stars. It suggests that frame-dragging might have a significant effect on the structure of rotating Skyrmion stars as P decreases since it is taking place against the background of a radially dependent, frame-dragging frequency (see Weber & Glendenning 1992 for example).
It has been noted by Hænsel & Zdunik (1989) and Friedman & Ipser (1992)
that the maximum spin rate for many neutron star EOS seems to be given by
Figure 3: Maximally rotating configurations: a) eccentricity versus gravitational mass for zero temperature Skyrmion stars in hydrostatic equilibrium. b) Kinetic to gravitational energy ratio versus gravitational mass. The dashed line is based on the empirical formula for the onset of the bar mode instability as described by Eq. (9). Configurations with above the dashed line might be unstable to the bar mode. |
In Fig. 4, the moment of inertia (Fig. 4a) and the angular momentum (Fig. 4b) are plotted. The sudden decrease in the moment of inertia for is the consequence of the mass reaching a "plateau" (at ) while the radius keeps decreasing (see Figs. 1a and 1b). The combined effects of I decreasing with increasing (Fig. 2a) explains then why the angular momentum stays constant for - using the simple dimensional analysis . Finally the height from surface of the last stable co-rotating orbit ( )^{} is shown in Fig. 4c. It can be seen from this figure that in general the stable orbits exists up to the surface of the star. The large radii calculated makes it very difficult for the stable orbit to be outside the star (when compared to 6GM/c^{2}; the inner most stable Kepler orbit around the non-rotating case). For the most massive configurations ( ), the boundary layer (the separation between the surface of the neutron star and its innermost stable orbit) can be as as high as 1.5 km for the maximum value. Lowering the spin frequency will eventually relax the above conclusions (see Sect. 3.3).
Skyrmion stars are
subject to rotational constraints
linked to gravitational-wave instabilities which make all rotating,
perfect fluid equilibria unstable
to modes with angular dependence
for sufficiently
large m (Friedman & Schutz 1978; Lindblom 1984).
These instabilities allow the star to convert
its rotational energy to gravitational energy waves (Glendenning 1997).
Fairly recent general relativistic calculations have shown that the
gravitational-radiation-driven bar-mode (f-mode) instability
sets in at values of
much lower than in the Newtonian limit.
For the case of realistic EOS, Morsink et al. (1999) found the following empirical formula for the
onset of the bar mode instability (the case of polytropes
can be found in Stergioulas & Friedman 1998):
The real question, however, is whether the f-mode instability is actually going to limit the spin period of Skyrmion stars. With the onset of the m=2 (f-mode), the r-mode^{} will soon become unstable (Friedman & Morsink 1998) and spin down the star to even slower rate. As a result, it is not expected that the f-mode will limit the spin-rate of any stars (Andersson et al. 1999), although it may become unstable and emit gravitational radiation in some cases.
Realistic compact stars are viscous, and the presence of viscosity will shift the onset of instability and if large enough will damp out the instability (Andersson et al. 1999). If the instability is driven by viscous dissipation, it was shown in Bonazzola et al. (1995) that for cold uniformly rotating NS only very stiff equations of state are likely to allow for spontaneous symmetry breaking. If we consider the stiffness of our EOS (when compared to standard EOS; see Fig. 2 in OB) then instability to a bar mode is a very likely plausibility in Skyrmion stars. So how viscous is a Skyrmion fluid and what is its temperature dependency?
The Skyrmion fluid can be looked at as made of fermionic soliton objects and in principle one should be able to calculate its physical parameters, among others its viscosity. The complication is linked to the fact that unlike a pure fermionic fluid, the structure of the Skyrmion fluid is highly non-linear (a consequence of the Skyrmion having structure). This makes the calculations not trivial and beyond the scope of this paper. There might still be the possibility that such a fluid is viscous enough that it will of course damp these instabilities altogether so that limits on rotation is set by the Kepler frequency - a notion to be confirmed.
Figure 5: Different spin frequencies: a) the ISCO frequency versus mass for Skyrmion stars rotating at 290, 360 and 580 Hz respectively. The dotted line represents the static models, . b) Height from the star surface of the innermost stable circular orbit (ISCO) versus mass for Skyrmion stars rotating at 290, 360 and 580 Hz respectively (see Table 1). |
The discovery of kilohertz Quasi-Periodic Oscillations (QPOs) in low mass X-ray binaries (LMXBs) with the Rossi X-Ray Timing Explorer has stimulated extensive studies of these sources (van der Klis 1998). One thing that has been suggested in the literature is that the highest frequency QPO observed could correspond to the orbital frequency at the inner stable circular orbit (ISCO) (Miller et al. 1998). The signature of the marginally stable orbit is a saturation in QPO frequency (assumed to track inner disk radius) versus mass accretion. Such a saturation where the frequency becomes independent of the chosen mass accretion rate indicator has been reported in 4U 1820-30 (Zhang et al. 1998; Kaaret et al. 1999).
R | e | I | j | T/W | ||||||
= 290.0 Hz | ||||||||||
3.0 | 1.35 | 1.46 | 15.67 | 0.00 | 0.00 | 0.31 | 2.36 | 0.49 | 0.016 | 0.31 |
5.0 | 2.28 | 2.59 | 16.47 | 1.93 | 6.26 | 0.28 | 5.09 | 1.06 | 0.011 | 0.49 |
6.9 | 2.60 | 3.02 | 16.13 | 4.89 | 9.31 | 0.24 | 5.83 | 1.21 | 0.009 | 0.59 |
= 360.0 Hz | ||||||||||
3.0 | 1.38 | 1.48 | 15.93 | 0.00 | 0.00 | 0.39 | 2.48 | 0.64 | 0.025 | 0.31 |
5.0 | 2.30 | 2.62 | 16.62 | 1.69 | 7.08 | 0.34 | 5.24 | 1.35 | 0.017 | 0.50 |
6.9 | 2.62 | 3.04 | 16.24 | 4.53 | 10.04 | 0.28 | 5.96 | 1.53 | 0.014 | 0.59 |
1.0 | 2.80 | 3.29 | 15.26 | 7.06 | 12.18 | 0.24 | 5.81 | 1.50 | 0.010 | 0.69 |
= 580.0 Hz | ||||||||||
3.0 | 1.53 | 1.66 | 17.43 | 0.00 | 3.33 | 0.64 | 3.25 | 1.35 | 0.069 | 0.34 |
5.0 | 2.43 | 2.76 | 17.42 | 1.31 | 10.17 | 0.53 | 6.05 | 2.51 | 0.048 | 0.52 |
6.9 | 2.72 | 3.16 | 16.78 | 3.58 | 12.72 | 0.47 | 6.58 | 2.73 | 0.038 | 0.61 |
1.0 | 2.86 | 3.36 | 15.57 | 5.85 | 14.29 | 0.39 | 6.18 | 2.56 | 0.028 | 0.70 |
In Fig. 5a we plotted ISCO frequency (for the
prograde case only) versus Mass for
Skyrmion star models rotating at frequencies 290, 360 and 580
Hz respectively.
These values allow us to cover the range of frequencies observed in LMXBs.
Figure 5b shows the corresponding gaps; that is the height between the
star surface and the ISCO (prograde case). In general, the Skyrmion star
mass range with an existing gap is calculated to be,
Star's angular velocity (10^{4} Hz) | |
Star's spin frequency, (Hz) | |
Central density (g cm^{-3}) | |
Gravitational mass ( ) | |
Baryonic mass ( ) | |
R | Radius of star, measured at equator (km) |
Radius of the innermost stable orbit (km) | |
Height from the star surface of innermost stable | |
prograde circular orbit (km) | |
Height from the star surface of innermost stable | |
retrograde circular orbit (km) | |
e | Eccentricity |
I | Moment of Inertia (10^{45} g cm^{2}) |
j | Angular momentum ( ) |
Rotational energy to the gravitational energy | |
Ratio of the inertial frame dragging at the | |
center of the star to the rotation rate | |
Polar redshift | |
Forward redshift | |
Backward redshift |
To first order in the rotation rate of the star ( Hz), the orbital frequency in the marginally stable orbit is (see Miller et al. 1998, for example). Consider, as an example, a moderately slowly rotating Skyrmion star, Hz, with mass . Using the value of j given in Table 1 we find Hz while the value in the non-rotating case is 8454 Hz (depicted by the dotted line in Fig. 5a).
We first apply our model to 4U 1820-30 which seems to present the strongest experimental evidence for the existence of the marginally stable orbit (Zhang et al. 1998; Kaaret et al. 1999). For Hz as the estimated frequency at the ISCO, and the spin frequency to near 300 Hz, Fig. 5a suggests a mass 2.35 . The radius is then calculated to be of the order of 16 km. These numbers remain to be confirmed from observations and furthet modelling of the system.
The highest known QPO has a frequency of 1329 Hz (van Straaten et al. 2000). The frequency of the corresponding star (4U 0614+09) is believed to be near 300 Hz. Simple estimates (van Straaten et al. 2000, Sect. 4.4; van der Klis 2000, Sect. 5.6) show the difficulty for extremely stiff EOS such as ours to account for the derived range in mass and radius ( and R < 15.2 km). We have noted that for our EOS kHz and R > 14 km. Recent modeling put even more constraints by suggesting a more compact 4U 0614+09 with a radius less than 10 km (Titarchuk & Osherovich 2000).
It is not trivial to account for such numbers in our model unless the 1329 Hz frequency does not actually correspond to the ISCO. Indeed, unlike in the 4U 1820-30 source, there is no obvious saturation of the kilo-hertz QPO with respect to mass accretion indicator in the 4U 0614+09 case (see Fig. 4 in van Straaten et al. 2000).
In this chapter we compare Skyrmion stars to the more traditional neutron stars. We chose three recent/modern EOSs: the first is described in Baldo et al. (1997) which uses the Argonne v14 (Av14) (Wiringa et al. 1984) two-body nuclear force; the second uses the Paris two-body nuclear force (Lacombe et al. 1980), implemented in both cases by the Urbana three-body force (TBF) (Carlson et al. 1983; Schiavilla et al. 1986) while the third one is that developed in Prakash et al. (1997) using a generalized Skyrme-like EOS. As in Datta et al. (1998), we refer to these EOSs as BBB1 (Av14+TBF), BBB2 (Paris+TBF) and BPAL32, respectively.
EOS | R | ||||
BBB1 | 3.09 | 1.788 | 2.082 | 9.646 | 15.845 |
BBB2 | 3.12 | 1.917 | 2.261 | 9.519 | 16.984 |
BPAL32 | 2.67 | 1.947 | 2.263 | 10.509 | 17.254 |
OB | 1.22 | 2.797 | 3.299 | 14.565 | 24.743 |
EOS | R | e | |||||
BBB1 | 2.56 | 2.135 | 2.471 | 13.129 | 13.490 | 0.703 | 1.095 |
BBB2 | 2.82 | 2.272 | 2.653 | 12.519 | 13.550 | 0.687 | 1.203 |
BPAL32 | 2.27 | 2.300 | 2.657 | 14.276 | 14.611 | 0.699 | 1.001 |
OB | 1.03 | 3.445 | 4.034 | 19.890 | 21.089 | 0.837 | 0.749 |
... | I | j | T/W | ||||
... | 2.428 | 3.019 | 0.120 | 0.764 | 0.690 | -0.330 | 1.975 |
... | 2.539 | 3.469 | 0.123 | 0.825 | 0.849 | -0.349 | 2.483 |
... | 3.005 | 3.416 | 0.113 | 0.771 | 0.679 | -0.328 | 1.933 |
... | 9.982 | 8.507 | 0.144 | 0.777 | 0.777 | -0.342 | 2.287 |
EOS | R | e | |||||
BBB1 | 2.44 | 2.133 | 2.468 | 13.264 | 13.558 | 0.706 | 1.079 |
BBB2 | 2.82 | 2.272 | 2.653 | 12.519 | 13.550 | 0.687 | 1.203 |
BPAL32 | 2.14 | 2.299 | 2.655 | 14.481 | 14.713 | 0.702 | 0.981 |
OB | 9.00 | 3.435 | 4.017 | 20.549 | 21.498 | 0.840 | 0.715 |
... | I | j | T/W | ||||
... | 2.465 | 3.021 | 0.120 | 0.756 | 0.677 | -0.328 | 1.935 |
... | 2.539 | 3.469 | 0.123 | 0.825 | 0.849 | -0.349 | 2.483 |
... | 3.070 | 3.419 | 0.114 | 0.760 | 0.661 | -0.326 | 1.881 |
... | 10.52 | 8.560 | 0.146 | 0.751 | 0.728 | -0.355 | 2.133 |
Figure 6: The M-R relation for non-rotating Skyrmion stars (OB) as compared to theoretical models of non-rotating neutron stars (UU, BBB1, BBB2, BPAL12, Hyp, and K^{-1}) and strange stars (SS1 and SS2). The data for the neutron stars and strange stars was kindly provided to us by Li et al. (1999). The Schwarzschild radius (2GM/c^{2}) is shown as a dotted line. Inside the triangle is the allowed range of M and R for 4U 1636-53 as modeled in Nath et al. (2001) using fits to X-ray bursts. |
Table 2 summarizes the non-rotating compact star structure parameters for the EOS models BBB1, BBB2, BPAL32 and ours (OB). The values listed correspond to the maximum stable mass configuration. Note that for this table and the rest of the tables, we consider only the co-rotating case (the cases where is non-existent or inside the star, is taken to be the Keplerian orbit radius at the surface of the star). The maximum mass is usually an indicator of the softness/stiffness of the EOS and its values as listed in Table 2 confirms our previous statements that our EOS is stiffer than the standard ones. The consequences of such a stiffness is also illustrated in Table 3 where we list the quantities corresponding to the maximum gravitational mass configurations. It can be seen that the gravitational mass of the maximum stable rotating Skyrmion star has a value of , while it is for the configurations constructed with the other EOS. That is, with our EOS we do not need to go to break up speeds to account for the mass predicted from analysis of LMXB observational data (Zhang et al. 1996). The values of these same quantities are listed in Table 4 for the maximum angular momentum models leading to similar conclusions.
In Fig. 6 we compare the M-R relation for Skyrmion stars (OB) to the theoretical M-R curve obtained using six recent realistic models for the EOS (UU, BBB1, BBB2, BPAL12, Hyp, and K^{-1}). The solid curves labeled SS1 and SS2 are for strange stars (the data was kindly provided to us by Li et al. 1999). The triangle depicts the mass-radius constraint from fits to X-ray bursts in 4U 1636-53. Inside the triangle is the allowed range of M and R which satisfies the compactness constraints as modeled in Nath et al. (2001; see their Fig. 4), and clearly favoring stiffer EOSs. Our modeled stars (OB) cross the triangle suggestive of 4U 1636-53 as a plausible Skyrmion star candidate. However, one should keep in mind the fact that modern EOSs can be modified as to also cross the triangle (Heiselberg & Hjorth-Jensen 1999).
With the RNS code, we constructed numerical models of rotating Skyrmion stars for a newly derived EOS of dense matter based on a fluid of Skyrmions. We calculated their masses and radii to be and , respectively. The spin period of the maximally rotating configurations are calculated to be . Owing to the stiffness of the Skyrmion fluid, Skyrmions stars are found to be on average heavier, to show higher equatorial radii, and to rotate slower than theoretical models of neutron stars based on modern realistic EOS.
Skyrmion stars might have little bearing on reality since there is still no guarantee that our EOS (OB) is a plausible representation of matter above nuclear densities. However, our model so far has been successful in reproducing the basic features of compact objects. An interesting consequence of our model, with its plausible applications to QPO systems, is the fact that massive Skyrmion stars ( ) can possess gaps with orbital frequencies in the kHz range ( ). These points suggest that the model warrants further study.
Acknowledgements
I am grateful to S. Morsink, M. Butler and G. Kälbermann for encouraging help and valuable discussions. I am also grateful to an anonymous referee for the remarks that helped improve this work. My thanks to N. Stergioulas for making his code available.