3 Quark stars with nearly uniform density

Quark stars - if they exist at all - are thought to be formed in (some) supernovae, or in a phase transition in a neutron star exceeding a critical central density, so typically their masses are expected to be comparable to those of neutron stars. However, unlike neutron stars which are unstable below a certain mass, quark stars (or planets) of arbitrarily small mass can exist, on the assumption that quark matter is self bound. It is not impossible that low-mass quark stars may form through fragmentation in a violent event, such as binary coalescence of a more massive quark star and a black hole (Lee & Kluzniak 2001).

Low-mass X-ray binaries are thought to contain stellar mass compact sources, but in most cases the mass has not actually been determined. In a source where much of the luminosity is released in an accretion disk, it would usually be difficult to discriminate directly between a star of mass $1.4~M_\odot$ with a 10 km radius and, say, a $0.1~M_\odot$ star of radius 5 km. One indication of a low mass could be a lower photon flux during an X-ray burst, when the luminosity is thought to reach the Eddington limit value (Margon & Ostriker 1973) $L_{\rm Edd}=9\times10^{37}(M/M_\odot)\,$erg/s.

Numerical models of static quark stars in general relativity have been constructed by Itoh (1970); Brecher & Caporaso 1976; Witten (1984); Alcock et al. (1986); and others. The first accurate, fully relativistic calculations of rotating quark stars were published by Gourgoulhon et al. (1999), and Stergioulas et al. (1999).

Using the Gourgoulhon code we have computed numerical models of uniformly rotating quark stars in general relativity. We assume quark matter to be self-bound and have used the simplest MIT-bag equation of state $p=(\rho-\rho_0)c^2/3$, to model it. The stars were constrained to be axisymmetric. We have found that for stellar masses much less than that of the sun ( $M\ll M_\odot$), the density is nearly uniform throughout the star and, as expected, close to the density $\rho_0$ at zero pressure of quark matter. We used $\rho_0=4.28\times10^{14}$ g/cm³, and obtained central densities of $4.287, 4.318, 4.479\times10^{14}$ g/cm³, for nonrotating stars with baryon masses of 0.001, 0.01, 0.1 $M_\odot$, respectively, and $4.283, 4.301, 4.399\times10^{14}$ g/cm³for models of the same baryon masses but rotating at a rate of 1 kHz.

Figure 1: The rotational frequency of Maclaurin spheroids as a function of their eccentricity (upper curve, the star indicates the point of onset of dynamical instability to a toroidal mode), and the Jacobi sequence (lower curve), after Chandrasekhar (1969). The circle indicates the spheroid for which the marginally stable orbit grazes the equator ( e=0.834583). Also shown (thick dots) are numerical models of uniformly rotating quark stars of mass $M=0.01~M_\odot$ calculated by us with the general relativistic spectral code of Gourgoulhon et al. (1999) (the value of "$\rho$'' on the vertical axis used to rescale $\Omega ^2$ is the volume averaged density of the non-rotating model).

Figure 2: The maximum frequency in stable circular orbits around the Maclaurin spheroids as a function of the spheroid eccentricity (continuous curves). The thin curve is the orbital frequency at the equator, Eq. (2), the dashed portion corresponds to unstable orbits at the equator. The thick curve is the orbital frequency in the marginally stable orbit, Eq. (3). The dotted portions of the curves correspond to spheroid eccentricities past the point of onset of dynamical instability, at e=0.95289. Note that the curves are Newtonian, the speed of light does not enter Eqs. (2), (3). Also shown (as thick dots) are the numerical models of quark stars presented in Fig. 1.

We have found that for $M\le 0.1~M_\odot$ the Maclaurin spheroid is a very good approximation to the quark star, at least in terms of gross stellar properties (Figs. 1-3). In the figures, we show (as dots) the results of our general-relativistic numerical computations for stars of baryon mass $0.01~M_\odot$. For such stars, the maximum orbital frequency in stable circular orbits (Fig. 3) is attained for the static model (e=0, $\Omega=0$), and is clearly given by the Keplerian expression $\omega(a)=\sqrt{4\pi G\bar\rho/3}$; we have used $\sqrt{3/4}$ times this value to rescale in Figs. 1, 2 the frequencies of all our numerical models. Here $\bar\rho$ is the volume averaged density of the static model. For every $e\le e_{\rm s}$ the maximum stable orbital frequency is attained in a circular orbit on the equator. For larger eccentricities the maximum value of orbital frequency is reached in the innermost stable orbit and it drops precipitously as $e\rightarrow1$. The numerical results are seen to agree with the Newtonian expressions of Eqs. (2), (3). Clearly, general-relativistic effects are negligible in the external metric of low-mass quark stars.

Because in Fig. 3 the dependence of $\omega$ on the stellar rotational frequency $(2\pi)^{-1}\Omega$ is shown, the curves are "double valued,'' for a given rotational frequency the larger value of orbital frequency corresponds to a spheroid/star of lower eccentricity, the smaller to the one for larger e (cf., Fig. 1). In Fig. 3 the intersection of the curves for $\omega_{\rm ms}$ and for $\omega(a)$ on the unstable branch (dashed curve) is a mirage caused by projection onto the $\Omega$ axis, in fact, values of about 1250 Hz are attained for different values of e in the case of $(2\pi)^{-1}~\omega_{\rm ms}$ and of $(2\pi)^{-1}~\omega(a)$. Compare Fig. 2, where it is clear that $\omega(a)$ and $\omega(r_{\rm ms})$ intersect only for $e=e_{\rm s}$ and e=1. Dynamical instability to a toroidal deformation sets in at e=0.95289 in the Newtonian theory (Chandrasekhar 1969), for this reason we do not extend the curves in Fig. 3 to low orbital frequencies.

Figure 3: The maximum orbital frequency as a function of the rotational frequency of the Maclaurin spheroid. All frequencies scale with the square root of the density. The numerical values shown correspond to $\rho =4.28\times 10^{14}\,$g/cm3, at this value of the density the marginally stable orbit is present for stellar rotational frequencies $\Omega /(2\pi )>945\,$Hz. All symbols have the same meaning as in Fig. 2, in particular, the steeply descending thick curve corresponds to the marginally stable orbit.

Finally, in Fig. 1, defining the eccentricity as $e=(1-r_{\rm p}^2/a^2)^{1/2}$, where $r_{\rm p}$ is the polar coordinate radius of the star and a is the r coordinate of the equator, we plot the rotational frequencies of the numerically computed stars, superimposed on Maclaurin's and Jacobi's sequence (Chandrasekhar 1969). We have found that even for higher mass stars (e.g., up to $1.5~M_\odot$) the rotational frequency in our numerical models departs by no more than several percent from the curve of Fig. 1, up to its maximum.

The circle in Fig. 1 marks the appearance of the marginally stable orbit at $e=e_{\rm s}$. This occurs above the bifurcation point (at eccentricity $e=e_{\rm J}=0.8127$) past which the Maclaurin spheroids are secularly unstable to deformation into a Jacobi ellipsoid. However, this does not necessarily imply that the Newtonian marginally stable orbit, discussed here for the Maclaurin spheroid, is irrelevant to quark stars or other stars.

First, a secular instability can grow only in the presence of dissipative mechanisms like viscosity (Roberts & Stewartson 1963) or gravitational radiation (Chandrasekhar 1970; Friedman & Schutz 1978; Friedman 1978). "By a suitable choice of the strength of viscosity relative to gravitational radiation, it is possible to stabilize the Maclaurin sequence all the way up to the point of dynamical instability'' (Shapiro & Teukolsky 1983).

Second, for a possibly realistic range of masses of compact stars, effects of general relativity may delay the onset of (viscosity driven) secular instability to non-axisymmetric deformations past the eccentricity where the Newtonian marginally stable orbit (related to rotation-induced stellar flattening) appears. What is the minimum mass of a quark star for which $e_{\rm J}>e_{\rm s}$? Calculations of Shapiro & Zane (1998) indicate that the bifurcation point occurs at $e_{\rm J}=0.835$ for GM/(Rc²)=0.013, where R is the radius of the non-rotating star of gravitational mass M. For the quark stars presented here, this would occur already for a star with $M=0.03~M_\odot$, for which R=3.2 km. It would appear then, that in the mass range $\sim$ $(0.03~M_\odot, 0.1~M_\odot)$ quark stars are well approximated by the Maclaurin spheroids and post-Newtonian effects may stabilize the quark star at eccentricities sufficiently high that the purely Newtonian marginally stable orbit is present outside the stellar surface, for $e>e_{\rm s}$.