Halder and colleagues studied the limiting mass for the rotating quark star, which is introduced here as the “Chandrasekhar limit for rotating quark stars”. They also studied whether quark star behave as a black hole or not.
The theoretical investigation for quark stars has turned out to be a worldwide enterprise after the first prediction of the quark core at the center of the core collapsed neutron stars. The idea of the quark star was brought to light by soviet physicists Ivanenko and Kurdgelaidze, about five years after the Gell-Mann prediction of quarks. Several recent observations of compact and comparatively cooler stars (SWIFTJ1749.4- 2807, RXJ185635-3754 and 3C58) also provide evidence in favor of the existence of quark stars. Unlike other compact astrophysical objects and main sequence stars, the quark stars are self-bound by strong interaction rather than by gravity alone. Apart from this unique characteristic, the density of the quark stars are remarkably high (even higher than nuclear density i.e. 2.4 × 1014 g/cm³) and hence gains immense importance as natural laboratory for quark matter.
The quark star is considered to be the very end product of the stellar evolution. As the fuel of a star (neutron star) tends to get exhausted, the radiation pressure of the star can no longer balance the self-gravity. As a result, the core part of the star starts collapsing due to its own gravity. However, in certain cases, the density turns out to be substantially high and therefore a quark core is spontaneously generated (even without strangeness) at the center of the star. Although initially, the collapse takes place in the core part, it grows over time and occupies the entire star eventually by capturing free neutrons from the vicinity of the surface in absence of the Coulomb barrier. Besides this process, a thin outer crust of comparatively lower density is also generated outside the quark core (∼ 1/1000 of the average density of the star), which does not contain any free neutrons. Consequently, the effect due to the outer crust of the quark star is ignorable in limiting mass as well as the limiting radius. The newly formed quark phase contains only 2 flavors of quarks (‘u’ and ‘d’ quark) as it is generated from the non-hyperonic baryons (mainly neutron). Later the strangeness can be developed in the 2 flavor quark state via weak interaction (ud → us, in excess of ‘d’ quark) by absorbing the energy of the quark star and thus transforms into a strange quark star (SQS). The SQS is basically composed of 3 flavors of quarks (contains strange quark in addition to the ‘u’ and ‘d’ quark) confined in a hypothetical large bag, which is characterized by the Bag Constant. The strange quark matter can be treated as the perfect ground state of the strongly interacting matter as predicted by E. Witten.
Now, Halder and colleagues, addressed a general solution for both types of quark stars (2-flavored normal quark star and 3-flavored strange quark star). They also studied the dependence of the limiting mass, radius and the total number of particles N on the bag constant as well as the rotational frequency. The rotational effect of the star is parameterized by the rotational frequency (ω), however, due to the extreme compactness and mass of the star (radius to mass ratio ≤ 2.2903 in geometric unit, see featured image), the general relativistic effects emerge prominently in the observed rotational frequency (ν) (as observed by a far away observer). As a result, a far away observer would observe a red shifted form (ν) of the actual frequency (ω), given by:
where, M and R are the mass and radius of the star respectively. Consequently, a significant deviation from ω is obtained in the observed frequency (ν) in the case of fast spinning stars as shown in (see Fig. 1). The variation of the limiting mass Mmax with observed red-shifted frequency (ν) and actual frequency (ω) is described in Fig. 1 for a chosen value of bag constant B = (145 MeV)⁴.
From Fig. 2 it is evident that, for each chosen value of bag constant, the limiting mass remains almost unchanged in the lower frequency range. But as the frequency (ν) reaches ∼ 300 Hz, the kinetic energy becomes sufficiently high to be taken into consideration for the evaluation of the effective mass. As a result Mmax starts increasing gradually with frequency, however it suffers a rapid increase above the frequency of 600 Hz. Beyond a certain limit of Mmax for a given bag constant (B) the star becomes so massive that observed frequency (ν) starts falling with increasing ω due to gravitational effect. This provides a limiting value of ν (i.e. νmax) (see Fig. 2) for each value of bag constant.
According to the best fit point of their χ2 analysis (B = (136.5 MeV)⁴), the maximum value of observed frequency (νmax) of quark star is ∼ 747.7346 Hz (agrees with observational evidences of fast spinning pulsar). However, the corresponding actual frequency (ωmax) for that case is ∼ 1248 Hz.
“Using bag model description, we found that the maximum mass of a rotating quark star is found to be depend on the rotational frequency apart from other fundamental parameters. The analytical results obtained agree with the result of several relevant numerical estimates as well as observational evidences.”— Told Halder, lead author of the study
Besides this, they also examined whether quark star behaves like a black hole or not. For a fast spinning star, the Schwarzschild radius cannot address the event horizon. So, in order to obtain the event horizon for such spinning bodies, they have taken Kerr space time into account, which provides a comparatively smaller horizon than that of the Schwarzschild metric system.
“From this one can conclude that a quark star can never behave as black hole.”— said Halder, lead author of the study
From the above study, it is evident that the compactness of the quark star is extremely high, but it cannot become a black hole even in the limiting case. Thus they (strange stars) and black hole could co-exist as candidates of cold dark matter. While their phenomenological signatures may not be sufficient to distinguish between them, their different signatures in the gravitational wave scenario may be interesting to study.
Featured image: Radius mass ratio with ν for different values of bag constant. © Halder et al.
Reference: Ashadul Halder, Shibaji Banerjee, Sanjay K. Ghosh, and Sibaji Raha, “Chandrasekhar limit for rotating quark stars”, Phys. Rev. C, 26 Feb 2021. DOI: https://doi.org/10.1103/PhysRevC.103.035806
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