Rotating Quark Stars in General Relativity

We have built quasi-equilibrium models for uniformly rotating quark stars in general relativity. The conformal flatness approximation is employed and the Compact Object CALculator (COCAL) code is extended to treat rotating stars with surface density discontinuity. In addition to the widely used MIT bag model, we have considered a strangeon star equation of state (EoS), suggested by Lai and Xu, that is based on quark clustering and results in a stiff EoS. We have investigated the maximum mass of uniformly rotating axisymmetric quark stars. We have also built triaxially deformed solutions for extremely fast rotating quark stars and studied the possible gravitational wave emission from such configurations.


Introduction
The gravitational-wave (GW) event GW170817 and the associated electromagnetic emission observations [1,2] from a binary neutron star (BNS) merger has announced the birth of a multimessenger observation era.Apart from enriching our knowledge on origins of short gamma-ray bursts [3] and heavy elements in the universe [4,5], it provides an effective way for us to constrain the equation of state ( EoS) of neutron stars (NSs).In addition to BNS systems , rapidly rotating compact stars are also important candidates of GW sources [6], which could be detected by ground-based GW observatories [7][8][9][10][11].Further, the properties of both uniformly and differentially rotating stars is tightly related to the evolution of the post-merger product during a BNS merger, for example, whether or not there will be a prompt collapse to a black hole.Hence, studying the properties of rotating compact stars has long been important and of great interest.
Following the first study on the equilibrium models of uniformly rotating, incompressible fluid stars in a Newtonian gravity scheme [12], various works have been done with more realistic EoSs and general relativity [13,14].Among those studies, quasi-universal relationship has been found for both uniformly rotating and differentially rotating NSs [15][16][17][18][19][20].Quasi-equilibrium figures of triaxially rotating NSs have also been created and studied in full general relativity [21,22].
However, it is worth noting that the EoS of compact stars is still a matter of lively debate as it originates from complicated problems in non-perturbative quantum choromodynamics (QCD).In addition to the conventional NS model, strange quark stars (QSs) are also suggested [23,24], after it was conjectured that strange quark matter (SQM) consists of up, down, and strange quarks that could be absolutely stable [25,26].Additionally, the small tidal deformability of QSs passes the test of GW170817 [27], which requires that a dimensionless tidal deformability of a 1.4 solar mass star is smaller than 800.A more detailed analysis based on the probability distribution of each star in the binary system also indicates that the strangeon star model is consistent with the observation GW170817 together with other EoSs such as APR4.Possible models are also suggested to explain the electromagnetic counterparts (c.f.[27,28]).
Following this possibility, we here use the Compact Object CALculator code, COCAL, to build general-relativistic rotating QS solution sequences using different EoS models.COCAL is a code to calculate general-relativistic equilibrium and quasi-equilibrium solutions for binary compact stars (black hole and NSs) as well as rotating NSs [21,22,[29][30][31][32].The EoS part of COCAL is modified to treat quark stars that have a surface density discontinuity.With the modified code, we have built a uniformly rotating axisymmetric and triaxial sequence for quark stars.

Maximum Mass of Axisymmetric Rotating Quark Stars
The maximum mass of a static spherical compact star and an axisymmetric rotating compact star depends on the EoS and is also closely related to the post-merger phase of a BNS merger.The total mass of the binary system could be obtained according to the GW observation.By comparing the total mass of the system with the maximum mass of a rotating star, it can be interpreted whether the post-merger product is a long-lived supramassive NS or short-lived hypermassive NS.
Various nuclear EoS models have been applied to build both uniform and differentially rotating NSs.It has been found that the maximum mass of uniformly rotating NSs, compared with the TOV maximum mass, depends very weakly on EoSs [15].More specifically, regardless of the EoS model, the star could support approximately 20% more mass by uniform rotation.Another universal relationship is also discussed by [16] for differentially rotating NSs.Such relations have been invoked to interpret the observation of GW170817, and constraints on the maximum mass of NSs have been set accordingly [33].In order to see whether this relationship still holds for rotating QSs, we have built axisymmetric rotating QS sequences for both the MIT bag model and the strangeon star model [34].We first build a TOV solution sequence for both EoSs with 24 successive central densities.From each of those TOV solutions, we construct a rotating QS solution sequence by fixing the central density and decrease the axis ratio R z /R x .The axis ratio parameter determines the rotation of the star and preserves axisymmetry at the same time.Those rotating QS solutions terminate at the mass shedding limit.In this way, we manage to explore the parameter space for rotating QSs for various central densities and angular velocities.
Once we have all the solutions ready, we can obtain the TOV maximum mass (M TOV ) and angular momentum at the mass shedding limit (J kep ) as well as the maximum mass for a certain angular momentum (M crit ).The relationship between normalized mass (M crit /M TOV ) and angular momentum (J/J kep ) can therefore be re-investigated for rotating QSs.The result is shown in Figure 1.As can be seen, the universal relationship for NSs no longer holds for QSs.Moreover, even for rotating QSs with different EoSs, the relation is quite different.
Although we couldn't extend the universal relationship or find a new one for rotating QSs, it does provide a potential to distinguish between NS and QS models from a BNS merger event.In particular, quark stars could be more massive when supported by uniform rotation compared with neutron stars (40% compared with 20%).Consequently, a post-merger phase might be longer before collapsing to a black hole if a QS is formed during the merger.Relationship between normalized critical mass and the normalized angular momentum for rotating eutron stars (NSs) and quark stars (QSs).Bottom black line is the quasi-universal relationship found by [15].Blue curve in the middle is the relationship for rotating QSs with strangeon star model and the top red line represents the relationship for the MIT bag model.The universal relationship cannot be extended for QSs easily.

Triaxial Rotating Quark Stars
Rotating QSs in a triaxial Jacobian sequence are another interesting type of QS and occur when the kinetic energy to potential ratio (T/|W|) is sufficient large.On the one hand, the post merger product in a BNS merger or a newly born compact star from a core collapse supernova possesses quite a large angular momentum, which might lead to a sufficiently large T/|W| ratio for the bifurcation for a triaxial Jacobian sequence to occur [35][36][37][38][39]. On the other hand, triaxially rotating compact stars is an effective GW radiator itself [40].
Unlike NSs, which are bound by gravity, QSs are self-bound by strong interaction.Consequently, rotating QSs can reach a much larger T/|W| ratio compared with NSs due to the finite surface density.Therefore, the triaxial instability can play a more important role [41][42][43] for QSs.The triaxial bar mode (Jacobi-like) instability for the MIT bag-model EoS has been investigated in a general relativistic framework [44].
Here, we build quasi-equilibrium constant rest mass sequences (axisymmetric and triaxial) for both the MIT bag-model EoS and the strangeon star model EoS.The surface fit coordinates used in COCAL allows us to treat the surface density discontinuity properly.We begin with the axisymmetric sequence in which we calculate solution sequences with varying parameters, i.e., the central rest-mass density (ρ c ) and the axis ratio (R z /R x ).We first impose axisymmetry as a separate condition and manage to reach the mass shedding limit for all the sequences.In order to access the triaxial solutions, we recompute the above sequence of solutions but this time without imposing axisymmetry.As the rotation rate increases (R z /R x decreases), the triaxial deformation (R y /R x < 1) is spontaneously triggered, since at a large rotation rate the triaxial configuration possesses a lower total energy and is therefore favored over the axisymmetric solution.
Overall, three main properties of triaxially rotating QSs are found according to our calculations.Firstly, QSs generally have triaxial sequences of solutions that are longer than those of NSs.In another words, QSs can see larger triaxial deformations before the sequence is terminated at the mass-shedding limit (c.f.Figures 2 and 3 for an example for the comparison with the n = 0.3 NS model), due to the much higher T/|W| ratio attained by rotating QSs.Secondly, when considering similar triaxial configurations, QSs are (slightly) more efficient GW sources because of the finite surface rest-mass density and hence larger mass quadrupole for QSs (c.f. Figure 4).Thirdly, triaxial supramassive solutions can be found for QSs.(labeled with gray curves).Solid curves are axisymmetric solution sequences, and dashed curves are triaxial solution sequences, which correspond to C = M/R = 0.2 (green curves), 0.15 (red curves), and 0.1 (blue curves), respectively.Note that M is the spherical ADM mass.Lower panel: Magnification of the region near the onset of the triaxial solutions marked with empty symbols.Filled symbols mark the models at the mass-shedding limit.Solutions labeled with "ML" are axisymmetric solutions (Maclaurin spheroids), while those labeled "JB" are triaxial solutions (Jacobi ellipsoids).Shown is the GW strain for the = m = 2 mode normalized by the distance and the ADM mass of the source.Triangle markers in blue and red stand for the triaxially rotating NS cases G4C010 and G4C025 models in [40], which indicates that triaxially deformed QSs are more effective graviational-wave (GW) sources compared with NSs.

Discussion
We have built both axisymmetric and triaxial solutions for uniformly rotating QSs.For axisymmetric rotating QSs, we have investigated the critical mass-angular momentum relation and find that it deviates from the universal relationship for rotating NSs.Especially, uniformly rotating QSs can be more massive compared with M TOV , indicating a quite different post-merger phase in binary merger events.For triaxially rotating QSs, we have identified the bifurcation point from axisymmetric solutions.Triaxial solutions have been constructed from the bifurcation point to the mass shedding limit.The GW emission from such a triaxial rotating QS is estimated with a quadrupole formula and is found to be more effective than that of an NS, which can also be tested with future GW observations.Additionally, since the spin period of a triaxially rotating star increases as the angular momentum increases, the spin frequency at the bifurcation point somehow represents a maximum spin frequency that can be attained by a pulsar when spun-up by accretion.Particularly, a solid QS model is suggested for the strangeon star model [45], which means that r-mode instability could be totally suppressed for such a star and a strangeon star might be spun up to the bifurcation frequency.With the construction of more power radio telescopes such as SKA and FAST, this limit could be tested by searching for faster spinning pulsars and might provide an important clue on the properties of the dense matter in compact stars.

Figure 1 .
Figure 1.Relationship between normalized critical mass and the normalized angular momentum for rotating eutron stars (NSs) and quark stars (QSs).Bottom black line is the quasi-universal relationship found by[15].Blue curve in the middle is the relationship for rotating QSs with strangeon star model and the top red line represents the relationship for the MIT bag model.The universal relationship cannot be extended for QSs easily.

Figure 2 .
Figure 2. Upper panel: T/|W| versus eccentricity e := 1 − (z/x) 2 (in proper length) for the MIT bag-model Equation of State (EoS) sequences as well as for NSs with n = 0.3 EoS reported in[21] (labeled with gray curves).Solid curves are axisymmetric solution sequences, and dashed curves are triaxial solution sequences, which correspond to C = M/R = 0.2 (green curves), 0.15 (red curves), and 0.1 (blue curves), respectively.Note that M is the spherical ADM mass.Lower panel: Magnification of the region near the onset of the triaxial solutions marked with empty symbols.Filled symbols mark the models at the mass-shedding limit.Solutions labeled with "ML" are axisymmetric solutions (Maclaurin spheroids), while those labeled "JB" are triaxial solutions (Jacobi ellipsoids).

Figure 3 .Figure 4 .
Figure 3.The same as 2 but for the LX EoS sequences.