Differentially rotating strange star in general relativity

Rapidly and differentially rotating compact stars are believed to be formed in binary neutron star merger events, according to both numerical simulations and the multimessenger observation of GW170817. Questions that have not been answered by the observation of GW170817 and remain open are whether or not a phase transition of strong interaction could happen during a binary neutron star merger event that forms a differentially rotating strange star as a remnant as well as the possibility of having a binary strange star merger scenario. The lifetime and evolution of such a differentially rotating star is tightly related to the observations in the postmerger phase. Various studies on the maximum mass of differentially rotating neutron stars have been done in the past, most of which assume the so-called $j$-constant law as the rotation profile inside the star. In this paper, we extend the studies to a more realistic differential rotation law and concentrate on bare quark star models. Significant differences are found between differentially rotating strange stars and neutron stars, with both the $j$-const law and the new rotation profile model. A moderate differential rotation rate for neutron stars is found to be too large for strange stars, resulting in a rapid drop in the maximum mass as the differential rotation degree is increased further from $\widehat{A}\sim 2.0$, where $\widehat{A}$ is a parameter characterizing the differential rotation rate for the $j$-const law. As a result, the maximum mass of a differentially rotating self-bound star drops below the uniformly rotating mass-shedding limit for a reasonable degree of differential rotation. The continuous transition to the toroidal sequence is also found to happen at a much smaller differential rotation rate and angular momentum than for neutron stars. In spite of those differences, $\widehat{A}$-insensitive relation between the maximum mass for a given angular momentum is still found to hold, even for the new differential rotation law. Astrophysical consequences of these differences and how to distinguish between strange star and neutron star models with future observations are also discussed.

Figure 1

Gravitational mass vs maximum density (in $G=M_{\odot }=c=1$ unit) diagram for the strangeon star model. The black curve is for the nonrotating case (TOV solution), while the red curve is for the mass-shedding limit for the uniformly rotating axisymmetric case. Curves with gradually changing color from green to blue represent the maximum mass configurations obtained for the differentially rotating case with $j$-const law. The $\widehat{A}$ parameter applied to those curves ranges from 1.8 to 0.8 as the color changes from green to blue (from top to the bottom). Note that, due to the existence of type C solutions mentioned in Sec. 4c, the maximum mass of the differentially rotating case could probably be found for the case in which the central density is not the maximum density inside the star. We have also shown the $\widehat{A}=5$ case, which corresponds to the maximum possible mass in our calculation in the yellow curve on top. For this particular differential rotation rate, type C solutions and the toroidal limit are only found for large central densities (i.e., ${\rho }_{\max }>1.12\times {10}^{-3}$). We label the part where only type A solutions exist by a dashed curve.

Figure 2

Gravitational mass vs maximum density diagram for MIT bag model SSs. The models for curves with different colors are exactly the same as in Fig. 1. We calculated one more model for the MIT bag model with $\widehat{A}=0.6$ as shown by the bottom blue curve. The yellow curve on the top, which corresponds to the maximum possible mass case for the MIT bag model, is with $\widehat{A}=3.0$.

Figure 3

The angular velocity profile for the strangeon star (blue) and MIT bag model (green) when the maximum mass becomes close to their rigidly rotating mass-shedding limit. This means $\widehat{A}=1.8$ for the strangeon star model and $\widehat{A}=0.8$ for the MIT bag model. The dashed horizontal line indicates the angular velocity for the mass-shedding limit of the rigid rotation case.

Figure 4

Gravitational mass vs maximum density diagram for the strangeon star model. The black and red curves are for the nonrotating and uniformly rotating mass-shedding limit cases, respectively. The other curves ranging from green to blue colors are differentially rotating solutions with constant axis ratio $R_z/R_x=0.75$, 0.5 and 0.25. The dashed curves are for models with the $j$-const law with $\widehat{A}=1.0$, and solid curves are for models with the new rotation law, Eq. (8).

Figure 5

The relationship between critical gravitational mass $M_{\mathrm{crit}}$ and angular momentum $J$ for strangeon stars (upper panel) and MIT bag model stars (lower panel). Both the rigid rotating case (solid blue line) and differentially rotating case (green dots for $\widehat{A}=3.0$ and red dots for $\widehat{A}=1.0$) are shown. The 1% error range for the relationship of the rigid rotating case is shown in dashed blue lines for comparison purposes. As can be seen, for both EoS even in the case of $\widehat{A}=1.0$, the relationship between $M_{\mathrm{crit}}$ and $J$ is still reasonably consistent with the rigid rotating case. We have also labeled the results from the new differential rotation law with black markers.

Figure 6

Stellar surface and rest-mass density contour of a differentially rotating strangeon star (upper panel) and its density and angular velocity profile (lower panel). Details about the solution shown in this figure can be found in the DR-LX-4 model in Table 3.