Stellar structure and stability of charged interacting quark stars and their scaling behavior

We explore the stellar structure and radial stability of charged quark stars composed of interacting quark matter in three classes of commonly used charge models. We adopt a general parametrization of interacting quark matter equation of state that includes the corrections from perturbative QCD, color superconductivity, and the strange quark mass into one parameter $\lambda$, or one dimensionless parameter $\overline{\lambda }={\lambda }^2/(4B_{\mathrm{eff}})$ after being rescaled with the effective bag constant $B_{\mathrm{eff}}$. We find that increasing charge tends to increase the mass and radius profiles, and enlarges the separation size in mass between the maximum mass point and the point where zero eigenfrequencies ${\omega }_0^2=0$ of the fundamental radial oscillation mode occur. The sign of the separation in central density depends on the charge model; this separation also has a dependence on $\lambda$ such that increasing $\lambda$ (which can occur for either large color superconductivity or small strange quark mass) tends to decrease this separation size for the first and third classes of charge models monotonically. Moreover, for the second and third classes of charge models, we manage to numerically and analytically identify a new kind of stellar structure with a zero central pressure but with a finite radius and mass. All the calculations and analysis are performed in a general dimensionless rescaling approach so that the results are independent of explicit values of dimensional parameters.

Figure 1

Plots of $\overline{M}$ vs $\overline{R}$ (upper graphs) and ${\overline{\rho }}_c$ (lower graphs) of charged interacting quark stars in model A for (left) $\alpha =(0,0.3,0.7)$ and (right) $\alpha =0.999$. The right and top axes in each plot are the corresponding dimensional parameters with $B_{\mathrm{eff}}=60\mathrm{MeV}/{\mathrm{fm}}^3$ for illustration. The black, blue, purple, and red curves, respectively, denote $\overline{\lambda }=(0,0.5,10,\infty )$ with the sign of $\lambda$ being positive. Darker shades correspond to increasing values of $\alpha$. The solid dots denote the maximum mass configurations, with the filled squares denoting where ${\overline{\omega }}_0^2=0$.

Figure 2

Plots of $\overline{M}$ vs $\overline{R}$ (upper graphs) and ${\overline{\rho }}_c$ (lower graphs) of charged interacting quark stars in model B for (left) $\overline{\beta }=(0,1.5,2.5)$ and (right) $\overline{\beta }=3.5$. The right and top axes in each plot are the corresponding dimensional parameters with $B_{\mathrm{eff}}=60\mathrm{MeV}/{\mathrm{fm}}^3$ for illustration. The black, blue, purple, and red curves, respectively, denote $\overline{\lambda }=(0,0.5,10,\infty )$ with the sign of $\lambda$ being positive. Darker shades correspond to increasing values of $\overline{\beta }$. The solid dots denote the maximum mass configurations, with the filled squares denoting where ${\overline{\omega }}_0^2=0$. In the right diagrams, the configurations denoted by the purple and red curves are radially unstable for all central densities.

Figure 3

$\overline{P}-\overline{r}$ plots of charged interacting quark stars in model B for (a) $(\overline{\lambda },\overline{\beta })=(0.5,2.5)$ and (b) $(\overline{\lambda },\overline{\beta })=(0.5,3.5),(10,2.5)$ with a tiny rescaled central pressure $\overline{P}=0.00001671$.

Figure 4

$\overline{M}-\overline{R}$ (left) and $\overline{M}-{\overline{\rho }}_c$ (right) plots of charged interacting quark stars for $\overline{Q}=(0,1.538,3.076)\times {10}^{-2}$. The right and top axes in each plot are the corresponding dimensional parameters with $B_{\mathrm{eff}}=60\mathrm{MeV}/{\mathrm{fm}}^3$ for illustration. The black, blue, and purple curves, respectively, denote $\overline{\lambda }=(0,0.5,10)$ with the sign of $\lambda$ being positive. Red curves are with $\overline{\lambda }=\infty$. Darker shades correspond to increasing values of $\overline{Q}$. The solid dots denote the maximum mass configurations, with filled squares denoting where ${\overline{\omega }}_0^2=0$.

Figure 5

Plots of $\overline{M}$ vs $\overline{R}$ (upper graphs) and ${\overline{\rho }}_c$ (lower graphs) of charged interacting quark stars in model A for (left) $\alpha =(0,0.3,0.7)$ and (right) $\alpha =0.999$. The right and top axes in each plot are the corresponding dimensional parameters with $B_{\mathrm{eff}}=60\mathrm{MeV}/{\mathrm{fm}}^3$ for illustration. The black, blue, and purple curves, respectively, denote $\overline{\lambda }=(0,0.5,10)$ with the sign of $\lambda$ being negative. Darker shades correspond to increasing values of $\alpha$. The solid dots denote the maximum mass configurations, with the filled squares representing where ${\overline{\omega }}_0^2=0$. Note that for the purple curves in the $\overline{M}-{\overline{\rho }}_c$ plots, we rescaled the axis as ${\overline{\rho }}_c\to {\overline{\rho }}_c/5$ for a clear illustration.

Figure 6

Plots of $\overline{M}$ vs $\overline{R}$ (upper graphs) and ${\overline{\rho }}_c$ (lower graphs) of charged interacting quark stars in model B for (left) $\overline{\beta }=(0,1.5,2.5)$ and (right) $\overline{\beta }=3.5$. The right and top axes in each plot are the corresponding dimensional parameters with $B_{\mathrm{eff}}=60\mathrm{MeV}/{\mathrm{fm}}^3$ for illustration. The black, blue, and purple curves, respectively, denote $\overline{\lambda }=(0,0.5,10)$ with the sign of $\lambda$ being negative. Darker shades correspond to increasing values of $\overline{\beta }$. The solid dots denote the maximum mass configurations, with the filled squares representing where ${\overline{\omega }}_0^2=0$. Note that for the purple curves in the $\overline{M}-{\overline{\rho }}_c$ plots, we rescaled the axis as ${\overline{\rho }}_c\to {\overline{\rho }}_c/5$ for a clear illustration.

Figure 7

$\overline{M}-\overline{R}$ (left) and $\overline{M}-{\overline{\rho }}_c$ (right) plots of charged interacting quark stars for $\overline{Q}=(0,1.538,3.076)\times {10}^{-2}$. The right and top axes in each plot are the corresponding dimensional parameters with $B_{\mathrm{eff}}=60\mathrm{MeV}/{\mathrm{fm}}^3$ for illustration. The black, blue, and purple curves, respectively, denote $\overline{\lambda }=(0,0.5,10)$ with the sign of $\lambda$ being negative. Darker shades correspond to increasing values of $\overline{Q}$. The solid dots denote the maximum mass configurations, with filled squares denoting where ${\overline{\omega }}_0^2=0$. Note that for the purple curves in the $\overline{M}-{\overline{\rho }}_c$ plots, we rescaled the axis as ${\overline{\rho }}_c\to {\overline{\rho }}_c/5$ for a clear illustration.