Viscosity driven instability in rotating relativistic stars

We investigate the viscosity driven instability in rotating relativistic stars by means of an iterative approach. We focus on polytropic rotating equilibrium stars and impose a $m=2$ perturbation in the lapse. We vary both the stiffness of the equation of state and the compactness of the star to study those effects on the value of the threshold. For a uniformly rotating star, the criterion $T/W$, where $T$ is the rotational kinetic energy and $W$ is the gravitational binding energy, mainly depends on the compactness of the star and takes values around $0.13–0.16$, which differ slightly from that of Newtonian incompressible stars ($\sim 0.14$). For differentially rotating stars, the critical value of $T/W$ is found to span the range $0.17-0.25$. This is significantly larger than the uniformly rotating case with the same compactness of the star. Finally we discuss a possibility of detecting gravitational waves from viscosity driven instability with ground-based interferometers.

Figure 1

Sketch of the computational procedure to determine the stability of rotating stars.

Figure 2

Diagnostic $\dot{q}/q$ as a function of iteration numbers $\mathcal{N}$ for sixteen different uniformly rotating stars of $\Gamma =2.3$ and $M/R=0.01$. Solid, dash-dotted, dashed, dotted lines denotes $T/W=(0.1306,0.1307,0.1308,0.1309)$ for $({\mathcal{N}}_{\theta },{\epsilon }_{\mathrm{amp}})=(17,1.00\times {10}^{-3})$, $T/W=(0.1309,0.1312,0.1313,0.1314)$ for $({\mathcal{N}}_{\theta },{\epsilon }_{\mathrm{amp}})=(17,1.00\times {10}^{-4})$, $T/W=(0.1309,0.1312,0.1313,0.1314)$ for $({\mathcal{N}}_{\theta },{\epsilon }_{\mathrm{amp}})=(17,1.00\times {10}^{-5})$, and $T/W=(0.1306,0.1307,0.1308,0.1310)$ for $({\mathcal{N}}_{\theta },{\epsilon }_{\mathrm{amp}})=(25,1.00\times {10}^{-5})$, respectively.

Figure 3

Stability of uniformly rotating stars of $\Gamma =2.3$, $M/R=0.01$ for four different parameters. Circle (open) and circle (filled) denotes stable and unstable to viscosity driven bar-mode perturbation, respectively. We fix the compactness for each parameter up to four digits. Note that there is a monotonic transition from stable to unstable when increasing $T/W$.

Figure 4

Diagnostic $\dot{q}/q$ as a function of iteration steps $\mathcal{N}$ for sixteen different uniformly rotating stars. Solid and dashed lines denotes the unstable and stable stars, respectively. Note that the $T/W$ for each stable star in the same compactness is 0.0001 lower than the critical value of that of an unstable star (see Table ).

Figure 5

Stability of uniformly rotating stars for sixteen different parameters. Circle (open) and circle (filled) denotes stable and unstable to viscosity driven bar-mode perturbation, respectively. We fix the compactness for each parameter up to four digits. Note that there is a monotonic transition from stable to unstable when increasing $T/W$.

Figure 6

Critical value of $T/W$ as a function of $\Gamma$ for four different compactness of uniformly rotating stars (see Table ). Circle (open), circle (filled), square (open), and square (filled) denotes the compactness ($M/R$) of 0.01, 0.05, 0.1, and 0.15, respectively. The star whose adiabatic index is ${\Gamma }_{\mathrm{low}}-0.1$, where ${\Gamma }_{\mathrm{low}}$ is the lowest $\Gamma$ of the unstable star in each compactness in the figure, is stable.

Figure 7

Diagnostic $\dot{q}/q$ as a function of iteration steps $\mathcal{N}$ for five different differentially rotating stars. Solid and dashed lines denotes the unstable and stable stars, respectively. Note that the $T/W$ for each stable star in the same compactness is 0.0001 lower than the critical value of that of an unstable star (see Table ).

Figure 8

Stability analysis of differentially rotating stars for five different parameters. Circle (open) and circle (filled) denotes stable and unstable to viscosity driven bar-mode perturbation, respectively. We fix the compactness for each parameter up to four digits. Note that there is a monotonic transition from stable to unstable when increasing $T/W$.

Figure 9

Relative deviation of the total angular momentum between two successive steps in the iteration process, as a function of the step number $\mathcal{N}$. Solid, dashed, dotted, dash-dotted lines represent ${\epsilon }_{\mathrm{omg}}=(0.5,0.6,0.7,0.8)\times {10}^{-4}$ for $M/R=0.01$ and $T/W=0.1828$, ${\epsilon }_{\mathrm{omg}}=(0.8,0.9,1.0,1.1)\times {10}^{-4}$ for $M/R=0.05$ and $T/W=0.1999$, ${\epsilon }_{\mathrm{omg}}=(1.1,1.2,1.3,1.4)\times {10}^{-4}$ for $M/R=0.1$ and $T/W=0.2186$, ${\epsilon }_{\mathrm{omg}}=(1.0,1.1,1.2,1.3)\times {10}^{-4}$ for $M/R=0.15$ and $T/W=0.2354$, and ${\epsilon }_{\mathrm{omg}}=(0.9,1.0,1.1,1.2)\times {10}^{-4}$ for $M/R=0.2$ and $T/W=0.2496$, respectively.

Figure 10

Diagnostic $\dot{q}/q$ as a function of iteration steps $\mathcal{N}$ for five different differentially rotating stars. Solid, dashed, dotted, and dashed line denotes $T/W=(0.1825,0.1828,0.1834,0.1844)$ and ${\epsilon }_{\mathrm{omg}}=0.8\times {10}^{-4}$ for $M/R=0.01$, $T/W=(0.1993,0.1996,0.1998,0.1999)$ and ${\epsilon }_{\mathrm{omg}}=0.9\times {10}^{-4}$ for $M/R=0.05$, $T/W=(0.2180,0.2184,0.2185,0.2186)$ and ${\epsilon }_{\mathrm{omg}}=1.3\times {10}^{-4}$ for $M/R=0.1$, $T/W=(0.2348,0.2352,0.2353,0.2354)$ and ${\epsilon }_{\mathrm{omg}}=1.1\times {10}^{-4}$ for $M/R=0.15$, and $T/W=$ $(0.2494,0.2495,0.2496)$ and ${\epsilon }_{\mathrm{omg}}=1.0\times {10}^{-4}$ for $M/R=0.2$, respectively. Note that $\dot{q}/q$ is always increasing around the critical value of $T/W$ in differentially rotating stars, which means the change of rotational profile due to viscosity unstabilizes the star.