Determining the stiffness of the equation of state using low $T/W$ dynamical instabilities in differentially rotating stars

We investigate the nature of low $T/W$ dynamical instabilities in various ranges of the stiffness of the equation of state in differentially rotating stars. Here $T$ is the rotational kinetic energy, while $W$ is the gravitational binding energy. We analyze these instabilities in both a linear perturbation analysis and a three-dimensional hydrodynamical simulation. An unstable normal mode of a differentially rotating star is detected by solving an eigenvalue problem along the equatorial plane of the star. The physical mechanism of low $T/W$ dynamical instabilities is also qualitatively confirmed by a scattering of sound waves between corotation and the surface caused by the corotation barrier. Therefore, we can draw a picture of existing pulsation modes unstabilized due to an amplified reflection of sound waves from the corotation barrier. The feature in the eigenfrequency and eigenfunction of the unstable mode in the linear analysis roughly agrees with that in the three-dimensional hydrodynamical simulation in Newtonian gravity. Moreover, the nature of the eigenfunction that oscillates between corotation and the surface for an unstable star requires reinterpretation of pulsation modes in differentially rotating stars. Finally, we propose a manner by which to constrain the stiffness of the equation of state by the direct detection of mode decomposed gravitational waveforms.

Figure 1

Comparison of a rest mass density, specific internal energy, and velocity between numerical and analytical results of the one-dimensional wall shock problem at $t=1.0x^{(\mathrm{bound})}/v_0$. Red and blue lines represent our computational and analytical results. We choose the parameter sets as $\mathrm{\Gamma }=2$, ${\kappa }_{\mathrm{s}}=1$ with grid space $\mathrm{\Delta }x=1.0\times {10}^{-3}{x}^{(\mathrm{bound})}$ and $v_0=7.07c_s^{(0)}$, where $c_s^{(0)}$ is the initial speed of sound at $t=0$.

Figure 2

The $m=1$ and $m=2$ eigenfunctions $|\delta U_m|$ for four low $T/W$ dynamically unstable stars in cylindrical models. The labels I(a), II(a), III(a), and IV(a), respectively, represent the equilibrium models in Table 1. Red, blue, green, magenta, brown, black, grey, and cyan, respectively, represent the node number between corotation and surface of $N=0$, 1, 2, 3, 4, 5, 6, and 7. The increasing number of nodes between corotation and the surface can clearly be seen in all eigenfunctions.

Figure 3

Same as Fig. 2, but for $m=2$ eigenfunctions $|\delta U_m|$ in spheroidal models.

Figure 4

The $M_2$ diagnostics for three low $T/W$ $n=1$ dynamically unstable stars. Red, blue, and green lines, respectively, represent models I-i(a), I-ii(a), and I-iii(a). We find an amplified oscillation in the diagnostics for all three cases.

Figure 5

Spectra of $M_2$ diagnostics ($|F_2{|}^2$) for three low $T/W$ $n=1$ dynamically unstable stars. Red, blue, and green lines, respectively, represent models I-i(a), I-ii(a), and I-iii(a). Only a single dominant peak in the spectrum shows that the unstable stars contain one dominant characteristic frequency.

Figure 6

Growth rate of $|M_2|$ diagnostics for three low $T/W$ $n=1$ dynamically unstable stars. Red, blue, and green lines, respectively, represent models I-i(a), I-ii(a), and I-iii(a). Dotted lines, respectively, represent the fitting formula $|M_2|=A_2\exp [B_{2}t/P_c]$ for each model ($[A_2,B_2]=[5.90\times {10}^{-3},7.82\times {10}^{-2}]$ for model I-i[a], $[5.31\times {10}^{-3},8.35\times {10}^{-2}]$ for model I-ii[a], and $[5.74\times {10}^{-3},4.20\times {10}^{-2}]$ for model I-iii[a]).

Figure 7

Scalar potentials $|U_2|$ for three low $T/W$ $n=1$ dynamically unstable stars in the equatorial plane. Red, blue, and green lines, respectively, represent models I-i(a) at $t=105.6P_c$, I-ii(a) at $t=88.5P_c$, and I-iii(a) at $t=102.1P_c$.

Figure 8

The $M_m$ diagnostics for four low $T/W$ dynamically unstable stars. Red, blue, green, and magenta lines, respectively, represent diagnostics $M_1$, $M_2$, $M_3$, and $M_4$.

Figure 9

Spectra of $M_m$ diagnostics ($|F_m{|}^2$) for four low $T/W$ dynamically unstable stars. Red, blue, green, and magenta lines, respectively, represent $m=1$, 2, 3, and 4. Only a single dominant peak in the spectrum shows that the unstable stars contain one dominant characteristic frequency.

Figure 10

Scalar potentials of $|U_1|$ (red line) and $|U_2|$ (blue line) diagnostics for four low $T/W$ dynamically unstable stars in the equatorial plane. The potentials are respectively plotted at $t=171.8P_c$ (model I[a]), at $t=219.3P_c$ (model II[a]), at $t=106.75P_c$ (model III[a]), and at $t=78.11P_c$ (model IV[a]).

Figure 11

Gravitational waveforms for four low $T/W$ dynamically unstable stars observed along the rotational axis of the equilibrium star. Red and blue lines, respectively, represent $+$ and $\times$ modes.

Figure 12

Spectra of gravitational waveforms observed along the rotational axis for four low $T/W$ dynamically unstable stars.

Figure 13

Same as Fig. 11, but along the principal axis in the equatorial plane.

Figure 14

Same as Fig. 12, but in the equatorial plane.

Figure 15

The $m=1$ and $m=2$ eigenfrequencies for four low $T/W$ dynamically unstable stars (models I[a], II[a], III[a], and IV[a]) in cylindrical models. Circles, squares, top triangles, and bottom triangles, respectively, denote the polytropic index of $n=1$, 1.5, 2, and 3. Opened and filled symbols, respectively, represent the $m=1$ and $m=2$ modes. Comparing the largest imaginary frequencies between $m=1$ and $m=2$ in each polytropic index, the dominant $m$ mode changes at the stiffness of the equation of state around $\mathrm{\Gamma }\approx 1.50$.

Figure 16

Allowed frequency region (white region) to treat the pulsation system for spheroidal models as an eigenvalue problem. The left panel represents the case of $l=2$, $m=1$, and the right one represents that of $l=2$, $m=2$.