Dynamical bar instability in a relativistic rotational core collapse

We investigate the rotational core collapse of a rapidly rotating relativistic star by means of a $3+1$ hydrodynamical simulation in conformally flat spacetime of general relativity. We concentrate our investigation to the bounce of the rotational core collapse, since potentially most of the gravitational waves from it are radiated around the core bounce. The dynamics of the star is started from a differentially rotating equilibrium star of $T/W\sim 0.16$ ($T$ is the rotational kinetic energy and $W$ is the gravitational binding energy of the equilibrium star), depleting the pressure to initiate the collapse and to exceed the threshold of dynamical bar instability. Our finding is that the collapsing star potentially forms a bar when the star has a toroidal structure due to the redistribution of the angular momentum at the core bounce. On the other hand, the collapsing star weakly forms a bar when the star has a spheroidal structure. We also find that the bar structure of the star is destroyed when the torus is destroyed in the rotational core collapse. Since the collapse of a toroidal star potentially forms a bar, it can be a promising source of gravitational waves which will be detected in advanced LIGO.

Figure 1

Evolution of quadrupole diagnostics in 6 different collapsing stars. Roman character in the panel corresponds to the model type in Table . Solid lines and dashed lines represent toroidal and spheroidal stars, which correspond to model $a$ and $b$ in Table , respectively.

Figure 2

Snapshot of the coordinate density along the $x$-axis for 6 different stars (see Table ). Solid lines and dashed lines represent $t/P_{\mathrm{c}}=$ I(a) (9.61, 16.0), I(b) (9.67, 15.3), II(a) (2.39, 12.0), II(b) (3.25, 12.0), III(a) (3.19, 7.77), III(b) (3.20, 7.77), respectively.

Figure 3

Density contour in the equatorial plane of 2 collapsing stars (Model I[a] and [b] of Table ). Snapshots are plotted at the parameter ($t/t_{\mathrm{dyn}}$, ${\rho }_{\max }^{*}$) $=$ I(a)-i ($9.61$, $9.13\times {10}^{-3}$), I(a)-ii ($16.0$, $6.04\times {10}^{-3}$), I(b)-i ($9.67$, $1.33\times {10}^{-2}$), I(b)-ii ($15.3$, $1.57\times {10}^{-2}$), respectively. The contour lines denote coordinate densities ${\rho }^{*}={\rho }_{\max }^{*}\times {10}^{-0.220(16-i)}(i=1,\cdots ,15)$.

Figure 4

Density contour in the equatorial plane of 2 collapsing stars (Model II[a] and [b] of Table ). Snapshots are plotted at the parameter ($t/t_{\mathrm{dyn}}$, ${\rho }_{\max }^{*}$) $=$ II(a)-i ($3.19$, $1.13\times {10}^{-3}$), II(a)-ii ($12.0$, $4.62\times {10}^{-3}$), II(b)-i ($3.25$, $3.39\times {10}^{-3}$), II(b)-ii ($12.0$, $8.71\times {10}^{-3}$), respectively. The contour lines denote coordinate densities ${\rho }^{*}={\rho }_{\max }^{*}\times {10}^{-0.200(16-i)}(i=1,\cdots ,15)$.

Figure 5

Density contour in the equatorial plane of 2 collapsing stars (Model III[a] and [b] of Table ). Snapshots are plotted at the parameter ($t/t_{\mathrm{dyn}}$, ${\rho }_{\max }^{*}$) $=$ III(a)-i ($3.19$, $2.70\times {10}^{-4}$), III(a)-ii ($7.77$, $4.02\times {10}^{-3}$), III(b)-i ($3.20$, $3.28\times {10}^{-4}$), III(b)-ii ($7.77$, $1.92\times {10}^{-3}$), respectively. The contour lines denote coordinate densities ${\rho }^{*}={\rho }_{\max }^{*}\times {10}^{-0.220(16-i)}(i=1,\cdots ,15)$.

Figure 6

Gravitational waveforms from a distant observer along the rotational axis of the equilibrium star. Roman character in the panel corresponds to the model type in Table . Solid lines and dashed lines represent toroidal and spheroidal stars, which correspond to $a$ and $b$ in Table , respectively.

Figure 7

Diagnostics of dynamical bar instability $\beta$ as a function of time. Roman character in the panel corresponds to the model type in Table . Solid lines and dashed lines represent toroidal and spheroidal star, which corresponds to $a$ and $b$ in Table .

Figure 8

Snapshot of the orbital angular velocity along the $x$-axis. Solid lines and dashed lines represent $t/P_{\mathrm{c}}=$ I(a) (9.61, 16.0), I(b) (9.67, 15.3), II(a) (2.39, 12.0), II(b) (3.25, 12.0), III(a) (3.19, 7.77), III(b) (3.20, 7.77), respectively.