Dynamical stability of quasitoroidal differentially rotating neutron stars

We investigate the dynamical stability of relativistic, differentially rotating, quasitoroidal models of neutron stars through hydrodynamical simulations in full general relativity. We find that all quasitoroidal configurations studied in this work are dynamically unstable against the growth of nonaxisymmetric modes. Both one-arm and bar mode instabilities grow during their evolution. We find that very high rest mass configurations collapse to form black holes. Our calculations suggest that configurations whose rest mass is less than the binary neutron star threshold mass for prompt collapse to black hole transition dynamically to spheroidal, differentially rotating stars that are dynamically stable, but secularly unstable. Our study shows that the existence of extreme quasitoroidal neutron star equilibrium solutions does not imply that long-lived binary neutron star merger remnants can be much more massive than previously found. Finally, we find models that are initially supra-Kerr ($J/M^2>1$) and undergo catastrophic collapse on a dynamical timescale, in contrast to what was found in earlier works. However, cosmic censorship is respected in all of our cases. Our work explicitly demonstrates that exceeding the Kerr bound in rotating neutron star models does not imply dynamical stability.

Figure 1

Projection of the solution space for a $\mathrm{\Gamma }=2$ polytrope in the $(r_p/r_e,\widehat{\beta })$ plane for a fixed value of the stellar maximum energy density ${\epsilon }_{\max }=0.12$ (in polytropic units) and several values of the degree of differential rotation ${\widehat{A}}^{-1}$. Each sequence is labeled by ${\widehat{A}}^{-1}$. The bold black line corresponds to the critical value of the degree of differential rotation ${\widehat{A}}_{\mathrm{crit}}^{-1}=0.75904$ [14], which here divides the solution space into three regions corresponding to type A (bottom right), B (left), and C (top) models. The color bar corresponds to the rest mass of models along each sequence of fixed ${\widehat{A}}^{-1}$, normalized by the maximum rest mass for a nonrotating model (i.e., the TOV limit rest mass) $M_{0,\max }^{\mathrm{TOV}}$.

Figure 2

Equatorial contours (i.e., on the $X\text{−}Y$ plane, where we scale the $X$ and $Y$ coordinate by the gravitational mass $M$) of the rest mass density ${\rho }_0$ at $t=0$ for the maximum rest mass A, B, and C type models (left, middle, and right, respectively) in Table 1. The color bar shows the value of the rest mass density scaled to the maximum value on a logarithmic scale.

Figure 3

Left panel: Maximum rest mass density as a function of time for the A, B, and C models in the case of zero initial perturbations. The green, blue, and red lines correspond to the A, B, and C models, respectively. Center panel: Evolution of the amplitude $|C_m|$ of nonaxisymmetric density modes for the B model [$m=1(\text{red line})$, $m=2(\text{blue line})$, $m=3(\text{green line})$, and $m=4(\text{yellow line})$] in the case of zero initial perturbations. Right panel: Same as the center panel but for the C model.

Figure 4

Snapshots of equatorial contours (i.e., on the $X\text{−}Y$ plane) of the rest mass density ${\rho }_0$, scaled to the maximum value at the start of simulation ${\rho }_{0,\max }(0)$ for the C model under equilibrium evolution. The dashed lines indicate the boundary of the regions within which the rest mass density satisfies ${\rho }_0\ge {\rho }_{0,\max }(0)$. The left and center panels show the development of the model at $t=8.07T_c$ and $t=10.85T_c$, respectively. The right panel shows the state of the model in the time near collapse at $t=12.68T_c$.

Figure 5

Same as Fig. 3 but for the case of pressure depletion.

Figure 6

Left panel: Maximum rest mass density as a function of time for the A, B, and C models in the case of nonaxisymmetric $m=1$ (solid lines) and $m=2$ (dash-dotted lines) initial rest mass density perturbations. The green, blue, and red lines correspond to the A, B, and C models, respectively. Center panel: Evolution of the dominant $m=1$ (solid lines) and $m=2$ (dashed lines) nonaxisymmetric density modes for the B (blue lines) and C (red lines) models in the case of an $m=1$ perturbation. Right panel: Same as the center panel but for an $m=2$ perturbation.

Figure 7

Left panel: Maximum rest mass density as a function of time for the ${\mathrm{B}}_{\mathrm{low}}$ and ${\mathrm{C}}_{\mathrm{low}}$ models in the case of zero initial perturbations. The orange and magenta lines correspond to the ${\mathrm{B}}_{\mathrm{low}}$ and ${\mathrm{C}}_{\mathrm{low}}$ models, respectively. Center panel: Evolution of nonaxisymmetric density modes for the ${\mathrm{B}}_{\mathrm{low}}$ model for the [$m=1(\text{red line})$, $m=2(\text{blue line})$, $m=3(\text{green line})$, and $m=4(\text{yellow line})$] modes in the case of zero initial perturbations. Right panel: Same as the center panel but for the ${\mathrm{C}}_{\mathrm{low}}$ model. Note that model ${\mathrm{B}}_{\mathrm{low}}$ has a central period which in units of $M$ is almost an order of magnitude longer than that of model ${\mathrm{C}}_{\mathrm{low}}$. Thus, the evolution of model ${\mathrm{B}}_{\mathrm{low}}$ is very long. It is the normalization with respect to $T_c$ that makes it appear that this evolution is short.

Figure 8

Same as Fig. 7 but for the case of pressure depletion.

Figure 9

Snapshots of equatorial contours (i.e., on the $X\text{−}Y$ plane) of the rest mass density ${\rho }_0$, scaled to the maximum value at the start of simulation ${\rho }_{0,\max }(0)$ for model ${\mathrm{B}}_{\mathrm{low}}$ under nonaxisymmetric initial perturbations. The top (bottom) panels show the evolution under an initial $m=1$ ($m=2$) perturbation. The right panels show the final states in both cases. The dashed curves outline the boundary of the regions within which the rest mass density satisfies ${\rho }_0\ge {\rho }_{0,\max }(0)$.

Figure 10

Left panel: Maximum rest mass density as a function of time for the ${\mathrm{B}}_{\mathrm{low}}$ and ${\mathrm{C}}_{\mathrm{low}}$ models in the case of nonaxisymmetric $m=1$ (solid lines) and $m=2$ (dash-dotted lines) rest mass density initial perturbations. The orange and magenta lines correspond to the ${\mathrm{B}}_{\mathrm{low}}$ and ${\mathrm{C}}_{\mathrm{low}}$ models, respectively. Center panel: Evolution of the dominant $m=1$ (solid lines) and $m=2$ (dashed lines) nonaxisymmetric density modes for the ${\mathrm{B}}_{\mathrm{low}}$ (orange lines) and ${\mathrm{C}}_{\mathrm{low}}$ models (magenta lines) in the case of an $m=1$ initial perturbation. Right panel: Same as the center panel but for an $m=2$ initial perturbation.

Figure 11

The real part of the Newman-Penrose scalar $s=-2$ spin-weighted spherical harmonic modes ${\mathrm{\Psi }}_4^{2,m}$ (multiplied by the coordinate radius of the extraction spherical surface $r$ and the ADM mass $M$) as a function of $t-r$ (scaled by the central period $T_c$) for two representative evolutions presented in this work. We focus on the contribution from the dominant $l=2$, $m=0$ (yellow lines), $m=1$ (red lines), and $m=2$ (blue lines) modes. The model and type of perturbation are shown at the top of each panel.

Figure 12

Contour of the ratio of thermal pressure to cold pressure on the equatorial plane (i.e., on the $X\text{−}Y$ plane) corresponding to the final state of the ${\mathrm{B}}_{\mathrm{low}}$ model under all initial perturbations. Only densities with ${\rho }_0\ge {10}^{-3}{\rho }_{0,\max }$ are shown, where ${\rho }_{0,\max }$ corresponds to the maximum rest mass density at the time of the snapshots. All snapshots correspond to $t=20.13T_c$; the second from the right and rightmost snapshots corresponds to the top and bottom right panels of Fig. 9 for the $m=1$ and $m=2$ perturbations, respectively.

Figure 13

Azimuthally and time-averaged angular velocity (normalized by the initial maximum angular velocity) vs distance on the equatorial plane in the case of equilibrium evolution (green line), pressure depletion (yellow line), $m=1$ initial perturbation (red line), and $m=2$ initial perturbation (blue line) near the end of the simulations. The dashed black line corresponds to the initial angular velocity profile.

Figure 14

Left panel: Convergence of $|{\mathrm{\Psi }}_4^{2,2}|$ as a function of $(t-r)/T_c$ in the case of the B model under an $m=2$ perturbation. The left panel shows the difference of $|{\mathrm{\Psi }}_4^{2,2}|$ between the medium and canonical resolution runs (cyan line, labeled “med., can.”), the medium and high resolution runs (green line, labeled “hi., med.”), and the high and canonical resolution runs (orange line, labeled “hi., can.”), scaled assuming second order convergence. Center panel: L2 norm of the Hamiltonian constraint times the squared ADM mass $M^2||\mathcal{H}||$ as a function of time for the B model under an $m=2$ perturbation. Shown are the canonical (blue line), medium (cyan line), and high (orange line) resolution runs. Right panel: Same as the center panel, but for the L2 norm of the momentum constraint $||\mathcal{M}||$.

Figure 15

Density mode decomposition for the shifted grid simulations in the case of the B model under pressure depletion. We show the two most dominant modes, which are always the $m=1$ (solid lines) or $m=2$ (dash-dotted lines) modes. The dark purple lines show the results for a small grid shift $\delta y_{\mathrm{low}}=0.0001$, the blue lines show the results for our standard simulations with grid shift $\delta y_{\text{stand}}=0.001$, the red lines show the results for a medium sized grid shift $\delta y_{\mathrm{med}}=0.005$, and the green lines show the results for a large grid shift $\delta y_{\mathrm{high}}=0.01$.