Dynamical bar-mode instability in rotating and magnetized relativistic stars

We present three-dimensional simulations of the dynamical bar-mode instability in magnetized and differentially rotating stars in full general relativity. Our focus is on the effects that magnetic fields have on the dynamics and the onset of the instability. In particular, we perform ideal-magnetohydrodynamics simulations of neutron stars that are known to be either stable or unstable against the purely hydrodynamical instability, but to which a poloidal magnetic field in the range of ${10}^{14}–{10}^{16}\mathrm{G}$ is superimposed initially. As expected, the differential rotation is responsible for the shearing of the poloidal field and the consequent linear growth in time of the toroidal magnetic field. The latter rapidly exceeds in strength the original poloidal one, leading to a magnetic-field amplification in the stars. Weak initial magnetic fields, i.e., $\lesssim {10}^{15}\mathrm{G}$, have negligible effects on the development of the dynamical bar-mode instability, simply braking the stellar configuration via magnetic-field shearing. On the other hand, strong magnetic fields, i.e., $\gtrsim {10}^{16}\mathrm{G}$, can suppress the instability completely, with the precise threshold being dependent also on the amount of rotation. As a result, it is unlikely that very highly magnetized neutron stars can be considered as sources of gravitational waves via the dynamical bar-mode instability.

Figure 1

Initial profiles of the rest-mass density $\rho$ (left panel), the angular velocity $\Omega$ (center panel) and of the $z$ component of the magnetic field (right panel) for models S8, S7, S6, S1, U3, U11 and U13. The profiles of the stable models are here drawn with blue solid lines, and those for the unstable models with red dot-dashed lines.

Figure 2

Representation of the initial models in a $(\beta ,{\beta }_{\mathrm{mag}})$ plane. Blue and red symbols mark models that are respectively bar-mode stable and bar-mode unstable at zero magnetizations, while the vertical red dashed line marks the stability threshold for zero magnetic fields. The red shaded area collects as a function of their magnetization models that the evolutions reveal to be bar-mode unstable; hence, red squares refer to initial models that develop a bar-mode instability, while red triangles refer to potentially bar-unstable models that are stabilized by the strong magnetic fields.

Figure 3

Evolution of the distortion parameters ${\eta }_{+}$ and $\eta$ for model U11 with four different values of the initial poloidal magnetic field: $B_{\max }^z{|}_{t,z=0}=1.0\times {10}^{14}$, $2.0\times {10}^{15}$, $4.0\times {10}^{15}$, and $1.0\times {10}^{16}\mathrm{G}$.

Figure 4

Snapshots of the rest-mass density on the $(x,y)$ plane for models U11-1.0e14 (top row), U11-4.0e15 (central row) and U11-1.0e16 (bottom row) at different times during the evolution, namely, $t=15.0\mathrm{ms}$ (left column), $t=22.5\mathrm{ms}$ (central column) and $t=30\mathrm{ms}$ (right column). The color code is defined in terms of ${log}_{10}(\rho )$, where $\rho$ is in cgs units ($\mathrm{g}{\mathrm{cm}}^{-3}$). Additionally, isodensity contours are shown for $\rho ={10}^{11}$, ${10}^{12}$, $5\times {10}^{12}$, ${10}^{13}$, $5\times {10}^{13}$ and ${10}^{14}\mathrm{g}{\mathrm{cm}}^{-3}$.

Figure 5

Spacetime diagrams of the evolution of the rest-mass density $\rho$ (left column), of the time component of the fluid four-velocity $u_t$ (central column), and of the angular velocity $\Omega$ (right column) along the $x$ axis. The models considered here are U11-1.0e14 (top row), U11-4.0e15 (central row), and U11-1.0e16 (bottom row). The color code is indicated to the right of each plot. In addition, all diagrams also report isodensity contours of the rest-mass density $\rho ={10}^6$, ${10}^{11}$, ${10}^{12}$, $5\times {10}^{12}$, ${10}^{13}$, $5\times {10}^{13}$ and ${10}^{14}\mathrm{g}{\mathrm{cm}}^{-3}$.

Figure 6

Snapshots of the total electromagnetic energy density $T_{\mathrm{em}}^{00}$, as measured in the Eulerian frame, on a horizontal plane at $z\simeq 1.5\mathrm{km}$ for model U11-1.0e14 (top row), U11-4.0e15 (central row), and U11-1.0e16 (right row), at different times during the evolution, namely, $t=15.0\mathrm{ms}$ (left column), $t=22.5\mathrm{ms}$ (central column), and $t=30\mathrm{ms}$ (right column). The color code is defined in terms of ${log}_{10}(T_{\mathrm{em}}^{00}/c^2)$ where $T_{\mathrm{em}}^{00}/c^2$ is in cgs units ($\mathrm{g}{\mathrm{cm}}^{-3}$). Additionally, magnetic field lines are shown with white solid lines.

Figure 7

Growth time of the bar-mode instability for the three unstable models U3 (blue), U11 (red) and U13 (black), shown as a function of the initial magnetization. The horizontal dashed lines report the growth times in the absence of magnetic fields, while the dotted lines represent the corresponding error bars.

Figure 8

Left panel: Evolution of the total magnetic energy $E_{\mathrm{mag}}$, normalized to its initial value, for models U11 and different values of the initial poloidal magnetic field. The black solid line refers to the less magnetized case, the blue solid line for the most magnetized case, and a red solid line for the last unstable model, before excessive magnetic tension suppresses the instability. Middle and right panels: The same as in the left panel, but for model U13 and model U3, respectively.

Figure 9

Evolution of the rotation parameter $\beta ≔T/|W|$ (top panel) and of the total magnetic energy normalized to its initial value (bottom panel) for models S1, S6, S7 and S8 which are stable against the bar-mode deformation in the unmagnetized case. In both panels the solid lines refer to models with $B_{\max }^z{|}_{t,z=0}={10}^{15}\mathrm{G}$, and the dash-dotted lines to models with $B_{\max }^z{|}_{t,z=0}={10}^{16}\mathrm{G}$.

Figure 10

Initial (i.e., shown with black lines at $t=0\mathrm{ms}$) and final (i.e., shown with red lines at $t=25\mathrm{ms}$) normalized profiles along the $x$ direction of the angular velocity (solid lines) and of the rest-mass density (dashed lines) for model S1 and the two different values of the magnetic field, i.e., ${10}^{15}\mathrm{G}$ (top panel) and ${10}^{16}\mathrm{G}$ (bottom panel).

Figure 11

Evolution of the distortion parameters ${\eta }_{+}$ (top panel) and $\eta$ (bottom panel) for model U11-1.0e15 when imposing a bitant symmetry (black solid line) or when using the full domain (red dot-dashed line).

Figure 12

Initial growth of the square root of the toroidal component of the magnetic energy $E_{\mathrm{mag}}^{\mathrm{tor}}$ [Eq. (3.13)] for different resolutions of the finest grid. The reference resolution, $\Delta x\simeq 0.550\mathrm{km}$, is shown with a black solid line. The inset shows instead the growth rate $\gamma$ as a function of the resolution and its fit with a quadratic function (dashed line). Note that the expected value $\gamma =1$ is approached in the limit of $\Delta x\to 0$.

Figure 13

Top panel: Evolution of the toroidal component of the magnetic energy $E_{\mathrm{mag}}^{\mathrm{tor}}$ for three different resolutions. Bottom panel: Order of the self-convergence test, ${\gamma }_c$, shown as a function of time. Note that a convergence order around 2 is measured before the bar-mode instability develops and shocks are produced (gray shaded area).