Neutron star instabilities in full general relativity using a $\mathrm{\Gamma }=2.75$ ideal fluid

We present results about the effect of the use of a stiffer equation of state, namely the ideal-fluid $\mathrm{\Gamma }=2.75$ one, on the dynamical bar-mode instability in rapidly rotating polytropic models of neutron stars in full general relativity. We determine the change on the critical value of the instability parameter $\beta$ for the emergence of the instability when the adiabatic index $\mathrm{\Gamma }$ is changed from 2 to 2.75 in order to mimic the behavior of a realistic equation of state. In particular, we show that the threshold for the onset of the bar-mode instability is reduced by this change in the stiffness and give a precise quantification of the change in value of the critical parameter ${\beta }_c$. We also extend the analysis to lower values of $\beta$ and show that low-beta shear instabilities are present also in the case of matter described by a simple polytropic equation of state.

Figure 1

Diagram of the pressure $P$ vs the energy density $e$ for two polytropic EOSs and two realistic EOSs for nuclear matter, namely: (1) the Shen proposal [22, 23]; (2) the unified SLy prescription [24].

Figure 2

Density profile (top panel) and differential rotation profile (bottom panel) of some representative models among all the ones that have been evolved in this paper.

Figure 3

Main features of the five sequences of initial models analyzed in the present work. On the $x$ axis we report the compactness of each stellar model, while its rotation parameter $\beta$ is on the $y$ axis. Squares denote models that are not subject to the bar-mode instability while circles represent the ones that are bar-mode ($m=2$) unstable.

Figure 4

Corotation bands for the five sequences of initial models studied in the present work. For each sequence is report the rotation frequency on the axis (where the mass of the sequence is also reported) and on the equator as a function of the rotational parameter $\beta$ using continuous-black, dotted-blue, dash-dash-dotted-green, dash-dotted-magenta, and dashed-green lines for the sequences of baryonic mass 0.5, 1.0, 1.5, 2.0, and $2.5M\odot$, respectively. We also shaded, with the corresponding color, the initial part of the corotation bands. For any value of the frequency of a mode between these two lines there is exactly one radius inside the star which is corotating with the mode.

Figure 5

Snapshots of the rest-mass density $\rho$ in the $(x,y)$ plane for M1.5b0.270 at different stages of the evolution, namely $t=8$ and 10 ms (top row), $t=12$ and 14 ms (central row), and $t=20$ and 30 ms (bottom row). The color code is defined in terms of $g/{\mathrm{cm}}^3$.

Figure 6

Mode dynamics for selected evolved models that are characterized by a value of the instability parameter $\beta$ between 0.250 and 0.272. All models with $\beta \ge 0.255$ show the typical dynamics one would expect when the dynamical $m=2$ bar-mode instability is the dominating phenomenon.

Figure 7

Main features of the dynamics of a representative example of an evolved stellar model that is unstable against the bar-mode instability, namely M1.5b0.270. In particular, we show the time evolution of the power of the cylindrical $m=1$, 2, 3, 4 modes of the matter density (upper panel), the time evolution of the $xy$ component of the quadrupole moment of the conserved density $Q^{xy}$ defined in Eq. (18) (center panel), and the time evolution of the rotational parameter $\beta =T/|W|$. In the shaded region, corresponding to $7\mathrm{ms}<t<10\mathrm{ms}$, there is a clear exponential growth of the $m=2$ azimuthal matter mode that later reaches a saturation at $t\approx 12\mathrm{ms}$. Eventually, the model seems to settle down around a new (less differentially rotating) configuration after $t\approx 20\mathrm{ms}$.

Figure 8

The same as Fig. 6 but for models that are characterized by a value of the $\beta$ parameter in the range $0.14\le \beta \le 0.24$.

Figure 9

The same as Fig. 6 but for model M2.5b0.200, that shows the presence of an $m=2$ shear instability at a low value of $\beta$. The dynamics enclosed in the shaded region has been selected to compute the characteristic frequency and growth time of the $m=2$ instability, that turned out to be $f_2=1.943\mathrm{kHz}$ and ${\tau }_2=1.02\mathrm{ms}$ respectively.

Figure 10

Here we show the frequency of the corotation band multiplied by 2 (shaded region) and the measured values for the coordinate frequency of the observed $m=2$ modes (circles) for all the models belonging to the sequence with baryon mass $M_0=2.5M_{\odot }$.

Figure 11

Critical diagram relating the growth time ${\tau }_2$ of each bar-mode unstable model to the value of the instability parameter $\beta$. Triangles represent the values corresponding to all the models listed in Table 1. More precisely, the quantity on the $x$ axis is actually expressed in terms of $1/{\tau }_2^2$, in order to highlight the very good fit, while the quantity on the $y$ axis is the value of $\beta$ at the beginning of the time interval chosen for performing the fit of $m=2$ mode growth reported in Table 1 as $\beta (t_i)$. For all the five constant rest-mass sequences considered we also report, with open circles, the extrapolated value ${\beta }_c$ and the fitted straight lines with Eqs. (26).

Figure 12

The open circles are the same as in Fig. 11, but here they are shown as a function of the baryonic mass $M_0$. The filled circle, on the other hand, marks the extrapolation for zero rest mass representing the limit of Newtonian gravity (or zero compactness). The result of this extrapolation is reported in Eq. (27).

Figure 13

Evolution of M1.5b0.270 for different values of the resolution on the finest grid. Note that the total computational cost of the simulation at resolution $dx=0.25$ is 16 times greater than a simulation at $dx=0.5$.

Figure 14

Here we show the time evolution of the $\beta$ parameter for different values of the employed resolution. The filled dots mark the time at which the values $\beta (t_i)$ used in the fits are evaluated.

Figure 15

The filled circles represent the values of $\beta$ at the time when the distortion parameter $\eta$ reaches 1% of its maximum value, as a function of the inverse of the square of the growth time, for model M1.5b0.270 at different resolutions for the finest grid. The straight line represents the result of the fit (as a function of $\beta$) for the whole sequence of models at fixed value of the conserved baryonic mass $M_0=1.5M_{\odot }$ and at resolution $dx=0.50$. The open triangles represent the values of two different models among them.

Figure 16

Resolution’s effect on the mode dynamics for model M2.5b0.200. In the case of models below the threshold the interplay of modes with different shapes does not show a clear convergent behavior on the dynamics.