Effects of the nuclear equation of state on the $r$-mode instability and evolution of neutron stars

I study the effect of nuclear equation of state on the $r$-mode instability of a rotating neutron star. I consider the case where the crust of the neutron star is perfectly rigid and I employ the related theory introduced by Lindblom et al. [Phys. Rev. D 62, 084030 (2000)]. The gravitational and the viscous time scales, the critical angular velocity, and the critical temperature are evaluated by employing a phenomenological nuclear model for the neutron-star matter. The predicted equations of state for the $\beta$-stable nuclear matter are parameterized by varying the slope $L$ of the symmetry energy at saturation density on the interval $72.5\mathrm{MeV}\le L\le 110\mathrm{MeV}$. The effects of the density dependence of the nuclear symmetry energy on $r$-mode instability properties and the time evolution of the angular velocity are presented and analyzed. A comparison of theoretical predictions with observed neutron stars in low-mass x-ray binaries and millisecond radio pulsars is also performed and analyzed. I estimate that it may be possible to impose constraints on the nuclear equation of state by a suitable treatment of observations and theoretical predictions of the rotational frequency and spin-down rate evolution of known neutron stars.

Figure 1

The energy per baryon of symmetric nuclear matter (a) and pure neutron matter (b), as a function of the baryon density $n$, of the present model (MDIM) in comparison with those originating from realistic calculations. More details for the models UV14 + TNI, UV14 + UVII, and AV14 + UVII can be found in Ref. [35] and for the models A18 + UIX and A18 + du + ${\mathrm{UIX}}^{*}$ in Ref. [36].

Figure 2

The nuclear symmetry energy (a) the proton fraction (b) and the fourth-order term $E_{\text{sym},4}\left(u\right)$ of the symmetry energy (c) as a function of the ratio $u=n/n_0$ for various values of the slope parameter $L$.

Figure 3

The transition baryon density $n_t$ (a) and the transition pressure $P_t$ (b) as a function of the slope parameter $L$. For more details about the linear fit, see text.

Figure 4

The energy density $E_{\text{ns}}$, taking into account both the fluid core and the solid crust matter, of neutron-star matter as a function of the pressure $P$ for various values of the slope parameter $L$. The data used for densities lower than the transition density $n_t$ have been taken from Refs. [24] and [25] (for more details, see text).

Figure 5

The mass-radius relations for the selected EOSs.

Figure 6

The values of the fundamental integral $I\left(R_c\right)={\int }_0^{R_c}\epsilon \left(r\right)r^{6}dr$ as a function of the slope parameter $L$ for the interval $72.5\mathrm{MeV}\le L\le 110\mathrm{MeV}$. For more details about the linear fit, see text.

Figure 7

The fiducial time scales ${\tilde{\tau }}_{\text{GR}},{\tilde{\tau }}_{ee},{\tilde{\tau }}_{nn}$, as well as ${\tilde{\tau }}_{\text{GR}}^{\text{approx}},{\tilde{\tau }}_{ee}^{\text{approx}},{\tilde{\tau }}_{nn}^{\text{approx}}$, as a function of the slope parameter $L$ for a neutron star with mass $M=1.4M_{\odot }$ (a) and $M=1.8M_{\odot }$ (b). In the case $M=1.4M_{\odot }$ the gravitational times scales are multiplied by a factor of 5, and in the case $M=1.8M_{\odot }$ they are multiplied by a factor of 20.

Figure 8

Temperature dependence of the critical angular velocity ratio ${\Omega }_c/{\Omega }_0$ for a neutron star with mass $M=1.4M_{\odot }$ (a) and $M=1.8M_{\odot }$ (b) constructed for the selected EOSs.

Figure 9

The critical temperature as a function of the slope parameter $L$ for a neutron star with mass $M=1.4M_{\odot }$ and $M=1.8M_{\odot }$.

Figure 10

The critical frequency temperature dependence for a neutron star with mass $M=1.4M_{\odot }$ (a) and $M=1.8M_{\odot }$ (b) constructed for the selected EOSs. The observed cases of LMXBs and MSRPs from Haskell et al. [21] are also included for a comparison. The horizontal vectors extending leftward exhibit the uncertainties of the core temperature. The cases IGR J00291 + 5934, XTE J1751-305, and SAX J1808-3658 with well-known observation spin-down rate, are also indicated [see also Fig. 12(c)].

Figure 11

The critical frequency temperature dependence for a neutron star with mass $M=1.4M_{\odot }$ for a fixed $E_{\text{sym}}\left(u\right)$, given by Eq. (62) and for three values of the incompressibility $K$.

Figure 12

(a) The time evolution of the spin frequency, (b) the spin-down rate evolution for a neutron star with mass $M=1.4M_{\odot }$ for the selected EOSs, and (c) the spin-down rate versus the spin frequency for the selected EOSs compared to three observed pulsars data [53, 55].