Shear oscillations in neutron stars and the nuclear symmetry energy

The shear and interface modes excited inside the neutron star due to the presence of elasticity depend on the properties of both the crust and core region. To examine how such eigenfrequencies depend on the neutron star properties, we solve the eigenvalue problem by adopting the relativistic Cowling approximation. Then, we confirm that the number of the interface modes excited in the star is generally equivalent to the number of the interface where the shear modulus discontinuously becomes zero, but we also find that the number of interface modes becomes smaller than that of the interface for the stellar model with lower or higher value of the slope parameter $L$. Furthermore, we derive the empirical relations for expressing the shear modes and one of the interface modes, which is the mode whose amplitude becomes dominant at the interface between the crust and envelopes and at the interface between the phases of slablike and cylindrical nuclei. At the end, we also show the possibility of identifying the higher quasiperiodic oscillation frequencies observed in GRB 200415A with the shear oscillations, as an alternative possibility instead of the torsional oscillations.

Figure 1

Radial profile of the amplitude of $i_i$-modes for the neutron star mode with $1.44M_{\odot }$ and 10.2 km using the EOS with $K_0=230$ and $L=42.6\mathrm{MeV}$. Here, $W$ and $V$ denote the Lagrangian displacement in the radial and angular directions. The vertical lines from left to right denote the boundaries between core and spherical-hole nuclei, between cylindrical-hole and slablike nuclei, between slablike and cylindrical nuclei, and between spherical nuclei and envelop. The eigenfunctions shown in the left panels correspond to the modes excited even without the pasta structure, while those in the right panels correspond to the modes additionally excited due to the presence of the phase composed of cylindrical-hole and spherical-hole nuclei.

Figure 2

Eigenfrequencies of the $i_i$-modes are shown as a function of the stellar compactness, $M/R$, for the neutron star models constructed using the original OI-EOSs with $K_0=230\mathrm{MeV}$ and $L=23.7\mathrm{MeV}$, 42.6 MeV, and 73.4 MeV in the left, middle, and right panel, respectively. We note the labeling of the $i_i$ modes for the stellar model shown in the middle panel is different from that in the previous study [42] (see text for details).

Figure 3

$f_{i_1}M$ for various stellar models are shown as a function of stellar compactness, where the circles, diamonds, squares, and triangles denote the results with stellar models constructed with original OI-EOS, OI-EOS, connected to the EOS with $\alpha =1/3$, 0.6, and 1, respectively. The solid lines are the fitting given by Eq. (7) and we denote the values of $(K_0,L)$ on each line.

Figure 4

Same as Fig. 3, but for $i_4$-mode frequencies on the neutron star models with $K_0=180$ and $L=31.0\mathrm{MeV}$.

Figure 5

Correlation between the coefficients in Eq. (7) and $L$ or the combination of $K_0$ and $L$, where the solid lines denote the fitting given by Eqs. (9) and (10). The panels from left to right correspond to $a_{ji}$ for $i=1$, 2, 3, 4, respectively.

Figure 6

The relative deviation, $\mathrm{\Delta }$, of the $i_i$-mode frequencies estimated with the empirical formulas from those determined via the eigenvalue problem, which is calculated with Eq. (11).

Figure 7

$f_{s_1}R$ for various stellar models are shown as a function of stellar compactness, where the circles, diamonds, squares, and triangles denote the results with stellar models constructed with original OI-EOS, OI-EOS connected to the EOS with $\alpha =1/3$, 0.6, and 1, respectively. The solid lines are the fitting given by Eq. (8) and we denote the values of $(K_0,L)$ on each line.

Figure 8

Coefficients, $b_{0i}$ and $b_{1i}$, in Eq. (8) are fitted as a function of $z_{0i}$ or $z_{1i}$ listed in Table 5, which is the combination of $K_0$ and $L$ with using the fitting formulas given by Eqs. (12) and (13). The coefficients in Eqs. (12) and (13), i.e., $b_{0i}^{(j)}$ and $b_{1i}^{(j)}$ for $j=0$, 1, 2, are listed in Table 6.

Figure 9

The relative deviation, $\mathrm{\Delta }$, of the $s_i$-mode frequencies estimated with the empirical formulas from those determined via the eigenvalue problem, which is calculated with Eq. (11).

Figure 10

The $f$- and $s_i$-mode frequencies are shown as a function of the stellar compactness for the stellar models constructed with $(K_0,L)=(230,42.6)$, together with the QPO frequencies observed in GRB 200415A [47]. The left and right panels correspond to the results with $\alpha =1/3$ and 0.6, respectively. The horizontal shaded bands denote the observed QPO frequencies, i.e., ${835.9}_{+77.3}^{-84.7}$, ${1443.7}_{+74.8}^{-68.7}$, ${2131.7}_{+148.2}^{-151.0}\mathrm{Hz}$, and ${4249.7}_{+116.0}^{-102.7}\mathrm{Hz}$ [47], while the marks denote the frequencies expected theoretically. The vertical lines denote the suitable stellar compactness for identifying the observed QPO frequencies with the $s_1$-, $s_2$-, $s_4$-, and $s_8$-modes, i.e., $M/R\simeq 0.183$ (0.195) for $\alpha =1/3$ (0.6).

Figure 11

Dependence of the $i$-mode frequencies on the surface density, ${\rho }_s$, for the $1.44M_{\odot }$ neutron star model constructed with $K_0=230$ and $L=42.6\mathrm{MeV}$. The transition densities between the phases with and without elasticity are fixed as in the text.