Imprints of the nuclear symmetry energy on gravitational waves from the axial $w$-modes of neutron stars

The eigenfrequencies of the axial $w$-modes of oscillating neutron stars are studied using the continued fraction method with an equation of state (EOS) partially constrained by the recent terrestrial nuclear laboratory data. It is shown that the density dependence of the nuclear symmetry energy $E_{\mathrm{sym}}(\rho )$ affects significantly both the frequencies and the damping times of these modes. Besides confirming the previously found universal behavior of the mass-scaled eigenfrequencies as functions of the compactness of neutron stars, we explored several alternative universal scaling functions. Moreover, the $w_{\mathit{II}}$-mode is found to exist only for neutron stars having a compactness of $M/R⩾0.1078$ independent of the EOS used.

Figure 1

The symmetry energy (a) and pressure (b) versus the baryon number density (baryon number density ρ is in units of saturation nuclear density ${\rho }_0$).

Figure 2

Mass-radius relation for static neutron stars.

Figure 3

Frequency of $w_I$-mode (a) and $w_{\mathit{II}}$-mode (b) as a function of the neutron star mass $M$.

Figure 4

Damping time of $w_I$-mode (a) and $w_{\mathit{II}}$-mode (b) as a function of the neutron star mass $M$.

Figure 5

Eigen-frequency, ω, of the $w_I$-mode scaled by the stellar mass $M$ as a function of compactness $M/R$. The fit is performed by employing the parameters of Tsui et al. [36].

Figure 6

Eigen-frequency, ω, of the $w_{I2}$-mode scaled by the stellar mass $M$ as a function of compactness $M/R$. The fit is performed by employing the parameters of Tsuiet al. [36].

Figure 7

Eigen-frequency, ω, of the $w_{\mathit{II}}$-mode scaled by the stellar mass $M$ as a function of compactness $M/R$.

Figure 8

Eigen-frequency, ω, of the $w_I$-mode scaled by the gravitational energy $\left|E_g\right|$ versus the metric function $e^{2\lambda }$ at the stellar surface.

Figure 9

Eigen-frequency, ω, of the $w_{I2}$-mode scaled by the gravitational energy $\left|E_g\right|$ versus the metric function $e^{2\lambda }$ at the stellar surface.

Figure 10

Eigen-frequency, ω, of the $w_{\mathit{II}}$-mode scaled by the gravitational energy $\left|E_g\right|$ versus the metric function $e^{2\lambda }$ at the stellar surface.

Figure 11

Real part of the scaled frequency $M\omega$ of $w_{\mathit{II}}$-mode versus the metric function $e^{2\lambda }$ at the stellar surface.

Figure 12

Frequency and damping time of the first, second and third axial w-mode ($w_I,w_{I2}$ and $w_{I3}$) and the axial $w_{\mathit{II}}$-mode for neutron star models of mass $M=1.4M_{\odot }$.