Neutron star asteroseismology and nuclear saturation parameter

Adopting various unified equations of state (EOSs), we examine the quasinormal modes of gravitational waves from cold neutron stars. We focus on the fundamental ($f$-), 1st pressure ($p_1$-), and 1st spacetime ($w_1$-) modes, and derive the empirical formulas for the frequencies and damping rate of those modes. With the resultant empirical formulas, we find that the value of $\eta$, which is a specific combination of the nuclear saturation parameters, can be estimated within $\sim 30\%$ accuracy, if the $f$-mode frequency from the neutron star whose mass is known would be observed or if the $f$- and $p_1$-mode frequencies would be simultaneously observed, even though this estimation is applicable only for the low-mass neutron stars. Additionally, we find that the mass and radius of canonical neutron stars can be estimated within a few per cent accuracy via the simultaneous observations of the $f$- and $w_1$-mode frequencies. We also find that, if the $f$-, $p_1$-, and $w_1$-mode frequencies would be simultaneously observed, the mass of canonical neutron stars can be estimated within 2% accuracy, while the radius can be estimated within 1% for the neutron star with $M\ge 1.6M_{\odot }$ or within 0.6% for the neutron star with $M\ge 1.4M_{\odot }$ constructed with the EOS constrained via the GW170817 event. Furthermore, we find the strong correlation between the maximum $f$-mode frequency and the neutron star radius with the maximum mass, between the minimum $w_1$-mode frequency and the maximum mass, and between the minimum damping rate of the $w_1$-mode and the stellar compactness for the neutron star with the maximum mass. With these correlations, one may constrain the upper limit of the neutron star radius with the maximum mass if a larger $f$-mode frequency would be observed, the lower limit of the maximum mass if a smaller $w_1$-mode frequency would be observed, or the lower limit of the stellar compactness for the neutron star with the maximum mass if a smaller value of the $w_1$-mode damping rate would be observed.

Figure 1

Mass-radius relation for neutron star models constructed with various EOSs. In the figure, the dotted, solid, and dashed lines correspond to the EOSs based on the relativistic framework, the Skyrme-type effective interactions, and derived with variational method, respectively, while the marks on each line denote the background stellar models for the linear analysis.

Figure 2

Complex frequency (quasinormal mode) for the neutron star model constructed with Togashi EOS, whose mass is $1.57M_{\odot }$. The real and imaginary parts of complex frequency correspond to the oscillation frequency, $f$, and damping rate, $1/\tau$, respectively. In particular, the $f$-, $p_1$-, and $w_1$-modes, which are discussed in this article, are shown here.

Figure 3

The frequency, $f$, and damping rate, $1/\tau$, of the $f$-, $p_1$-, and $w_1$-modes for the neutron star models constructed with Togashi EOS are shown as a function of the square root of the normalized stellar average density, $x$, in the top and bottom panels, respectively. In the both panels, the value of $x_{\mathrm{AC}}$, where the avoided crossing between the $f$- and $p_1$-modes happens, is shown with the vertical line.

Figure 4

In the left panel, as an example, the $f$- and $p_1$-mode frequencies for the neutron star model constructed with Togashi EOS are shown as a function of the central density, ${\rho }_c$, normalized by the saturation density, ${\rho }_0$. The arrow indicates the neutron star model at the avoided crossing between the $f$- and $p_1$-mode frequencies. In the right panel, the enlarged view is shown together with the fitting lines, where the solid and dashed lines correspond to the fitting given by Eqs. (2) and (3), respectively.

Figure 5

The $f$-mode frequency, $f_{f,\mathrm{AC}}$, and the square root of the normalized stellar average density, $x_{\mathrm{AC}}$, for the neutron star model at the avoided crossing between the $f$- and $p_1$-mode frequencies are shown as a function of $\eta$ in the top and bottom panels, respectively. The thick-solid lines correspond to the fitting formulas given by Eqs. (4) and (5). For reference, we also show the fitting lines with the Cowling approximation with the dashed lines, which correspond to Eqs. (A1) and (A2) in Ref. [24].

Figure 6

Relative deviation of the value of $x_{\mathrm{AC}}$ and $f_{f,\mathrm{AC}}$ obtained in the previous study with the relativistic Cowling approximation [24] from those with full perturbation analysis in this study. The value of $\mathrm{\Delta }{x}_{\mathrm{AC}}$ and $\mathrm{\Delta }{f}_{f,\mathrm{AC}}$ are calculated with Eqs. (6) and (7), where the filled and open marks denote $\mathrm{\Delta }{x}_{\mathrm{AC}}$ and $\mathrm{\Delta }{f}_{f,\mathrm{AC}}$, respectively.

Figure 7

In the left panel, the $f$-mode frequencies for various EOSs are shown as a function of the square root of the normalized average density. For reference, the empirical formula derived in Ref. [17] is shown with the thick-dotted line. In the middle panel, the shifted $f$-mode frequencies, $f_f-f_{f,\mathrm{AC}}$, are shown as a function of the shifted square root of the normalized average density, $x-x_{\mathrm{AC}}$, where the thick-solid line denotes the fitting formula given by Eq. (8). In the right panel, the relative deviation, $\mathrm{\Delta }{f}_f$, of the $f$-mode frequency estimated with Eqs. (9) and (10) from that calculated for each stellar model is shown as a function of $x-x_{\mathrm{AC}}$ for various EOSs, where $\mathrm{\Delta }{f}_f$ is calculated with Eq. (11).

Figure 8

In the left panel, the $f$-mode frequencies for various EOSs are shown as a function of $M/R$. In the middle panel, the $f$-mode frequencies multiplied by the normalized stellar mass, $f_f(M/1.4M_{\odot })$, are shown as a function of $M/R$, where the thick-solid line denotes the fitting formula given by Eq. (12). In the right panel, the relative deviation, $\mathrm{\Delta }{f}_f$, of the $f$-mode frequency estimated with Eq. (12) from that determined via the eigenvalue problem is shown as a function of $M/R$, where $\mathrm{\Delta }{f}_f$ is calculated with Eq. (11), using $f_f(\mathcal{C},M)$ instead of $f_f(x,\eta )$.

Figure 9

In the left panel, the $p_1$-mode frequencies for various EOSs are shown as a function of $x$. In the middle panel, the shifted $p_1$-mode frequencies, $f_{p_1}-f_{f,\mathrm{AC}}$, are shown as a function of $x-x_{\mathrm{AC}}$, where the thick line denote the fitting formulas given by Eq. (13). In the right panel, the relative deviation of the $p_1$-mode frequencies for various EOSs are shown as a function of $x-x_{\mathrm{AC}}$, where $\mathrm{\Delta }{f}_{p_1}$ is calculated with Eq. (15).

Figure 10

Ratio of the $p_1$- to the $f$-mode frequency is shown as a function of the $f$-mode frequency for various EOSs. The minimum value of the ratio corresponds to the stellar model at the avoided crossing between the $f$- and $p_1$-mode frequencies, while the maximum value corresponds to that at the avoided crossing between the $p_1$- and $p_2$-mode frequencies.

Figure 11

In the left panel, the $p_1$-mode frequencies, $f_{p_1}$, are shown as a function of the stellar compactness, $M/R$, for various EOSs. In the middle panel, the $p_1$-mode frequencies multiplied by the normalized stellar mass, $f_{p_1}(M/1.4M_{\odot })$, are shown as a function of $M/R$, where the thick-solid and thick-dotted lines denote the expectation with the fitting formula given by Eq. (16) and with the fitting derived in Ref. [17], respectively. In the right panel, the relative deviation, $\mathrm{\Delta }{f}_{p_1}$, of the $p_1$-mode frequency estimated with Eq. (16) from that calculated for each stellar model, where $\mathrm{\Delta }{f}_{p_1}$ is calculated with Eq. (15), using $f_{p_1}(\mathcal{C},M)$ instead of $f_{p_1}(x,\eta )$.

Figure 12

In the left panel, the $w_1$-mode frequencies are shown as a function of the stellar compactness, $M/R$, for various EOSs. In the middle panel, the $w_1$-mode frequencies multiplied by the normalized stellar radius, $f_{w_1}(R/10\mathrm{km})$, are shown as a function of $M/R$, where the thick-solid and thick-dotted lines correspond to our fitting formula given by Eq. (17) and the fitting derived in Ref. [17], respectively. In the right panel, the relative deviation, $\mathrm{\Delta }{f}_{w_1}$, of the $w_1$-mode frequencies expected with Eq. (17) from that calculated for each stellar model is shown as a function of $M/R$, where $\mathrm{\Delta }{f}_{w_1}$ is calculated with Eq. (18).

Figure 13

In the left panel, the maximum frequency of the $f$-mode gravitational wave is shown as a function of the minimum radius of the neutron star constructed with each EOS. In the right panel, the minimum frequency of the $w_1$-mode gravitational wave is shown as a function of the maximum mass of the neutron star constructed with each EOS. The thick-solid lines in the left and right panels denote the fitting formulas given by Eqs. (19) and (20), respectively.

Figure 14

In the left panel, the damping rate of the $f$-mode, $1/{\tau }_f$, is shown as a function of the stellar compactness, $M/R$, for various EOSs. In the middle panel, the damping rate of the $f$-mode multiplied by the normalized stellar mass, $1/{\tau }_f(M/1.4M_{\odot })$, is shown as a function of $M/R$, where the thick-solid line corresponds to our fitting formula given by Eq. (21). In the right panel, the relative deviation, $\mathrm{\Delta }1/{\tau }_f$, of the damping rate of the $f$-mode expected with Eq. (21) from that calculated for each stellar model is shown as a function of $M/R$, where $\mathrm{\Delta }1/{\tau }_f$ is calculated with Eq. (22). The extended figure is also shown inside the right panel.

Figure 15

The damping rate of the $p_1$-mode, $1/{\tau }_{p_1}$, is shown as function of the stellar compactness for various EOSs.

Figure 16

In the left panel, the damping rate of the $w_1$-mode, $1/{\tau }_{w_1}$, is shown as a function of the stellar compactness, $M/R$, for various EOSs. In the middle panel, the damping rate of the $w_1$-mode multiplied by the normalized stellar radius, $1/{\tau }_{w_1}(R/10\mathrm{km})$, is shown as a function of $M/R$, where the thick-solid line corresponds to our fitting formula given by Eq. (23). In the right panel, the relative deviation, $\mathrm{\Delta }1/{\tau }_{w_1}$, of the damping rate of the $w_1$-mode expected with Eq. (23) from that calculated for each stellar model is shown as a function of $M/R$, where $\mathrm{\Delta }1/{\tau }_{w_1}$ is calculated with Eq. (24).

Figure 17

The minimum damping rate of the $w_1$-mode gravitational wave is shown as a function of the stellar compactness for the neutron star with the maximum mass. The thick-solid line denotes the fitting formula given by Eq. (25).

Figure 18

In the top panel, with using the empirical formulas of $M(u_c,\eta )$ and $f_f(u_c,\eta )$, the expected value of $\eta$ is shown as a function of $f_f$ from the neutron star with $M/M_{\odot }=0.5$, 0.7, 1.174, and 1.4 with various lines, where the lines from left to right correspond to the case with $M/M_{\odot }=0.5$, 0.7, 1.174, and 1.4. The various marks denote the concrete numerical results obtained via the eigenvalue problem for various EOSs. For reference, the plausible value of $\eta$ constrained from the terrestrial experiments is shown by shaded region. In the bottom panel, the relative error, $\mathrm{\Delta }\eta$, between the exact value of $\eta$ and the value of $\eta$ estimated with the empirical formulas is shown, when $f_f$ from the neutron star with the known mass would be observed. $\mathrm{\Delta }\eta$ is calculated with Eq. (26).

Figure 19

In the left panel, with using the empirical formulas of $f_f(x,\eta )$ and $f_{p_1}(x,\eta )$, the estimated value of $\eta$ is shown as a function of $f_f$ for the case that the ratio of $f_{p_1}/f_f$ is constant, where the solid, dashed, and dotted lines denote the case with $f_{p_1}/f_f=2.0$, 2.5, and 3.0, respectively. The various marks denote the concrete value of $\eta$ with the numerical results determined via the eigenvalue problem. In the right panel, we show the relative error, $\mathrm{\Delta }\eta$, is shown, where $\mathrm{\Delta }\eta$ is calculated with Eq. (26).

Figure 20

The constraint on the relation between the mass and radius, given by $f_f(\mathcal{C},M)=\text{const}$. (solid line), $f_{p_1}(\mathcal{C},M)=\text{const}$. (dashed line), and $f_{w_1}(\mathcal{C},R)=\text{const}$. (dashed line) for a specific stellar model constructed with Togashi EOS with $1.89M_{\odot }$. The mark denotes the exact value of the corresponding mass and radius.

Figure 21

With using the empirical formulas of $f_f(\mathcal{C},M)$, $f_{p_1}(\mathcal{C},M)$, and $f_{w_1}(\mathcal{C},R)$, the relative error in mass, $\mathrm{\Delta }M$, and in radius, $\mathrm{\Delta }R$, between the exact value and the value expected with the empirical formulae is shown for various EOSs as a function of stellar mass, when $f_f$ and $f_{w_1}$ (left panel), $f_{p_1}$ and $f_{w_1}$ (middle panel), and $f_f$ and $f_{p_1}$ (right panel) would be simultaneously observed. $\mathrm{\Delta }M$ and $\mathrm{\Delta }R$ calculated with Eqs. (27) and (28), respectively.

Figure 22

The relative error in mass, $\mathrm{\Delta }M$, and in radius, $\mathrm{\Delta }R$, with using the estimations given by Eqs. (27) and (28), assuming that $\xi =6/7$.