Gravitational wave asteroseismology for low-mass neutron stars

The fundamental ($f$-) mode gravitational waves from cold low-mass neutron stars are systematically studied with various equations of state (EOSs) characterized by the nuclear saturation parameters, especially focusing on the phenomena of the avoided crossing with the first pressure ($p_1$-) mode. We find that the $f$-mode frequency and the average density for the neutron star at the avoided crossing can be expressed as a function of the parameter, $\eta$, which is the specific combination of the nuclear saturation parameters. Owing to these relations, we can derive the empirical formula expressing the $f$-mode frequency for a low-mass neutron star, whose central density is larger than that for the neutron star at the avoided crossing, as a function of $\eta$ and the square root of the stellar average density, $x$. On the other hand, we also derive the empirical formula expressing the $f$-mode frequency for a neutron star, whose central density is less than that for the neutron star at the avoided crossing, as a function of $x$ independently of the adopted EOS. Furthermore, adopting the empirical formula of $x$ as a function of $\eta$ and $u_c$, which is the ratio of the stellar central density to the saturation density, we can also rewrite our empirical formula for the $f$-mode frequency to a function of $\eta$ and $u_c$. So, by observing the $f$-mode gravitational wave from a low-mass neutron star, whose mass or gravitational redshift is known, one could evaluate the values of $\eta$ and $u_c$, which enables us to severely constrain the EOS for neutron star matter.

Figure 1

Mass-radius relation with various EOSs for ${\rho }_c\le 2{\rho }_0$. For reference, we also show the radius of a $1.4M_{\odot }$ neutron star constrained from GW170817.

Figure 2

For ${\rho }_c/{\rho }_0=1.5$ and 2.0, the square root of the normalized stellar average density, $x\equiv (M/1.4M_{\odot }{)}^{1/2}(R/10\mathrm{km}{)}^{-3/2}$, constructed by various EOSs is shown as a function of $\eta$. The thick-solid line denotes the fitting line with Eq. (4).

Figure 3

The coefficients in Eq. (4) are shown as a function of $u_c\equiv {\rho }_c/{\rho }_0$. The thick-solid line in each panel is fitting line given by Eqs. (5)–(7).

Figure 4

For the low-mass neutron star constructed with OI-EOSs ($K_0=180$ and $L=31.0\mathrm{MeV}$), the frequencies of the $f$-, $p_1$-, and $p_2$-modes are shown as a function of the central density normalized by the saturation density.

Figure 5

For low-mass neutron stars constructed with various EOSs, the $f$-mode frequency is shown as a function of the central density normalized by the saturation density.

Figure 6

With OI-EOSs ($K_0=180$ and $L=31.0\mathrm{MeV}$), the $f$-mode frequencies before and after the avoided crossing are respectively fitted with Eqs. (8) and (9).

Figure 7

The $f$-mode frequencies at the avoided crossing, $f_{f,\mathrm{AC}}$, for various EOSs are shown as a function of $\eta$ (left), ${\rho }_{c,\mathrm{AC}}/{\rho }_0$ (middle), and $x_{\mathrm{AC}}$ (right), where $x_{\mathrm{AC}}$ denotes the value of $x\equiv (M/1.4M_{\odot }{)}^{1/2}(R/10\mathrm{km}{)}^{-3/2}$ for the neutron star model at the avoided crossing between the $f$- and $p_1$-modes. In each panel, the thick-solid line denotes the fitting formulas given by Eqs. (10), (11), and (12) respectively.

Figure 8

The values of ${\rho }_{c,\mathrm{AC}}/{\rho }_0(\equiv u_{c,\mathrm{AC}})$ (top), $x_{\mathrm{AC}}$ (middle), and $(M_{\mathrm{c}}/R_{\mathrm{c}})/(M/R{)}_{\mathrm{AC}}$ (bottom) for various EOSs are shown as a function of $\eta$. The thick-solid line denotes the fitting formulae given by Eqs. (13), (14), and (15), respectively. Note that $(M_{\mathrm{c}}/R_{\mathrm{c}})/(M/R{)}_{\mathrm{AC}}$ is the ratio of the compactness for the core region, $M_{\mathrm{c}}/R_{\mathrm{c}}$, to stellar compactness, $M/R$, for the neutron star model at the avoided crossing between the $f$- and $p_1$-modes, where $M_{\mathrm{c}}$ and $R_{\mathrm{c}}$ denote the mass and radius of the core region of neutron star.

Figure 9

Same as in Fig. 5, but as a function of square root of the normalized average density, $(M/1.4M_{\odot }{)}^{1/2}(R/10\mathrm{km}{)}^{-3/2}$. The thick-solid line denotes the fitting formula for the $f$-mode frequency before the avoided crossing given by Eq. (17), while the thick-dotted line denotes the fitting formula proposed by Andersson and Kokkotas given by Eq. (16) [16].

Figure 10

$f_f-f_{f,\mathrm{AC}}$ is shown as a function of $x-x_{\mathrm{AC}}$ for various EOSs. The thick-solid and thick-dotted lines denote the fitting formulas given by Eqs. (18) and (19), respectively.

Figure 11

Relative deviation, $\mathrm{\Delta }f/f_f$, of the $f$-mode frequency from the empirical formula given by Eqs. (17) and (20) is shown as a function of $x-x_{\mathrm{AC}}$, where $\mathrm{\Delta }f/f_f$ is calculated by Eq. (21).

Figure 12

The $f$-mode frequencies for $0.5M_{\odot }$ (solid line) and $0.7M_{\odot }$ (dashed line) neutron star models are shown as a function of $\eta$ with using the methodology proposed in this study, i.e., with using the fitting formulae for the $f$-mode frequency and the mass as a function of $u_c$ and $\eta$. The circles and squares are the $f$-mode frequencies for the neuron star models constructed with the EOS considered in this study. On each line, the asterisk denotes the neutron star model, where $u_c$ becomes 2. Since we derive the fitting formulae by considering the region for $u_c<2$, the left-hand part of the asterisk is incredible.