Gravitational wave asteroseismology with protoneutron stars

We examine the time evolution of the frequencies of the gravitational wave after the bounce within the framework of relativistic linear perturbation theory using the results of one-dimensional numerical simulations of core-collapse supernovae. Protoneutron star models are constructed in such a way that the mass and the radius of the protoneutron star become equivalent to the results obtained from the numerical simulations. Then we find that the frequencies of gravitational waves radiating from protoneutron stars strongly depend on the mass and the radius of protoneutron stars, but almost independently of the profiles of the electron fraction and the entropy per baryon inside the star. Additionally, we find that the frequencies of gravitational waves can be characterized by the square root of the average density of the protoneutron star irrespective of the progenitor models, which are completely different from the empirical formula for cold neutron stars. The dependence of the spectra on the mass and the radius is different from that of the $g$-mode: the oscillations around the surface of protoneutron stars due to the convection and the standing accretion-shock instability. Careful observation of these modes of gravitational waves can determine the evolution of the mass and the radius of protoneutron stars after core bounce. Furthermore, the expected frequencies of gravitational waves are around a few hundred hertz in the early stages after bounce, which must be a good candidate for the ground-based gravitational wave detectors.

Figure 1

Time evolutions of mass shell and shock radius of LS220M15.0. The position of the shock front is indicated by the thick solid line. The evolution of mass shells are plotted by thin solid lines ($\mathrm{\Delta }M=0.01M_{\odot }$). Specifically, the mass shell whose enclosed mass is $1.4M_{\odot }$ is plotted with the thick dotted line.

Figure 2

Time evolutions of $M_{\mathrm{PNS}}$ and $R_{\mathrm{PNS}}$ for the case of $M_{\mathrm{pro}}=15M_{\odot }$ and LS220, and its fitting with Eqs. (1) and (2). Two different lines correspond to the mass and the radius of PNSs for a surface density of $10^{11}$ and $10^{12}\mathrm{g}/{\mathrm{cm}}^3$, while the dotted lines denote the fitting to each case.

Figure 3

Mass-radius relation constructed with an LS220 EOS for constant values of the entropy per baryon and $Y_e$ inside the star, where $s=1.5(k_{\mathrm{B}}/\text{baryon})$ for various values of $Y_e=0.01$, 0.1, 0.2, and 0.3. For reference, the mass of J1614-2230 [2] is also shown in the figure.

Figure 4

Distributions of electron fraction ($Y_e$) and entropy per baryon ($s$) as a function of the mass coordinate normalized by the mass of the PNS determined with ${\rho }_s=10^{12}\mathrm{g}/{\mathrm{cm}}^3$. The three different lines correspond to the profiles of the various times after bounce, i.e., $t=100$, 200, and 500 ms.

Figure 5

Distributions of the electron fraction $Y_e$ and entropy $s$ as a function of the mass coordinate $m$, which simply imitates the results shown in Fig. 4. The distributions are expressed as Eqs. (4) and (5). In the right panel, $t_1$ and $t_2$ represent the evolution time, where $t_1<t_2$.

Figure 6

Distributions of the electron fraction $Y_e$ and entropy $s$ as a function of mass coordinate $m$, which simply imitates the results by Roberts [49]. The distributions are expressed as Eqs. (6) and (7). In the right panel, $t_1$ and $t_2$ represent the evolution time, where $t_1<t_2$.

Figure 7

Values of $s_m$ and ${ϵ}_c$ for each time step to realize the mass and the radius of PNSs for the case of $M_{\mathrm{pro}}=15M_{\odot }$ and LS220, where the left and right panels correspond to the combination of ($Y_e$, $s$) shown in Figs. 5 and 6, respectively. In the left panel, the value of $s_0$ is also shown for reference.

Figure 8

For the PNS model with $M_{\mathrm{pro}}=15M_{\odot }$ and LS220, time evolutions of eigenfrequencies are plotted where the left, middle, and right panels correspond to $f$-, $p_1$-, and $p_2$-modes. In the figure, the different lines denote the results with different ($Y_e$, $s$) distributions inside the star; i.e., the solid and broken lines correspond to the distributions shown in Figs. 5 and 6, respectively.

Figure 9

For the PNS model with $M_{\mathrm{pro}}=15M_{\odot }$ and LS220, the $f$-mode eigenfrequencies are plotted as a function of the PNS mass (left panel) and radius (right panel). We remark that the PNS mass and radius are time dependent, as shown in Fig. 2. As in Fig. 8, the different lines denote the results with different ($Y_e$, $s$) distributions inside the star.

Figure 10

The frequencies of $f$-, $p_1$-, and $p_2$-modes are shown as a function of the normalized square root of the average density of PNSs for $M_{\mathrm{pro}}=15M_{\odot }$ and LS220, where the normalized square root of the average density is defined by $(M_{\mathrm{PNS}}/1.4M_{\odot }{)}^{1/2}(R_{\mathrm{PNS}}/10\mathrm{km}{)}^{-3/2}$. In the left panel, the $f$-mode frequencies expected for cold neutron stars with the same square root of the average density, which are given by Eq. (18), are shown with the dotted line.

Figure 11

For the various progenitor models, the frequencies of the $f$-, $p_1$-, and $p_2$-modes are shown as a function of the normalized square root of the average density of the PNSs, where the normalized square root of the average density is defined by $(M_{\mathrm{PNS}}/1.4M_{\odot }{)}^{1/2}(R_{\mathrm{PNS}}/10\mathrm{km}{)}^{-3/2}$. The thick solid line in each panel corresponds to the universal relation, shown as Eq. (19).

Figure 12

Relative deviation of the eigenfrequencies for each PNS model constructed with the different EOSs from the universal relation given by Eq. (19). The left, middle, and right panels correspond to the relative deviation for the $f$-, $p_1$-, and $p_2$-modes. In the figure, the meaning of the lines is the same as in Fig. 11.

Figure 13

Time evolution of frequencies of the $f$- and $g$-modes radiating from the PNSs for $M_{\mathrm{pro}}=15M_{\odot }$ and LS220.