Determination of properties of protoneutron stars toward black hole formation via gravitational wave observations

We examine the frequencies of the fundamental ($f$) and the gravity ($g_i$) mode in gravitational waves from accreting protoneutron stars (PNSs) toward black hole formation. For this purpose, we analyze numerical results of gravitational collapse of massive stars for two different progenitors with three equations of state. We adopt profiles of central objects obtained from the numerical simulations by solving general relativistic neutrino-radiation hydrodynamics under the spherical symmetry. Using a series of snapshots as a static configuration at each time step, we solve the eigenvalue problem to determine the specific frequencies of gravitational waves from the evolving PNSs with accretion from core bounce to black hole formation by the relativistic Cowling approximation. We find that the frequency of the $f$-mode gravitational waves can be expressed as a function of the average density of the PNS at each time step almost independently of the progenitor models in accord with the previous studies on ordinary PNSs. We also find that the ratio of the $g_1$-mode frequency to the $f$-mode frequency is insensitive to the progenitor but strongly depends on the compactness of the PNSs. Having a simultaneous observation of the $f$- and $g_1$-mode gravitational waves, it would provide the information of the average density and compactness of PNS, which can reconstruct the evolution of mass and radius of the PNSs as a constraint on the equation of state at high densities. Furthermore, we show that the highest average density of the PNS with the maximum mass can be estimated with the detection of the $f$-mode gravitational waves together with the help of the neutrino observations.

Figure 1

Rest-mass density (left panel), temperature (middle panel), and electron fraction profiles (right panel) are shown for the PNS model of W40-Shen at $T_{\mathrm{pb}}=50$, 300, 700, and 1300 ms. For reference, ${\rho }_s={10}^{11}$, $10^{10}$, and $5\times {10}^9\mathrm{g}/{\mathrm{cm}}^3$ are shown by the horizontal lines in the left panel.

Figure 2

Evolution of the radius (left) and the gravitational mass (right) of PNS models of W40-Shen with different surface density, where the circles, diamonds, and squares correspond to ${\rho }_s=5\times {10}^9$, $10^{10}$, and $10^{11}\mathrm{g}/{\mathrm{cm}}^3$. In addition, for the case with ${\rho }_s={10}^{11}\mathrm{g}/{\mathrm{cm}}^3$, the radius by fitting with Eq. (2) for $T_{\mathrm{pb}}\ge 400\mathrm{ms}$ is shown by the solid thick line in the left panel, where the radius for $T_{\mathrm{pb}}<400\mathrm{ms}$ is the same as the data with squares without fitting. In the right panel, the gravitational mass inside the radius shown by the solid thick line in the left panel is also shown by the solid thick line.

Figure 3

Evolution of the PNS radius (left), the gravitational mass (center), and the average density (right) for various PNS models, where the surface density is fixed to be $10^{11}\mathrm{g}/{\mathrm{cm}}^3$.

Figure 4

Evolution of the eigenfrequencies for the PNS model of W40-Shen. The right panel is just an enlarged view of the left panel. The $f$, $p_i$, and $g_i$ modes are shown with the diamonds, squares, and circles.

Figure 5

Evolution of the frequency of the $f$-mode gravitational waves for various PNS models.

Figure 6

Comparison between $f_f$ obtained by eigenvalue problems (marks) and the expectation with the fitting formula given by Eq. (4) (solid thick lines). The left and right panels correspond to the cases for the PNS models with W40 and T50, respectively.

Figure 7

Frequencies of the $f$-mode gravitational waves from various PNS models are shown as a function of the square root of the PNS average density for each time step. The solid thick line denotes the fitting formula given by Eq. (5), using the data for PNSs with the square root of the average density larger than 0.1.

Figure 8

Left: evolution of the frequency of the $g_1$-mode gravitational wave for various PNS models. Right: ratio of the $g_1$-mode frequency to the $f$-mode frequency, $f_{g_1}/f_f$, as a function of the PNS compactness.

Figure 9

Evolutions of the eigenfrequencies from the PNS models produced by the various progenitor models. The left and right panels correspond to the results with W40 and T50, respectively. The top, middle, and bottom panels correspond the PNS constructed with Shen, LS180, and LS220, respectively. That is, the top-left panel is the same as the left panel of Fig. 4.

Figure 10

The frequency ratio of the $f$ mode to the $p_1$ mode is shown as a function of $Q_0$ given by Eq. (b1) for the PNS models discussed in this study. The thick solid line denotes the fitting line [Eq. (b2)] derived in Ref. [40].