Long-term gravitational wave asteroseismology of supernova: from core collapse to 20 seconds postbounce

Abstract

We use an asteroseismology method to calculate the frequencies of gravitational waves in a long-term core-collapse supernova simulation, with a mass of 9.6 $M_\odot$. The simulation, which includes neutrino transport in general relativity is performed from core-collapse, bounce, explosion and cooling of protoneutron stars (PNSs) up to 20 s after the bounce self-consistently. Based on the hydrodynamics background, we calculate eigenmodes of the PNS oscillation through a perturbation analysis on fluid and metric. We classify the modes by the number of nodes and find that there are several eigenmodes. In the early phase before 1 s, there are a low-frequency g-mode around 0.5 kHz, a mid-frequency f-modes around 1 kHz, and high-frequency p-modes above them. Beyond 1 s, the g-modes drop too low in frequency and the p-modes become too high to be detected by ground-based interferometers. However, the f-mode persists at 1 kHz. We present a novel fitting formula for the ramp-up mode, comprising a mixture of g-mode and f-mode, using postbounce time as a fitting parameter. Our approach yields improved results for the long-term simulation compared to prior quadratic formulas. We also fit frequencies using combinations of gravitational mass and radius of the PNS. We test three types of fitting variables: compactness, surface gravity, and average density. We present results of the time evolution of each mode and the fitting for three different ranges, from 0.2 to 1, 4, and 20 s for each formula. We compare the deviation of the formulas from the eigenmodes to determine which fitting formula is the best. In conclusion, any variable fits the eigenmodes well to a similar degree. Comparing 3 variables in detail, the fitting with compactness is slightly the best among them. We also find that the fitting using less than 1 s of simulation data cannot be extrapolated to the long-term frequency prediction.