# Dependence of outer boundary condition on protoneutron star asteroseismology with gravitational-wave signatures

## TL;DR

This study explores how the choice of outer boundary conditions in linear perturbation analysis affects the eigenfrequencies of protoneutron star oscillations and their gravitational wave signatures in core-collapse supernovae.

## Contribution

It demonstrates the significant impact of boundary density on PNS eigenfrequencies and links fundamental oscillations to gravitational wave features, considering different boundary placements.

## Key findings

- Eigenfrequencies depend strongly on boundary surface density.
- Surface g-mode frequencies relate to PNS mass and radius.
- Eigenfrequencies near SASI frequencies when including surrounding regions.

## Abstract

To obtain the eigenfrequencies of a protoneutron star (PNS) in the postbounce phase of core-collapse supernovae (CCSNe), we perform a linear perturbation analysis of the angle-averaged PNS profiles using results from a general relativistic CCSN simulation of a $15 M_{\odot}$ star. In this work, we investigate how the choice of the outer boundary condition could affect the PNS oscillation modes in the linear analysis. By changing the density at the outer boundary of the PNS surface in a parametric manner, we show that the eigenfrequencies strongly depend on the surface density. By comparing with the gravitational wave (GW) signatures obtained in the hydrodynamics simulation, the so-called surface $g$-mode of the PNS can be well ascribed to the fundamental oscillations of the PNS. The frequency of the fundamental oscillations can be fitted by a function of the mass and radius of the PNS similar to the case of cold neutron stars. In the case that the position of the outer boundary is chosen to cover not only the PNS but also the surrounding postshock region, we obtain the eigenfrequencies close to the modulation frequencies of the standing accretion-shock instability (SASI). However, we point out that these oscillation modes are unlikely to have the same physical origin of the SASI modes seen in the hydrodynamics simulation. We discuss possible limitations of applying the angle-averaged, linear perturbation analysis to extract the full ingredients of the CCSN GW signatures.