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To obtain the eigenfrequencies of a protoneutron star (PNS) in the postbouncephase of core-collapse supernovae (CCSNe), we perform a linear perturbationanalysis of the angle-averaged PNS profiles using results from a generalrelativistic CCSN simulation of a $15 M_{\odot}$ star. In this work, weinvestigate how the choice of the outer boundary condition could affect the PNSoscillation modes in the linear analysis. By changing the density at the outerboundary of the PNS surface in a parametric manner, we show that theeigenfrequencies strongly depend on the surface density. By comparing with thegravitational wave (GW) signatures obtained in the hydrodynamics simulation,the so-called surface $g$-mode of the PNS can be well ascribed to thefundamental oscillations of the PNS. The frequency of the fundamentaloscillations can be fitted by a function of the mass and radius of the PNSsimilar to the case of cold neutron stars. In the case that the position of theouter boundary is chosen to cover not only the PNS but also the surroundingpostshock region, we obtain the eigenfrequencies close to the modulationfrequencies of the standing accretion-shock instability (SASI). However, wepoint out that these oscillation modes are unlikely to have the same physicalorigin of the SASI modes seen in the hydrodynamics simulation. We discusspossible limitations of applying the angle-averaged, linear perturbationanalysis to extract the full ingredients of the CCSN GW signatures.

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