Proto-neutron star oscillations including accretion flows

TL;DR

This work addresses how proto-neutron star oscillations, observable as gravitational waves in core-collapse supernovae, are affected by realistic accretion flows and a surrounding stalled shock. It develops a general-relativistic, linear perturbation framework in the Cowling approximation, including advection and shock boundary conditions, and solves the resulting eigenvalue problem via Chebyshev spectral collocation. The authors demonstrate robust exponential convergence, recover known buoyancy-driven p- and g-modes in the appropriate limits, and reveal how accretion and shock RH conditions reshape mode frequencies and stability, including SASI-like dynamics under certain BCs. This framework advances PNS asteroseismology by enabling more accurate mode calculations in the presence of accretion, with potential to constrain neutron-star properties and high-density EOS from gravitational-wave signals.

Abstract

The gravitational wave signature from core-collapse supernovae (CCSNe) is dominated by quadrupolar oscillation modes of the newly born proto-neutron star (PNS), and could be detectable at galactic distances. We have developed a framework for computing the normal oscillation modes of a PNS in general relativity, including, for the first time, the presence of an accretion flow and a surrounding stalled accretion shock. These new ingredients are key to understand PNS oscillation modes, in particular those related to the standing-accretion-shock instability (SASI). Their incorporation is an important step towards accurate PNS asteroseismology. For this purpose, we perform linear and adiabatic perturbations of a spherically symmetric background, in the relativistic Cowling approximation, and cast the resulting equations as an eigenvalue problem. We discretize the eigenvalue problem using collocation Chebyshev spectral methods, which is then solved by means of standard and efficient linear algebra methods. We impose boundary conditions at the accretion shock compatible with the Rankine-Hugoniot conditions. We present several numerical examples to assess the accuracy and convergence of the numerical code, as well as to understand the effect of an accretion flow on the oscillation modes, as a stepping stone towards a complete analysis of the CCSNe case.

Figures (9)

• Figure 1: The profiles of the lapse function, $\alpha$, density, $\rho$, and the Brunt–Väisälä frequency, $\mathcal{N}^2$, with respect to $r$ for $NS1$. For larger values of $\mathcal{N}^2_0$, the density gradient becomes sharper. The density profile resembles the one of a neutron star, decreasing outwards.
• Figure 2: Eigenfrequencies with respect to the value of $\mathcal{N}_0^2$ are plotted in logarithmic scale for three different models NS1, NS2 and WD. The first ten analytical solutions for $\mathcal{N}^2_0 = 0$ are shown as black solid lines. The two families of modes, p- and g-modes, seem to be naturally separated for small values of $\mathcal{N}^2_0$, while for higher values, there appear avoided crossings. The dashed black line represents the frequencies $\mathcal{N}_0/(\alpha c_s)$. The g-modes scale with the the Brunt–Väisälä frequency, as expected theoretically.
• Figure 3: Radial profiles of the eigenfuctions of the two first p-modes (f and p$_1$) and two g-modes (g$_1$ and g$_4$) are plotted for different values of $\mathcal{N}^2_0$. As $\mathcal{N}^2_0$ increases the amplitude of the eigenfunctions is decreasing, while close to the avoided crossing the eigenfunctions are deformed. Although their shape changes, their number of nodes remains constant. This could be evidence that we are tracing the modes correctly after their interaction.
• Figure 4: Convergence test for the first ten p-modes in the absence of buoyancy, $\mathcal{N}^2=0$. In the upper figure, the absolute relative errors of the frequencies are plotted with respect to the total points used for all domains. In the lower one, the square root of the sums of the mean square errors of the eigenfunctions are plotted. In both graphs, the exponential convergence is evident. In addition, the dashed lines in the plot correspond to the convergence of the code GREAT, which is of quadratic order.
• Figure 5: Eigenvalues for the case of plane waves in an one-dimensional flow. The solid lines represent the analytical solution for constant $\upsilon$, while the dots are the numerical ones. The upper/lower graph shows the real/imaginary part of the frequency as a function of the background velocity at the shock, $\upsilon_{\rm int}$.