Tidal Asteroseismology: Possible Evidence of Non-linear Mode Coupling in an Equilibrium State in Kepler Eclipsing Binary KIC 3230227

TL;DR

This paper investigates non-linear mode coupling in the pulsations of the binary star KIC 3230227, providing evidence for an equilibrium state in mode interactions through detailed frequency and phase analysis.

Contribution

It offers the first detailed analysis supporting the equilibrium state of non-linear mode coupling in tidally excited stellar oscillations.

Findings

Modes are stable over four years, indicating equilibrium or long-period limit cycles.

Mode coupling explains observed frequency detuning and phase stability.

The steady-state relation of mode parameters is approximately satisfied.

Abstract

Previously, a series of tidally-excited oscillations were discovered in the eccentric eclipsing binary KIC 3230227. The pulsation amplitudes and phases suggest the observed oscillations are prograde quadruple modes. In this paper, we refine the analysis and extract more oscillation frequencies. We also study the temporal variations of amplitudes and phases and show that almost all modes have stable phases and amplitudes. We then focus on the non-orbital-harmonic oscillations. We consider two formation mechanisms: 1) nonlinear response of the surface convective layer, and 2) nonlinear three/multi-mode coupling. Although the former can explain some of the observed features, we find the latter mechanism is more probable. Assuming that these are coupled modes, the constant amplitude/phase over four years can be explained by either an equilibrium state in the mode coupling or modes undergoing limit cycles with very long periods. The observed frequency detuning and the calculated damping rates of the daughter modes favor the equilibrium-state interpretation. This is verified by integrating the amplitude equations of three-mode coupling. We find that the steady-state relation derived in Weinberg et al., which relates the observed frequency detuning, phase detuning, and mode damping rates, is approximately satisfied for one mode triplet. We also try to identify the azimuthal number of the modes based on the observed mode amplitude ratios and the selection rules in nonlinear three-mode coupling. We discuss further implications of these observations on nonlinear tidal asteroseismology.