Gravity-Modes in ZZ Ceti Stars: IV. Amplitude Saturation by Parametric Instability

Abstract

ZZ Ceti stars exhibit small amplitude photometric pulsations in multiple gravity-modes. We demonstrate that parametric instability, a form of resonant 3-mode coupling, limits overstable modes to amplitudes similar to those observed. In particular, it reproduces the observed trend that longer period modes have larger amplitudes. Parametric instability involves the destabilization of a pair of stable daughter modes by an overstable parent mode. The 3-modes must satisfy exact angular selection rules and approximate frequency resonance. The lowest instability threshold for each parent mode is provided by the daughter pair that minimizes $(\delta {\omega }^2+{\gamma }_d^2)/{\kappa }^2$, where $\kappa$ is the nonlinear coupling constant, $\delta \omega$ is the frequency mismatch, and ${\gamma }_d$ is the energy damping rate of the daughter modes. The overstable mode's amplitude is maintained at close to the instability threshold value. Although parametric instability defines an upper envelope for the amplitudes of overstable modes in ZZ Ceti stars, other nonlinear mechanisms are required to account for the irregular distribution of amplitudes of similar modes and the non-detection of modes with periods longer than $1,200\s$. Resonant 3-mode interactions involving more than one excited mode may account for the former. Our leading candidate for the latter is Kelvin-Helmholtz instability of the mode-driven shear layer below the convection zone.

16 pages with 10 figures, abstract shortened, submitted to ApJ

References in corpus (1)

• Evolutionary Timescale of the DAV G117-B15A: The Most Stable Optical Clock Known

Cited by in corpus (54)

• Saturation of the R-mode Instability
• Pulsating white dwarfs: new insights
• KIC 4544587: an Eccentric, Short Period Binary System with delta Sct Pulsations and Tidally Excited Modes
• Dynamical Tides in Eccentric Binaries and Tidally-Excited Stellar Pulsations in KEPLER KOI-54
• Nonlinear Tides in Close Binary Systems
• Origin of Tidal Dissipation in Jupiter: II. the Value of Q
• Orbital decay of hot Jupiters due to nonlinear tidal dissipation within solar-type hosts
• Amplitude and frequency variations of oscillation modes in the pulsating DB white dwarf star KIC~08626021: The likely signature of nonlinear resonant mode coupling
• Stability analysis of a tidally excited internal gravity wave near the centre of a solar-type star
• Observations of Non-radial Pulsations in Radio Pulsars
• The Everchanging Pulsating White Dwarf GD358
• An instability due to the nonlinear coupling of p-modes to g-modes: Implications for coalescing neutron star binaries
• Low-frequency variability in massive stars: Core generation or surface phenomenon?
• Signatures of nonlinear mode interactions in the pulsating hot B subdwarf star KIC 10139564
• Nonlinear Saturation of g-modes in Proto-Neutron Stars: Quieting the Acoustic Engine
• The stability of tidally deformed neutron stars to three- and four-mode coupling
• On an excitation mechanism for trapped inertial waves in discs around black holes
• Empirical Determination of Convection Parameters in White Dwarfs I : Whole Earth Telescope Observations of EC14012-1446
• A second case of outbursts in a pulsating white dwarf observed by Kepler
• It takes a village to raise a tide: nonlinear multiple-mode coupling and mode identification in KOI-54
• Resonant Excitation of White Dwarf Oscillations in Compact Object Binaries: 1. The No Back Reaction Approximation
• Energy Dissipation through Quasi-Static Tides in White Dwarf Binaries
• The Local Instability of Steady Astrophysical Flows with non Circular Streamlines with Application to Differentially Rotating Disks with Free Eccentricity
• Growth rate of the tidal p-mode g-mode instability in coalescing binary neutron stars
• HD 183648: a Kepler eclipsing binary with anomalous ellipsoidal variations and a pulsating component
• Weak nonlinear coupling between epicyclic modes in slender tori
• Nonlinear dynamical tides in white dwarf binaries
• Outbursts in two new cool pulsating DA white dwarfs
• Accumulated Tidal Heating of Stars Over Multiple Pericenter Passages Near SgrA*
• The unusual pulsation spectrum of the cool ZZ Ceti star HS 0507+0434B
• Time-resolved optical spectroscopy of the pulsating DA white dwarf HS 0507+0434B: New constraints on mode identification and pulsation properties
• Amplitudes of low frequency modes in rotating B type stars
• Nonlinear Couplings Between r-modes of Rotating Neutron Stars
• Limits on Mode Coherence in Pulsating DA White Dwarfs Due to a Non-static Convection Zone
• DAVs: Red edge and Outbursts
• A Non-Radial Oscillation Model for Pulsar State Switching
• High Resolution Spectroscopy of the Pulsating White Dwarf G29-38
• Axisymmetric simulations of the convective overstability in protoplanetary discs
• Tidal Asteroseismology: Possible Evidence of Non-linear Mode Coupling in an Equilibrium State in Kepler Eclipsing Binary KIC 3230227
• Deciphering the Pulsations of G 29-38 with Optical Time Series Spectroscopy
• How Cassini Can Constrain Tidal Dissipation in Saturn
• Equation of State Dependence of Nonlinear Mode-tide Coupling in Coalescing Binary Neutron Stars
• Nonlinear Mixed Modes in Red Giants
• Memoirs of a giant planet
• The 2:3:6 QPO structure in GRS 1915+105 and cubic subharmonics in the context of relativistic diskoseismology
• Pulsating or not? A search for hidden pulsations below the red edge of the ZZ instability strip
• Non-linear Three-mode Coupling of Gravity Modes in Rotating Slowly Pulsating B Stars: Stationary Solutions and Modeling Potential
• The former companion of the hyper-velocity star S5-HVS1
• Non-linear excitation of low frequency modes by overstable convective modes in rotating stars
• Nonlinear mode-tide coupling in coalescing binary neutron stars with relativistic corrections
• Relativistic Corrections in White Dwarf Asteroseismology
• White-dwarf asteroseismology: an update
• Nonlinear coupling between magnetar QPOs
• Damping of Oscillations in Red Giants by Resonant Mode Coupling