Tidal excitations of oscillation modes in compact white dwarf binaries - I. Linear theory

Abstract

We study the tidal excitation of gravity modes (g‐modes) in compact white dwarf binary systems with periods ranging from minutes to hours. As the orbit of the system decays via gravitational radiation, the orbital frequency increases and sweeps through a series of resonances with the g‐modes of the white dwarf. At each resonance, the tidal force excites the g‐mode to a relatively large amplitude, transferring the orbital energy to the stellar oscillation. We calculate the eigenfrequencies of g‐modes and their coupling coefficients with the tidal field for realistic non‐rotating white dwarf models. Using these mode properties, we numerically compute the excited mode amplitude in the linear approximation as the orbit passes though the resonance, including the back reaction of the mode on the orbit. We also derive analytical estimates for the mode amplitude and the duration of the resonance, which accurately reproduce our numerical results for most binary parameters. We find that the g‐modes can be excited to a dimensionless (mass‐weighted) amplitude up to 0.1, with the mode energy approaching 10−3 of the gravitational binding energy of the star. Therefore the low‐frequency (≲10−2 Hz) gravitational waveforms produced by the binaries, detectable by LISA, are strongly affected by the tidal resonances. Our results also suggest that thousands of years prior to the binary merger, the white dwarf may be heated up significantly by tidal interactions. However, more study is needed since the physical amplitudes of the excited oscillation modes become highly non‐linear in the outer layer of the star, which can reduce the mode amplitude attained by tidal excitation.

Three Hypervelocity White Dwarfs in Gaia DR2: Evidence for Dynamically Driven Double-degenerate Double-detonation Type Ia Supernovae

“…These values represent upper limits to the effects of tidal heating, as the companion will not be able to maintain complete synchronicity with the orbit. Recent work has found that tidal effects are dominated by the excitation of gravity waves within the companion, which deposit their energy and angular momentum near the surface of the WD (Fuller & Lai 2011;Burkart et al 2013). These surface layers can be kept near synchronous rotation all the way to the onset of mass transfer, but since the energy is not deposited deep within the core, the timescale for this excess heat to be radiated away is shorter than the typical > 10 6 yr between the explosion and the present day for nearby runaway WDs (Section 2.5), and thus the WD would cool and approach the luminosity of a dim, isolated WD.…”

Finding binaries from phase modulation of pulsating stars with Kepler: V. Orbital parameters, with eccentricity and mass-ratio distributions of 341 new binaries

“…Redistribution of angular momentum likely proceeds via dynamical oscillations (Zahn 1975;Witte & Savonije 2001;Fuller 2017). These modes can be excited to large amplitudes in a variety of systems (Fuller & Lai 2011;Fuller et al 2013;Hambleton et al 2015), and have been observed in abundance in Kepler data of eccentric ellipsoidal variables ('heartbeat stars'; Thompson et al 2012). Our sample contains the longestperiod heartbeat stars in the Kepler data set, which lie at the upper-left envelope of the period-eccentricity diagram ( Fig.…”

Heartbeat stars, tidally excited oscillations and resonance locking

“…If a heartbeat star exhibits a very large amplitude TEO (e.g., KIC 8164262, see companion papers Hambleton et al 2017;Fuller 2017) that is unlikely to stem from a chance resonance, it is a good candidate to be a resonantly locked mode. Resonantly locked modes arise from feedback between stellar evolution and TEOs, exciting a mode to large amplitude such that it increases the tidal dissipation rate, causing the orbital frequency to evolve such that the mode remains resonant.…”

“…Observed systems which defy theoretical expectations can then be used to improve theoretical understanding of tidal interactions in eccentric binaries. In two companion papers Hambleton et al (2017); Fuller (2017), we compare these theories with data for KIC 8164262. We measure the parameters of the system and the frequencies and amplitudes of its TEOs, showing that most of them can be explained by chance resonances with g modes, with the exception of its highest amplitude pulsation, which can instead by explained by resonance locking.…”