Nonlinear Damping of Oscillations in Tidal-Capture Binaries

We calculate the damping of quadrupole f- and low-order g-modes (primary modes) by nonlinear coupling to other modes of the star. Primary modes destabilize high-degree g-modes of half their frequency (daughter modes) by 3-mode coupling in radiative zones. For Sun-like stars, the growth time ≡η⁻¹≈4E-½₀,₄₂ days, where E₀,₄₂ is the initial energy of the primary mode in units of 10⁴² ergs, and the number of daughter modes N ∼ 10¹⁰E5/4₀,₄₂. The growth rate is approximately equal to the angular frequency of the primary mode times its dimensionless radial amplitude, δR/R* ≈ 0.002E-½₀,₄₂. Although the daughter modes are limited by their own nonlinearities, collectively they absorb most of the primary mode's energy after a time ∼10ɛ⁻¹ provided E0 > 10⁴⁰ ergs. This is orders of magnitude smaller than usual radiative damping time. In fact, nonlinear mode interaction may be the dominant damping process if E₀ ≥ 10³⁷ ergs. These results have obvious application to tidally captured main-sequence globular cluster stars of mass ≥ 0.5 Msun; the tidal energy is dissipated in the radiative core of the star in about a month, which is less than the initial orbital period. Nonlinear mode coupling is a less efficient damping process for fully convective stars, which lack g-modes. In convective stars, most of the tidal energy is in the quadrupole f-modes, which nonresonantly excite high-order p-modes of degree 0, 2, and 4. The resultant short-wavelength waves are more efficiently dissipated. The nonlinear damping time for f-modes is shown to be proportional to 1/E₀; this damping time is about 30 days for E₀ ≈ 10⁴⁵ ergs expected in tidal captures. However, at such a large energy the system is very nonlinear: 4-mode and higher order couplings are as important as 3-mode couplings.