Nonlinear Damping of Oscillations in Tidal-Capture Binaries

We calculate the damping of quadrupole fand 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 sunlike stars, the growth time ≡ η ≈ 4E −1/2 0,42 days, where E0,42 is the initial energy of the primary mode in units of 10 erg, and the number of daughter modes N ∼ 10E 5/4 0,42. The growth rate is approximately equal to the angular frequency of the primary mode times its dimensionless radial amplitude, δR/R∗ ≈ 0.002E 1/2 0,42. Although the daughter modes are limited by their own nonlinearities, collectively they absorb most of the primary mode’s energy after a time ∼ 10η provided E0 > 10 40 erg. This is orders of magnitude smaller than usual radiative damping time. In fact nonlinear mode interaction may be the dominant damping process if E0 ∼> 10 37 erg. These results have obvious application to tidally captured main sequence globular cluster stars of mass ≥ 0.5M⊙; 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 ≈ 10 45 erg expected in tidal captures. However, at such a large energy the system is very nonlinear: four-mode and higher-order couplings are as important as three-mode couplings.