We consider the tidal excitation of modes in a binary system of arbitrary eccentricity. For a circular orbit, the modes generally undergo forced oscillation with a period equal to the orbital period (T). For an eccentric orbit, the amplitude of each tidally excited mode can be written approximately as the sum of an oscillatory term that varies sinusoidally with the mode frequency and a "static" term that follows the time dependence of the tidal forcing function. The oscillatory term falls off exponentially with increasing bα (defined as the ratio of the periastron passage time to the mode period), whereas the "static" term is independent of bα. For small-bα modes (bα≈1), the two terms are comparable, and the magnitude of the mode amplitude is nearly constant over the orbit. For large-bα modes (bα ≳ a few), the oscillatory term is very small compared with the "static" term, in which case the mode amplitude, like the tidal force, varies as the distance cubed. For main-sequence stars, p-, f-, and low-order g-modes generally have large bα, and hence small amplitudes of oscillation. High-overtone g-modes, however, have small overlap with the tidal forcing function. Thus, we expect an intermediate overtone g-mode with bα ∼ 1 to have the largest oscillation amplitude. In addition, we find that the mode amplitude is independent of the dissipation rate except when the mode frequency is very close to orbital resonance or the damping time is less than T; both conditions are unlikely. Morever, orbital evolution causes a resonant mode to move off resonance with time. This severely limits the amplitude of modes near resonance. Rotation of the star shifts the mode frequencies but otherwise has little effect on the mode amplitude (provided that the rotation rate is small). Hence, tidally excited modes have amplitudes and phases that are periodic with period T, making them readily distinguishable from oscillations excited by other mechanisms. We apply our work to the SMC radio pulsar PSR J0045-7319, which is believed to be in a highly eccentric orbit with a 10 M⊙ B star. We find that the g7 mode (with period 1.07 days) of the B star has the largest oscillation amplitude, with a flux variation of 2.3 mmag and a surface velocity of 70 ms-1. The flux variation at periastron, summed over all modes, is about 10 mmag; in addition, we propose that the shape of the light curve can be utilized to determine the orbital inclination angle. The apsidal motion of this system, calculated without the usual static approximation, is larger than that predicted by the classical apsidal formula by about 1%. For the PSR B1259-63 system, the tidal amplitude of the Be star companion is smaller by a factor of 70 because of its larger periastron distance. To understand the dependence of tidal excitation on stellar structure, detailed numerical calculations of modes of a general polytropic star are also presented.
All Science Journal Classification (ASJC) codes
- Astronomy and Astrophysics
- Space and Planetary Science
- Binaries: close
- Pulsars: general
- Pulsars: individual (PSR J0045-7319)
- Stars: oscillations