Ellipsoidal Oscillations Induced by Substellar Companions: A Prospect for the Kepler Mission

Hundreds of substellar companions to solar-type stars will be discovered with the Kepler satellite. Kepler's extreme photometric precision gives access to low-amplitude stellar variability contributed by a variety of physical processes. We discuss in detail the periodic flux modulations arising from the tidal force on the star due to a substellar companion. An analytic expression for the variability is derived in the equilibrium-tide approximation. We demonstrate analytically and through numerical solutions of the linear, nonadiabatic stellar oscillation equations that the equilibrium-tide formula works extremely well for stars of mass <1.4 M_☉ with thick surface convection zones. More massive stars with largely radiative envelopes do not conform to the equilibrium-tide approximation and can exhibit flux variations ≳10 times larger than naive estimates. Over the full range of stellar masses considered, we treat the oscillatory response of the convection zone by adapting a prescription that A. J. Brickhill developed for pulsating white dwarfs. Compared to other sources of periodic variability, the ellipsoidal light curve has a distinct dependence on time and system parameters. We suggest that ellipsoidal oscillations induced by giant planets may be detectable from as many as ~100 of the 10⁵ Kepler target stars. For the subset of these stars that show transits and have radial-velocity measurements, all system parameters are well constrained, and measurement of ellipsoidal variation provides a consistency check, as well as a test of the theory of forced stellar oscillations in a challenging regime.