Non–adiabatic tidal oscillations induced by a planetary companion

Abstract

We calculate the dynamical tides raised by a close planetary companion on nonrotating stars of 1 M and 1.4 M . Using the Henyey method, we solve the fully nonadiabatic equations throughout the star. The horizontal Lagrangian displacement is found to be 10 to 100 times larger than the equilibrium tide value in a thin region near the surface of the star. This is because non-adiabatic effects dominate in a region that extends from below the outer edge of the convection zone up to the stellar surface, and the equilibrium tide approximation is inconsistent with non-adiabaticity. Although this approximation generally applies in the low frequency limit, it also fails in the parts of the convection zone where the forcing frequency is small but larger than the Brunt-Väisälä frequency. We derive analytical estimates which give a good approximation to the numerical values of the magnitude of the ratio of the horizontal and radial displacements at the surface. The relative surface flux perturbation is also significant, on the order of 0.1% for a system modelled on 51 Pegasi b. Observations affected by the horizontal displacement may therefore be more achievable than previously thought, and brightness perturbations may be the result of flux perturbations rather than due to the radial displacement. We discuss the implication of this on the possibility of detecting such tidally excited oscillations, including the prospect of utilising the large horizontal motion for observations of systems such as 51 Pegasi.

On a new formulation for energy transfer between convection and fast tides with application to giant planets and solar type stars

“…The component of the Reynolds stress which couples to this shear is u r u ϕ . This is zero when there is no dissipation, as u r and u ϕ are π/2 out of phase in that case, but this could become significant in regions where dissipation is large, as this introduces an additional phase shift (e.g., Bunting, Papaloizou, & Terquem 2019). Dissipation of inertial waves in the convective envelope has also been considered as a possible explanation for the observed circularization periods.…”

“…The equilibrium tide approximation is actually rather poor in convective regions where the Brunt-Väisälä frequency is not very large compared to the tidal frequency, and this yields to an over-estimate of tidal dissipation by a factor of a few for close binaries (Terquem et al 1998;Barker 2020). It also does not apply in a thin region near the surface of the convective envelope (Bunting, Papaloizou, & Terquem 2019). However, given all the uncertainties in estimating tidal dissipation here, the equilibrium tide approximation is sufficient.…”

Tidally induced stellar oscillations: converting modelled oscillations excited by hot Jupiters into observables

“…We follow Bunting et al (2019) and assume a non-rotating star with polar coordinates (r, θ * , φ * ) centred on the star, with the planetary companion existing in a circular orbit with θ * = π/2. In this frame, the observer is taken to be in the direction given by (θ 0 , φ 0 ).…”

“…The equilibrium tide approximation, whilst found to be reasonable throughout the bulk of the star (Pfahl et al 2008), breaks down at the stellar surface, where nonadiabatic effects become prominent (Henyey et al (1965); Savonije & Papaloizou (1983); Arras et al (2012); Houdek et al (2017); Fuller (2017)). The fully non-adiabatic stellar oscillation equations are solved here for the case of a periodic, tidal perturbation, as set out in Bunting et al (2019), with particular focus given to modelling the response at the surface.…”

“…Some alternative methods for observing the tidal oscillations are discussed in section 3. The predicted observable signals are explored for a test case in section 4, beginning with a summary of relevant results from Bunting et al (2019) in section 4.1, then addressing behaviour for long periods (4.2), short periods (4.3), resonances (4.4), general trends in behaviour (4.5) and variation with stellar mass (4.6). The observable signals for observed systems are presented in section 5, and discussed in section 6, whilst section 7 concludes this work.…”