The Thermal Phase Curve Offset on Tidally and Nontidally Locked Exoplanets: A Shallow Water Model

Abstract

Using a shallow water model with time-dependent forcing, we show that the peak of an exoplanet thermal phase curve is, in general, offset from the secondary eclipse when the planet is rotating. That is, the planetary hot spot is offset from the point of maximal heating (the substellar point) and may lead or lag the forcing; the extent and sign of the offset are functions of both the rotation rate and orbital period of the planet. We also find that the system reaches a steady state in the reference frame of the moving forcing. The model is an extension of the well-studied Matsuno–Gill model into a full spherical geometry and with a planetary-scale translating forcing representing the insolation received on an exoplanet from a host star. The speed of the gravity waves in the model is shown to be a key metric in evaluating the phase curve offset. If the velocity of the substellar point (relative to the planet’s surface) exceeds that of the gravity waves, then the hot spot will lag the substellar point, as might be expected by consideration of forced gravity wave dynamics. However, when the substellar point is moving slower than the internal wave speed of the system, the hottest point may lead the passage of the forcing. We provide an interpretation of this result by consideration of the Rossby and Kelvin wave dynamics, as well as, in the very slowly rotating case, a one-dimensional model that yields an analytic solution. Finally, we consider the inverse problem of constraining planetary rotation rate from an observed phase curve.