# The thermal phase curve offset on tidally- and non-tidally-locked exoplanets: A shallow water model

## TL;DR

This study uses a shallow water model to analyze how the thermal phase curve offset on exoplanets depends on rotation, orbital period, and wave dynamics, revealing conditions for hot-spot leading or lagging relative to the substellar point.

## Contribution

The paper extends the Matsuno-Gill model to spherical geometry with moving insolation forcing, providing new insights into phase curve offsets influenced by gravity wave speeds and planetary rotation.

## Key findings

- Hot-spot offset depends on the ratio of substellar point speed to gravity wave speed.
- Hot-spot can lead or lag the substellar point based on wave dynamics and rotation.
- Analytic solutions relate phase curve offset to planetary rotation and wave properties.

## 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 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 is a function 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 hotspot 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 wavespeed of the system the hottest point can 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.