Physically-motivated basis functions for temperature maps of exoplanets

¹ Center for Space and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
e-mail: morrisbrettm@gmail.com
² Department of Physics, Astronomy & Astrophysics Group, University of Warwick, Coventry CV4 7AL, UK
³ Ludwig Maximilian University, University Observatory Munich, Scheinerstrasse 1, Munich 81679, Germany
⁴ Institut de recherche sur les exoplanètes, Département de physique, Université de Montréal, 2900 boul. Édouard-Montpetit, Montreal, QC H3T 1J4, Canada
⁵ Lund Observatory, Department of Astronomy and Theoretical Physics, Lunds Universitet, Solvegatan 9, 222 24 Lund, Sweden

Received: 2 September 2021
Accepted: 22 October 2021

Abstract

Thermal phase curves of exoplanet atmospheres have revealed temperature maps as a function of planetary longitude, often by sinusoidal decomposition of the phase curve. We construct a framework for describing two-dimensional temperature maps of exoplanets with mathematical basis functions derived for a fluid layer on a rotating, heated sphere with drag/friction, which are generalisations of spherical harmonics. These basis functions naturally produce physically-motivated temperature maps for exoplanets with few free parameters. We investigate best practices for applying this framework to temperature maps of hot Jupiters by splitting the problem into two parts: (1) we constrain the temperature map as a function of latitude by tuning the basis functions to reproduce general circulation model outputs, since disk-integrated phase curve observations do not constrain this dimension; and (2) we infer the temperature maps of real hot Jupiters using original reductions of several Spitzer phase curves, which directly constrain the temperature variations with longitude. The resulting phase curves can be described with only three free parameters per bandpass – an efficiency improvement over the usual five or so used to describe sinusoidal decompositions of phase curves. Upon obtaining the hemispherically averaged day side and night side temperatures, the standard approach would be to use zero-dimensional box models to infer the Bond albedo and redistribution efficiency. We elucidate the limitation of these box models by demonstrating that negative Bond albedos may be obtained due to a choice of boundary condition on the night side temperature. We propose generalized definitions for the Bond albedo and heat redistribution efficiency for use with two-dimensional (2D) temperature maps. Open-source software called kelp is provided to efficiently compute the 2D temperature maps, phase curves, albedos and redistribution efficiencies.