Generalizations for Insolation and Albedo to Adapt an Energy Balance Model to Other Planets

Abstract

Interest in modeling the climates of other planets has been stimulated by observations of the Pluto-Charon system and seven Earth-sized planets orbiting the nearby star TRAPPIST-1. Furthermore, as of March 2019, over four thousand planets outside of our solar system have been discovered. Scientists are interested in what these planets might be like and if they could support life as we know it, but there is very little empirical information that they can collect in order to learn more about them. For this reason, scientists must rely on models to study climate on these planets. Because so little is known about our planetary neighbors compared to Earth, and even less is known about planets outside of our solar system, it is hard to faithfully model their climates using complex models such as such as the class of models referred to as General Circulation Models (GCMs). Instead, conceptual climate models may be preferred because the small number of state variables and parameters (relative to GCMs) make it easier to quantify possible behaviors of the system. Adapting well known conceptual models for Earth to extraterrestrial and extrasolar planets raises issues whose solutions draw from the fields of celestial mechanics, harmonic analysis and nonsmooth systems. This work focuses on a main component of conceptual climate models---incoming radiation absorbed by the planet---and the mathematical considerations for and implications of adapting this component to planets other than Earth. We generalize both the distribution of insolation and location of different albedos on the planet's surface. We find that the insolation distribution for slowly rotating planets approaches a rapid rotation distribution like the reciprocal of the rotation rate. Additionally, we show that it is possible to have stable, asymmetric configurations of ice in an energy balance model of Pluto.