Analytical Modelling of Exoplanet Transit Specroscopy with Dimensional Analysis and Symbolic Regression

Abstract

The physical characteristics and atmospheric chemical composition of newly discovered exoplanets are often inferred from their transit spectra which are obtained from complex numerical models of radiative transfer. Alternatively, simple analytical expressions provide insightful physical intuition into the relevant atmospheric processes. The deep learning revolution has opened the door for deriving such analytical results directly with a computer algorithm fitting to the data. As a proof of concept, we successfully demonstrate the use of symbolic regression on synthetic data for the transit radii of generic hot Jupiter exoplanets to derive a corresponding analytical formula. As a preprocessing step, we use dimensional analysis to identify the relevant dimensionless combinations of variables and reduce the number of independent inputs, which improves the performance of the symbolic regression. The dimensional analysis also allowed us to mathematically derive and properly parametrize the most general family of degeneracies among the input atmospheric parameters which affect the characterization of an exoplanet atmosphere through transit spectroscopy.

Transverse Vector Decomposition Method for Analytical Inversion of Exoplanet Transit Spectra

“…The analytical formulas typically rely on simplifying assumptions and approximations,.e.g, spherical symmetry, isothermal atmosphere, isobaric approximation, constant specific gravity, etc. Nevertheless, these theoretical results agree well with the numerical forward models and allow for better understanding of the underlying physics and of the intrinsic parameter degeneracies present in the problem (Griffith 2014;Heng & Kitzmann 2017;Welbanks & Madhusudhan 2019;Matchev et al 2021).…”

“…The first simplification arises due to standard dimensional analysis (Langhaar 1951;Barenblatt 1996). Matchev et al (2021) showed that, again without any loss of generality, the analytical expression behind the forward model ( 5) must be of the form…”

Unsupervised Machine Learning for Exploratory Data Analysis of Exoplanet Transmission Spectra

“…This implies that the five degrees of freedom are reduced effectively to only two: the temperature T and the total opacity 𝜅. One can reach similar conclusions on the basis of dimensional analysis as in Matchev et al (2021). As an illustration, in Figure 2 we show scatter plots of one representative feature (𝑀 (𝜆 1 ) for 𝜆 1 = 0.867 𝜇m) versus the two relevant degrees of freedom: the temperature 𝑇 (left panel) and a convenient naive proxy for the total atmospheric opacity in eq.…”

“…2014; Heng & Kitzmann 2017;Welbanks & Madhusudhan 2019;Matchev et al 2021); transparent and reproducible results.…”

“…The data set of Márquez-Neila et al ( 2018) is a well-known benchmark in the literature, which has already been used in several related studies (Márquez-Neila et al 2018;Cobb et al 2019;Welbanks & Madhusudhan 2019;Matchev et al 2021). Alternative benchmark data sets can be found in Goyal et al (2019Goyal et al ( , 2020.…”