A Closed-Form Analytical Theory of Non-Isobaric Transmission Spectroscopy for Exoplanet Atmospheres

TL;DR

This paper develops a closed-form analytical model for exoplanet transmission spectra that accounts for pressure-dependent opacity, improving interpretation of high-precision observations and aiding retrieval methods.

Contribution

It generalizes the classical isobaric model by incorporating a pressure-dependent opacity law and derives a closed-form expression for the effective transit radius.

Findings

Model fits empirical spectra of Earth and WASP-39b better than previous models.

Provides a physically interpretable framework linking opacity gradients to spectral features.

Enables efficient semi-analytical retrievals for upcoming JWST and ARIEL data.

Abstract

Analytical models are essential for building physical intuition and guiding the interpretation of exoplanet observations by clarifying the dependencies that shape atmospheric signatures. We present a generalization of the classical isothermal, isobaric transmission model by allowing the opacity to vary with pressure as a power law, $\kappa \propto P^{n}$, and explicitly defining the reference opacity $\kappa_{0}$ at a chosen pressure $P_{0}$. By treating the slant optical depth as an Abel transform of the radial absorption coefficient, we derive a closed-form expression for the effective transit radius in a hydrostatic, isothermal atmosphere with pressure-dependent opacity. The solution provides a compact framework for exploring non-isobaric effects and explicitly links the vertical opacity gradient to observable spectral features. We benchmark the model against empirical transmission spectra of Earth and the hot Jupiter WASP-39b, finding a significantly improved fit relative to the isobaric formula. This generalized expression offers a physically interpretable foundation for analyzing high-precision spectra from JWST and upcoming ARIEL observations, and can serve as a basis for semi-analytical retrieval approaches optimized for computational efficiency.