A Non-isothermal Theory for Interpreting Sodium Lines in Transmission Spectra of Exoplanets

TL;DR

This paper develops a non-isothermal theoretical framework for interpreting sodium lines in exoplanet transmission spectra, enabling the inference of atmospheric temperature gradients and densities from observational data.

Contribution

It introduces a novel algebraic method to jointly interpret transit radius and absorption depth measurements, accounting for temperature gradients in exoplanet atmospheres.

Findings

Confirmed upper atmospheric heating in HD 189733b

Derived a temperature gradient of approximately 0.44 K/km

Estimated sodium densities ranging from 1 to 10^4 cm^-3

Abstract

We present a theory for interpreting the sodium lines detected in transmission spectra of exoplanetary atmospheres. Previous analyses employed the isothermal approximation and dealt only with the transit radius. By recognising the absorption depth and the transit radius as being independent observables, we develop a theory for jointly interpreting both quantities, which allows us to infer the temperatures and number densities associated with the sodium lines. We are able to treat a non-isothermal situation with a constant temperature gradient. Our novel diagnostics take the form of simple-to-use algebraic formulae and require measurements of the transit radii (and their corresponding absorption depths) at line center and in the line wing for both sodium lines. We apply our diagnostics to the HARPS data of HD 189733b, confirm the upper atmospheric heating reported by Huitson et al. (2012), derive a temperature gradient of $0.4376 \pm 0.0154$ K km$^{-1}$ and find densities $\sim 1$ to $10^4$ cm$^{-3}$.