Exploring A Photospheric Radius Correction to Model Secondary Eclipse Spectra for Transiting Exoplanets

Abstract

We highlight a physical effect that is often not considered that impacts the calculation of model spectra of planets at secondary eclipse, affecting both emission and reflection spectra. The radius of the emitting surface of the planet is not merely one value measured from a transit light curve, but is itself a function of wavelength, yet it is not directly measurable. At high precision, a similar effect is well-known in transit "transmission spectroscopy" but this related effect also impacts emission and reflection. As is well-appreciated, the photospheric radius can vary across $\sim$4-8 atmospheric scale heights, depending on atmospheric opacity and spectral resolution. This effect leads to a decreased weighting in model calculations at wavelengths where atmospheric opacity is low, and one sees more deeply into the atmosphere, to a smaller radius. The overall effect serves to mute emission spectra features for atmospheres with no thermal inversion but to enhance features for atmospheres with a thermal inversion. While this effect can be ignored for current \emph{Hubble} observations, it can lead to wavelength-dependent 10-20\% changes in planet-to-star flux ratios in the infrared at $R\sim~200-1000$ (readily achievable for JWST) for low-gravity hot Jupiters, although values of 5\% are more typical for the population. The effect is mostly controlled by the ratio of the atmospheric scale height to the planet radius, and can be important at any planetary temperature. Of known planets, the effect is largest for the cool "super-puffs" at very low surface gravity, where it can alter calculated flux ratios by over 100\%. We discuss complexities of including this photospheric radius effect in 1D and 3D atmosphere models.