JWST spectral retrieval of cold directly imaged planet WD0806 b and the first measurement of altitude-dependent K$_{zz}$ in exoplanet atmospheres

Abstract

WD0806 b is a rare exoplanet companion orbiting a white dwarf, currently with a projected orbital distance of 2500 au. The Spitzer mid-IR photometry suggests that the temperature is as cold as 350K, making it one of the coldest directly imaged exoplanets. In this paper, we present the Near-infrared Camera (NIRCam) F150W2, F200W, F356W, and F444W broadband photometry and a 3--5\um Near-Infrared spectroscopy (NIRSpec) G395M spectrum obtained with the James Webb Space Telescope (JWST). We develop a new retrieval framework based on the open-source PICASO software that includes additive and multiplicative systematic parameters. Our retrieval results reveal bounded abundances of H$_2$S, CO$_2$, CO, NH$_3$, H$_2$O, and CH$_4$. We present a new chemical analysis framework that utilizes retrieved abundances to measure altitude-dependent eddy diffusion coefficients (K$_{\mathrm zz}$) at multiple quenched pressures. We find that the eddy diffusion coefficients decrease from around $10^4$ to $10^2$ $\rm cm^2/s$ as the atmospheric pressure decreases from from 50 to 20 bars. To our knowledge, this is the first study to report altitude-dependent vertical mixing (or, equivalently, quenched-species-dependent vertical mixing) based on the measured molecular abundances of CO, CH$_4$, and CO$_2$. With the 1--21\um NIRCam, NIRSpec and the previously published MIRI data, we measure the bolometric luminosity to be log(L/L$_{\odot}$) = $-6.75\pm0.01$ and derive the mass to be $8\pm 1 \mathrm{M_J}$. The retrieval results suggest that \target has an elevated C/O ratio of 0.76, or 1.3$\times$ solar, sub-solar metallicity ([M/H ]= -0.25), and a nearly solar C/S ratio (1.17x solar).