p-winds: an open-source Python code to model planetary outflows and upper atmospheres

Atmospheric escape is considered to be one of the main channels for evolution in sub-Jovian planets, particularly in their early lives. While there are several hypotheses proposed to explain escape in exoplanets, testing them with atmospheric observations remains a challenge. In this context, high-resolution transmission spectroscopy of transiting exoplanets for the metastable helium triplet (He 2$^3$S) at $1\,083$ nm has emerged as a reliable technique to observe and measure escape. To aid in the prediction and interpretation of metastable He transmission spectroscopy observations, we developed the code p-winds. This is an open-source, fully documented, scalable Python implementation of the one-dimensional, purely H+He Parker wind model for upper atmospheres coupled with ionization balance, ray-tracing, and radiative transfer routines. We demonstrate an atmospheric retrieval by fitting p-winds models to the observed metastable He transmission spectrum of the warm Neptune HAT-P-11 b, and take into account the variation of the in-transit absorption caused by transit geometry. For this planet, our best fit yields a total atmospheric escape rate of approximately $2.5 \times 10^{10}$ g s$^{-1}$ and wind temperature of $7200$ K. The range of retrieved mass loss rates increases significantly when we let the H atom fraction be a free parameter, but the posterior distribution of the latter remains unconstrained by He observations alone. The stellar host limb darkening does not have a significant impact in the retrieved escape rate or outflow temperature for HAT-P-11 b. Based on the non-detection of escaping He for GJ 436 b, we are able to rule out total escape rates higher than $3.4 \times 10^{10}$ g s$^{-1}$ at 99.7% (3$\sigma$) confidence.