Detecting atmospheres in the presence of stellar contamination: an empirically calibrated framework applied to four JWST/NIRSpec PRISM transits of TRAPPIST-1c

Transmission spectroscopy of rocky exoplanets orbiting late M dwarfs is limited by stellar contamination rather than photon noise. We present a framework for this contamination-dominated regime, fitting all transit epochs jointly with a Gaussian process per epoch for the time-varying stellar contamination and a single atmospheric model shared across epochs. We use pure CO${}_2$, H${}_2$O, and CH${}_4$ models scaled by their peak-to-trough amplitude, allowing us to measure how large a signal must be to be recovered. We calibrate this framework using injection-recovery tests and 17 archival JWST/PRISM transits of the airless sibling TRAPPIST-1 b to quantify the spurious signals that stellar contamination alone produces. Applied to four JWST/NIRSpec PRISM transits of TRAPPIST-1 c, our framework recovers injected CO${}_2$ signals without bias and reaches detection thresholds of 146 (171) ppm for moderate (strong) evidence, compared with 96 ppm if there were no stellar contamination. CO${}_2$ and CH${}_4$ produce no spurious signals on the airless control planet, whereas H${}_2$O shows a persistent stellar floor of $+55\pm 34$ ppm. TRAPPIST-1 c is consistent with a featureless, airless baseline for all three molecules. Our 95 per cent upper limit of 90 ppm for CO${}_2$-shaped signals effectively rules out clear, isothermal pure-CO${}_2$ atmospheres ($\ge$0.1 bar) and N${}_2$-dominated atmospheres with 1 per cent CO${}_2$ ($\ge$1 bar) at the equilibrium temperature. An empirical scaling study using 2 to 10 transits of TRAPPIST-1 b shows the CO${}_2$ detection threshold decreasing with the number of transits $k$ as $k^{-0.6}$, reaching 80 ppm at ten transits, while water searches become limited by the stellar floor beyond about six transits. CO${}_2$ atmospheres, by contrast, are detectable when we utilize our framework to account for stellar activity at the cost of roughly 1.5 to 2 times as many transits.