A Single Power Law for the TRAPPIST-1 Flare Distribution across 4 Orders of Magnitude in Energy

TRAPPIST-1 is an ultracool dwarf that flares frequently. These flares shape the surrounding planets’ high-energy irradiation environment, with consequences for atmospheric chemistry and escape, and they can contaminate transmission spectroscopy of those planets. A quantitative flare-frequency distribution (FFD) spanning the full energy range is therefore essential for both interpreting JWST spectra and modeling the planets’ irradiation histories. Here we present a unified FFD over 4 orders of magnitude in energy by jointly analyzing ≈87 hr of JWST/NIRISS and JWST/NIRSpec time-series spectroscopy together with ≈74 days of Kepler Space Telescope (Kepler) K2 photometry. To enable a consistent comparison across these heterogeneous datasets, we convert all events to energies in the Transiting Exoplanet Survey Satellite (TESS) bandpass. For the Kepler-to-TESS conversion we adopt a cooler flare continuum appropriate for ultracool dwarfs ( T flare = 3500 K). After correcting for flare-detection sensitivities, the combined JWST + K2 cumulative FFD is consistent with a single power law, N ( ≥ E TESS ) ∝ E TESS − β , with β = 0.753 over E TESS ≃ 10 29 –10 33 erg. The slope of the distribution indicates that the time-averaged flare energy budget is dominated by rare, high-energy events rather than by the more numerous low-energy flares. This bandpass-consistent FFD provides a practical basis for JWST transit-spectroscopy planning and for modeling the flare-driven irradiation environment of the TRAPPIST-1 planets.