Abstract; Variability in the far-ultraviolet (FUV) emission produced by stellar activity affects photochemistry and heating in orbiting planetary atmospheres. We present a comprehensive analysis of the FUV variability of GJ 436, a field-age M2.5V star (P; rot ≈ 44 days) that is orbited by a warm Neptune-sized planet (M ≈ 25 M; ⊕, R ≈ 4.1 M; ⊕, P ; orb ≈ 2.6 days). Observations at three epochs from 2012 to 2018 span nearly a full activity cycle, sample two rotations of the star and two orbital periods of the planet, and reveal a multitude of brief flares. From 2012 to 2018, the star’s 7.75 ± 0.10 yr activity cycle produced the largest observed variations, 38% ± 3% in the summed flux of the major FUV emission lines. In 2018, the variability due to rotation was 8% ± 2%. An additional 11% ± 1% scatter at a cadence of 10 minutes, which is treated as white noise in the fits, likely has both instrumental and astrophysical origins. Flares increased time-averaged emission by 15% over the 0.88 days of cumulative exposure, peaking as high as 25× quiescence. We interpret these flare values as lower limits given that flares too weak or too infrequent to have been observed likely exist. GJ 436’s flare frequency distribution at FUV wavelengths is unusual compared to other field-age M dwarfs, exhibiting a statistically significant dearth of high-energy (>4 × 1028 erg) events, which we hypothesize to be the result of a magnetic star–planet interaction (SPI) triggering premature flares. If an SPI is present, GJ 436 b’s magnetic field strength must be ≲100 G to explain the statistically insignificant increase in the orbit-phased FUV emission.
