Detecting stellar flares in the presence of a deterministic trend and stochastic volatility

We develop a new and powerful method to analyze time series to rigorously detect flares in the presence of an irregularly oscillatory baseline, and apply it to stellar light curves observed with TESS. First, we remove the underlying non-stochastic trend using a time-varying amplitude harmonic model. We then model the stochastic component of the light curves in a manner analogous to financial time series, as an ARMA+GARCH process, allowing us to detect and characterize impulsive flares as large deviations inconsistent with the correlation structure in the light curve. We apply the method to exemplar light curves from TIC13955147 (a G5V eruptive variable), TIC269797536 (an M4 high-proper motion star), and TIC441420236 (AU Mic, an active dMe flare star), detecting up to $145$, $460$, and $403$ flares respectively, at rates ranging from ${\approx}0.4$--$8.5$~day$^{-1}$ over different sectors and under different detection thresholds. We detect flares down to amplitudes of $0.03$%, $0.29$%, and $0.007$% of the bolometric luminosity for each star respectively. We model the distributions of flare energies and peak fluxes as power-laws, and find that the solar-like star exhibits values similar to that on the Sun ($α_{E,P}\approx1.85,2.36$), while for the less- and highly-active low-mass stars $α_{E,P}>2$ and $<2$ respectively.