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

ABSTRACT We develop a new and powerful method to analyse time series to rigorously detect flares in the presence of an irregularly oscillatory baseline, and apply it to stellar light curves observed with the Transiting Exoplanet Survey Satellite. 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 TIC 13955147 (a G5V eruptive variable), TIC 269797536 (an M4 high-proper motion star), and TIC 441420236 (AU Mic, an active dMe flare star), detecting up to 145, 460, and 403 flares, respectively, at rates ranging from ${\approx }0.4~\mathrm{ to}~8.5$ d$^{-1}$ over different sectors and under different detection thresholds. We detect flares down to amplitudes of 0.03 per cent, 0.29 per cent, and 0.007 per cent 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 those on the Sun ($\alpha _{E,P}\approx 1.85,2.36$), while for the less- and highly-active low-mass stars $\alpha _{E,P}\gt2$ and $\lt 2,$ respectively.