Time-variable electromagnetic star-planet interaction: The TRAPPIST-1 system as an exemplary case.

Exoplanets sufficiently close to their host star can in principle couple electrodynamically to the star. This process is known as electrodynamic star-planet interaction (SPI). The expected emission associated with this coupling is however difficult to observe due to the bright intrinsic stellar emission. Identification of time-variability in the stellar lightcurve is one of the most promising approaches to identify SPI. In this work we therefore systematically investigate various mechanisms and their associated periods, which generate time-variability to aid the search for SPI. We find that the synodic and half the synodic rotation periods of the stars as measured in the rest frames of the orbiting exoplanets are basic periods occurring in SPI. We apply our findings to the example of TRAPPIST-1 with its seven close-in planets for which we investigate the possibility of SPI and the associated time-variabilities. We show that especially TRAPPIST-1b and c, are very likely subject to sub-Alfv\'{e}nic interaction, a necessary condition for SPI. Both planets are therefore expected to generate Alfv\'{e}n wings, which can couple to the star. The associated Poynting fluxes are on the order of $10^{11}$ to $10^{15}$ W and thus can hardly be the direct source of currently observable time-variability from TRAPPIST-1. However these Poynting fluxes might trigger flares on the star. We find correlations between the observed flares and the expected planetary induced signals, which could be due to SPI but our findings are not conclusive and warrant further observations and modelling.