Planet-Planet Tides in the TRAPPIST-1 System

The star TRAPPIST-1 hosts a system of seven transiting, terrestrial exoplanets apparently in a resonant chain, at least some of which are in or near the Habitable Zone. Many have examined the roles of tides in this system, as tidal dissipation of the orbital energy of the planets may be relevant to both the rotational and orbital dynamics of the planets, as well as their habitability. Generally, tides are calculated as being due to the tides raised on the planets by the star, and tides raised on the star by the planets. I write this research note to point out a tidal effect that may be at least as important as the others in the TRAPPIST-1 system and which is so far unremarked upon in the literature: planet-planet tides. Under some reasonable assumptions, I find that for every planet $p$ in the TRAPPIST-1 system there exists some other planet $q$ for which the planet-planet dynamical tidal strain is within an order of magnitude of the stellar eccentricity tidal strain, and that the effects of planet $f$ on planet $g$ are in fact greater than that of the star on planet $g$. It is thus not obvious that planet-planet tides can be neglected in the TRAPPIST-1 exoplanetary system, especially the tides on planet $g$ due to planet $f$, if the planets are in synchronous rotation.