How Flat Can a Planetary System Get? I. The Case of TRAPPIST-1

The seven planets orbiting TRAPPIST-1 in a compact near-resonant chain offer a unique case to study in planet formation theory. We demonstrate in this paper that the remarkable flatness of the system, exceeding that of any other known planetary system, is an important constraint on the mass of the gaseous disk in which it formed and attained its current configuration. We use three-dimensional hydrodynamic simulations of the gas and planets to study specific formation models. In particular, we report simulations motivated by the model proposed by Ormel et al.—in this model, the dispersal of the gas disk pushes the planets from an initial resonant chain into their present configuration. We find that a disk with the mass used in this model is consistent with the flatness of the TRAPPIST-1 system, but a more massive disk is not, with the transition occurring between 15 and 50 times the mass of the Ormel et al. disk. This upper limit on mass rules out certain models of the formation of the system, namely in situ formation and disk migration on long timescales.