Architecture of Three-planet Systems Predicted from the Observed Protoplanetary Disk of HL Tau

Abstract

A number of protoplanetary disks (PPDs) observed with the Atacama Large Millimeter/submillimeter Array potentially provide direct examples of initial conditions for planetary systems. In particular, the HL Tau disk has been intensively studied, and its rings/gaps are conventionally interpreted to be a result of unseen massive planets embedded in the gaps. Based on this interpretation, we carried out N -body simulations to investigate the orbital evolution of planets within the PPD and after disk dispersal. Before disk dispersal, our N -body simulations include both migration and mass growth of the planet coupled with the evolution of the disk. By varying the disk parameters, we produce a variety of widely separated planetary systems consisting of three super-Jupiters at the end of disk dispersal. We found that the outer planet is more massive than the inner one, and the migration of the innermost planet is inefficient due to the accretion of outer planet(s). We also showed how the final configuration and the final planetary mass depend on disk parameters. The migration is found to be convergent, and no planet pair has a period ratio less than 2. After disk dispersal, we switch to purely gravitational N -body simulations and integrate the orbits up to 10 Gyr. Most simulated systems remain stable for at least 10 Gyr. We discuss the implications of our result in terms of the observed widely separated planetary systems HR 8799 and PDS 70.