Traditional formation scenarios fail to explain 4:3 mean motion resonances

At least two multi-planetary systems in a 4:3 mean motion resonance have been found by radial velocity surveys.¹ These planets are gas giants and the systems are only stable when protected by a resonance. Additionally the Kepler mission has detected at least four strong candidate planetary systems with a period ratio close to 4:3. This paper investigates traditional dynamical scenarios for the formation of these systems. We systematically study migration scenarios with both N-body and hydrodynamic simulations. We investigate scenarios involving the in situ formation of two planets in resonance. We look at the results from finely tuned planet-planet scattering simulations with gas disc damping. Finally, we investigate a formation scenario involving isolation-mass embryos. Although the combined planet-planet scattering and damping scenario seems promising, none of the above scenarios is successful in forming enough systems in 4:3 resonance with planetary masses similar to the observed ones. This is a negative result but it has important implications for planet formation. Previous studies were successful in forming 2:1 and 3:2 resonances. This is generally believed to be evidence of planet migration. We highlight the main differences between those studies and our failure in forming a 4:3 resonance. We also speculate on more exotic and complicated ideas. These results will guide future investigators towards exploring the above scenarios and alternative mechanisms in a more general framework.