How to reach the orbital configuration of the inner three planets in HD 40307 planet system?

The formation of the present configuration of three hot super-Earths in the planet system HD 40307 is a challenge to dynamical astronomers.With the two successive period ratios both near and slightly larger than 2,the system may have evolved from pairwise2:1 mean motion resonances(MMRs).In this paper,we investigate the evolutions of the period ratios of the three planets after the primordial gas disk was depleted.Three routines are found to probably result in the current configuration under tidal dissipation with the center star,they are:(i)through apsidal alignment only;(ii)out of pairwise 2:1 MMRs,then through apsidal alignment;(iii)out of the 4:2:1 Laplace Resonance(LR),then through apsidal alignment.All the three scenarios require the initial eccentricities of planets～0.15,which implies a planetary scattering history during and after the gas disk was depleted.All the three routines will go through the apsidal alignment phase,and enter a state with near-zero eccentricities finally.We also find some special characteristics for each routine.If the system went through pairwise 2:1 MMRs at the beginning,the MMR of the outer two planets would be broken first to reach the current state.As for routine(iii),the planets would be out of the LR at the place where some high-order resonances are located.At the high-order resonances 17:8 or 32:15 of the planets c and d,the system will possibly enter the current state as the final equilibrium.