The orbital evolution of resonant chains of exoplanets incorporating circularisation produced by tidal interaction with the central star with application to the HD 158259 and EPIC 245950175 systems

We study orbital evolution of multi-planet systems that form a resonant chain, with nearest neighbours close to first order commensurabilities, incorporating orbital circularisation produced by tidal interaction with the central star. We develop a semi-analytic model applicable when the relative proximities to commensurability, though small, are large compared to epsilon^(2/3) , with epsilon being a measure of the characteristic planet to central star mass ratio. This enables determination of forced eccentricities as well as which resonant angles enter libration. When there are no active linked three body Laplace resonances, the rate of evolution of the semi-major axes may also be determined. We perform numerical simulations of the HD 158259 and EPIC 245950175 systems finding that the semi-analytic approach works well in the former case but not so well in the latter case on account of the effects of three active three body Laplace resonances which persist during the evolution. For both systems we estimate that if the tidal parameter, Q', significantly exceeds 1000, tidal effects are unlikely to have influenced period ratios significantly since formation. On the other hand if Q' < ~ 100 tidal effects may have produced significant changes including the formation of three body Laplace resonances in the case of the EPIC 245950175 system.