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 \(Q' < \sim 100\) tidal effects may have produced significant changes including the formation of three body Laplace resonances in the case of the EPIC 245950175 system.

我们研究了形成共振链的多行星系统的轨道演化，其中最近的邻居接近一阶可公度，并结合了由与中心恒星的潮汐相互作用产生的轨道圆化。我们开发了一个半解析模型，当与\(\epsilon ^{2/3},\)与\(\epsilon \)作为特征行星与中心恒星质量比的量度。这使得能够确定受迫偏心以及哪些共振角进入振动。当不存在有源链接的三体拉普拉斯共振时，也可以确定半长轴的演化速率。我们对 HD 158259 和 EPIC 245950175 系统进行了数值模拟，发现半解析方法在前一种情况下效果很好，但在后一种情况下效果不佳，因为在演化过程中持续存在的三个有源三体拉普拉斯共振的影响. 对于这两个系统，我们估计如果潮汐参数\(Q',\)显着超过 1000，潮汐效应不太可能自形成以来显着影响周期比。另一方面，如果\(Q' < \sim 100\)潮汐效应可能已经产生了显着的变化，包括在 EPIC 245950175 系统的情况下形成三个拉普拉斯体共振。