On the formation of hot Jupiters in two-planet systems within open clusters

Within the dynamical environment of open clusters, we investigate the formation of hot Jupiters in two-planet systems through long-term N-body simulations． The systems are initialized with two Jupiter-mass planets on coplanar circular orbits, with the inner planet fixed at 5 ua and the outer planet placed between 10 and 100 ua． The orbital evolution is followed over a timescale of 1 Gyr while including perturbations from neighbouring stars． The results show that stellar companions play a key role in triggering high-eccentricity migration and promoting hot-Jupiter formation, whereas planet–planet scattering exhibits both positive and negative effects depending on the initial orbital configuration． When the outer planet is initially located at small separations, enhanced scattering tends to disrupt the long-term growth of orbital eccentricity and reduces the formation efficiency of hot Jupiters． The formation rate reaches its maximum when the initial semi-major axis of the outer planet is around 30 ua and remains higher than that of the single-planet reference systems． As the orbital separation increases further, the evolutionary behaviour of two-planet systems gradually approaches that of the single-planet case． These results suggest that hot-Jupiter formation in open clusters is jointly regulated by companion perturbations, long-term gravitational interactions, and planet–planet scattering, with the final outcome determined by the competition among these dynamical processes．