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The discovery of high incidence of hot Jupiters in dense clusters challengesthe field-based hot Jupiter formation theory. In dense clusters, interactionsbetween planetary systems and flyby stars are relatively common. This has asignificant impact on planetary systems, dominating hot Jupiter formation. Inthis paper, we perform high precision, few-body simulations of stellar flybysand subsequent planet migration in clusters. A large parameter spaceexploration demonstrates that close flybys that change the architecture of theplanetary system can activate high eccentricity migration mechanisms:Lidov-Kozai and planet-planet scattering, leading to high hot Jupiter formationrate in dense clusters. Our simulations predict that many of the hot Jupitersare accompanied by "ultra-cold Saturns", expelled to apastra of thousands ofAU. This increase is particularly remarkable for planetary systems originallyhosting two giant planets with semi-major axis ratios $\sim$ 4 and the flybystar approaching nearly perpendicular to the planetary orbital plane. Theestimated lower limit to the hot Jupiter formation rate of a virialized clusteris $\sim 1.6\times10^{-4}({\sigma}/{\rm 1kms^{-1}})^5({a_{\rm p}}/{\rm 20AU})({M_{\rm c}}/{\rm 1000M_\odot})^{-2}$Gyr$^{-1}$ per star, where $\sigma$ isthe cluster velocity dispersion, $a_{\rm p}$ is the size of the planetarysystem and $M_{\rm c}$ is the mass of the cluster. Our simulations yield a hotJupiter abundance which is $\sim$ 50 times smaller than that observed in theold open cluster M67. We expect that interactions involving binary stars, aswell as a third or more giant planets, will close the discrepancy.

Hot Jupiter and ultra-cold Saturn formation in dense star clusters