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Forming Habitable Planets around Dwarf Stars: Application to OGLE-06-109L

Dwarf stars are believed to have small protostar disk where planets may grow up. During the planet formation stage, embryos undergoing type I migration are expected to be stalled at inner edge of magnetic inactive disk ($a_{\rm crit} \sim 0.2-0.3 ~$AU). This mechanism makes the location around $a_{\rm crit}$ a sweet spot of forming planets. Especially, $a_{\rm crit}$ of dwarf stars with masses $\sim 0.5 M_\odot$ is roughly inside the habitable zone of the system. In this paper we study the formation of habitable planets due to this mechanism with a model system OGLE-06-109L. It has a $0.51 M_\odot$ dwarf star with two giant planets in 2.3 and 4.6 AU observed by microlensing. We model the embryos undergoing type I migration in the gas disk with a constant disk accretion rate ($\dot M$). Giant planets in outside orbits affect the formation of habitable planets through secular perturbations at the early stage and secular resonance at the later stage. We find that the existence and the masses of the habitable planets in OGLE-06-10L system depend on both $\dot M$ and the speed of type I migration. If planets formed earlier so that $\dot M$ is larger ($\sim 10^{-7} M_\odot $ yr$^{-1}$), terrestrial planet can not be survived unless the type I migration rate is an order of magnitude less. If planets formed later so that $\dot M$ is smaller ($\sim 10^{-8} M_\odot $ yr$^{-1}$), single and high mass terrestrial planets with high water contents ($\sim 5% $) will be formed by inward migration of outer planet cores. A slower speed migration will result in several planets by collisions of embryos, thus their water contents are low ($\sim 2%$). Mean motion resonances or apsidal resonances among planets may be observed if multiple planets survived in the inner system.