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H. Tanaka (Tokyo Inst. of Tech./SwRI), T. Takeuchi (Stockholm Observatory)
Linear calculations of the tidal interaction between a planet and a three-dimensional gas disk has been done to obtain the tidal torque exerted on the planet and the radial migration speed of a planet. A rapid migration (~105 orbit periods) of a planet due to the tidal torque is one of the serious problems in the theory of planet formation. Most of the previous studies on the disk-planet interaction assumed that the disk is infinitely thin (e.g., Ward 1986, Icarus 67, 164-180; Korycansky & Pollack 1993, Icarus 102, 105-165). But, this assumption would not be valid because a torque on a planet is exerted mainly by the gas located within a disk scale-height from the planet. Generally, the inner disk exerts the positive torque on a planet while the outer one exerts the negative torque. The sum of these positive and negative torques leads to the net tidal torque on the planet. Thus, the three-dimensional effect can significantly change the net torque and the migration speed.
Recently, the three-dimensional calculations of the disk-planet tidal interaction were done by some authors (e.g., Takeuchi & Miyama 1998, Publ.\ Astron.\ Soc.\ Japan 50,141-148; Miyoshi et al. 1999, ApJ 516, 451-464). But they could not obtain the net torque on a planet, since they adopted a local approximation. In our calculation, keeping terms neglected in their calculations, we enabled to obtain the net tidal torque. Our obtained torque by 3D disk is about 30% of that by 2D disk for the minimum-mass solar nebula model. In spite of this reduction due to the three-dimensional effect, the net torque (or the migration speed) is still large. The migration problem in planet formation is still unsolved.