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Kelvin-Helmholtz Instabilities in Multi-Sized Dust Layers

We examine the effect of the dust size distribution on Kelvin-Helmholtz instabilities in the protoplanetary disk with dust sedimentation. With newly taking into account the dust size distribution, the growth rate of the Kelvin-Helmholtz instability is calculated using the linear stability analysis with the dust density distribution consistent with sedimentation. Dust abundance required for gravitational instabilities before the Kelvin-Helmholtz instability is derived from the linear stability analysis, and it is found that the required dust abundance significantly coincides with that estimated from the Richardson number. It is also found that when the dust size distribution is taken into account, the critical Richardson number for the onset of the Kelvin-Helmholtz instability tends to increase with dust abundance. This result is different from that in the case without the dust size distribution.