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Notes:


This is really the bad news for planetesimals formation from small size particles. In order to form planetesimals from direct gravitational collapse, an extraordinary high dust density is required. In other words, for given metallicity, the dust has to settle into an unrealistically thin layer in order to become gravitational unstable.

For example, taken the critical density as rhoroche, the resulting profile is this dashed curve, which indicates the thickness is two orders of magnitude smaller than gas disk scaleheight; however, take the rhoc as the midplane density,

We find the thickness of this layer is further reduced by 3 orders of magnitude . So a little bit turbulence in the disk would easily prevent such thin layer to form at the first place even if the dust abundance get enriched by a factor of a few.

So the conclusion is that : it is extremely difficult, even more difficult that previously thought, to form planetesimals from small size paritcles by direct gravitational collapse.

To get GI of small size particles, Qd<=1, the dust layer has to be this thin, more than 5 orders of magnitude smaller than the scaleheight of the gas disk. For comparison, lee+ found 10^-2 is the least for KH stable..