Secular Gravitational Instability of a Dust Layer in Shear Turbulence
MICHIKOSHI, Shugo, KOKUBO, Eiichiro,(NAOJ), INUTSUKA Shu-Ichiro (Nagoya University)
In the standard scenario of planet formation, planetesimals are the precursors of planets, the sizes of which
are on the order of kilometer. Their formation process
is one of the unsolved problems of the planet formation theory. From micron-sized dust grains dust particles grow to centimeter size in a protoplanetary disk bycollisional agglomeration. The least understood growthphase is growth from centimeter-sized dust to kilometersized planetesimals.
One of the possible models in this stage is the gravitational instability model. A very thin and dense layerof settled dust aggregates in the mid-plane of the protoplanetary disk may be gravitationally unstable. Thenthe gravitational collapse of the dust layer occurs, andkilometer-sized planetesimals are formed directly. This
scenario has the advantage of a very rapid formation
timescale, which is on the order of the Keplerian time.
However, the various hydrodynamic instabilities cause
the turbulence in the protoplanetary disk. In this case,the gravitational instability may be suppressed since thedust layer can not become thin due to the turbulence [1].
We investigated the motion of dust particles in turbulent gas and derived the surface density evolution using
the stochastic model [2]. We derived the advection diffusion equation with the strong coupling approximation.
We solved the advection diﬀusion equation coupled with
the Poisson equation for gravity using the linear approximation. As a result, we found that the dust layer is always unstable regardless of the turbulence strength, but
its timescale becomes longer as the turbulence becomesstronger. The dust density increases monotonically because of the instability.
The physical meaning of the secular gravitational instability can be understood in terms of the responses tothe density perturbation. We suppose that the density
fluctuation exists. The gravitational potential is inducedby the density fluctuation. Due to the resultant potential
gradient, the matter moves to the local maximum pointof the surface density with the terminal velocity. Therefore, the density fluctuation increases monotonically.
We applied this instability to the protoplanetary disk.
This instability always occurs, but its timescale is typically very long. If the radial drift of dust particles due togas drag is faster than the secular gravitational instability, the secular instability seems to be ineﬃcient. In thiscase, the particle pileup mechanism due to the radial drift

enhances the surface density of dust [3]. Thus, we should
compare the timescales of the secular gravitational instability and the radial drift. Figure 1 shows the critical fg
value, where fg is the enhancement factor of the surface
density of gas, as a function of the distance from the Sun.
The value Jc is the critical Richardson number. Here weassume the quasi-equilibrium shear turbulence model. If
fg is larger than the critical value, the secular gravitational instability is faster than the radial drift. Whenfg > 3 and Jc = 0.1, the secular gravitational instability
is faster than the radial drift in the entire disk. If thiscondition is satisfied, axisymmetric high-density patterns
form. When the dust surface density becomes large dueto this instability, the classical gravitational instability
may occur finally [4]. We will investigate this process by
numerical simulations in the future work.
65Secular GI
4fg

Jc=0.25
3Jc=0.1

21

Radial drift

00.1

1a (AU)

10Figure 1: The critical fg values as a function of the distancefrom the Sun a.

References[1] Weidenschilling, S. J. 1980, Icarus, 44, 172[2] Michikoshi, S., Kokubo, E., & Inutsuka, S.-i. 2012,ApJ, 746, 35[3] Youdin, A. N., & Shu, F. H. 2002, ApJ, 580, 494[4] Sekiya, M. 1998, Icarus, 133, 298

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