Forming Planetesimals by Gravitational Instability
FORMING PLANETESIMALS BY GRAVITATIONAL INSTABILITY. I. THE ROLE OF THE RICHARDSON NUMBER IN TRIGGERING THE KELVIN–HELMHOLTZ INSTABILITY [ADS]
FORMING PLANETESIMALS BY GRAVITATIONAL INSTABILITY. II. HOW DUST SETTLES TO ITS MARGINALLY STABLE STATE [ADS]
Aaron T. Lee, Eugene Chiang, Xylar Asay-Davis, and Joseph Barranco
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In the most venerable of planet-forming scenarios, planetesimals--the km-sized building blocks of rocky planets and gas giant cores--form in the mass-rich midplane of a circumstellar disk through the gravitational collapse of m-sized boulders. However, as mass settles toward the midplane, shear between the dust-rich midplane and the poorer regions above and below produce a shear that can potentially overturn this layer. Additionally, these m-sized objects experience significant drag with the gas, resulting in radial migration time scales of 102 years, which is significantly shorter than the 106 year time scale for planet formation.
How then do we create planetesimals?
The Richardson Number
Where the individual fails the collective may succeed. If instead we can collect enough mm-sized pebbles and smaller dust grains together in the midplane, their collective gravitational pull could cause a collapse directly into planetesimals, skipping the questionable m-sized regime all together. Dust must then be able to settle without the shear upending the material out of the midplane. A necessary condition for this to occur is characterized by the Richardson number, the ratio of the Brunt oscillation frequency to the shearing frequency. For non-rotating, 2D, unmagnetized gasses, values below the critical value of 0.25 are susceptible to instability.
Does this criterion still hold in a rotating disk? Can we use this number to determine whether planetesimal formation via collective collapse is possible?
If the heavier materials in the disk gently settle toward the midplane, then it is conceivable that the profile of the dust settles into a state with a nearly constant Richardson number. As the dust becomes more centrally condensed, the value of Ri will decrease. Ultimately the shear will become too strong and turnover the midplane, reducing the dust density there.
But if instead the dust density could continue increasing, it is possible the dust density could exceed a critical value and go gravitationally unstable, where then the midplane would fragment and the dust would collapse, potentially forming planetesimals directly from the mm-sized and smaller dust grains.
Our study evolved dust layers that are initially in equilibrium with profiles defined by having a nearly constant Richardson number. We assessed whether or not these layers remain stable against shear. If so, that suggests that the dust could continue settling to even higher densities at the midplane. What you see plotted here is the dust-to-gas ratio (mu_0). For gravitationally instability to occur in the simplest of disks (disks where planet formation is nearly ~100% efficient, i.e., there is no extra disk mass available besides the mass that will go into the planets), mu values of ~30 are necessary.
Can we get high enough so that gravity makes the midplane fragment?
This movie shows an example simulation where the dust layer goes unstable over ~10 orbital periods. While the resolution appears coarse, the use of spectral methods (solving the fluid equations in Fourier space rather than Cartesian space) gives us resolution several times that which is shown.
The line of critical stability
Monitoring whether or not dust layers embedded in a gaseous disk remain stable to shear, we mapped out the region of parameter space where stability occurs (black points on the figure). For various midplane dust-to-gas ratios (mu_0), we selected several Richardson numbers and assessed whether stability was possible. The classical Ri = 0.25 value is not a universal critical value when in a rotating circumstellar disk; instead, the critical value scales with the midplane dust-to-gas ratio. Large midplane values are possible only with larger Richardson numbers, which means that the dust layer is also fairly spread out around the midplane. This suggests that there may be more overall dust and heavier elements in these disks compared to those who have lower midplane dust values but achieve smaller Richardson numbers.
However, these choice of parameters is a bit too abstract. We can translate these results into a more meaningful figure, which shows how dense the midplane can become depending on the global properties of the disk.
Conditions for Planetesimal formation
If we know the overall vertical distribution of the dust and gas, we can compute the overall "metallicity" of the disk by integrating over these layers. This has the benefit of being an observable quantity and it relates to the overall properties of the disk.
If we take the best-fit line of our results as the divide between stability and instability, this suggests the dust layer can remain stable up until mu_0 ~ 30 (the threshold for gravitational instability) if the overall metallicity of the disk is only a few times the metallicity of the Sun.
This is conceivable in circumstellar disks today. The solar metallicity value would assume that a disk around a star like the Sun has its dust and gas equally spread across the disk, where in fact radial motion of the dust can result in pileups where the dust content Sigma_d can be increased relative to gas content Sigma_g, and there local regions of enhanced metallicity can undergo gravitational collapse.
I have also been assuming here that planet formation is efficient, which is unlikely the case. In reality, the disks where these planets are forming are likely composed of much more mass than that which ends up in the form of planets. This extra mass may end up on the star or get blown away from the stellar winds. If instead we assume the overall mass of the disk is larger than the minimum mass needed to form planets, the overall gravitational tug of the dust+gas is larger, and the threshold value of mu_0 reduces from 30 to even lower values. A disk that is 3x more massive than the minimum mass needed to form planets has a threshold value of ~10 instead of ~30. It becomes even easier to form planetesimals this way!
Is it self-consistent?
In the first paper we explored dust layers that had presumably settled to an equilibrium state. A more self-consistent simulation would allow this dust to settle into this state. In the second paper we approximated this settling and tested whether our results in Paper I were robust. We started with a diffuse dust layer, which slowly settled toward the midplane. As the dust settled, we simulated the dust layer to test whether shear would disrupt the midplane. As found in Paper I, we found that more massive disks or more metal-rich disks could indeed go gravitationally unstable, where a multitude of small dust particles can collapse into a planetesimal-sized object.