# Harvard Astronomy 201b

## Formation of planetesimals

In Uncategorized on April 11, 2011 at 9:09 pm

Introduction

Terrestrial planets are thought to be built up through the collisions of many smaller objects. Our story begins in the remains of the stellar accretion disk which, for a sun-like star, is 99% gaseous hydrogen and helium by mass.  Terrestrial planets (and it is thought the cores of giant planets) are created from the remaining 1% of mass, which consists of tiny solid grains.  Collisions of these micrometer dust grains eventually can create planets like Earth.

Much of the following is based on the review article “Terrestrial Planet Formation” by John Chambers (2010) which can be found in Exoplanets, edited by Sara Seager and an article by Andrew Youdin (2008).

Artist's conception of a protoplanetary disk from Keck (http://keckobservatory.org/blog/zooming_in_on_infant_planetary_systems/)

The second half of the story: from planetesimals to planets (reference: Chambers 2010)

A key step along the way is the formation of planetesimals; these 100-1000m piles of debris are massive enough that their gravitational fields significantly influence the behavior of nearby objects.  After planetesimals have been formed, we think we know the story: planetesimals grow by accreting smaller passing objects.  Their orbits are circularized and flattened by dynamical interactions with other planetesimals and gas drag and their range of influence grows.  Because the growth rate increases with mass, large planetesimals grow faster than smaller ones–this is called the “runaway growth” stage.  Eventually, some planetesimals grow massive enough to exert a significant influence on the velocities of smaller objects:  with the relative velocities of planetesimals set by the most massive object, the “oligarchic growth” stage is reached.  In this phase, the most massive planetesimal, now called a planetary embryo, dominates evolution in its part of the disk and grows rapidly by accretion.  At this point, we can get something akin to Mars, but the story is far from over.  The planetary embryos continue to grow, atmospheres can be gained and lost, tidal effects and resonances modify the embryos’ orbits and planets may undergo migration.

That was the second half of the story.  Many questions remain unanswered: what is the principle cause of planet migration?  How much migration occurs?  How do the dynamics of planetesimals evolve?

The first half of the story: from dust to planetesimals

But what about the first chapters: how do we even get planetesimals in the first place?  How do tiny dust particles build up to create compact solids a kilometer in diameter?  This is one of the major questions remaining in planet formation research and although much progress has been made, the first half of the book as yet to be written.  Here, I will talk about some of the ideas for planetesimal formation.

Collision of dust grains (reference: Youdin 2008)

In this theory, collisions between dust grains build up larger and larger objects.  In a simple theoretical model, this can be understood by the following representation taken directly from Youdin (2008).  When a grain collides with another particle, its kinetic energy increases due to binding energy and because it is falling into a potential well:

$KE_{impact}=KE_{init}+BE$

Some energy is dissipated, decreasing the total energy.  Here, $f_{diss}$ is the fraction of energy turned into heat during the collisions.

$KE_{after}=(1-f_{diss})KE_{impact}=(1-f_{diss})(KE_{init}+BE)$

After the collision, the final kinetic energy is reduced by the binding energy as it leaves the potential well.  For the grain to remain bound to the particle, the final kinetic energy must be less than 0:

$KE_{final}=KE_{after}-BE=(1-f_{diss})(KE_{init}+BE)-BE < 0$

$KE_{init} < BE\frac{f_{diss}}{1-f_{diss}}$

Thus, low collisions speeds (which lower the initial kinetic energy), high binding energies (which increase the final energy barrier) and high energy dissipation during impact (lowering $E_{grain}$) facilitate particle growth.  The electrostatic and gravitational forces are both possible binding energies but the latter only become important when planetesimals are very massive.  The electrostatic interaction involved is typically the van der Waals force; the possibility of charged grains has been discussed but not explored.  A collision between two charged particles would have a higher binding energy, making sticking easier.  An ice coating also helps (see Alyssa’s post)!

Figure and caption from Beitz et al. 2011 (http://arxiv.org/abs/1102.4441)

Chambers (2010) discusses the laboratory observations that have helped us to understand this process, although it is difficult to know exactly what conditions were like in the protoplanetary disk.  At very low speeds (much less than 1 m/s), collisions between micrometer-sized dust grains tend to result in the grains loosely sticking together, at intermediate speeds (on the order of 1-10 m/s) more compact aggregates are formed, while at high speeds the grains tend to rebound and growth does not occur.  Now, we have conglomerations that have reached millimeter to centimeter sizes and the a different behavior is observed: a collision between a grain and a compact aggregate at speeds less than 10 m/s results in rebound, moderate-speed collisions result in growth of the aggregate, while collisions at high speeds can fragment it.

Growth of planetesimals gets increasingly difficult as sizes approach a meter because binding energies decline while relative velocities increase.  According to Chambers (2010), with turbulence disruptive collisions between a meter-sized and a much smaller planetesimal are frequent because relative speeds of 100 m/s are often reached and it is difficult to get larger bodies.  Youdin (2008) also discusses other issues (both theoretical and observation) with growing large bodies via collisions.  Since we need kilometer-sized planetesimals to proceed with the second half of our story, we have a problem which is usually called the “meter-size barrier” in the literature.

Gravitational instability (reference: Youdin 2008)

When the Toomre Q parameter is describes the stability of a thin disk to gravitational collapse (this is quoted exactly from Youdin 2008):

$Q_T=\frac{c \Omega}{\pi G \Sigma}$

where $c$ is the random velocity (the sound speed for a gas), $\Omega$ is the orbital frequency and $\Sigma$ is the surface density.  When $Q_T < 1$, the disk is unstable.  Thus, small speeds (both random and orbital) and large mass surface densities are conducive to collapse.  (A different approach to characterizing instability is the Roche criterion, where self-gravity is greater than tidal forces.)  With turbulence in the mid-plane of the disk, the surface mass density does not become high enough for gravitational instability because particles are prevented from settling there.  Particles themselves generate mid-plane turbulence so there is no way around this problem; however, it might not be the final nail in the coffin for gravitational instability.  First, planets are more likely to be formed around metal-rich stars and thus presumably out of more metal-rich disks.  There is a limit to the amount of material turbulence can prevent from settling so in very metal-rich disks most of the solids present will be subject to gravitational instabilities.  Second, turbulence can create temporary overdensities which facilitate local collapse.

The streaming instability: concentrations of boulders in a simulation by Johansen et al. 2007. The colors represent column density. An animation is linked in the references.

There are many possible ways to create local concentrations.  Points of maximum pressure are stable and could be generated by large density waves or pressure fluctuations.  If centimeter and meter-sized bodies can be formed, they rapidly migrate to pressure maxima; once an over-density develops, more large objects are attracted to the region, thus further increasing the surface density and facilitating the gravitational collapse of kilometer-size planetesimals.  This is called the streaming instability and has been studied, for example,  by Johansen & others (2006) and Youdin & Goodman (2005).  Although one of the most promising theories, it does require meter-sized planetesimals to have been formed early in the disk, at odds with observations of meteorites (essentially left-over planetesimals) which are composed of millimeter-size particles, not meter-sized ones (Chambers 2010).  Other possibilities include particles being trapped in vortices and gas drag at boundary layers.  The latter is called the secular instability has been explored, for example, by Goodman & Pindor (2000) and Youdin (2011).

Current lines of research seem to support gravitational instability with particle concentration by turbulence as the method for making kilometer-sized planetesimals.  This is a relatively new area of research and clearly many issues remain.

References

Chambers, J. 2010, “Terrestrial Planet Formation” in Exoplanets (ed: Sara Seager)
Youdin, A. 2008, “From grains to planetesimals”

Further reading

Ian’s post on Astrobites, which is a review of protoplanetary disks from an observational perspective.
Youdin & Shu 2002, simulations of planetesimal formation via gravitational instability
Rafikov 2003, dynamic evolution of planetesimals
Hillenbrand 2005, observational constrains on disk lifetimes
Johansen & others 2006, magnetorotational turbulence and planetesimals
Johansen et al. 2007, turbulence and planet formation, including this animation
Youdin 2011, simulations of the streaming instability
Bai & Stone 2011, simulations of the streaming instability

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1. A recent paper on aggregation in low-velocity collisions from a microgravity experiment: http://arxiv.org/abs/1106.4760

2. […] bump into each other and stick together, eventually conglomerating to form larger bodies – planetesimal seeds for the planets we observe many years later. This one-sentence summary hides a plethora of […]