Download Formation of planetesimals in collapsing particle clouds

Formation of planetesimals in
collapsing particle clouds
Karl W. Jansson & Anders Johansen, Department of Astronomy and Theoretical Physics, Lund University, Sweden
Dust Coagulation and Planetesimal Formation
The first step of planet formation in a protoplanetary disk (PPD) is coagulation of μm-sized dust
particles. This works fine up to ~mm-cm sized particles when the contact forces begin to be too small
to hold the particles together. Also at high relative speeds (≥1 m/s, Güttler et al. 2010) collisions tend
to result in fragmentation instead of coagulation.
One way to solve this is to have a self-gravitating cloud of pebbles that is in virial equilibrium. The total
energy of the cloud is:
In the cloud, particles would move around and collide with each other. Collisions dissipate away energy
from the cloud (unless they are completely elastic) and increase the binding energy. Due to the negative
heat capacity nature of a self-gravitating system the cloud ‘heats’ up when it loses energy. As the cloud
loses energy it contracts and the particles start to move faster and faster. This means that the collision
rate between particles increases, the cloud loses energy even faster and you get a runaway collapse.
This is nice but is there a way to form the cloud from the beginning? The answer to that question is
yes. One difference between the gas and the particles in a PPD is that the gas feels an outwards force
from the pressure gradient in the disk. This means that the gas can orbit with a speed lower than
Keplerian and still balance the gravity from the star. The particles, in return, will feel a headwind, lose
energy and drift towards the star. If the particles clump together, however, they will drift slower (cf.
flying geese and cyclists). They will then catch single particles coming from outside and the clump will
grow larger. This is the so called streaming instability (Youdin & Goodman 2005).
Figure: Formation of gravitationally bound clouds of cm-sized
pebbles by the streaming instability (Johansen, Youdin & Mac Low
2009). Unresolved pebble clouds with planetesimal masses are
formed in only a few orbital periods. These could later collapse to
solid bodies by energy dissipation in inelastic collisions between
pebbles.
Representative Particle Approach
One Pluto mass split into cm-sized pebbles results in N ~ 1024 pebbles. This means that there are too many pebbles to
keep track of every individual particle but instead you can use a statistical approach to the problem. To investigate the
evolution of self-gravitating pebble clouds I use the representative particle approach of Zsom & Dullemond (2008).
The underlying idea is that you follow the evolution of a smaller number, N’ ≪ N, of representative particles instead of
all physical particles. The number of these representative particles still has to be large enough such that they mirror
the true distribution of particles. One can think of the representative particles as swarms of identical particles.
To follow the evolution of a particle clump we need to find the rates with which particles collide. The figure to the
right shows the idea behind a collision between representative particle i and a physical particle from swarm k. The rate
can be written as
where nk is the number density of particles k, σik is the cross-section and Δvik is the relative velocity. From the total
collision rate (sum over all possible collision pairs) and a random number you get the time until next collision. With
more random numbers and the individual collision rates you can get the two swarms involved in the collision. Next
you calculate the outcome of the collision. Do the particles stick, fragment or bounce? How much energy is
dissipated? What are the new particle properties (size, velocity, ...)? One only follows the evolution of the
representative particle so it is only the particles in swarm i that change properties. This is corrected for when
representative particle k collides with a physical particle in swarm i and only the properties of the particles in swarm k
is changed. With this method you resolve every single collision for the N’ representative particles.
Figure: Collision between two swarms of particles i
and k. The rate with which the representative particle
(green dot) collides with any physical particle (blue
dots) is used for this particular collision pair. If the
collision occur the outcome is ‘applied’ to all particles in
the red swarm. Note that a representative particle can
collide with its own swarm of particles, i.e. i = k.
Evolution of a particle clump in virial equilibrium
Size of the particle cloud as function of time.
Best-fit power-law, log(RCloud) = b*log(1-t/a) + log(R0) also included (2/7 5 0.286).
By assuming dissipative bouncing as the only collisional outcome the problem becomes analytically solvable.
The size, R, of the cloud as function of time, t, can be written as
10000
Simulated RCloud
Analytic RCloud, a=0.733; b=2/7
Fitted RCloud, a=0.763; b=0.301
where R0 is the original size of the cloud and tcrit is the time it takes for the cloud to collapse to a single
point. For a Pluto-mass cloud of cm-sized pebbles at Pluto’s distance from the Sun the cloud collapses in
tcrit ~ 0.73 years, i.e. much shorter than it’s orbital period. The figure to the right shows the result of a
simulation and comparison to the analytic results. The bottom line here is that if you can form your pebbles
in some way and that the bouncing assumption is correct it only takes a few orbits to make the cloud by
the streaming instability and after that you have an almost immediate collapse because of inelastic collisions.
The problem is that particles don’t only bounce. They also stick to and fragment each other which adds
complexity to the problem. Fragmentation can cause large problems with simulation time if you get very
small particles since you resolve all collisions for the representative particles.
References
Güttler, Blum, Zsom, Ormel & Dullemond (2010) A&A 513 A56
Youdin & Goodman (2005) ApJ 620 459
Johansen,Youdin & Mac Low (2009) ApJ 704 75
Zsom & Dullemond (2008) A&A 489 931
Email
[email protected]
Cloud size, RCloud, (RPluto)
8000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
Time (yrs)
Figure: Evolution of the size of a particle cloud that is
contracting due to dissipation of energy in particle
collisions. The line are the analytic expression for the
collapse and a fit to it. The initial size of the cloud is one
Pluto Hill sphere and it is assumed to be self-gravitating.
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