Planet Formation
Topic:
Formation of rocky planets from planetesimals
Lecture by: C.P. Dullemond

Standard model of rocky planet formation
1. Start with a sea of planetesimals (~1...100 km)
2.
Mutual gravitational stirring, increasing „dynamic temperature“ of the planetesimal swarm.
3. Collisions, growth or fragmentation, dependent on the impact velocity, which depends on dynamic temperature.
4. If velocities low enough: Gravitational focusing:
Runaway growth: „the winner takes it all“
5. Biggest body will stir up planetesimals: gravitational focusing will decline, runaway growth stalls.
6.
Other „local winners“ will form: oligarchic growth
7. Oligarchs merge in complex Nbody „dance“

Gravitational stirring of planetesimals by each other and by a planet

Describing deviations from Kepler motion
We can describe an inclined elliptic orbit as an in-plane circular orbit with a „perturbation“ on top: v body
= v
K
+ D v body
For the z-component we have:
D v
( z ) body
( t )
= sin i v
K sin(
W t )
So the mean square is:
v
( z ) body
1
2 sin i v
K
1
2 v
K i
For bodies at the midplane (maximum velocity):
v
( z ) body midplane
sin i v
K
v
K i

Describing deviations from Kepler motion
We can describe an inclined elliptic orbit as an in-plane circular orbit with a „perturbation“ on top: v body
= v
K
+ D v body
For the x,y-components we have epicyclic motion.
D v
( x ) body
D v
( y ) body
( t )
» e v
K
( t )
»
2 e v
K sin(
W t ) cos(
W t ) guiding center epicycle
But notice that compared to the local (shifted) Kepler velocity
(green dashed circle in diagram), the y-velocity is lower:
D v
( y ) body,local
( t )
»
1
2 e v
K cos(
W t )

„Dynamic temperature“ of planetesimals
If there are sufficient gravitational interactions between the bodies they „thermalize“. We can then compute a dynamic „temperature“:
D v
2 body
= k
B
T dyn m body
Example: 1 km planetesimals at <i>=0.1, <e>=0.2, have a dynamic temperature around 10 44 Kelvin!
Now that is high-energy physics! ;-)
Most massive bodies have smallest Δv. Thermalization is fast.
So if we have a planet in a sea of planetesimals, we can assume that the planet has e=i=0 while the planetesimals have e>0, i>0.

Gravitational stirring
When the test body comes very close to the bigger one, the big one can strongly „kick“ the test body onto another orbit.
This leads to a jump in a, e and i . But there are relations between the „before“ and „after“ orbits:
From the constancy of the Jacobi integral one can derive the
Tisserand relation, where a p is the a of the big planet: before after
T
º a p
2 a
+ a
(1
e 2 ) cos i
» a p
2 a '
+ a a p
'
(1
e ' 2 ) cos i ' a p
Conclusion: Shortrange „kicks“ can change e, i and a

Gravitational stirring
Orbit crossings: Close encounters can only happen if the orbits of the planet and the planetesimal cross.
Given a semi-major axis a and eccentricity e, what are the smallest and largest radial distances to the sun?
r ( v )
= a (1
e
2
)
1
+ e cos( v )
= a (1
e )(1
+ e )
1
+ e cos( v ) r min r max
= a (1
e )
= a (1
+ e )

Gravitational stirring
No close encounter possible
Can have close encounter
No close encounter possible
Figure: courtesy of Sean Raymond

e 2
+ i 2
Gravitational stirring
Lines of constant
Tisserand number
Ida & Makino 1993 a

e 2
+ i 2
Gravitational stirring
Lines of constant
Tisserand number
Ida & Makino 1993 a

Ida & Makino 1993
Gravitational stirring

Gravitational stirring: Chaotic behavior

Gravitational stirring: resonances
We will discuss resonances later, but like in ordinary dynamics, there can also be resonances in orbital dynamics. They make stirring particularly efficient.
Movie: courtesy of Sean Raymond

Limits on stirring: The escape speed
A planet can kick out a small body from the solar system by a single „kick“ if (and only if): v esc,planet
º
2 GM
R planet planet
> v esc,system
º
2 GM
* a planet
Jupiter can kick out a small body from the solar system, but the Earth can not.

Collisions and growth

Feeding the planet
Feeding dynamically
„cool“ planetesimals.
D v
£ v
Hill v
Hill
=
GM p
R
Hill
The „shear-dominated regime“

Close encounters and collisions
Hill Sphere
Greenzweig & Lissauer 1990

Feeding the planet
Feeding dynamically
„warm“ planetesimals.
v
Hill
£ D v
£ v esc v
Hill
=
GM p
R
Hill v esc
=
2 GM p
R p
The „dispersion-dominated regime“ with gravitational focussing (see next slide).
D v>v esc dominated regime: no gravitational focussing („hot“ planetesimals).

Gravitational focussing m
M
Due to the gravitational pull by the (big) planet, the smaller body has a larger chance of colliding. The effective cross section becomes: s eff
= p (
R
+ r
) 2 £
£
£
£
1
+
£ v esc
D v
£ 2
£
£
£
£
Where the escape velocity is: v esc
=
2
( + m
)
( R
+ r )
Slow bodies are easier captured! So: „keep them cool“!

Collision
Collision velocity of two bodies: v c
= D v
2 + v
2 esc
Rebound velocity:
 v c with

1: coefficient of restitution.
 v c
 v e
Two bodies remain gravitationally bound: accretion
 v c
 v e
Disruption / fragmentation
Slow collisions are most likely to lead to merging.
Again: „Keep them cool!“

Example of low-velocity merging
Formation of Haumea (a Kuiper belt object)
Leinhardt, Marcus & Stewart (2010) ApJ 714, 1789

Example of low-velocity merging
Formation of Haumea (a Kuiper belt object)
Leinhardt, Marcus & Stewart (2010) ApJ 714, 1789

Growth of a planet
Increase of planet mass per unit time: Gravitational focussing
 sw
= mass density of swarm of planetesimals
M = mass of growing protoplanet
 v = relative velocity planetesimals r = radius protoplanet

= Safronov number dr dt
= r sw
4 r
D v p
(1
+ q
)
 p
= density of interior of planet dM
=
4 p r
2 r p dr

Growth of a planet
Estimate properties of planetesimal swarm:
Assuming that all planetesimals in feeding zone finally end up in planet
R = radius of orbit of planet

R = width of the feeding zone
 z = height of the planetesimal swarm
Estimate height of swarm:

Growth of a planet
Remember: dr dt
= r sw
4 r
D v p
(1
+ q
)
Note: independent of
 v!!
For M<<M p one has linear growth of r dr dt
= v
K
(1
+ q
)( M p
-
M )
16 p
R
2 D
R r p dM dt
µ
M 2/3 (1
+ q
æ
) 1
è
-
M
M p
ö
ø

Growth of a planet dr dt
= v
K
(1
+ q
)( M p
-
M )
16 p
R
2 D
R r p
Case of Earth: v k
= 30 km/s,

=6, M p
1 AU,

R = 0.5 AU,
 p
= 6x10
= 5.5 gr/cm
27
3 gr, R = dr
=
15 cm/year dt t growth
=
40 Myr
Earth takes 40 million years to form (more detailed models: 80 million years).
Much longer than observed disk clearing time scales. But debris disks can live longer than that.

Runaway growth dM dt
µ
M
2/3
(1
+ q
)
=
M
2/3
£
£
1
+
£ v esc
D v
£ 2
£
£
£ v
2 esc
=
2 GM
µ
M
2/3
R
So for Δv<<v esc we see that we get: dM
µ
M
4/3 dt
The largest and second largest bodies separate in mass: d dt lg
ç
æ
è
M
M
2
1
ö
ø
÷µ
M
1
1/3 -
M
2
1/3 >
0
So: „The winner takes it all“!

End of runaway growth: oligarchic growth
Once the largest body becomes planet-size, it starts to stir up the planetesimals. Therefore the gravitational focussing reduces eventually to zero, so the original geometric cross section is left: dM
µ
M 2/3 (1
+ q
)
®
M 2/3 dt
Now we get that the largest and second largest planets approach each other in mass again: d dt lg
æ
è
M
M
1
2
ö
ø
M
1
-
1/3 -
M
-
1/3
2
<
0
Will get locallydominant „oligarchs“ that have similar masses, each stirring its own „soup“.

Gas damping of velocities
• Gas can dampen random motions of planetesimals if they are < 100 m - 1 km radius (at 1AU).
• If they are damped strongly, then:
– Shear-dominated regime (  v <
 r
Hill
– Flat disk of planetesimals (h << r
Hill
)
)
• One obtains a 2-D problem (instead of 3-D) and higher capture chances.
• Can increase formation speed by a factor of 10 or more.
This can even work for pebbles (cm-size bodies):
“pebble accretion” is a recent development.

Isolation mass
Once the planet has eaten up all of the mass within its reach, the growth stops.
M iso
= S m
( t
=
0)
B
ö 1/ 3
ø with B
=
3
1/ 3
M
*
1/ 3
2 p bR
2 b = spacing between protoplanets in units of their Hill radii. b

5...10.
Some planetesimals may still be scattered into feeding zone, continuing growth, but this depends on presence of scatterer (a Jupiter-like planet?)