### Rocky planet formation

```Planet Formation
Topic:
Formation of
rocky planets from
planetesimals
Lecture by: C.P. Dullemond
Standard model of rocky planet formation
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 N-body „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 + Dv body
For the z-component we have:
Dv
(z)
body
(t) = sini v K sin(Wt)
So the mean square is:
Dv
(z)
body
= 12 sini v K » 12 v K i
For bodies at the midplane (maximum velocity):
Dv
(z)
body midplane
= sini 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 + Dv body
For the x,y-components we have epicyclic
motion.
Dv
Dv
( x)
body
( y)
body
(t) » ev K sin(Wt)
(t) » 2ev K cos(Wt)
But notice that compared to the local (shifted) Kepler velocity
(green dashed circle in diagram), the y-velocity is lower:
Dv
( y)
body,local
(t) » 12 ev K cos(Wt)
guiding
center
epicycle
„Dynamic temperature“ of planetesimals
If there are sufficient gravitational interactions between the bodies
they „thermalize“. We can then compute a dynamic „temperature“:
Dv
2
body
=
kBTdyn
mbody
Example: 1 km planetesimals at <i>=0.1, <e>=0.2, have a
dynamic temperature around 1044 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
ap is the a of the big planet:
before
ap
ap
a
a'
2
Tº
+
(1- e ) cosi »
+
(1- e'2 ) cosi'
2a
ap
2a'
ap
Conclusion: Short-range „kicks“ can change e, i and a
after
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?
a(1- e2 )
a(1- e)(1+ e)
r(v) =
=
1+ ecos(v)
1+ ecos(v)
rmin = a(1- e)
rmax = a(1+ e)
Gravitational stirring
Can have close encounter
No close
encounter
possible
Figure: courtesy of Sean Raymond
No close
encounter
possible
Gravitational stirring
Lines of constant
Tisserand number
e2 + i 2
Ida & Makino 1993
a
Gravitational stirring
Lines of constant
Tisserand number
e2 + i 2
Ida & Makino 1993
a
Gravitational stirring
Ida & Makino 1993
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
2GM planet
Rplanet
2GM *
> v esc,system º
aplanet
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.
Dv £ v Hill
GM p
v Hill =
RHill
The „shear-dominated regime“
Close encounters and collisions
Hill Sphere
Greenzweig & Lissauer 1990
Feeding the planet
Feeding dynamically
„warm“ planetesimals.
vHill £ Dv £ vesc
GM p
v Hill =
RHill
v esc =
2GM p
Rp
The „dispersion-dominated regime“
with gravitational focussing (see
next slide).
Note: if Dv>vescwe would be in the ballistic dispersion
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
2ù
é
æ v esc ö
2
= p ( R + r ) ê1+ ç
÷ú
êë è Dv ø úû
Where the escape velocity is:
v esc =
2G ( M + m)
(R + r)
Slow bodies are easier captured! So: „keep them cool“!
Collision
Collision velocity of two bodies:
2
vc = Dv 2 + vesc
Rebound velocity: vc with 1: coefficient of restitution.
vc  ve
Two bodies remain gravitationally
bound: accretion
vc  ve
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:
sw
M
v
r

Gravitational
focussing
= mass density of swarm of planetesimals
= mass of growing protoplanet
= relative velocity planetesimals
= Safronov number
dr rsw Dv
=
(1+ q )
dt
4rp
p = density of interior of planet
dM = 4p r 2 rp 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 rsw Dv
=
(1+ q )
dt
4rp
dr v K (1+ q )(M p - M )
=
dt
16p R 2 DR rp
Note: independent of v!!
For M<<Mp one has linear growth of r
æ
dM
M ö
2/3
÷÷
µ M (1+ q ) çç1dt
è Mp ø
Growth of a planet
dr v K (1+ q )(M p - M )
=
dt
16p R 2 DR rp
Case of Earth:
vk = 30 km/s,
=6,
1 AU,
R = 0.5 AU,
dr
= 15 cm/year
dt
Mp = 6x1027 gr, R =
p = 5.5 gr/cm3
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
2ù
é
æ
ö
dM
v
µ M 2/3 (1+ q )= M 2/3 ê1+ ç esc ÷ ú
dt
êë è Dv ø úû
2GM
v =
µ M 2/3
R
2
esc
So for Δv<<vesc we see that we get:
dM
µ M 4/3
dt
The largest and second largest bodies separate in mass:
d æ M1 ö
1/3
1/3
lg ç
÷ µ M1 - M 2 > 0
dt è M 2 ø
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 æ M1 ö
-1/3
-1/3
lg ç
µ
M
M
<0
÷
1
2
dt è M 2 ø
Will get locally-dominant „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 < rHill)
– Flat disk of planetesimals (h << rHill)
• 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
æ Sm (t = 0) ö1/ 3
=ç
÷
è
ø
B
with
31/ 3 M*1/ 3
B=
2p 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?)
```