Planet Formation
Topic:
Formation of
gas giant planets
Lecture by: C.P. Dullemond
Two main theories
• Gravitational instability of the gas disk
• Core accretion scenario
Giant Planet Formation by
Gravitational Instability
Gravitational fragmentation of a gas disk
From earlier chapters we
know that a disk with
Q<1 will fragment into
clumps.
Image: Quinn et al.
From: http://www.psc.edu/science/quinn.html
Will a clump stay bound?
The big discussion: Can a clump cool quickly enough to stay bound?
Let‘s take a clump of polytropic gas of radius R and squeeze it:
If gravity increases faster than the opposing pressure forces:
it will continue to collapse.
1 dP
P
apressure =
»µ -r g -1R-1
r dR
rR
GM
agravity = - 2
R
Will a clump stay bound?
1 dP
P
apressure =
»µ -r g -1R-1
r dR
rR
GM
agravity = - 2
R
Approximate relation between mass and density:
4p
M»
r R3 ® r µ R-3
3
apressure µ-R-3g +2
So the gravity wins out over pressure acceleration upon contraction if:
4
g<
3
Since most astrophysical gases have γ>4/3 they will be stable
against gravitational collapse, UNLESS the gas cools (and thus
the gas deviates from the strictly polytropic EOS)!
Will a clump stay bound?
But cooling timescale must be shorter than 1 orbit, otherwise a
clump of gas will be quickly dispersed again.
Let‘s calculate the cooling time of a gravitationally unstable (Q=1)
protoplanetary disk at radial coordinate R:
dE
E
qcool = 2s SBT = »
dt t cool
4
csWK
WK æ kT ö
çç
÷÷
Q =1 =
=
p GS p GS è m m p ø
1/2
S
k 1
t cool =
2s SB (g -1) m m p T 3
1 kT
E=
S
g -1 m m p
WK æ kT ö
çç
÷÷
S=
p G è mmp ø
1/2
Will a clump stay bound?
-1/2
æ R ö
T (R) = 400K ç
÷
è 1AU ø
In outer
disk: Can
fragment
and form
Gas Giant
Exoplanets: Direct imaging
HR 8799
Credit: Marois et al (2010)
Which mass planets will form?
Since the disk muss be massive to become self-gravitating, the
odds are, that the planet will be massive too:
WK æ kT ö
çç
÷÷
S=
p G è mmp ø
1/2
Mplanet
c
M clump » Sp h = Sp
W
2
p
2
s
2
K
But many clumps can form
a planet:
M planet » Sp (0.5R)2
Typically more massive than Jupiter!
Mclump
Giant Planet Formation by
Core accretion
Core accretion main idea
• First form a rocky planet (a „core“)
• As the rocky core‘s mass increases, it will attract a
hydrogen atmosphere from the disk. A given core mass
yields a given atmosphere thickness.
• The core mass can grow when the core+atmosphere
accretes planetesimals or pebbles and/or when the
atmosphere can cool and thus shrink.
• As the core‘s mass increases further, the mass of the
atmosphere will grow faster than linear with core mass.
• Eventually become similar to the core‘s mass, so the
additional mass of the gas will attract new gas, which will
attract further gas etc: runaway gas accretion!
Attracting a hydrogen atmosphere
Smallest core mass to attract a hydrogen atmosphere:
Bondi radius is the radius from the
planet (core) at which the escape
speed equals the sound speed of the gas
2GM core
RBondi =
cs2
If RBondi < Rcore, then no atmosphere can be kept bound to the
core.
æ 3 M core ö
Rcore = ç
÷
è 4p rcore ø
1/3
æ 1 ö æ 3 ö 3
M core,atmo,min = ç ÷ ç
÷ cs
è 2G ø è 4prcore ø
3/2
1/2
Typically: 10-3...10-2 Mearth
Atmosphere structure
The equations for the atmosphere are very similar to those for
stellar structure, just with a fixed core mass added:
dP
GM r
=- 2
dr
r
kT
P=r
mmp
M(r) =
ò
r
0
ˆ rˆ
4p rˆ2 r (r)d
If the atmosphere is thick enough, and if it is continuously
bombarded with planetesimals (=heating), then to good
approximation it can be regarded as adiabatic: P = K r g
Outer boundary: R=RBondi. Boundary condition: density
and temperature equal to disk density and temperature.
Atmosphere structure
Varying the
mass of the
core
From: Bachelor thesis
Gianni Klesse
Atmosphere structure
Varying the rate
of accretion of
pebbles and/or
planetesimals
From: Bachelor thesis
Gianni Klesse
Formation of a Gas Giant Planet
Total
Gas
Solids
Original: Pollack et al. 1996;
Here: Mordasini, Alibert, Klahr
& Henning 2012
Formation of a Gas Giant Planet
Growth by accretion of
planetesimals until the
local supply runs out
(isolation mass).
Total
Gas
Solids
Original: Pollack et al. 1996;
Here: Mordasini, Alibert, Klahr
& Henning 2012
Formation of a Gas Giant Planet
Slow accretion of gas
(slow, because the
gas must radiatively
cool, before new gas
can be added).
Speed is limited by
opacities.
Total
Gas
Solids
If planet migrates, it
can sweep up more
solids, accellerating
this phase.
The added gas
increases the mass,
and thereby the size
of the feeding zone.
Hence: New solids
are accreted.
Original: Pollack et al. 1996;
Here: Mordasini, Alibert, Klahr
& Henning 2012
Formation of a Gas Giant Planet
Once Mgas > Msolid, the
core instability sets in:
accelerating accretion of
more and more gas
Total
Gas
Solids
Original: Pollack et al. 1996;
Here: Mordasini, Alibert, Klahr
& Henning 2012
Formation of a Gas Giant Planet
A hydrostatic envelope
smoothly connecting
core with disk no longer
exists. Planet envelope
detaches from the disk.
Total
Gas
Solids
Original: Pollack et al. 1996;
Here: Mordasini, Alibert, Klahr
& Henning 2012
Formation of a Gas Giant Planet
Something ends the gas
accretion phase, for
example: strong gap
opening. „Normal“ planet
evolution starts.
Total
Gas
Solids
Original: Pollack et al. 1996;
Here: Mordasini, Alibert, Klahr
& Henning 2012
Population synthesis
• Put this model into varying disks, at varying positions
(Monte Carlo)
• Allow the planet to migrate (which means, incidently,
that it can sweep up more solids than before)
•  Obtain a statistical sample of exoplanets and
compare to observed statistics.
East-Asian Models: Ida & Lin Toward a Deterministic Model of Planetary Formation
I...VI (2004...2010)
Bern Models: Mordasini, Alibert, Benz et al. Extrasolar planet population synthesis
I...IV (2009...2012)
Kornet et al. (2001...2005), Robinson et al. (2006)
Thommes et al. (2008) [multi-planet: with full N-body]
Predicted initial mass function
Growth by accretion of planetesimals
until the local supply runs out (isolation
mass). Note: effect caused by reduced
type I migration rate.
Once the faster type II migration
sets in, the core can sweep up
fresh material from further
inward
Runaway gas accretion
„Failed
cores“
Ice giants
Gas giants
Mordasini, Alibert, Benz &
Naef 2009
Lots of added complexities
Accretion of gas onto GP
is a complex 3-D
problem
Lubow, Seibert & Artymowics (1999)