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Beyond Oligarchy:
The Final Chapter of Planet Formation
Yoram Lithwick
Peter Goldreich
Re’em Sari
Observations
30 Hill
Radii
0.72 AU
50 Hill
Radii
1 AU
1.5 AU
15 Hill
Radii
19 AU
30 AU
Questions
When does oligarchy end, and what happens
next?
What sets the spacing between the planets?
Why are their orbits circular and coplanar?
How long did it take them to form?
Oligarchy
Big bodies:
e
have comparable masses
are spaced by a few Hill radii
(From Kokubo & Ida ‘98)
Accretion
During Oligarchy
Accretion Rate:
1 dM
∼
M dt
!v
esc
u
"2 ! σΩ "
ρR
｝ ｝
｝
100 Myr for Earth
gravitational geometric
Gyr for Jupiter’s core
focusing
accretion ⇒ 1
1000 Gyr for Neptune
factor
rate
Heating and Cooling
During Oligarchy
cooling
cooling
by inelastic collisions by dynamical friction
heating
by viscous stirring
Heating
!
ΣΩ v
On small bodies: ρR
esc
u
= Cooling
"4
σΩ
∼
ρs
⇒ u ∼ vesc
!
Σs
σR
Very small s ⇒ very small u ⇒ very fast accretion
"1/4
Heating and Cooling
During Oligarchy
cooling
cooling
by inelastic collisions by dynamical friction
heating
by viscous stirring
Heating
On big bodies:
ΣΩ ! v
esc
ρR
v
"4
= Cooling
∼
σΩ ! v
esc
ρR
u
"4
! "1/4
Σ
⇒v∼u
σ
Heating and Cooling
During Oligarchy
cooling
cooling
by inelastic collisions by dynamical friction
heating
by viscous stirring
Heating
On big bodies:
ΣΩ ! v
esc
ρR
v
"4
= Cooling
σΩ ! v
esc
∼
X
u
> ρR Xv
"4
! "1/4
Σ
⇒
u away
⇒ vv ∼
runs
σ
Beyond Oligarchy: Unrestrained Heating
When Σ > σ, heating of big bodies exceeds cooling
⇒ heating runs away
Beyond Oligarchy: Unrestrained Heating
When Σ > σ, heating of big bodies exceeds cooling
⇒ heating runs away
For Uranus and Neptune
vescape > vorbital , so
Some big bodies
Oort
cloud
or
beyond
! ejected"to
2
M"
in time 1 Ω−1
∼ 1 Gyr
10
MNeptune
Remaining big bodies cooled by small bodies
Continue until no large-scale chaos ⇒ no heating
Remaining small bodies probably not accreted
For Venus, Earth, and Mars
vescape < vorbital , so
Ejection is impossible
Remaining
! big
"−1bodies collide and coalesce
in time σΩ
∼ 100 Myr
ρR
Continue until no large-scale chaos. Residual small
bodies can cool planets.
Answers
When does oligarchy end, and what happens next?
When Σ=σ, heating runs away
Answers
What sets the spacing between planets?
Stability ⇒ no large-scale chaos
Answers
Why are their orbits circular and coplanar?
Small bodies damp velocities
Answers
How long did it take them to form?
In inner Solar System: ∼ 100 Myr
In outer Solar System:
but to eject neighbours:
100 Myr,
1 Gyr
The End