Planet formation

Comments

Transcription

Planet formation
Planet formation
Star formation Lecture Series
19 December 2012
Andrea Stolte
Outline of lectures
Oct. 10th : Practical details & Introduction
Oct. 17th : Physical processes in the ISM (I): gas + dust radiative processes, solving radiative transfer
Oct. 24th : Physical processes in the ISM (II): thermal balance of the ISM, heating/cooling mechanisms
Oct. 31st : Interstellar chemistry
Nov. 7th : ISM, molecular clouds
Nov. 14th : Equilibrium configuration and collapse
Nov. 21th : Protostars
Nov. 28th : Pre-main sequence evolution
Dec. 5th : Dies Academicus
Dec. 12th : Discs
Dec. 19st : Planet formation
Jan. 9th : Formation of high-mass stars
Jan. 16th : IMF and star formation on the galactic scale
Jan. 23th : Extragalactic star formation
Jan. 30th : Visit of Effelsberg 10 am!!!
Andrea Stolte
Planet formation
Andrea Stolte
Literature on Planet Formation
Literature:
Theory of Planet Formation
General review
Mordasini, Klahr, Aliberg, Benz, Dittkrist, 2010, Proceedings “Circumstellar disks and planets”, Kiel
Building Terrestrial Planets
General review
Morbidelli, Lunine, O'Brien, Raymond, Walsh 2012, Annual Reviews of Planetary Sciences, 40 , 251
Scientific papers on the subject:
Accumulation of a Swarm of Small Planetesimals
Wetherill & Stewart 1989, ICARUS, 77, 330
Scattering of Planetesimals by a Protoplanet: Slowing Down of Runaway Growth
Ida & Makino 1993, ICARUS, 106, 210
Orbital evolution of Protoplanets embedded in a Swarm of Planetesimals
Kokubo & Ida 1995, ICARUS, 114, 247
Oligarchig Growth of Protoplanets
Kokubo & Ida 1998, ICARUS, 131, 171
Formation of Protoplanets from Planetesimals in the Solar Nebula
Kokubo & Ida 2000, ICARUS, 143, 15
Andrea Stolte
Outline of today’s lecture
Planet Formation
Clues from the solar system
Terrestrial planets & gas giants
Small bodies in the inner solar system
Small bodies in the outer solar system
Planet formation scenarios
Planet formation Timeline
Disc instabilities as the seeds for planet formation
Planetesimal growth via inelastic collisions
Runaway growth
Oligarchic growth
Final mass accumulation phase
Gas Giants: Accretion of gaseous envelopes
Terrestrial Planets: Collisional growth between protoplanets
Open questions
Andrea Stolte
Circumstellar discs around young stars
Motivation:
• Young, circumstellar discs are the places of
vigorous planet formation.
Andrea Stolte
Planet formation - Clues from the solar system
Jupiter
Saturn
Mars
Earth
Venus
Mercury
Uranus
Neptune
Ice giants
rocky
planet
zone Terrestrials
Snow line
Densities:
~ 5 g cm-2
gaseous “giant” planet zone - Jovian planets
1.3 g cm-2
0.7 g cm-2
Dwarf planets
Pluto
Eris
Ceres
Haumea
Makemake
1.3 g cm-2 1.6 g cm-2
Snow line: 2.7 AU from the Sun, temperatures are low enough that molecules appear in the
form of icy grains, and dust grains have sufficiently low temperatures to capture
ice molecules in their mantles.
The Snow line is crucial for the planet formation process, as ice-covered grains are suggested to
have transported the gaseous material for the gas giants atmospheres onto their rocky cores.
The ices have also formed icy asteroids, which likely provided the source of water on Earth.
Andrea Stolte
Planet formation - Clues from the solar system
What is a planet? Definition of the International Astronomical Union (2006):
A celestial body that is
(a) in orbit around the Sun,
(b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes
a hydrostatic equilibrium (nearly round) shape, and
(c) has cleared the neighbourhood around its orbit.
=> characterisation based on the formation process!
The end product of secondary disk accretion is a
small number of relatively large bodies (planets)
in either non-intersecting or resonant orbits,
which prevent collisions between them.
Asteroids and comets, including KBOs [Kuiper belt objects],
differ from planets in that they can collide with each other
and with planets.
Soter 2006
HST composite of Shoemaker-Levy 9
image courtesy: NASA/JPL
Andrea Stolte
Planet formation - Clues from the solar system
Image credit: Lunar & Planetary Institute
Rocky/Icy core:
0-18 MEarth
9-12 MEarth
Andrea Stolte
14 MEarth
17 MEarth
Planet formation - Clues from the solar system
Sun
Jupiter
Saturn
Uranus
Neptune
Rocky/Icy core:
0-18 MEarth
9-12 MEarth
12 MEarth
14 MEarth
Total mass
318 MEarth
95 MEarth
14 MEarth
17 MEarth
10-15 %
10-15%
80-85 %
80-85 %
H + He mass
Higher elements
71.0 + 27.1 %
71 + 24 %
1.9 %
5%
The increase in metallicity
with distance from the Sun
indicates that the gas giants
have not fragmented out of
the “Urnebel” or the
“Minimum Solar Nebula”.
The material had to be enriched
in heavier elements before the
planets were formed.
Andrea Stolte
he
e
s
rea
c
n
i
avy
me
e
l
e
nts
Clues from small bodies in the inner solar system
Asteroid Belt: A successful model of solar system formation also has to explain the asteroid belt,
and how small bodies could become “trapped” in this zone. The origin of asteroids from the early
solar nebula, and their composition, provide clues on planet system formation.
Andrea Stolte
Clues from small bodies in the outer solar system
Kuiper Belt: At 30 - 55 AU, Kuiper Belt objects including the dwarf planets are found on orbits
ranging from almost circular to highly eccentric. Beyond the Kuiper Belt, the scattered disc
can reach -- like the Oort cloud -- orbital distances of several 100 AU from the Sun.
Kuiper Belt objects might be related to the time of heavy bombardment in the late phase
of Solar System formation.
Andrea Stolte
Clues from small bodies in the outer solar system
Kuiper Belt: The Kuiper belt contains a large
number of hundreds of thousands of “planetesimals”,
also called “trans-neptunian objects” due to their
location outside the major planetary orbits.
These objects are likely the most pristine remnants
of solar-system formation.
Oort Cloud: The hypothetical Oort cloud is a
spherical cloud of icy objects out to 50,000 AU
or 1 lightyear, believed to be the source of longperiod comets. It is believed to be composed of
comets that were ejected from the inner Solar
system by gravitational interactions with the
outer planets.
text & images: Wikipedia.org
Andrea Stolte
Angular momentum distribution in the solar system
The angular momentum L of an object of mass m
moving in a circle of radius r, with orbital period
P = 2pi r / v is given by
The rotational angular momentum of a solid homogeneous
sphere of mass m and radius r with rotational rate p is
given by
L = 4π m r2 / 5 p
L = mvr = 2π m r2 / p
Rotational Angular Momentum
Orbital Angular Momentum
Body
orb radius period
(km)
(days)
mass
(kg)
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
58.e6
108.e6
150.e6
228.e6
778.e6
1429.e6
2871.e6
4504.e6
5914.e6
3.30e23
4.87e24
5.97e24
6.42e23
1.90e27
5.68e26
8.68e25
1.02e26
1.27e22
87.97
224.70
365.26
686.98
4332.71
10759.50
30685.00
60190.00
90800
all planets (including Pluto)
L
(kg m2/s)
9.1e38
1.8e40
2.7e40
3.5e39
1.9e43
7.8e42
1.7e42
2.5e42
3.6e38
3.1e43
Body
radius rota period
(km)
(days)
mass
(kg)
Sun
Earth
Jupiter
695000
6378
71492
1.99e30
5.97e24
1.90e27
24.6
0.99
0.41
L
(kg m2/s)
1.1e42
7.1e33
6.9e38
The total angular momentum in the
Solar System is -- by far -- dominated
by the planetary orbits.
(and not by planetary rotation!)
The angular momentum of the Sun
is less than 4 % of the total orbital
angular momentum in the Solar System.
S. White 2001
http://www.zipcon.net/~swhite/docs/astronomy/Angular_Momentum.html
Andrea Stolte
Planet formation - Clues from the solar system
Summary
Early evolution of the solar system:
* inner planets = rocky
* outer planets = gaseous/icy
* gaseous planets are enriched in heavy elements compared to Sun
Mass accumulation:
* 99.86 % in Sun
* 0.14 % in planets and small bodies
of these: 92 % in Jupiter & Saturn
> 99 % in 8 major planets
Angular momentum distribution:
* in contrast to mass, almost all angular momentum is distributed to the planets,
predominantly in their orbital motion
Small objects in orbit around major planets:
* icy moons with large inclinations have to be attracted during the latest stage
of planet formation
Andrea Stolte
Outline of today’s lecture
Planet Formation
Clues from the solar system
Terrestrial planets & gas giants
Small bodies in the inner solar system
Small bodies in the outer solar system
Planet formation scenarios
Planet formation Timeline
Disc instabilities as the seeds for planet formation
Planetesimal growth via inelastic collisions
Runaway growth
Oligarchic growth
Final mass accumulation phase
Gas Giants: Accretion of gaseous envelopes
Terrestrial Planets: Collisional growth between protoplanets
Open questions
Andrea Stolte
Planet formation Scenarios
“Gas instability model”
Direct fragmentation from the protoplanetary disc
* disc becomes gravitationally unstable & forms massive clumps
problem 1: gas giants are enriched in metals
* gas giants do not have the same composition as the Sun,
hence the material from which they formed must have been processed
problem 2: Neptune & Uranus are mostly made of heavy elements
* if they were formed directly from the fragmenting disc, this is not expected
problem 3: Rocky planets are made almost entirely from heavy elements
* clumps forming in the disc cannot explain the origin of rocky planets
Andrea Stolte
Planet formation Scenarios
“Core nucleated accretion model”
Rocky, Earth-like cores form first for all planets, including gas giants
* after a rocky core of several Earth masses is formed,
planetesimals are accreted from the disc
=> planets are clearing their orbits
* rocky planets are nearer to the star than the Snow line
=> most disc material is in gaseous form and will not be accreted
* gas giants are outside of the Snow line in the Ice Zone
=> gaseous material is frozen out onto grain mantles
=> accreted grains carry icy gas onto the protoplanet,
from which an atmosphere can form
Andrea Stolte
The “rocky core” planet formation model
“Core nucleated accretion model”
Circumstellar disc
Timeline:
few 100 1000 yrs
1. Dust settles in the disc midplane
planetesimals with a ≲ 1 meter-size form through co-agulation
2. Planetesimals grow via pair-wise inelastic collisions
3. Gravity takes over when vescape > vthermal → accretion!
* accretion is more efficient for the most massive planetesimals
* envelope has to radiate away accretion energy & will contract
=> contraction allows more material to become gravitationally bound
=> early phase: run-away growth
* slow-down after large planetesimals have been cleared from the orbit
=> later phases: oligarchic growth
4. Final mass accumulation:
* Gas giants: accretion of gaseous envelope
* Terrestrial planets: growth from large-body collisions
Andrea Stolte
The “rocky core” planet formation model
Phase 1: Larger dust grains settle in the disc midplane,
while small grains move with the gas (gas-dust coupling)
=> surface density of the compacting central plane increases until the disc becomes
instable to gravitational fragmentation
Williams & Cieza 2011
Andrea Stolte
When does the disc become unstable?
Phase 1: The fragmentation phase -- “seeds” for planetesimal growth
Gravitational instability in a rotating, thin gaseous disc
* protostars: we discussed the Jeans instability for a collapsing sphere
Toomre 1964: Estimation of the stability of an infinitely thin, uniformly rotating sheet with
constant surface density.
Step 1: consider a small circular patch with radius ΔR
Mpatch = π (ΔR)2 Σ
Σ
Σ = surface density of the disc/sheet
π (ΔR)2 = area of circular patch
ΔR
we assume that the gravity in the thin sheet is the only force
reduce the area of the patch by a tiny perturbation:
=> pressure perturbation:
p1(1-α) = FG / A = p0
Δp = p1 - p0
⇔
A’ = A(1-α)
α << 1
where: p0 = F G / A
p1 = FG / A(1-α)
p1 - p0 = α p1
the perturbation is very small, hence we assume for the thermal pressure change:
Δp = α p1 ≈ α p0 = α cs2 Σ
c s : sound speed
Andrea Stolte
When does the disc become unstable?
Phase 1: The fragmentation phase -- “seeds” for planetesimal growth
Σ
Step 2: determine the force balance in the circular test patch
* the extra pressure leads to an outward force
Fp = - ∇p / Σ
for a sheet with constant surface density Σ
FG
Fp
in cylindrical coordinates:
∇p = 1/ΔR d(ΔR p) / d(ΔR) + 1/ΔR d (pφ) / dφ +
d(pz) / dz
when shrinking the circular area of the patch uniformly, we have exerted a pressure force in
the plane, and in the R-direction: the φ and z components do not experience a pressure force
Fp = ‒ 1/(ΔR Σ) d(ΔR Δp) / d(ΔR)
with
(ΔR Δp) = ΔR α c s2 Σ
Fp = ‒ 1/(ΔR Σ) α cs2 Σ = - α cs2 / ΔR
Stability criterion: the amount of the outward pressure force has to be balanced by the inward
gravitational force.
FG ≈ α GMpatch / (ΔR)2 = α G π Σ
where we have used: Σ = M / (π ΔR 2 )
Andrea Stolte
When does the disc become unstable?
Phase 1: The fragmentation phase -- “seeds” for planetesimal growth
Step 2: determine the force balance in the circular test patch
the sheet
- is stable if the outward force exceeds gravity: |F p| > |FG|
- experiences gravitational collapse if
|FG| > |Fp|
Stability criterion:
α cs2 / ΔR > α G π Σ
⇔
ΔR < cs2 / (π G Σ) ≡ ΔRlower
implies a gravitationally stable disc
So far, we have not used any rotation, just a sheet geometry.
Compare this to the Jeans instability (for a uniform density sphere):
Jeans length:
λ2 > λJ2 = π cs2 / (G ρ0)
causes gravitational collapse of the sphere!
Andrea Stolte
When does the disc become unstable?
Phase 1: The fragmentation phase -- “seeds” for planetesimal growth
Step 3: a disc is rotating, hence there is also a centrifugal force:
* the spin angular momentum (per unit mass) in the patch has to be conserved
S = J / Mpatch = Ω ΔR2
Ω: angular velocity
* above: increase in inward gravitational force
* analoguously: compression causes an increase in the outward centrifugal force
|Fc| = Ω2 ΔR = S2 / ΔR3
where S2 has to be conserve
S2 = ΔR3 Fc = (ΔR / π ) A Fc = (ΔR / π) A (1-α) (Fc + ΔFc)
with A = π ΔR2 , A’ = (1-α)A
⇔ ΔFc = α Fc = α Ω2 ΔR
* this rotating sheet is stable if the outwards centrifugal force
|Fc| > |FG|
α Ω2 ΔR > α π G Σ
⇔ ΔR > G π Σ / Ω2 ≡ ΔRupper
implies a centrifugally stable disc
Andrea Stolte
The “rocky core” planet formation model
Phase 1: The fragmentation phase -- “seeds” for planetesimal growth
Step 4: combine internal pressure and outwards centrifugal forces
* now we have two size regimes, where the disc is stable:
- small regions are stable through internal pressure balance
ΔR < cs2 / (G π Σ) ≡ ΔRlower
- large regions are stable through centrifugal forces
ΔR > G π Σ / Ω2 ≡ ΔRupper
Case 1: ΔRupper < ΔRlower : the disc is always stable!
cs Ω / (π G Σ) > 1
Toomre stability criterion
Case 2: ΔRupper > ΔRlower : there is a size regime, where gravitational instabilities
are unavoidable:
cs Ω / (π G Σ) < 1 the disc will produce small-scale fragments
During this early phase, small planetesimals (a < 1 m) are fragmented out of the disc.
The planetesimals have similar, but not identical masses, which leads to a mass
spectrum of planetesimals. Some planetesimals are more massive than the mean.
Andrea Stolte
The “rocky core” planet formation model
Phase 1: Co-agulation & gravoturbulent planetesimal growth
Here, we have neglected the effects of turbulence, which change the pressure support.
Today, the most convincing model for solar system formation is the “gravoturbulent
planetesimal formation” scenario.
animation courtesy: Chris Butler
Andrea Stolte
The “rocky core” planet formation model
“Core nucleated accretion model”
Circumstellar disc
Timeline:
few 100 1000 yrs
1. Dust settles in the disc midplane
planetesimals with a ≲ 1 meter-size form through co-agulation
2. Planetesimals grow via pair-wise inelastic collisions
few 10^3 10^5 yrs
3. Gravity takes over when vescape > vthermal → accretion!
* accretion is more efficient for the most massive planetesimals
* envelope has to radiate away accretion energy & will contract
=> contraction allows more material to become gravitationally bound
=> early phase: run-away growth
* slow-down after large planetesimals have been cleared from the orbit
=> later phases: oligarchic growth
4. Final mass accumulation:
* Gas giants: accretion of gaseous envelope
* Terrestrial planets: growth from large-body collisions
Andrea Stolte
?
The “rocky core” planet formation model
Phase 2: “Orderly growth”
* Planetesimals grow via pair-wise inelastic collisions
* determined by the velocity dispersion of planetesimals
* equiparition in the mass spectrum leads to velocity segregation/stratification
=> during interactions,
- higher-mass planetesimals will loose energy to lower-mass planetesimals
=> they will sink & condense in the midplane even more
- lower-mass planetesimals are partially ejected to larger radii
Planetesimals continue to grow via inelastic collisions to km-size bodies.
This phase sets the stage to allow for collisions between more massive particles.
The low relative velocities of massive planetesimals also facilitate accretion of
lower-mass planetesimals (low relative velocity = high interaction cross section).
Andrea Stolte
The “rocky core” planet formation model
“Core nucleated accretion model”
Circumstellar disc
Timeline:
1. Dust settles in the disc midplane
planetesimals with a ≲ 1 meter-size form through co-agulation
2. Planetesimals grow via pair-wise inelastic collisions
3. Gravity takes over when vescape > vthermal
→
few 100 1000 yrs
few 10^3 10^5 yrs
accretion!
* accretion is more efficient for the most massive planetesimals
* envelope has to radiate away accretion energy & will contract
=> contraction allows more material to become gravitationally bound
=> early phase: run-away growth
* slow-down after large planetesimals have been cleared from the orbit
=> later phases: oligarchic growth
4. Final mass accumulation:
* Gas giants: accretion of gaseous envelope
* Terrestrial planets: growth from large-body collisions
Andrea Stolte
few 10^5 10^6 yrs
The runaway growth phase
Phase 3 a: “Run-away growth”
This step is the crucial step to form a few massive planetary cores.
Dynamical interactions between more & less massive planetesimals:
* small planetesimals gain kinetic energy
* large planetesimals loose kinetic energy
* the largest particle has the smallest velocity, hence relative velocity to all others
=> largest gravitational cross section with m bigger & v smaller
* gravitational focussing increases the effective cross section even more
Andrea Stolte
The runaway growth phase
Phase 3 a: Runaway growth
of
km planetesimals → 1000 km protoplanets
Assumption: a large protoplanet (or protoplanets) is immersed in a Sea of small planetesimals
represented by the average planetesimal mass <m> and eccentricities e, inclinations i
Safronov described the growth rate as a rate equation:
dMpp = πRpp2 Ω Σpl FG
dt
with
geometric cross section
“Effective collisional cross section”:
Rpp = radius of protoplanet
Ω = Kepler frequency of protoplanet
Ω2 = GM* / a3
a = semi-major axis
Σpl = surface mass density of planetesimals
FG = gravitational focussing factor
πRpp2 FG
Gravitational focussing FG determines whether growth is
- runaway
dMpp / dt >> dMpl / dt
- oligarchic
dMpp / dt ~ same for all protoplanets (independent of mass)
- “orderly”
dMpp / dt ~ dMpl / dt uniform for all particles
Andrea Stolte
The runaway growth phase
Phase 3 a: Runaway growth
of
km planetesimals → 1000 km protoplanets
vpl
Simplest approximation of gravitational focussing:
- the faster a planetesimal moves past the protoplanet, the smaller the interaction
time, the higher its chances to escape.
FG
⇒ FG must scale with the relative velocities of both particles:
FG = 1 + vesc2 / σpl2
with σpl = velocity dispersion of planetesimals
vpl
FG
Safronov equation in this case:
dMpp = πRpp2 Ω Σpl + πRpp2 Ω Σpl vesc2
dt
σpl2
vesc2 = 2GMpp from the protoplanet
Rpp
dMpp = geom. term + 4 π2 Ω Σpl Rpp 2Gρpp Rpp3
dt
3
σ pl2
Assumption: Mpp = ρpp 4/3 π Rpp3
~ Rpp4
highly non-linear!
with ~ constant mass density
Larger (and more massive) planetesimals grow disproportaionally faster than smaller particles.
⇒ Runaway growth!
Andrea Stolte
The runaway growth phase
Phase 3 a: Runaway growth
of
km planetesimals → 1000 km protoplanets
Relative growth rate per particle mass:
1 dMpp ~ Mpp1/3 Σpl em-2
Mpp dt
early phase: mean eccentricity em and surface density Σpl
are determined by planetesimals, independent of Mpp
At some point, the protoplanet mass is large enough to influence the planetesimals:
Mpp > 50 <mpl>
=>
σpl ~ Mpp1/3
and
σpl = (em2 + im2)1/2 vcirc ~ em
The particle eccentricity increases rapidly with increasing mass: em2 ~ Mpp2/3
Relative growth rate per particle mass now scales inversely with protoplanet mass:
1 dMpp ~ Mpp1/3 Σpl em-2
Mpp dt
~
Mpp-1/3
As soon as the protoplanets are massive enough to influence the sea of planetesimals,
the runaway growth phase ends.
At this point, the more massive protoplanets grow slower (because they pump up the
eccentricities of their neighbouring planetesimals more), and the lower-mass protoplanets
catch up.
Andrea Stolte
The runaway growth phase - Summary
Phase 3 a: Runaway growth
of
km planetesimals → 1000 km protoplanets
a) Runaway growth:
- larger planetesimals grow much more rapidly than smaller planetesimals
⇒ detachment of the largest bodies from the Sea of planetesimals
Few massive bodies on exclusive orbits win the accretion/collision race over the large
amount of small bodies.
What stops runaway growth?
- the more massive a protoplanet, the more effective it will scatter planetesimals
- as Mpp increases, σpl increases (along with orbital eccentricities and inclinations)
and planetesimals are less likely to be captured
⇒ the higher the protoplanet’s mass, the sooner the growth rate slows down
⇒ self-regulated particle accretion: the most massive protoplanets slow down first
⇒ Consequence: protoplanets will grow in parallel with similar growth rates
Andrea Stolte
The “rocky core” planet formation model
“Core nucleated accretion model”
Circumstellar disc
Timeline:
1. Dust settles in the disc midplane
planetesimals with a ≲ 1 meter-size form through co-agulation
2. Planetesimals grow via pair-wise inelastic collisions
3. Gravity takes over when vescape > vthermal
→
few 100 1000 yrs
few 10^3 10^5 yrs
accretion!
* accretion is more efficient for the most massive planetesimals
* envelope has to radiate away accretion energy & will contract
=> contraction allows more material to become gravitationally bound few 10^5 10^6 yrs
=> early phase: run-away growth
* slow-down after large planetesimals have been cleared from the orbit
=> later phases: oligarchic growth
4. Final mass accumulation:
* Gas giants: accretion of gaseous envelope
* Terrestrial planets: growth from large-body collisions
Andrea Stolte
few x 10^6
yrs
The oligarchic growth phase
Phase 3 b: “Oligarchic growth”
* Here, a protoplanetary system is already in place.
* each of the “oligarchs” grows within its own orbital zone of influence
A protoplanet orbiting its host star comprises a rotating 2-body problem:
- in the co-rotating frame, the conserved quantity is the Jacobi integral
EJ = 1/2 v2 + Φ(x) + 1/2 |Ω x X|2
rotation term: centrifugal & Coriolis forces
= 1/2 v2 + Φeff(x)
Physical solutions are given if
EJ - Φeff(x) = 1/2 v2 > 0
⇔
EJ > Φeff(x)
Equi-potential surfaces display the Lagrange points:
(= stationary points in the co-rotating frame)
Only when a particle enters the planets sphere of
gravitational influence between L1 and L2, can the
particle be captured by the protoplanet.
Andrea Stolte
The oligarchic growth phase
Phase 3 b: “Oligarchic growth”
The sphere of influence of the protoplanet can be estimated from L1,
which is a sattle point in the potential:
d Φeff [ x = xm‒rL1] = 0
dx
to first order, with assumptions m << M, rL1 << semi-major axis of protoplanetary orbit
m
r L1 ∼
M (3+ m/ M )
(
)
1/ 3
a
where m: mass of protoplanet
M: mass of star
a: semi-major axis of protoplanet’s orbit
here: called either the Jacobi limit, or the “Hill radius”
M pp
rH∼
3M star
(
1 /3
)
a
with Mpp << Mstar with Mpp: mass of protoplanet
Dynamical simulations show that the sphere of influence of a protoplanet appears
to be larger than rH, and gravitational interactions take place for all planetesimals
within
rpl ≲ 5 rH
and a particle can be captured if additionally, EJ > 0
Andrea Stolte
The oligarchic growth phase
Phase 3 b: “Oligarchic growth”
Dimensional argument for the Hill radius:
The radius of gravitational influence of a protoplanet on a planetesimal can be considered
as the region where the gravity from the protoplanet exceeds the gravity from the star.
Then, at the Hill radius:
Gravitational influence of star on planetesimal = influence of protoplanet on planetesimal:
Ωpp = Ωstar
M pp
rH∼
3M star
(
⇔
√(GMpp/RH3 ) = √(G Mstar / a3 )
1 /3
)
a
with a: semi-major axis of the protoplanet
Note: The Hill radius is not a constant with time.
As the protoplanet grows, the feeding zone expands, and eventually neighbouring
protoplanets can be captured in the most massive planet’s feeding/ejection zone.
Andrea Stolte
The oligarchic growth phase
Phase 3 b: “Oligarchic growth”
Kokudo & Ida 2000
A planetary system emerges with protoplanets at comparable distances from each other .
Andrea Stolte
The oligarchic growth phase
Phase 3 b: “Oligarchic growth”
- comparably sized “oligarchs” grow on un-disturbing orbits
- each oligarch accretes from its gravitational zone of influence:
rH = ( Mpp / 3M* )1/3 a
Feeding zone:
with
Mpl = 5 x π 2arH Σpl
rH = Hill radius
a = protoplanet’s semi-major axis
with rH << a
- each oligarch clears planetesimals from ~ 5 RH
- when the disc is depleted in this radial region, the oligarch reaches its isolation mass
outer planet
a
Miso = ( 5 x 2π a2 Σpl )3/2
( 3M* )1/2
inner planet
10RH
10RH
Feeding area increases
with increasing distance from the star.
Isolation mass >> at large semi-major axis, << at small semi-major axis.
Andrea Stolte
The oligarchic growth phase - Summary
Phase 3 b: “Oligarchic growth”
What stops oligarchic growth?
Isolation mass larger at large semi-major axis, smaller at small semi-major axis:
inner planetary system: many oligarchs with Mpp ~ 0.01 - 0.1 MEarth
outer planetary system: protoplanets reach Mpp ~ 1 - 10 MEarth
Solar system:
Boundary between “inner” and “outer” system:
“Snowline” or “Iceline”
where water, ammonia, methane, and other hydrogen compounds can freeze out
as solid ices, and hence can stick on grain surfaces
Snowline in the Solar System: a = 2.7 AU
where T~ 150 K
which is in the middle of the asteroid belt.
Rapid growth to large protoplanet masses occurs outside the snowline!
With masses of 10 MEarth, rocky protoplanets can accrete gas/grains from the disc.
Andrea Stolte
The “rocky core” planet formation model
“Core nucleated accretion model”
Circumstellar disc
Timeline:
few 100 1000 yrs
1. Dust settles in the disc midplane
planetesimals with a ≲ 1 meter-size form through co-agulation
few 10^3 10^5 yrs
2. Planetesimals grow via pair-wise inelastic collisions
3. Gravity takes over when vescape > vthermal → accretion!
* accretion is more efficient for the most massive planetesimals
* envelope has to radiate away accretion energy & will contract
=> contraction allows more material to become gravitationally bound
=> early phase: run-away growth
* slow-down after large planetesimals have been cleared from the orbit
=> later phases: oligarchic growth
few 10^5 10^6 yrs
few x 10^6
yrs
4. Final mass accumulation:
* Gas giants: accretion of gaseous envelope
* Terrestrial planets: growth from large-body collisions
...until the disc is dissipated.
Andrea Stolte
< 10^7 yrs
Planetary System
Gas giants: accretion of gaseous atmospheres
Phase 4 a: Gas Giants: accretion of gaseous envelopes
After the rocky core is formed, accretion occurs in 2 stages:
I. a gaseous envelope is accreted from the circumstellar disc
⇒ core continues to grow slowly as matter is accreted
(ices can be coupled to dust grains)
⇒ core contraction leads to further instreaming of gas & accretion
II. Runaway accretion
Menv ≈ Mcore
⇒ radiative losses from envelope can not be compensated by accretion luminosity
⇒ envelope contracts & radiative losses increase further
Andrea Stolte
Gas giants: accretion of gaseous atmospheres
Phase 4 a: Gas Giants: accretion of gaseous envelopes
II. Runaway accretion
Menv ≈ Mcore
⇒ radiative losses from envelope can not be compensated by accretion luminosity
⇒ envelope contracts & radiative losses increase further
Reminder:
Accretion is driven by the Kelvin-Helmholtz timescale, which decreases rapidly:
tKH = ΔEg
L pp
where
tKH = 3 G Mpp2
10 Rpp Lpp
ΔEg = Einitial - Efinal
~ Rpp5
to first order: Mpp = ρpp 4/3 π Rpp3
As the protoplanet contracts, the timescale for further accretion becomes very short.
⇒ Runaway accretion
(not to be confused with the Runaway growth phase)
Andrea Stolte
Gas giants: accretion of gaseous atmospheres
Phase 4 a: Gas Giants: accretion of gaseous envelopes
II. What stops runaway accretion?
- accretion rate is limited by the external disc reservoir:
⇒ all matter flowing towards the protoplanet is accreted instantaneously
- the radius of the gas giant shrinks below its Hill sphere
⇒ gas giant detaches from the disc
The envelope of the gas giant has to collapse (slowly, hydrostatically),
and the giant shrinks to its final Rplanet with accumulated Mplanet.
The timescale of this last phase of gas giant formation is ~ 10 5 years.
Andrea Stolte
Gas giants: accretion of gaseous atmospheres
Phase 4 a: Gas Giants: accretion of gaseous envelopes
Runaway accretion
Oligarchic growth
Runaway growth
Runaway growth
Runaway accretion
Mordasini et al. 2010
Andrea Stolte
Terrestrial planets: interactions between protoplanets
Phase 4 b: Terrestrial planets: collisions & orbit mixing
After oligarchic growth, planetesimals are on the order of the isolation mass at 0-few AU.
Isolation mass smaller at small semi-major axis:
inner planetary system: many oligarchs with Mpp ~ 0.01 - 0.1 MEarth
Dense gas disc damps eccentricities & prohibits interactions between rocky cores.
Depletion of the gas disc (either from stellar radiation or grain growth):
interactions between oligarchs increase eccentricities.
A planetesimal entering the feeding zone of a more massive planetesimal
has two likely fates:
1. it is ejected from the system (eccentricity pumping)
2. it is accreted onto the more massive planetesimal (i.e. the protoplanet)
Collisions occur until a stable orbit configuration is reached for all planets:
Δa ≳ 5 RH
for all remaining planets & their respective Hill radii
The final configuration has to be stable over Gyr timescales!
Andrea Stolte
Terrestrial planets: interactions between protoplanets
Phase 4 b: Terrestrial planets: collisions & orbit mixing
Earth isotope dating suggests the collision episode to last for
Δt ≈ 50-100 Myr
water-bearing
planetesimals
Resonances with Jupiter
N-body simulation of terrestrial growth:
Ingredients:
Jupiter
- 100 oligarchs with 0.01 - 0.1 MEarth
- background planetesimals
- Jupiter and Saturn
After 200 Myr:
- 4 terrestrial planets have formed
- most left planetesimals are ejected
- terrestrial planets are in
Non-interacting stable orbits
Raymond et al (2009)
Andrea Stolte
Terrestrial planets: interactions between protoplanets
Phase 4 b: Terrestrial planets: collisions & orbit mixing
Raymond et al (2006)
Andrea Stolte
Disc depletion & the end of the accretion phase
Planet formation & final mass accretion phase ends when:
* planet has cleared a gap, and Ṁacc is limited by planetesimal transport through
the disc until it eventually stops (no more particle transport through gap)
Mars-mass protoplanet colliding with Moon-mass protoplanet (1:10 mass ratio collision)
Animation courtesy: Craig Agnor 2005
Andrea Stolte
Outline of today’s lecture
Planet Formation
Clues from the solar system
Terrestrial planets & gas giants
Small bodies in the inner solar system
Small bodies in the outer solar system
Planet formation Timeline
Disc instabilities as the seeds for planet formation
Planetesimal growth via inelastic collisions
Runaway growth
Oligarchic growth
Final mass accumulation phase
Gas Giants: Accretion of gaseous envelopes
Terrestrial Planets: Collisional growth between protoplanets
Open questions
Andrea Stolte
Open questions not answered by current theories
of planet formation
Problems not easily explained by present planet formation theories:
* most gas giants are found with < 1 AU of their host stars
* only planets very near < 0.1 AU their host star are circularised
* median eccentricity of extra-solar planets: e = 0.25
=> not comparable to solar system
Andrea Stolte
Solar system
Migration influences terrestrial planet formation
Migration of Jupiter-like planets might explain
* the large eccentricities observed in extrasolar systems
* the small semi-major axis at which massive planets are observed
Raymond et al (2006)
Andrea Stolte
Summary
There are 4 important stages of planet formation:
Phase 1: disc instability & fragmention
Phase 2: orderly growth of planetesimals by inelastic collitions
Phase 3: runaway growth - the most massive protoplanets win!
Phase 4: oligarchic growth - and they take over the “protoplanetary system”
The final mass accumulation shapes the type of planet:
Case 1: Gas giants form outside the Snowline by accreting massive envelopes
Case 2: Terrestrial planets form inside the Snowline by colliding with other protoplanets
The rocky-core planet formation model can explain:
* the existence of a few dominating massive planets
* the formation of the asteroid belt from Jupiter resonance orbits
* the formation of the Kuiper Belt & Oort cloud
* the period of “Late Heavy Bombardment” might have led to the formation of the
Earth-Moon System
Merry Christmas & a happy New Year 2013!!!
Andrea Stolte