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PLANETARY MIGRATION
in protoplanetary discs
and
OUTER SOLAR SYSTEM
ARCHITECTURE
Aurélien CRIDA 1,
A. MORBIDELLI, K. TSIGANIS
H. LEVISON, R. GOMES
( 1 Institüt für Astronomie und Astrophysik, Universität Tübingen, GERMANY)
Introduction :
Proto-planetary disks :
Planets form in it.
Size : several 100s A.U.
Life time : ~ 3-5 106 years.
Aspect Ratio : H/r ~ 0.05 + flaring
Viscosity : n = a cs H (10-3<a<10-1)
Mass : < M* / 10
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Planet-disk interactions :
Wake formation :
Angular Momentum exchanges :
A planet on a fixed circular orbit launches
a spiral wake by gravitational perturbation
Positive torque exerted by the
planet on the outer disk.
Negative torque on the inner disk.
Net result for the planet :
differential Lindblad torque
(negative), type I migration.
(Ward, 1986, 1997 )
Migration rate ~ planet mass.
Migration time scale :
≈ 105 years for 10 M at 5 AU.
≈ disk life time / 100.
( Animation by Frédéric Masset (C.E.A),
using FARGO (Masset 2000a,b) )
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Planet-disk interactions :
The planet repels the inner disk
inward, and the outer disk
outward.
If the planet is massive enough :
Gap opening.
Gap formation :
Condition 1 : Planetary
torque < viscous torque :
q = Mp/M* > 40n / rp²Ωp
Condition 2 : Angular
momentum not taken away
by the wave : RHill > H
Unified criterion :
3/4 H/RH + 50n / qrp²Ωp < 1
( Frédéric Masset again, corotating frame )
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Planet-disk interactions :
Type II migration :
After gap opening :
The planet is no longer
inside the gas disk, it
cannot drift with respect
to the disk.
Locked in the gap, the planet follows the disk viscous evolution
(accretion onto the central star and spreading, Lynden-Bell & Pringle, 1974)
Migration rate α viscosity n ; Migration time scale <~ disk life time.
Inner disk
Outer disk
star
The disk global evolution is the key.
It drives the giant planet close to the central star.
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Questions and Summary
Type II migration explains
the hot Jupiters,
but not Jupiter !
In the Solar System, no giant planet passed
through the Main Asteroid Belt or the Kuiper Belt.
Jupiter, Saturn, Uranus, Neptune didn’t migrate significantly.
How to explain this ?
SUMMARY :
1) Migration of a pair of giant planets (+ 2 ice giants)
2) The « Nice model »
3) The « Nice model », revisited in agreement with 1
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1) Migration of Jupiter & Saturn
Two planets in their own gaps migrate in parallel.
Inner disk
Outer disk
star
Two planets in a same gap approach each other → MMR.
Inner disk
Outer disk
star
If the planets have different masses, the pair of planets is not in
equilibrium. → gas passes the gap ; decoupling from disk evolution.
Migration of a pair of planets ≠ migration of one planet.
Lighter outer planet → outward migration.
(Masset & Snellgrove, 2001, MNRAS)
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1) Migration of Jupiter & Saturn
Dependence on
viscosity n :
Start with aSaturn = 1.4 aJupiter,
fixed planets (for gap opening).
At t ≈ 500, release the planets.
n<10-5 (black, red) : They approach,
lock in MMR at t ≈ 1000,
and then migrate together.
Low n :
outward migration rate increases with n :
- Jupiter feels a stronger positive torque (αn),
- corotation torque increases.
n = 2.10-5 : Saturn migrates outward (strong
corotation torque), then parallel migration in
separated gaps : Mechanism broken.
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1) Migration of Jupiter & Saturn
Dependence on
aspect ratio H/r :
Start with aSaturn = 1.4 aJupiter,
n = 10 -5.5,
fixed planets (for gap opening).
At t ≈ 500, release the planets.
They approach,
lock in MMR at t ≈ 1000,
and then migrate together.
H/r = 0.05 :
stationary solution.
The smaller H/r,
the deeper Saturn’s gap,
the more Jupiter pushed
outward.
(Morbidelli & Crida, 2007)
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1) Migration of Jupiter & Saturn
Dependence on
the masses :
Jupiter and Saturn :
stationary solution.
Planets of same mass :
slowed down inward migration.
More massive outer planet :
accelerated migration.
3 times more massive planets :
perturbations, scattering, then 2:1
MMR and migration stopped.
(Morbidelli & Crida, 2007)
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1) Migration of Jupiter, Saturn, Uranus, Neptune
Add Uranus below Saturn orbit :
Uranus migrates inward (type I), and is
caught in MMR with Saturn (3:2 or 4:3).
Add Neptune :
After inward migration, Neptune is
caught in 3:2, 4:3, or 5:4 MMR with
Uranus.
The 4 planets in this resonant
configuration avoid migration in
the disc.
(Morbidelli et al, 2007)
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1) Migration of Jupiter, Saturn, Uranus, Neptune
In this model, the 4 giant
planets of the Solar System
avoid migration in a disc with
reasonable parameters.
→ no perturbation of the inner
Solar System nor the MAB.
BUT: This configuration has nothing to do with the present one:
the outer Solar System is fully resonant and too compact.
→ An other model requires a compact configuration of the
outer SS after de gas disk phase : the “Nice model”…
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2) The “Nice model”
Questions :
~650 Myr after Solar System birth, a spike of
asteroïd bombardment occured, creating the
moon bassins (Late Heavy Bombardment).
The four giant planets, particularly Jupiter and Saturn, have a non
negligible eccentricity, while planet formation in a gas disk should lead to
circular orbits.
Idea :
A late instability in the planets dynamics excited the planets eccentricities
and destabilized a reservoir a of small bodies, leading to the LHB.
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2) The “Nice model”
The Nice model (Tsiganis et al, Gomes et al, 2005) :
After the gas disk disappearance, the four giant planets were initially
- on circular orbits
- in a compact configuration (within 17 A.U., with J & S inside their 2:1 MMR)
- surrounded by a disk of planetesimals (ancestor of the Kuiper Belt). (a)
Planetesimals
scattering
makes
Neptune, Uranus Saturn move slowly
outward, and Jupiter inward (b).
At some point, the 1J:2S is reached,
which increases their eccentricity and
destabilises the whole system, leading
to the LHB (c).
It clears the planetesimal disc and
causes a major change of the planets
orbits (d).
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2) The “Nice model”
During the instability,
- the planets have close encounters,
- migration through planetesimal scattering
runs away,
- eccentricities are damped by dynamical
friction.
Finally, the planets reach their present
orbits, while the quantity of small bodies
crossing the terrestrial orbit is in good
agreement with estimates of the LHB.
This model also explains :
- the capture of the Jupiter trojans on inclined
orbits (Morbidelli et al 2005),
- the orbital distribution of the irregular satellites of Saturn, Uranus, Neptune
(Nesvorny et al 2007),
- the main properties of the Kuiper Belt models (Levison et al 2007).
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2) The “Nice model”
Conclusion on the Nice model :
It explains a lot of characteristics of our Solar System,
thanks to a late instability in the outer planets dynamics,
with crossing of the 1J:2S MMR.
It relies on : a compact configuration, stable over hundreds of
millions of years in the absence of perturbation, that can lead
to instability if perturbed.
The initial condition assumed in the original Nice model is
arbitrary and somehow ad hoc. Can the planets form in the
disc in this configuration ?
Our goal : bridge the gap between disc phase and early
dynamics of the SS (Nice model).
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3) The “Nice model”, revisited
Through planet-disc simulations, six
resonant configurations of the 4 giant
planets can be achieved, that prevent
migration :
2J:3S - 2S:3U - 2U:3N, 3U:4N, 4U:5N.
2J:3S - 3S:4U - 2U:3N, 3U:4N, 4U:5N.
Test their stability on long term with Nbody simulations, after having smoothly
removed the disc.
Only two are stable over several
hundreds of millions of years :
2J:3S - 2S:3U - 2U:3N
2J:3S - 2S:3U - 3U:4N (figure)
(Morbidelli et al 2007)
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3) The “Nice model”, revisited
A first attempt :
Take 2J:3S - 2S:3U - 3U:4N.
Add a random small inclination, and a
planetesimal disc close beyond Neptune
(50 or 65 M)
→ 24 Initial Conditions.
→ 13 yield to a new stable configuration
that resembles closely to the one of the
outer planets of the Solar System.
Here, the instability is triggered by the
3J:5S MMR. Then, everything goes like
in the Nice model. In particular, Jupiter
and Saturn cross their 1:2 MMR, which
gives them their present eccentricities.
(Morbidelli et al 2007)
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3) The “Nice model”, revisited
A late instability is required :
Changing the initial setup of the planetesimal disc (that was
artificially close to Neptune), the instability can be delayed by
200 million years.
At 140 My, Neptune
leaves the 3U:4N.
At 190 My, crossing of
the 5U:7N.
Then, crossing of the
3J:5S and global
instability.
(Tsiganis et al, in prep.)
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3) The “Nice model”, revisited
Some work is still needed to improve the statistics on
the final outcome about a, e, the close encounters
between Saturn (or even Jupiter) and the ice giants…
Check if al the properties of the Nice model are kept.
(Tsiganis et al, in prep.)
Note that Thommes et al 2007 also studied fully resonant
configurations, but with Jupiter and Saturn in the 1:2 (and
without hydro simulations).
Fully resonant configurations may be a frequent outcome
of the disc phase, and a global instability may be a step
of the evolution of many planetary systems.
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CONCLUSION
Planets migrate in gaseous discs, and then interact
→ They not necessarily formed where they orbit now.
1) To prevent type II migration : use a pair of planets in
MMR, with the lighter one out (ex: Jupiter & Saturn).
2) From a compact configuration, slowly perturbed by an
outer planetesimal disc, a global instability can arise,
explaining the Late Heavy
Bombardment, the
eccentricities of the giant planets…
(Nice model)
3) From a fully resonant configuration, compatible with
the gas disc phase also.
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THE END
Thank you for your attention.
ENDE
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FIN
Merci de votre attention.