1. in protoplanetary discs and OUTER SOLAR SYSTEM ARCHITECTURE
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)

2. Introduction : Proto-planetary disks : Planets form in it.
Size : several 100s A.U.
Life time : ~ years.
Aspect Ratio : H/r ~ flaring
Viscosity : n = a cs H (10-3<a<10-1)
Mass : < M* / 10
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3. Planet-disk interactions :
Wake formation :
A planet on a fixed circular orbit launches a spiral wake by gravitational perturbation
Angular Momentum exchanges :
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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4. 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 ( Crida et al, 2006 )
( Frédéric Masset again, corotating frame )
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5. 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.
Outer disk
Inner disk
star
The disk global evolution is the key. It drives the giant planet close to the central star.
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6. Type II migration explains the hot Jupiters, but not Jupiter !
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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7. 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.
Outer disk
Inner 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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8. 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.
= : Saturn migrates outward (strong corotation torque), then parallel migration in separated gaps : Mechanism broken.
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9. 1) Migration of Jupiter & Saturn
Dependence on aspect ratio H/r :
Start with aSaturn = 1.4 aJupiter, n = , fixed planets (for gap opening).
At t ≈ 500, release the planets.
They approach, lock in MMR at t ≈ 1000, and then migrate together.
The smaller H/r, the deeper Saturn’s gap, the more Jupiter pushed outward.
(Morbidelli & Crida, 2007)
H/r = 0.05 : stationary solution.
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10. 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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11. 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)
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.
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12. 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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13. 2) The “Nice model” Questions : Idea :
~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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14. 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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15. 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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16. 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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17. 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 N-body 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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18. 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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19. 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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20. 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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21. 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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22. Thank you for your attention.
THE END
Thank you for your attention.
ENDE
Danke für Ihre Aufmerksamkeit.
FIN
Merci de votre attention.