Planetary Migration and Extrasolar Planets in the

2:1 Mean-Motion Resonance(short review)

Renate Zechner

im Rahmen des

Astrodynamischen Seminars

basierend auf den Arbeiten von

C. Beaugé, S. Ferraz-Mello undT. A. Michtchenko

Wien, am 03.06.2004

Exoplanets and Planetary Formation Theories

single planetsplanetary systems

semi-major axis [AU]

ecc

ent

rici

ty

Mercury

Theories predict giant planets (M* M๏)

with e ~ 0and a > 4 AU

We observea 4 AUe ~ 0.1 – 0.8

=> Exoplanets do not fit into classical theories!

2 possible explanations Present cosmogonic theories are wrong -> formation mechanism was completely differentExoplanets formed far from the central star and migrated inwards = Hypothesis of Planetary Migration

2 conditions must be metExistence of a plausible driving mechanismConcrete evidence that exoplanets did undergo such an evolution

Planetary Migration

Hypothesis of Planetary Migration

1. Interaction with planetesimal disk (Murray et al. 1998)

Initial setup:Formation of proto-planets initially far away from central star immersed in remnant planetesimal disk

Evolution: Ejection of planetesimals caused orbital decay of planets

Problems: Very large disk mass is necessary (0.1 M๏)Primordial eccentricity would be preserved

Advantage:Migration stops when all planetesimals are ejected

2. Interaction with gaseous disk(Goldreich & Tremaine 1979, Ward 1997)

Initial setup:Formation of proto-planet initially far away from central star immersed in gaseous disk

Evolution: Planet excites density waves in disk -> Inward migration of proto-planet

Problem: How to stop migration?

Advantage:Several simulations indicate that this mechanism works reasonably well

Hypothesis of Planetary Migration

Resonant Exoplanets in 2:1 MMR?Analyze whether extrasolar planetary systems are in MMR Check those planetary systems with

-> 6 Systems

System P2/P1

GJ 876 2:1

HD 82943 2:1

55 Cnc 3:1

47 UMa 7:3

HD 160691 2:1

Orbits not well determined

Secular Res. Ups And

Configuration System3a

a

1

2

Evidence of Migration?

Observational data seems inexact

Indirect feature to study orbital characteristics of resonant planets

Corotation

Apsidal Corotation for the 2:1 MMR

Assumptionm1, m2 located in the vicinity of a resonance ni (i = 1,2): n1/n2 (p+q)/p Resonant Anglesq1 = (p + q)1 - p1 - q1 q2 = (p + q)2 - p2 - q2

with:i = qi

Apsidal Corotation (Ferraz-Mello et al. 1993)Simultaneous libration of both resonant angles 1, 2

Libration of the difference in longitudes of pericenter Semimajor axis of the planets is aligned/anti-aligned

1 - = q(1 - 2) = qbzw.2 - = 1 - 2 =

with = = 22 – - 1

(2:1 MMR)

Apsidal (zero-amplitude) corotation depends onThe masses only through m2/m1

-> Independent of sin(i) Semimajor axes only through a1/a2

-> Independent of a1, a2

For a given resonance and mass ratioWe can plot all the families in the plane of eccentricities (e1,e2) as level curves of 1, andm2/m1

Extremely general solutions -> Valid for any planetary system (independently of real masses and distance from the central star)

Families of Periodic Orbits

4 types of corotational solutions

Aligned apsidal corotation

1

Anti-aligned apsidal corotation 1

Asymmetric apsidal corotation1

Apsidal corotation for very high values of e1 and e2 1

Families of Corotations (2:1 MMR)No solutions in this region!!

e.g. (0,0) (=0, =0)

e1

e 2

=0

Asymmetric Apsidal Corotation for 1 and

e1

e 2

1 = 0

collision curve

1 =const.

= 0

=const.

Level Curves of Constant Mass Ratio for Stable Corotation (2:1 MMR)

=const.m2

m1

e1

e 2

e1

m2/m1 > 1

e 2

m2/m1 < 1

Numerical Simulationsof the Planetary Migration

Beaugé et al. are studying Process of resonance trappingPosterior evolution inside the resonance

Initial conditionsa1 = 5.2 AU, a2 = 8.5 AU, e = 0, m2/m1 = const.Adoption of various types of forces

tidal interaction, interaction with planetesimal disk, disk torques,...

ResultsAll runs ended in apsidal corotations! Duration of the migration: 105 – 107 years

ConclusionsTrapped bodies must show apsidal corotationsFamilies of apsidal corotations show the possible location of the system in the vicinity of the 2:1 MMR and their evolutionary tracks!

Orbital Evolution inside the 2:1 MMR

Results of all Numerical Simulations(„Evolutionary Curve“)

Asymmetric

Solutio

n Aligned Configurati

on

No Solution

Anti-Aligned Configurati

on

= 1.5m2

m1

A [10-6,10-4] and E [10-11,10-4] with a(t)=a0 exp(-At), e(t)=e0 exp(-Et)

)vv(Cdt

rdc2

2

Stokes-type non-conservative force of

the type:

A = 2C (1 - ) E = C

(Non-) Adiabatic MigrationAdiabaticMigration(A = 10-

6)

Non-AdiabaticMigration (A = 10-4)

SimilarEvolutionary

Tracks

All these interpretations are valid for adiabatic migration

when the driving mechanism is sufficiently slow: a = mig » cor

Numerical simulation shows corotational solutions

for m2/m1 > 1: (e.g. m2/m1 = 3 for GJ 876)System is still adiabatic with: mig ~ 104 years

for m2/m1 < 1: Migration must be slow: mig ~ 105 – 106 years

What about known planetary systems?

Evolutionary Tracks for GJ 876

GJ876

Asym

2 different possible orbits

Keck+Lick: (e1, e2) = (0.27, 0.10)

Keck alone: (e1, e2) = (0.33, 0.05)

Observational fits lie very close to the zero-amplitude solution ->

Fit is consistent with apsidal corotation!

?

Asym

Old fit of HD 82943

Observational Data m2/m1 = 1.9

(e1, e2) = (0.54, 0.41)

Stabile configuration only for (,)-corotation

Problem -> obital fit is not correct!

New fit for HD 82943

HD 82943

New analysis ofMayor et al. (2004)

m2/m1 1 1.9 (e1, e2)=(0.38, 0.18) (0.54, 0.41)

Fit is more consistent with apsidal corotation

No (,)-Corotations

AsymmetricSolution

Results

GJ 876Shows apsidal corotation in the 2:1 MMR

HD 82943Problem with old orbital fit but: New orbital determination is completely compatible with corotational solutions

HD 160691Problems due to uncertainties in the fits -> Existence of the exterior planet is questionable

ConclusionOrbital characteristics of exoplanets can only be explained through:

Planetary formation completely different from oursPlanetary migration

Evidence for migrations are planetary systems in MMR!

Hydrodynamical and numerical simulations predict corotations in 2:1 MMRCurrent orbits of GJ 876 and HD 82943 are consistent Non-consistent orbits of HD 160691 (and old fit of HD 82943):

Systems did not undergo migrationMigration process was non-adiabaticUncertainties in orbital determination

The End

Content

IntroductionPlanetary Migration & Driving MechanismFamilies of Corotations (2:1 MMR)Numerical SimulationsPlanetary Systems in the 2:1 MMRResultsConclusions

Confirmed Migration in our Solar System

Outer Planets Migration due to interaction with a remnant planetesimal disk

Planets are not exactly in resonance -> random-walk characteristics of driving mechanismMigration doesn‘t necessarily imply MMR but:Massive bodies in MMR do imply migration

Planetary Satellites Migration due to tidal effects of the central mass

Galilean satellites are in exact MMR due toGravitational perturbation + resonance trapping

Apsidal Corotation

Aligned Apsidal Corotation(Gliese 876)

Anti-Aligned Corotation(Galilean satellites)

= 2 2 - 1 - 1

Libration resonant angle Libration = 1 - 2 +COROTATION

Numerical Simulationof GJ 876: Laughlin & Chambers (2001)

Numeric Simulation:(None-) Adiabatic Migration

AdiabaticMigration(A = 10-

6) Non-AdiabaticMigration

(A = 10-4)

SimilarEvolutionary

Tracks

Similar Symmetric

Apsidal Corotations

AsymmetricApsidal

Corotations

No solutions in this region!!

Domains of Different Types of Corotational Solution (2:1 MMR)

e.g. (0,0) (=0, =0)

HD 82943

Analysis of the new dataNumerical integration for 1 million years100 different initial conditions

Results80% unstable orbits (T = 106 years)20% stable orbits

15 are in a stable large-amplitude apsidal corotation 5 systems show an apparent libration of 1 but with a circulation of

HD 160691

Orbital characteristics (Jones et al. 2002)(e1, e2) = (0.31, 0.80)

m2/m1 = 0.6

Dynamical analysis (Bois et al. 2003)Confirmation of apsidal corotation

ProblemNo explanation for these values of (e1,e2) with such a m2/m1

Possible solution (Mayor et al.)Outer planet is probably not existent