planetary migration and extrasolar planets in the 2:1 mean-motion resonance (short review) renate...
TRANSCRIPT
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