1. 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 und T. A. Michtchenko Wien, am 03.06.2004

2. Exoplanets and Planetary Formation Theories single planets planetary systems semi-major axis [AU] eccentricity Mercury Theories predict giant planets (M *  M๏) with e ~ 0 and a > 4 AU We observe a  4 AU e ~ 0.1 – 0.8 => Exoplanets do not fit into classical theories!

3. 2 possible explanations Present cosmogonic theories are wrong -> formation mechanism was completely different Exoplanets formed far from the central star and migrated inwards = Hypothesis of Planetary Migration 2 conditions must be met Existence of a plausible driving mechanism Concrete evidence that exoplanets did undergo such an evolution Planetary Migration

4. 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

5. 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

6. Resonant Exoplanets in 2:1 MMR? Analyze whether extrasolar planetary systems are in MMR Check those planetary systems with -> 6 Systems System P 2 /P 1 GJ 8762:1 HD 829432:1 55 Cnc3:1 47 UMa7:3 HD 1606912:1 Orbits not well determined Secular Res. Ups And Configuration System

7. Evidence of Migration? Observational data seems inexact Indirect feature to study orbital characteristics of resonant planets Corotation

8. Apsidal Corotation for the 2:1 MMR Assumption m 1, m 2 located in the vicinity of a resonance n i (i = 1,2): n 1 /n 2  (p+q)/p Resonant Angles q  1 = (p + q) 1 - p 1 - q  1 q  2 = (p + q) 2 - p 2 - q  2 with:  i = q  i 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 ) = q  bzw.  2 -   =  1 -  2 =  with  =  = 2 2 –  -  1 (2:1 MMR)

9. Apsidal (zero-amplitude) corotation depends on The masses only through m 2 /m 1 -> Independent of sin(i) Semimajor axes only through a 1 /a 2 -> Independent of a 1, a 2 For a given resonance and mass ratio We can plot all the families in the plane of eccentricities (e 1,e 2 ) as level curves of  1,  and  m 2 /m 1 Extremely general solutions -> Valid for any planetary system (independently of real masses and distance from the central star) Families of Periodic Orbits

10. 4 types of corotational solutions Aligned apsidal corotation  1  Anti-aligned apsidal corotation  1  Asymmetric apsidal corotation  1  Apsidal corotation for very high values of e 1 and e 2  1  Families of Corotations (2:1 MMR) No solutions in this region!! e.g. (0,0)  (  =0,  =0)

11. e 1 e 2  =0 Asymmetric Apsidal Corotation for  1 and  e 1 e 2  1 = 0 collision curve  1 =const.  = 0  =const.

12. Level Curves of Constant Mass Ratio for Stable Corotation (2:1 MMR) =const. m2m1m2m1 e 1 e 2 e 1 m 2 /m 1 > 1 e 2 m 2 /m 1 < 1

13. Numerical Simulations of the Planetary Migration Beaugé et al. are studying Process of resonance trapping Posterior evolution inside the resonance Initial conditions a 1 = 5.2 AU, a 2 = 8.5 AU, e = 0, m 2 /m 1 = const. Adoption of various types of forces tidal interaction, interaction with planetesimal disk, disk torques,... Results All runs ended in apsidal corotations! Duration of the migration: 10 5 – 10 7 years Conclusions Trapped bodies must show apsidal corotations Families of apsidal corotations show the possible location of the system in the vicinity of the 2:1 MMR and their evolutionary tracks!

14. Orbital Evolution inside the 2:1 MMR Results of all Numerical Simulations („Evolutionary Curve“) Asymmetric Solution Aligned Configuration No Solution Anti-Aligned Configuration = 1.5 m2m1m2m1 A  [10 -6,10 -4 ] and E  [10 -11,10 -4 ] with a(t)=a 0 exp(-At), e(t)=e 0 exp(-Et) Stokes-type non-conservative force of the type: A = 2C (1 -  ) E = C 

15. (Non-) Adiabatic Migration Adiabatic Migration (A = 10 -6 ) Non-Adiabatic Migration (A = 10 -4 ) Similar Evolutionary 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 m 2 /m 1 > 1: (e.g. m 2 /m 1 = 3 for GJ 876) System is still adiabatic with:  mig ~ 10 4 years for m 2 /m 1 < 1: Migration must be slow:  mig ~ 10 5 – 10 6 years What about known planetary systems?

16. 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!

17. ? Asym Old fit of HD 82943 Observational Data m 2 /m 1 = 1.9 (e1, e2) = (0.54, 0.41) Stabile configuration only for ( ,  )-corotation Problem -> obital fit is not correct!

18. New fit for HD 82943 HD 82943 New analysis of Mayor et al. (2004) m 2 /m 1  1 1.9 (e 1, e 2 )=(0.38, 0.18) (0.54, 0.41) Fit is more consistent with apsidal corotation No ( ,  )-Corotations Asymmetric Solution

19. Results GJ 876 Shows apsidal corotation in the 2:1 MMR HD 82943 Problem with old orbital fit but: New orbital determination is completely compatible with corotational solutions HD 160691 Problems due to uncertainties in the fits -> Existence of the exterior planet is questionable

20. Conclusion Orbital characteristics of exoplanets can only be explained through: Planetary formation completely different from ours Planetary migration Evidence for migrations are planetary systems in MMR! Hydrodynamical and numerical simulations predict corotations in 2:1 MMR Current 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 migration Migration process was non-adiabatic Uncertainties in orbital determination

21. The End

22. Content Introduction Planetary Migration & Driving Mechanism Families of Corotations (2:1 MMR) Numerical Simulations Planetary Systems in the 2:1 MMR Results Conclusions

23. 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 mechanism Migration 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 to Gravitational perturbation + resonance trapping

24. Apsidal Corotation Aligned Apsidal Corotation (Gliese 876) Anti-Aligned Corotation (Galilean satellites)

25.  = 2 2 - 1 -  1 Libration resonant angle  Libration  =  1 -  2 + COROTATION Numerical Simulation of GJ 876: Laughlin & Chambers (2001) 

26. Numeric Simulation: (None-) Adiabatic Migration Adiabatic Migration (A = 10 -6 ) Non-Adiabatic Migration (A = 10 -4 ) Similar Evolutionary Tracks Similar Symmetric Apsidal Corotations Asymmetric Apsidal Corotations

27. No solutions in this region!! Domains of Different Types of Corotational Solution (2:1 MMR) e.g. (0,0)  (  =0,  =0)

28. HD 82943 Analysis of the new data Numerical integration for 1 million years 100 different initial conditions Results 80% unstable orbits (T = 10 6 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  

29. HD 160691 Orbital characteristics (Jones et al. 2002) (e 1, e 2 ) = (0.31, 0.80) m 2 /m 1 = 0.6 Dynamical analysis (Bois et al. 2003) Confirmation of apsidal corotation Problem No explanation for these values of (e 1,e 2 ) with such a m 2 /m 1 Possible solution (Mayor et al.) Outer planet is probably not existent