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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

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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!

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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

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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

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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

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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 :1 55 Cnc3:1 47 UMa7:3 HD :1 Orbits not well determined Secular Res. Ups And Configuration System

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

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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)

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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

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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)

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e 1 e 2 =0 Asymmetric Apsidal Corotation for 1 and e 1 e 2 1 = 0 collision curve 1 =const. = 0 =const.

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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

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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!

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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

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(Non-) Adiabatic Migration Adiabatic Migration (A = ) Non-Adiabatic Migration (A = ) 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?

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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!

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? Asym Old fit of HD 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!

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

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

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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 are consistent Non-consistent orbits of HD (and old fit of HD 82943): Systems did not undergo migration Migration process was non-adiabatic Uncertainties in orbital determination

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The End

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

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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

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

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

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Numeric Simulation: (None-) Adiabatic Migration Adiabatic Migration (A = ) Non-Adiabatic Migration (A = ) Similar Evolutionary Tracks Similar Symmetric Apsidal Corotations Asymmetric Apsidal Corotations

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

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HD 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

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HD 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

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