1. Predicting the Future of the Solar System: Nonlinear Dynamics, Chaos and Stability Dr. Russell Herman UNC Wilmington

2. Outline Chaos in the Solar System The Stability of the Solar System Linear and Nonlinear Oscillations Nonspherical Satellite Dynamics Numerical Studies Summary

3. The Solar System Planet Orbit Parameters DistancePeriod Inclination (degrees) Eccentricity Compared to Earth Mercury0.3870.24170.206 Venus0.7230.6153.390.007 Earth1.00 00.017 Mars1.5241.881.850.093 Jupiter5.20311.861.30.048 Saturn9.53929.462.490.056 Uranus19.18840.770.047 Neptune30.06164.81.770.009 Pluto39.53247.717.150.248

4. Chaos in the Solar System

5. Chaos in the News

6. Kirkwood Gaps http://ssd.jpl.nasa.gov/a_histo.html Daniel Kirkwood -1886 Few asteroids have an orbital period close to 1/2, 1/3, or 2/5 that of Jupiter Due to Mean Motion Resonances 3:1 Resonance - the asteroid completes 3 orbits for every 1 orbit of Jupiter

8. Celestial Mechanics – from Aristotle to Newton Aristotle 384-322 BCEAristotle Hipparchus of Rhodes 190-120 BCE – season errorsHipparchus of Rhodes Claudius Ptolemy 85- 165 – epicyclesClaudius Ptolemy Nicolaus Copernicus 1473-1543 – heliocentricNicolaus Copernicus Tycho Brahe 1546-1601 – planetary dataTycho Brahe Galileo Galilei 1564-1642 – kinematicsGalileo Galilei Johannes Kepler 1571-1630 – Planetary LawsJohannes Kepler Sir Isaac Newton 1642-1727 – Gravity/Motion Robert Hooke 1635-1703 – Inverse Square?Sir Isaac Newton Robert Hooke Edmond Halley 1656-1742 - CometsEdmond Halley … Euler, Laplace, Lagrange, Jacobi, Hill, Poincare, Birkhoff...

9. The Stability of the Solar System King Oscar II of Sweden - Prize: How stable is the universe? Jules Henri Poincaré (1854-1912) – Sun (large) plus one planet (circular orbit) Stable –Added 3 rd body – not a planet! Strange behavior noted … not periodic! –But there is more …

10. Sensitivity to Initial Conditions "A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still know the situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible...". (Poincaré)

11. Can one predict the motion of a single planet a billion years from now? Laplace and Lagrange – Yes Poincare’ – No Lyapunov – speed neighboring orbits diverged Lorenz – 1963 – “Butterfly Effect”

12. Solar System Simulations Sun plus 7 planets – 21 degrees of freedom Numerical Studies –Mitchtchenko and Ferraz-Mello 2004 35 Gyr – 660 MHz Alpha 21264A – 15 weeks of CPU time –1988 – Sussman and Wisdom Lyapunov time - 10 Myrs –Laskar, et. Al. 8 planets w/corrections – 5 Myrs 1 km error = 1 au error in 95 Myrs Planets –Pluto – chaotic –Inner Planets – chaotic –Earth – stabilizer Klavetter – 1987 –Observations of Hyperion wobbling

13. Nonlinear Dynamics Continuous Systems Simple Harmonic Motion Phase Portraits Damping Nonlinearity Forced Oscillations Poincaré Surface of Section

14. Linear Oscillations

15. Phase Portrait for Equilibrium: Classification by Eigenvalues: System:

16. Damped Oscillations System: Classification by Eigenvalues:

17. Nonlinear Pendulum Integrable Hamiltonian System Separatrix Perturbations – entangle stable/unstable manifolds

18. Damped Nonlinear Pendulum No Damping vs Damping

19. Forced Oscillations System: Resonance

20. Phase Plots – Forced Pendulum No Damping vs Damping

21. Poincaré Surface of Section System: Regular orbit movie (Henon-Heiles equations)

22. Damped, Driven Pendulum No Damping vs Damping

23. The Onset of Chaos Lorenz Equations, Strange Attractors, Fractals …

24. Nonspherical Satellites Hyperion Rotational Motion Orbital Mechanics Nonlinear System Phase Portraits http://www.solarviews.com/cap/ast/toutat9.htm

25. Hyperion MPEG (no audio) http://www.planetary.org/saturn/hyperion.html http://www.nineplanets.org/hyperion.html

26. The Hyperion Problem

27. Rotational Motion

28. Computing Torque I

29. Computing Torque II

30. Computing Torque III

31. Summary

32. Orbital Motion

33. Constants of the Motion

34. Equation of the Orbit

35. Orbit as a Function of Time

36. Kepler’s Equation I

37. The Anomalies

38. Kepler’s Equation II

39. The Reduced Problem

40. The System of Equations

41. Dimensionless System

42. Numerical Results

43. Spin-Orbit Resonance Satellite moves about Planet –triaxial (A<B<C) –Keplerian Orbit Nearly Hamiltonian System –Oblateness Coefficient –Orbital Eccentricity Resonance T rev /T rot = p/q –1:1 – Synchronous – like Moon-Earth –Mercury 3:2

44. Moon e = 0.0549,  = 0.026

45. Mercury e = 0.2056,  = 0.017

46. e = 0.02

47. e= 0.04

48. e = 0.06

49. e = 0.08

50. e = 0.10

51.  = 0.1

52.  = 0.3

53.  = 0.5

54.  = 0.7

55.  = 0.9

56. Summary Chaos in the Solar System The Stability of the Solar System Linear and Nonlinear Oscillations Nonspherical Satellite Dynamics Numerical Studies Where now?

57. More in the Fall …

58. References