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

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Predicting the Future of the Solar
SystemNonlinear Dynamics, Chaos and
StabilityDr. Russell HermanUNC Wilmington
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Outline

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

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The Solar System
Planet Orbit Parameters Orbit Parameters Orbit Parameters Orbit Parameters
Planet Distance Period Inclination(degrees) Eccentricity
Planet Compared to Earth Compared to Earth Compared to Earth Eccentricity
Mercury 0.387 0.241 7 0.206
Venus 0.723 0.615 3.39 0.007
Earth 1.00 1.00 0 0.017
Mars 1.524 1.88 1.85 0.093
Jupiter 5.203 11.86 1.3 0.048
Saturn 9.539 29.46 2.49 0.056
Uranus 19.18 84 0.77 0.047
Neptune 30.06 164.8 1.77 0.009
Pluto 39.53 247.7 17.15 0.248
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Chaos in the Solar System
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Chaos in the News
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Kirkwood Gaps
Daniel Kirkwood -1886 Few asteroids have an
orbital period close to1/2, 1/3, or 2/5 that of
Jupiter Due to Mean Motion Resonances 31
Resonance - the asteroid completes 3 orbits for
every 1 orbit of Jupiter
http//ssd.jpl.nasa.gov/a_histo.html
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Celestial Mechanics from Aristotle to Newton

• Aristotle 384-322 BCE
• Hipparchus of Rhodes 190-120 BCE season errors
• Claudius Ptolemy 85- 165 epicycles
• Nicolaus Copernicus 1473-1543 heliocentric
• Tycho Brahe 1546-1601 planetary
  data
• Galileo Galilei 1564-1642 kinematics
  
• Johannes Kepler 1571-1630 Planetary Laws
  
• Sir Isaac Newton 1642-1727
  Gravity/MotionRobert Hooke 1635-1703
  Inverse Square?
• Edmond Halley 1656-1742 - Comets
• Euler, Laplace, Lagrange, Jacobi, Hill,
  Poincare, Birkhoff ...

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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 3rd body not a planet!
• Strange behavior noted
• not periodic!
• But there is more

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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)

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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

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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

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Nonlinear DynamicsContinuous Systems

• Simple Harmonic Motion
• Phase Portraits
• Damping
• Nonlinearity
• Forced Oscillations
• Poincaré Surface of Section

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Linear Oscillations
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Phase Portrait for
System
Equilibrium
Classification by Eigenvalues
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Damped Oscillations
System
Classification by Eigenvalues
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Nonlinear Pendulum

• Integrable Hamiltonian System
• Separatrix
• Perturbations entangle stable/unstable
  manifolds

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Damped Nonlinear Pendulum
No Damping vs Damping
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Forced Oscillations
System
Resonance
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Phase Plots Forced Pendulum
No Damping vs Damping
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Poincaré Surface of Section
System
Regular orbit movie (Henon-Heiles equations)
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Damped, Driven Pendulum
No Damping vs Damping
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The Onset of Chaos

• Lorenz Equations, Strange Attractors, Fractals

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Nonspherical Satellites

• Hyperion
• Rotational Motion
• Orbital Mechanics
• Nonlinear System
• Phase Portraits

http//www.solarviews.com/cap/ast/toutat9.htm
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Hyperion
  MPEG (no audio)
http//www.planetary.org/saturn/hyperion.html
http//www.nineplanets.org/hyperion.html
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The Hyperion Problem
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Rotational Motion
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Computing Torque I
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Computing Torque II
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Computing Torque III
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Summary
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Orbital Motion
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Constants of the Motion
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Equation of the Orbit
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Orbit as a Function of Time
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Keplers Equation I
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The Anomalies
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Keplers Equation II
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The Reduced Problem
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The System of Equations
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Dimensionless System
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Numerical Results
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Spin-Orbit Resonance

• Satellite moves about Planet
• triaxial (AltBltC)
• Keplerian Orbit
• Nearly Hamiltonian System
• Oblateness Coefficient
• Orbital Eccentricity
• Resonance Trev/Trot p/q
• 11 Synchronous like Moon-Earth
• Mercury 32

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Moon e 0.0549, w 0.026
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Mercury e 0.2056, w 0.017
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e 0.02
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e 0.04
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e 0.06
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e 0.08
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e 0.10
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w 0.1
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w 0.3
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w 0.5
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w 0.7
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w 0.9
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Summary

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

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More in the Fall
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References