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

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

The Solar System Planet Orbit Parameters DistancePeriod Inclination (degrees) Eccentricity Compared to Earth Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

Chaos in the Solar System

Chaos in the News

Kirkwood Gaps Daniel Kirkwood 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

Celestial Mechanics – from Aristotle to Newton Aristotle BCEAristotle Hipparchus of Rhodes BCE – season errorsHipparchus of Rhodes Claudius Ptolemy – epicyclesClaudius Ptolemy Nicolaus Copernicus – heliocentricNicolaus Copernicus Tycho Brahe – planetary dataTycho Brahe Galileo Galilei – kinematicsGalileo Galilei Johannes Kepler – Planetary LawsJohannes Kepler Sir Isaac Newton – Gravity/Motion Robert Hooke – Inverse Square?Sir Isaac Newton Robert Hooke Edmond Halley CometsEdmond Halley … Euler, Laplace, Lagrange, Jacobi, Hill, Poincare, Birkhoff...

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

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)

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”

Solar System Simulations Sun plus 7 planets – 21 degrees of freedom Numerical Studies –Mitchtchenko and Ferraz-Mello 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

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

Linear Oscillations

Phase Portrait for Equilibrium: Classification by Eigenvalues: System:

Damped Oscillations System: Classification by Eigenvalues:

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

Damped Nonlinear Pendulum No Damping vs Damping

Forced Oscillations System: Resonance

Phase Plots – Forced Pendulum No Damping vs Damping

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

Damped, Driven Pendulum No Damping vs Damping

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

Nonspherical Satellites Hyperion Rotational Motion Orbital Mechanics Nonlinear System Phase Portraits

Hyperion MPEG (no audio)

The Hyperion Problem

Rotational Motion

Computing Torque I

Computing Torque II

Computing Torque III

Summary

Orbital Motion

Constants of the Motion

Equation of the Orbit

Orbit as a Function of Time

Kepler’s Equation I

The Anomalies

Kepler’s Equation II

The Reduced Problem

The System of Equations

Dimensionless System

Numerical Results

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

Moon e = ,  = 0.026

Mercury e = ,  = 0.017

e = 0.02

e= 0.04

e = 0.06

e = 0.08

e = 0.10

 = 0.1

 = 0.3

 = 0.5

 = 0.7

 = 0.9

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

More in the Fall …

References