Predicting the Future of the Solar System:Nonlinear Dynamics, Chaos and Stability

Dr. Russell HermanUNC 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

Distance PeriodInclination(degrees) Eccentricity

Compared to Earth

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

Chaos in the Solar System

Chaos in the News

Kirkwood Gaps

http://ssd.jpl.nasa.gov/a_histo.html

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

3:1 Resonance - the asteroid completes 3 orbits for every 1 orbit of Jupiter

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/Motion

Robert Hooke 1635-1703 – Inverse Square?• Edmond Halley 1656-1742 - Comets • … 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 (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 …

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 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

Nonlinear DynamicsContinuous Systems

• Simple Harmonic Motion

• Phase Portraits

• Damping

• Nonlinearity

• Forced Oscillations

• Poincaré Surface of Section

Linear Oscillations

2

2d x

m kxdt

2

21d qq

LCdt

2

2d g

Ldt

22

2d x

xdt

Phase Portrait for

0 1

0

x xdk

v vdtm

dxv

dtdv k

xdt m

2

2d x

m kxdt

Equilibrium:

0 0, 0dx dv

x vdt dt

Classification by Eigenvalues:

2 0k

m

-5 0 5-5

-4

-3

-2

-1

0

1

2

3

4

5y(2) vs y(1)

y(1)

y(2)

System:

Damped Oscillations

2

2d x dx

m kx bdtdt

0 1

/ /

x xd

v k m b m vdt

2 0m b k

System:

Classification by Eigenvalues:

2 4

2 2

b b mk

m m

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Damped Pendulum

xy

Nonlinear Pendulum 2

2sin

d x gx

Ldt

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5v vs x

x

v

•Integrable Hamiltonian System

•Separatrix

•Perturbations – entangle stable/unstable manifolds

Damped Nonlinear Pendulum

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5v vs x

x

v

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5v vs x

x

v

No Damping vs Damping

Forced Oscillations2

2cos 2

d xm kx A ftdt

0 1 0

/ 0 cos 2

x xd

v k m v A ftdt

System:

Resonance

Phase Plots – Forced Pendulum

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5v vs x

x

v

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5v vs x

x

v

No Damping vs Damping

Poincaré Surface of Section2

2cos 2

d xm kx A ftdt

System:

cos

2

dxv

dtdv k

x Adt md

fdt

Regular orbit movie (Henon-Heiles equations)

Damped, Driven Pendulum

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5v vs x

x

v

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5v vs x

x

v

No Damping vs Damping

The Onset of Chaos

Lorenz Equations, Strange Attractors, Fractals …

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

yPoincare Section of the Duffing System

Nonspherical Satellites

• Hyperion

• Rotational Motion

• Orbital Mechanics

• Nonlinear System

• Phase Portraits

http://www.solarviews.com/cap/ast/toutat9.htm

Hyperion

  MPEG (no audio)

http://www.planetary.org/saturn/hyperion.html

http://www.nineplanets.org/hyperion.html

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 Trev/Trot = p/q– 1:1 – Synchronous – like Moon-Earth– Mercury 3:2

Moon e = 0.0549, = 0.026

Mercury e = 0.2056, = 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