predicting the future of the solar system: nonlinear dynamics, chaos and stability dr. russell...
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![Page 1: Predicting the Future of the Solar System: Nonlinear Dynamics, Chaos and Stability Dr. Russell Herman UNC Wilmington](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e3a5503460f94b2c591/html5/thumbnails/1.jpg)
Predicting the Future of the Solar System:Nonlinear Dynamics, Chaos and Stability
Dr. 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
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
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Chaos in the Solar System
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Chaos in the News
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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
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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/Motion
Robert 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
2
2d x
m kxdt
2
21d qq
LCdt
2
2d g
Ldt
22
2d x
xdt
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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:
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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
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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
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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
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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
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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
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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)
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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
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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
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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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Kepler’s Equation I
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The Anomalies
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Kepler’s 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 (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
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Moon e = 0.0549, = 0.026
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Mercury e = 0.2056, = 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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= 0.1
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= 0.3
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= 0.5
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= 0.7
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= 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