Chaos in the Solar System

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# Chaos in the Solar System - PowerPoint PPT Presentation

Chaos in the Solar System. The structure of the solar system is controlled by gravity F=-GMm/r 2 Motion of a planet distance r from the Sun can be found by solving Newton’s laws mdv/dt=F. Calculating the orbit with a computer. Coordinates (x,y) give position

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Presentation Transcript
Chaos in the Solar System
• The structure of the solar system is controlled by gravity

F=-GMm/r2

• Motion of a planet distance r from the Sun can be found by solving Newton’s laws

mdv/dt=F

Calculating the orbit with a computer
• Coordinates (x,y) give position
• x-motion described by

dx/dt=v

dv/dt=Fx=-GMmx/r3

• dx/dt = rate of change of x with time or velocity
• Approximately:

dx/dt=(xn+1-xn)/t

t secs between xn+1 and xn

More equations ...
• The equations for x look like

xn+1=xn+vnt

vn+1=vn-tGMmxn/rn3

• Similar for y-motion …
• Look like (non-linear) map !
• Iterate using computer to generate (x,y) as functions of time
Orbit Shape
• For this simplest situation (neglect F due to other planets) we find
• orbit is an ellipse
• Thus motion is regular and predictable
• Not typical …
• Consider Hyperion - a moon of Saturn ...
Moons
• Most of larger planets have them.
• Two parts to their motion:
• Almost every moon we know has it spin rotation rate = orbital rotation rate.
• Present always the same face to the planet.
Why is this ?
• Initially spins faster
• Tidal gravitational forces dissipate energy through friction
• slow down the spin
• When the spin rate = rotation rate this effect is zero
• spin-orbit resonance
• no alternate stretching/compression
Hyperion
• Hyperion feels gravitational force of a Saturn and is in orbit about it.
• What is different ?
• Hyperion is a very odd shape
• in a very elongated elliptical orbit
• Rotation is not in sync with orbit
• Chaotic rotational motion !
Simulation
• Model Hyperion as a dumbbell
• Not accurate for detailed predictions of its motion
• will allow us to understand chaotic nature.
• Use Newton’s laws to find rotational and orbital motion
• Plot rotation rate versus time
Motion
• See chaotic tumbling of Hyperion
• only for elliptical orbit
• Plot (angle,rotation rate)
• see fractal-like attractor
• If start model off with 2 different initial conditions see that motions rapidly diverge.
• Chaos!
More chaos in the Solar System
• If we have just 2 bodies eg Earth and Sun
• elliptic orbits
• What about 3 bodies eg. Earth, Sun and Jupiter ?
• Not soluble except via computer
• Find that effect of Jupiter is to cause a precession of orbit.
• Too large a mass - destroys orbit completely !
Precession ..
• Axis of ellipse rotates slowly with time - motion does not exactly repeat every orbit
• Same effect seen when Einstein’s theory of gravity used

F=-GMm/r2(1+a/r)

• Precession of Mercury - test of General Relativity.
Asteroids ...
• Titus-Bode law for planets around the Sun
• missing planet ?
• Jupiter’s gravity is so strong it prohibits formation of planet
• asteroid belt only
• Kirkwood gaps -- asteroids never found at certain radii
• resonance with Jupiters motion
• Motion near gap - chaotic