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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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## PowerPoint Slideshow about ' Chaos in the Solar System' - ciara

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Presentation Transcript

• 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

• 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

• 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

• 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 ...

• 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.

• 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 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 !

• 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

• 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!

• 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 !

• 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.

• 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