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Chaos in the Solar SystemPowerPoint Presentation

Chaos in the Solar System

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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:
- orbit about planet
- spin about an axis

- 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

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