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/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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Chaos in the solar system
Chaos in the Solar System

  • The structure of the solar system is controlled by gravity


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


Calculating the orbit with a computer
Calculating the orbit with a computer

  • Coordinates (x,y) give position

  • x-motion described by



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

  • Approximately:


    t secs between xn+1 and xn

More equations
More equations ...

  • The equations for x look like



  • Similar for y-motion …

  • Look like (non-linear) map !

  • Iterate using computer to generate (x,y) as functions of time

Orbit shape
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 ...


  • 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
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 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!

More chaos in the solar system
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


  • 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