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

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
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
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
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
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
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
  • 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
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
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
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
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
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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