1 / 12

Chaos in the Solar System

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

ciara
Download Presentation

Chaos in the Solar System

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. 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

  3. 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

  4. 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 ...

  5. 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.

  6. 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

  7. 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 !

  8. 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

  9. 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!

  10. 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 !

  11. 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.

  12. 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

More Related