Phy107Lect7 - From Last Time Gravitational forces are...

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Sep. 20, 2004 Physics 107, Lecture 7 From Last Time… • Gravitational forces are apparent at a wide range of scales. • Obeys F gravity ( Mass of object 1) × (Mass of object 2) square of distance between them F = 6.7 × 10 -11 m 1 × m 2 d 2 Gravitational Constant
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Sep. 20, 2004 Physics 107, Lecture 7 The deterministic solar system
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Sep. 20, 2004 Physics 107, Lecture 7 Simple Planetary motion http://galileoandeinstein.phys.virginia.edu/ more_stuff/flashlets/kepler6.htm
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Sep. 20, 2004 Physics 107, Lecture 7 Newtonian Determinism • Newton’s laws seem to determine all future motion. • All future behavior exactly known.
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Sep. 20, 2004 Physics 107, Lecture 7 The ‘three-body’ problem • Newton could not solve any problem past a single planet orbiting the sun. • Prize offered for solution of 3-body problem. • Poincare in 1896 showed problem not analytically solvable.
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Sep. 20, 2004 Physics 107, Lecture 7 Complicated motion of 3 bodies
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Sep. 20, 2004 Physics 107, Lecture 7 Sensitivity to initial conditions in the 3-body problem • Flash simulation http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Chaos/ ThreeBody/ThreeBody.html • Two planets start out with almost identical positions and velocities. • Resulting motions due to gravitational attraction to the two suns are very different • Sensitive to initial conditions
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Sep. 20, 2004 Physics 107, Lecture 7 How can we summarize this motion? The three-body problem exhibits all of the hallmarks of chaos. In particular, the outcome of any given interaction depends sensitively on the initial conditions. The following image shows how the final state of a scattering encounter between a binary star system and another star depends on the initial phase (horizontal axis) of the binary and the impact parameter (vertical axis) of the incomer. Color represents the angle at which the star that eventually escapes leave the interaction region. Note the alternating regions of regular (smooth) and irregular (chaotic, resonant) behavior. Each pixel in this image coresponds to a complete three-body encounter. The particular series of calculations shown here has relative velocity at infinity equal to 10% of the binary orbit speed. Encounters such as these are believed to be important in determining the dynamical evolution of globular star clusters in the Milky Way galaxy.
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20, 2004 Physics 107, Lecture 7 Dynamical Systems • The system evolves in time according to a set of rules. • The present conditions determine the future.
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This note was uploaded on 01/05/2012 for the course PHYS 107 taught by Professor N/a during the Spring '08 term at University of Wisconsin Colleges Online.

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Phy107Lect7 - From Last Time Gravitational forces are...

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