Astro 201: Sept. 9, 2010 Do on-line practice quiz #2 by 9/14 Turn in HW 2 in box in front of the room Reading: Hester Chapter 4 Today: String Theory Determinism v. Chaos; Fractals Light

STRING THEORY everything is ultimately made of strings (sub-sub-sub atomic particles) How big are strings? Smaller than a Planck length, which is about 10-33 centimeters or about a millionth of a billionth of a billionth of a billionth of a centimeter.

Strings vibrate Closed string Open string

In many string theories the Universe is 10 dimensional, with the "extra" dimensions COMPACTIFIED. All HS math geeks read a book called FLATLAND: A Romance of Many Dimensions, by Edwin A. Abbott (1884) FLATLAND: a "first person" account of life of a 2-dimensional society a radical named Arthur Square figures out that space is really 3 dimensional.

So what the string theorists say is that in ordinary life we think we live in 3 dimensions, and we have to think of ways to detect the other 7. The extra-dimensions in string theory are Calabi-Yau figures:

Knitted Calabi-Yau Figures

Kepler: Empirical description of the motion of the planets Newton: Law of Gravity. Developed Calculus, derived orbits of the planets Solved the “Two-body” problem: Sun + one planet Couldn’t solve the “Three-body” problem Mechanical Universe: In Newtonian physics, objects move in perfectly determined ways

Orrery Mechanical models of planetary motions in the solar system But is the real solar system accurately described by an orrery?

"We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes." — Marquis Pierre Simon de Laplace (1749-1827)

The Equations Many thanks To Scott Tremaine’s Notes from April 2006

King Oscar II of Sweden (1829- 1907) - Prize: How stable is the universe? Jules Henri Poincaré (1854-1912) Sun (large) plus one planet (circular orbit) Stable Added 3rd body Strange behavior Not periodic

Modern approach: Solve many-body problem with computer calculations Take a distribution of mass Figure out the gravitational force on each part F=ma gives you the acceleration on each part Compute velocity of each part Move the parts a little Repeat

Kuiper belt objects Plutinos (3:2) Centaurs comets as of March 8 2006 (Minor Planet Center)

Sensitivity to Initial Conditions "A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still know the situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. But is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible...". (Poincaré)

Can we predict the motion of a single planet a billion years from now? Laplace and Newton – Yes Poincare’ – No Lorenz – 1963 – “Butterfly Effect” If a butterfly flaps its wings in Brazil, does it result in a tornado in Kansas?

Two kinds of dynamical systems Regular highly predictable, “well-behaved” e.g. baseball, golf, simple pendulum, all problems in mechanics textbooks, planetary orbits on short timescales Chaotic difficult to predict, “erratic” appears regular on timescales short compared to Liapunov time e.g. roulette, dice, pinball, weather, billiards, double pendulum The Solar System

Double Pendulum: a chaotic system

Consequences of chaos Positions of planets are not predictable on timescales longer than 100 Myr the solar system is a poor example of a deterministic universe The solar system is “Chaotic”

Fractals Geometric forms Define by a recursive rule Same on all scales Benoit Mandelbrot

Serpinski Triangle

Von Koch Snowflake

Fractals are “Self-Similar”: same when you zoom in

The Julia Set Gaston Julia, French mathematician

The Mandelbrot Set

Fractals in Nature