1. 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

2. 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 centimeters
or about a millionth of a billionth of a billionth
of a billionth of a centimeter.

3. Strings vibrate
Closed string
Open string

4. 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.

6. 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:

9. Knitted Calabi-Yau Figures

11. 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

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

13. "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 ( )

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

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

16. 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

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

18. 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)

19. 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?

20. 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

21. Double Pendulum: a chaotic system

22. 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”

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

24. Serpinski Triangle

25. Von Koch Snowflake

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

27. The Julia Set
Gaston Julia, French mathematician

28. The Mandelbrot Set

29. Fractals in Nature