1. Five of the six worlds orbiting Kepler-11, a sunlike star 2,000 light-years distant in the constellation Cygnus, orbit closer to their parent star than the Mercury-Sun distance. All six are larger than Earth and are likely composed of mixtures of rocky material and gas.
The Kepler mission has now identified over 1200 exoplanet candidates in a field of view that covers only about 1/400th of the sky.

2. Homework #3 is due Tuesday, Feb. 15, 5:00 pm.
Exam 1: Wednesday, Feb. 16
Review session: 7:00-9:00 pm, Tuesday Feb. 15 Swain West 007

3. It did a reasonably good job explaining these motions.
What causes the observed motions of the stars, sun, moon, and planets in the sky?
The Greeks developed a model for the Universe that lasted for nearly 15 centuries.
It did a reasonably good job explaining these motions.

4. Geocentric Model Earth is at center Sun orbits Earth
Planets orbit on small circles (epicycles) whose centers orbit the Earth on larger circles
Planet orbits lie in approximately the same plane
Inferior planet epicycles were fixed to the Earth-Sun line
Geocentric Model

5. Over the years since Ptolemy, astronomers struggled to make his model more accurate, leading to an increasingly complicated model:
moved the earth slightly off the center of the solar system to avoid needing have the planets move at varying speeds along their orbits.
a different type of epicycle was used to explain the motions of Mercury and Venus than that used for Mars, Jupiter, and Saturn
model got extremely complex for Mercury and Mars, e.g., Mercury riding on an epicycle on an epicycle on an epicycle, with constant speed measured from a point well off-center!!

6. “If I had been present at the creation, I would have recommended a simpler design for the universe”.
- Spanish monarch Alphonso X
(13th century)

7. “Hereafter, when they come to model Heav'n, And calculate the Stars, How will they wield The mighty frame, how build, unbuild, contrive, To save appearances, how grid the Sphere With Centric and Eccentric scribbl'd o'er, Cycle and Epicycle, Orb in Orb”.
- John Milton, Paradise Lost

8. The center moves…

9. The Revolution Begins!

10. The Copernican Revolution
Copernicus, Tycho, Kepler, and Galileo.
Kepler’s three laws of planetary motion

11. Nicolaus Copernicus (1473-1543)
He thought Polemy’s model was contrived
Yet he believed in circular motion
De Revolutionibus Orbium Coelestium
Replace picture with Fig

12. Copernicus’ Heliocentric Model
Sun is at center of the Universe
Earth orbits the Sun like any other planet
Earth rotates
Circular orbits for all planets
Inferior planet orbits are smaller
Planets move at constant velocities in their orbits
Retrograde motion occurs when we “lap” Mars & the other superior planets

13. Copernicus’ Heliocentric Model
Retrograde motion occurs when we “lap” Mars & the other superior planets

14. (1) Planets, including the Earth, orbit the Sun
Retrograde Motion
(1) Planets, including the Earth, orbit the Sun
(2) Planets closer to the Sun have shorter orbital periods than planets farther from the Sun

15. As we “pass” a planet, it appears to move backwards (as seen from Earth)

16. Simpler, more “elegant” than the model of Ptolemy
BUT
it still required some epicycles in order to make accurate predictions
it was still wedded to Aristotle's circular orbit paradigm
predictions were not much better than those of Ptolemy

17. Right track?
Perhaps,
but still not quite there.

18. Tycho Brahe (1546-1601) • Charted accurate positions of planets
Greatest observer of his day
• Charted accurate positions of planets
(accurate positions of the planets were not fully available)

19. Tycho Brahe…
was motivated by inadequacy of existing predictions
made very accurate observations of positions (this was prior to the development of the telescope)
advocated a model in which Sun orbits Earth because he could not observe stellar parallax

20. The parallax problem troubled the Greeks and Tycho
The parallax problem troubled the Greeks and Tycho. It led both to reject a heliocentric universe.

21. The problem was that stars are too distant to produce a parallax large enough to be seen with the technology of those time.

22. 1600 – Tycho brought Johannes Kepler to bear on problem
1600 – Tycho brought Johannes Kepler to bear on problem. He assigned him the task of understanding the motions of Mars.
Kepler had great faith in Tycho's measurements; they placed strong constraints on model

23. Suggested webpage to visit for more insight into Tycho Brahe, Johannes Kepler, and the development of Kepler’s Laws:

24. Johannes Kepler (1571-1630) Greatest theorist of his day a mystic
there were no heavenly spheres
forces made the planets move
Developed his three laws of planetary motion

25. Kepler’s First Law
Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.

26. Ellipse: defined by points located such that the sum of the distances from the two foci is constant

27. The circle is a special form of an ellipsey
x2/a2 + y2/b2 = 1
focus
Semimajor axis = a X
Semiminor axis = b
Eccentricity
e2 = 1 - b2/a2

28. Kepler’s Second Law
A planet moves along its orbit with a speed that changes in such a way that a line from the planet to the Sun sweeps out equal areas in equal intervals of time.

29. Consequence - planets move faster when they are closer to the sun and planets spend more time in the more distant parts of their orbits

30. Kepler’s Third Law a3 / P2 = constant
The ratio of the cube of a planet’s average distance from the Sun “a” to the square of its orbital period “P” is the same for each planet.
a3 / P2 = constant

31. a3 / P2 = constant
Consequence: Planets with larger orbits have longer orbital periods.
Earth: a = 1 AU, P = 1 year
So, if we use distance in AU and time in years, the constant in the 3rd Law is 1 AU3 yr-2
Jupiter: a = AU,
P = years

32. Kepler’s Laws are extremely accurate in their predictions of planetary motions.
They are “empirical”, i.e., they are derived from experiment, experience, and observation rather than from theory or logic
Isaac Newton subsequently demonstrated that Kepler’s laws are the natural outcome of gravity.

33. Kepler’s laws were determined empirically, i. e. , by observation
Kepler’s laws were determined empirically, i.e., by observation. No theoretical basis existed for these laws. Now, we will determine an A100 analogue:
Consider a number of fixed points on a phonograph record that is spinning. Inspired by Kepler’s 3rd Law, determine the relationship between the distance from the center of the record (“a”) and the length of time it takes for that point to make one complete circuit around the center (“P”).
Think!!