Ptolemy's Geocentric Model and Kepler's Laws of Planetary Motion

1. Notes & Reminders Homework #1 is due Thursday, September 6. #5) Need to calculate the semi-major axis of Eris’s orbit from the information given, not simply look it up.

2. Ptolemy’s Geocentric Model • Earth is at center • Sun orbits Earth • Planets orbit on small circles whose centers orbit the Earth on larger circles • This view of the Solar System held for 1500 years….

3. How was Greek knowledge preserved through history? • The Muslim world preserved and enhanced the knowledge they received from the Greeks. • Al-Mamun’s House of Wisdom in Baghdad was a great center of learning around A.D. 800. • With the fall of Constantinople (Istanbul) in 1453, Eastern scholars headed west to Europe, carrying knowledge that helped ignite the European Renaissance.

4. How did Copernicus, Tycho, and Kepler challenge the Earth-centered model? Nicolaus Copernicus • Proposed a Sun-centered model (published 1543) • Used model to determine layout of solar system (planetary distances in AU) But . . . • The model was no more accurate than the Ptolemaic model in predicting planetary positions, because it still used perfect circles (and still used epicycles). Copernicus (1473-1543)

5. Tycho Brahe • Compiled the most accurate (one arcminute) naked eye measurements ever made of planetary positions. • Still could not detect stellar parallax, and thus still thought Earth must be at center of solar system (but recognized that other planets go around Sun). • Hired Johannes Kepler, who used Tycho’s observations to discover the truth about planetary motion. Tycho Brahe (1546-1601)

6. Johannes Kepler • Kepler first tried to match Tycho’s observations with circular orbits -- no success • An 8-arcminute discrepancy led him eventually to use ellipses rather than circles • “If I had believed that we could ignore these eight minutes [of arc], I would have patched up my hypothesis accordingly. But, since it was not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy.” Johannes Kepler(1571-1630) Good scientist!

7. Ellipses An ellipse is the locus of points where the sum of distances from two specified points is the same.

8. What are Kepler’s three laws of planetary motion? Kepler’s First Law: The orbit of each planet around the Sun is an ellipse with the Sun at one focus. perihelion – nearest to Sun aphelion – farthest from Sun

9. What are Kepler’s three laws of planetary motion? Kepler’s First Law: The orbit of each planet around the Sun is an ellipse with the Sun at one focus.   planet Sun You will NEVER find an orbit that looks like this!

10. Kepler’s Second Law: As a planet moves around its orbit, it sweeps out equal areas in equal times. This means that a planet travels faster when it is nearer to the Sun and slower when it is farther from the Sun.

11. Kepler’s Third Law More distant planets orbit the Sun at slower average speeds, obeying the relationship p2 = a3 p = orbital period in years a = avg. distance from Sun in astronomical units (AUs) We will derive this relation in a later chapter from the law of gravity. Kepler knew this only as an empirical relation he derived from the data (he didn’t know why it worked).

12. Thought QuestionAn asteroid orbits the Sun at an average distance a = 4 AU. How long does it take to orbit the Sun? • 4 years • 8 years • 16 years • 64 years

13. Thought QuestionAn asteroid orbits the Sun at an average distance a = 4 AU. How long does it take to orbit the Sun? • 4 years • 8 years • 16 years • 64 years This form of the law only works if the period is in years and the distance in AU around the Sun (would not work for a moon around a planet, for example)

14. How did Galileo solidify the Copernican revolution? Galileo overcame major objections to the Copernican view. Three key objections rooted in Aristotelian view were: Earth could not be moving because objects in air would be left behind. Non-circular orbits are not “perfect” as heavens should be. If Earth were really orbiting Sun, we’d detect stellar parallax.

15. Overcoming the first objection (nature of motion): Galileo’s experiments showed that objects in air would stay with Earth as it moves. • Aristotle thought that all objects naturally come to rest. • Galileo showed that objects will stay in motion unlessa force acts to slow them down (Newton’s first law of motion).

16. Overcoming the second objection (heavenly perfection): • Tycho’s observations of a comet and supernova already challenged this idea. • Using his telescope, Galileo saw: • Sunspots on Sun (“imperfections”) • Mountains and valleys on the Moon (proving it is not a perfect sphere) Galileo’s 2-inch diameter telescope

17. Overcoming the third objection (parallax): • Tycho thought he had measured stellar distances, so lack of parallax seemed to rule out an orbiting Earth. • Galileo showed stars must be much farther than Tycho thought — in part by using his telescope to see the Milky Way has countless individual stars. • If stars were much farther away, then lack of detectable parallax was no longer so troubling.

18. (only crescent phase) (crescent and gibbous phases) Galileo’s telescopic observations of the phases of Venus proved that it orbits the Sun and not Earth (Mercury shows phases also).

19. Galileo also saw four moons orbiting Jupiter, proving that not all objects orbit Earth.

20. Summary of Renaissance Cast of Characters Copernicus – first to suggest a Sun-centered solar system - still insisted on circular orbits, epicycles Tycho Brahe – best observer of his day - lost his nose, had a drunk moose Kepler – first to propose elliptical orbits  Bingo! - used Brahe’s excellent data - developed 3 laws of planetary motion Galileo – first to point a telescope at the night sky - final nail in the coffin of Earth-centered cosmologies

21. A Scientific Theory is… • an established model repeatedly tested with observations and experiments • quantities involved are well known • abstract explanation of observations Contrast this with a hypothesis, which is only an unverified educated “guess”. A theory is a comprehensive explanation of a large set of observations.

22. Scientific Theory A theory is the highest “honor” that an idea can obtain. A hypothesis can become a theory after extensive testing, but a theory will never “graduate” to become a fact. It is always subject to revision in light of new evidence. E.g., Fact: a ball falls when dropped. Theory: gravitational attraction between the ball and the Earth caused the ball to fall toward the Earth

23. The Scientific Method • Question • Hypothesis • a tentative explanation • Prediction • Test • Result – becomes a theory after repeated confirmation. confirm reject, or modify

24. Hallmarks of Science A theory must not only explain all known existing observations, but needs to make new predictions that are verifiable (or falsifiable). Example of a good theory: Big Bang Theory It is not enough to just explain the previously known facts, a good theory must also predict future observations  important concept

25. Big Bang Theory Fit the available data at the time it was developed (e.g., galaxies in the Universe moving away from each other). Predicted that the radiation left over by the Big Bang should be redshifted into the microwave part of the electromagnetic spectrum in the form of a cosmic microwave background. This is exactly what was detected in 1965 by Penzias and Wilson with a microwave antenna.

26. Hallmarks of Science • Seeks explanations based on natural causes • The simplest models are favored (Occam’s Razor) • Models must be predictive and falsifiable by anyone A model that is not testable is not a useful model, nor is a model that is unfalsifiable.

27. Pseudoscience/Non-Science Fails to make testable predictions, or does not yield results beyond that expected from chance: Failed PredictionsUntestable Astrology Cryptozoology ESP/psychic ability Aliens built the pyramids Homeopathy Anthropomorphic fallacy Numerology Often, these ideas have no theoretical foundation on why they should work. These ideas cannot be proven right or wrong.

28. Is Pseudoscience Alive Today? $3 billion spent on homeopathy by Americans in 2007

29. Is Pseudoscience Alive Today? $3 billion spent on homeopathy by Americans in 2007 38 people followed Marshall Applewhite to their deaths in an attempt to hitch a ride on a spaceship behind Comet Hale-Bopp in 1997

30. Is Pseudoscience Alive Today? $3 billion spent on homeopathy by Americans in 2007 38 people followed Marshall Applewhite to their deaths in an attempt to hitch a ride on a spaceship behind Comet Hale-Bopp in 1997 December 21, 2012, the Mayan calendar, arrival of Planet X/Nibiru, Rapture dates see conspiracy websites: abovetopsecret.com godlikeproductions.com

31. Chapter S1Celestial Timekeeping and Navigation

32. Length of a Day • Sidereal day: Earth rotates once on its axis in 23 hours, 56 minutes, and 4.07 seconds. • This is the time it takes a star to be in the same position from night to night.

33. Length of a Day • Solar (or synodic) day: The Sun makes one circuit around the sky in 24 hours. • Therefore, the stars rise 3.93 minutes earlier each day  Sun appears to move thru the 12 (13) zodiacal constellations throughout the course of 1 year.

34. Length of a Day • A solar day is longer than a sidereal day by about 1/360 because Earth moves about 1in orbit each day.

35. Length of a Month • Sidereal month: Moon orbits Earth in 27.3 days. Earth and Moon travel 30 around Sun during that time (30/360 = 1/12). • Synodic month: A cycle of lunar phases; takes about 29.5 days, 1/12 longer than a sidereal month.

36. Length of a Year • Sidereal year: Time for Earth to complete one orbit of Sun 365.256363004 days • Tropical (or synodic) year: Time for Earth to complete one cycle of seasons. Tropical year is about 20 minutes (1/26,000) shorter than a sidereal year because of precession.

37. Planetary Periods • Opposition – defined only for planets outside of Earth’s orbit – Sun  Earth  Planet (outer planets only) – represents planet’s closest approach to Earth

38. Planetary Periods • Conjunction – for an outer planet defined as: Planet  Sun  Earth – represents planet’s farthest distance from Earth

39. Planetary Periods • Superior Conjunction – for inner planets only – Planet  Sun  Earth – represents planet’s farthest distance from Earth – planet is behind Sun

40. Planetary Periods • Inferior Conjunction – for inner planets only – Sun  Planet  Earth – represents planet’s closest approach to Earth – planet is in front of Sun

41. Planetary Periods • Greatest eastern/western elongation – for inner planets only – planet is largest angular distance east/west of Sun

42. Planetary Periods • Planetary periods can be measured with respect to stars (sidereal) or to apparent position of Sun (synodic).

43. Planetary Periods – Outer Planets • Difference between a planet’s orbital (sidereal) and synodic period depends on how far planet moves in 1 Earth year. 1/S = abs(1/E – 1/P) Where P = sidereal period of the planet you are observing E = sidereal period of the planet you are on (not necessarily Earth) S = synodic period of the planet you are observing

44. Planetary Periods – Outer Planets • Difference between a planet’s orbital (sidereal) and synodic period depends on how far planet moves in 1 Earth year. For inner planets: exact same formula! 1/S = abs(1/E – 1/P)

45. Planetary Periods – An Example Mars orbits the Sun every 1.88 years. How long does it take to go from one opposition to the next? 1/S = abs(1/1 year - 1/1.88 years)  S = 2.14 years (We see Mars go from opposition to opposition every 2.14 years.) What if you are given S instead of P (or E)? 1/S = abs(1/E – 1/P) 1/P = 1/E – 1/S for a superior planet 1/P = 1/E + 1/S for an inferior planet