Homework 1

1. Unit 2. Problems 13, 16, 18, Unit 3. Problems 9. 18, 19, 20 Homework 1 Unit 8. Problem 20 Unit 10. Problem 17, 18 For Honors: special assignment (talk with me after the lecture if you have not done this) Reading: We covered first week Units 1-9 Jan. 30 Units 10-12

2. During the winter the temperature is lower because the Sun • A. Stops moving • B. has lower temperature • C. is farther away from the Earth • D. does not rise as high in the sky

3. Size of the Earth • Eratosthenes (296-195 b.c.e.) wanted to know the size of the Earth • He noted that the sun could be seen from the bottom of a well in Syene, so the Sun must be directly overhead • Then he measured the angle the Sun made with the horizon in Alexandria (7 degrees) • Calculated a diameter of 13,000 km, almost exactly correct!

4. In Astronomy, we will frequently estimate the sizes of planets, etc. To do this, we measure the angle that the object makes in the sky. We say that an object subtends an angle (A) in the sky For example, the moon subtends 0.5 degrees. The Sun also subtends 0.5 degrees, which is why solar eclipses are so beautiful! Measuring Angular Diameter

5. Measuring Linear Diameter • If we measure the angle subtended by an object in the sky (A), and we know the distance to it (d), we can calculate its actual, linear diameter (L)!

6. The Motion of the Planets • Because the planets’ orbits all lie in more or less the same plane, the paths of the planets through the sky all lie close to the ecliptic, appearing to move through the constellations of the zodiac • Only Pluto seems to move far from the ecliptic

7. Retrograde Motion • As the Earth catches up to the orbital position of another planet, that planet seems to move backwards through the sky. • This is called retrograde motion • Posed a frustrating problem to the ancients – if all planets moved in perfect circles, how could they move backwards, and why only occasionally?

8. Geocentric Models • Models in which everything revolves around the Earth are called Geocentric models. • From earliest Greek times, this kind of model was used to describe the heavens • Planets and stars resided on their own spheres, each tipped slightly relative to each other. This reproduced the motion of the planets and Sun through the sky. • Did not explain retrograde motion!

9. Ptolemy (100-170 C.E.) improved the geocentric models by including epicycles Planets were attached to small circles (epicycles) that rotated. These epicycles were attached to a larger circle, centered on Earth This can be visualized as a planet attached to a Frisbee, attached to a bicycle wheel with the Earth at the center. Did a fair job of reproducing retrograde motion. Epicycles

10. Heliocentric Models • Nicolas Copernicus devised a heliocentric (Sun-centered) model in which everything, including the Earth, revolves around the Sun • Retrograde motion is a natural result of these models! • Copernicus was also able to measure the relative distances between the Sun and the planets

11. It was found that Mercury and Venus were closer to the Sun than the Earth, as they were never found very far from the Sun in the sky Mercury’s greatest elongation, or angular separation from the Sun, is never more than 28 degrees Venus’s greatest elongation is never more than 47 degrees Mercury is therefore closer to the Sun than Venus Mercury and Venus

12. Built instruments to measure the positions of planets very accurately (~1 arc minute) Found that comets moved outside of the Earth’s atmosphere Witnessed a supernova and concluded that it was much farther away than any celestial sphere As he could detect no parallax motion in the stars, he held that the planets go around the Sun, but the Sun, in turn, orbits around the Earth Tycho Brahe (1546-1601 C.E.)

13. Using Tycho Brahe’s data, discovered that planets do not move in circles around the Sun, rather, they follow ellipses with the Sun located at one of the two foci! Johannes Kepler (1571-1630

14. Kepler’s First Law • Planets move in elliptical orbits with the Sun at one focus of the ellipse • Developed a heliocentric (Sun-centered) model • Did not agree with the ancients (or Brahe!) • The shape of the ellipse is described by its semi-major and semi-minor axes.

15. Kepler’s Second Law • The orbital speed of a planet varies so that a line joining the Sun and the planet will sweep out equal areas in equal time intervals • That is, planets move faster when near the Sun, and slower when farther from the Sun • Explained the non-circular behavior of the planets!

16. Kepler’s Third Law • The amount of time a planet takes to orbit the Sun (its period) P is related to its orbit’s size, a, by P2 = a3 • Kepler’s Laws describe the shape of a planet’s orbit, its orbital period, and how far from the Sun the planet is positioned. • These were empirical relationships, found from observation rather than the logic of the ancients.

17. Using a Dutch-designed telescope that he built himself, he made several startling observations that disproved ancient thinking about the Universe Found sunspots, showing that the Sun was not a perfect sphere Found craters on the Moon, showing that the Moon was not a perfect sphere Discovered four moons of Jupiter, showing that not everything revolved around the Sun Observed the rings of Saturn Observed that Venus passed through all phases, just as the Moon does. In a geocentric model, the phases of Venus were limited to crescents. One of the principal founders of the experimental method for studying scientific problems. Galileo Galilei (1564-1642)

18. Isaac Newton (1642-1727) • Isaac Newton described the fundamental laws covering the motion of bodies • Had to invent his own mathematics (Calculus) to do it! • His work is used even today in calculating everything from how fast a car stops when you apply the brakes, to how much rocket fuel to use to get to Saturn! • And he did most of it before his 24th birthday…