Astro Lecture 8 The Copernican Revolution (Cont d)

Transcription

1 Astro Lecture 8 The Copernican Revolution (Cont d) or the revolutionaries: Nicolas Copernicus ( ) Tycho Brahe ( ) Johannes Kepler ( ) Galileo Galilei ( ) Isaac Newton ( ) who toppled Aristotle s cosmos Astro Lecture 8 1

2 Johannes Kepler ( ) In the interplay between quantitative observation and theoretical construction that characterizes the development of modern science, Brahe was the master of the first but was deficient in the second. The next great development in the history of astronomy was the theoretical intuition of Johannes Kepler ( ), a German who went to Prague to become Brahe's assistant. Astro Lecture 8 2

3 Kepler and the Elliptical Orbits Unlike Brahe, Kepler believed firmly in the Copernican system. Kepler realized that the orbits of the planets were not the circles but were instead the "flattened circles" called ellipses The difficulties with the Martian orbit derive precisely from the fact that the orbit of Mars was the most elliptical of the planets for which Brahe had extensive data. Astro Lecture 8 3

4 What is an ellipse? An ellipse looks like an elongated circle. Astro Lecture 8 4

5 Eccentricity of an Ellipse Eccentricity and Semimajor Axis of an Ellipse Astro Lecture 8 5

6 Kepler s three laws of planetary motions Astro Lecture 8 6

7 Kepler s First Law: The orbit of each planet around the Sun is an ellipse with the Sun at one focus. [Greek: near the Sun] [Greek: away from the Sun] Astro Lecture 8 7

8 Kepler s Second Law: As a planet moves around its orbit, it sweeps out equal areas in equal times. A planet travels faster when it is nearer to the Sun and slower when it is farther from the Sun. Astro Lecture 8 8

9 Kepler's 2nd Law Astro Lecture 8 9

10 Kepler s Third Law The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes: Choosing subscript 1 for the Earth, the relation can be rewritten as: p 2 = a 3 with p = orbital period in years and a = average distance from Sun in AU Astro Lecture 8 10

11 Kepler s Third Law Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. More distant planets orbit the Sun more slowly than the ones that are closer - Mercury, the innermost planet, takes only 88 days to orbit the Sun - the outermost planet (Pluto) requires 248 years to do the same. Astro Lecture 8 11

12 Planets period Astro Lecture 8 12

13 Kepler s Third Law (Cont d) The only thing that affects the orbital period p of the planets is the semimajor axis a The mass, and orbital eccentricity, do not matter Astro Lecture 8 13

14 Kepler s Third Law Astro Lecture 8 14

15 Graphical version of Kepler s Third Law This graph shows that Kepler s 3 rd law hold true. The graph shows the planets that were known during Kepler s time This graph shows how the orbital speeds of the planets depend on their distances from the Sun: More distant planets orbit the Sun more slowly. Astro Lecture 8 15

16 Clicker Question An asteroid orbits the Sun at an average distance a = 4 AU. How long does it take to orbit the Sun? A. 4 years B. 8 years C. 16 years D. 64 years (Hint: Remember that p 2 = a 3.) Astro Lecture 8 16

17 An asteroid orbits the Sun at an average distance a = 4 AU. How long does it take to orbit the Sun? A. 4 years B. 8 years C. 16 years D. 64 years Clicker Question We need to find p so that p 2 = a 3. Since a = 4, a 3 = 4 3 = 64. Therefore p = 8, p 2 = 8 2 = 64. Astro Lecture 8 17

18 Clicker question: Planetary orbits When we say that a planet has a highly eccentric orbit, we mean that: 1. it is spiraling in toward the Sun. 2. its orbit is an ellipse with the Sun at one focus. 3. in some parts of its orbit it is much closer to the Sun than in other parts. Astro Lecture 8 18

19 Clicker question: Planetary orbits When we say that a planet has a highly eccentric orbit, we mean that: 1. it is spiraling in toward the Sun. 2. its orbit is an ellipse with the Sun at one focus. 3. in some parts of its orbit it is much closer to the Sun than in other parts. Astro Lecture 8 19

20 Clicker question: Comets Suppose a comet orbits the Sun on a highly eccentric orbit with an average (semimajor axis) distance of 1 AU. How long does it take to complete each orbit, and how do we know? 1. It depends on the eccentricity of the orbit, as described by Kepler's second law year, which we know from Kepler's third law. 3. Each orbit should take about 2 years, because the eccentricity is so large. 4. It depends on the eccentricity of the orbit, as described by Kepler's first law. Astro Lecture 8 20

21 Clicker question: Comets Suppose a comet orbits the Sun on a highly eccentric orbit with an average (semimajor axis) distance of 1 AU. How long does it take to complete each orbit, and how do we know? 1. It depends on the eccentricity of the orbit, as described by Kepler's second law year, which we know from Kepler's third law. 3. Each orbit should take about 2 years, because the eccentricity is so large. 4. It depends on the eccentricity of the orbit, as described by Kepler's first law. Astro Lecture 8 21

22 Problem 1 The recently discovered object Eris, which is slightly larger than Pluto, orbits the Sun every 560 years. What is its average distance (or semimajor axis) from the Sun? Astro Lecture 8 22

23 Problem 1: solution Use Kepler s 3 rd law to find the period: p 2 = a 3 Solve for a: a = p 2/3 Take p = 560 yr a = 67.9 AU Astro Lecture 8 23

24 Problem 2 Halley s comet orbits the Sun every 76 years and has an orbital eccentricity of 0.97 Find its average distance to the Sun (i.e. its semimajor axis) Astro Lecture 8 24

25 Problem 2: solution Use Kepler s 3 rd law to find the period: p 2 = a 3 Solve for a: a = p 2/3 Take p = 76 yr a = 17.9 AU Astro Lecture 8 25

26 Problem 3 Halley s orbit is very eccentric (stretchedout ellipse), so that at perihelion it is only about 90 million km from the Sun, compared to more than 5 billion km at aphelion. Does Halley s comet spend most of its time near its perihelion, aphelion, or halfway between? Astro Lecture 8 26

27 Problem 3: Solution Halley s comet spends most of its time far from the Sun near aphelion; since Kepler s second law says that bodies move faster when they are closer to the Sun than when they are farther away. Halley s comet moves most slowly at aphelion. Since it is moving most slowly there, it spends more time in that part of the orbit Astro Lecture 8 27

28 Galileo Galilei ( ) The main objections of the Aristotle view to a Sun-centered Universe were: Earth could not be moving because objects in air (birds, clouds,..) would be left behind as Earth moved along its way Noncircular orbits are not perfect as heavens should be If Earth were really orbiting the Sun, we would detect stellar parallax Astro Lecture 8 28

29 Overcoming the first objection (nature of motion): Galileo s experiments with rolling balls and dropping objects from a height showed that objects in air would stay with a moving Earth. Aristotle thought that all objects naturally come to rest. Galileo showed that objects will stay in motion unless a force acts to slow them down (Newton s first law of motion). Astro Lecture 8 29

30 Overcoming the second objection (heavenly perfection) Tycho s observations of a comet and a 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) Astro Lecture 8 30

31 Overcoming the third objection (parallax) Tycho thought he had measured stellar distances, so lack of parallax seemed to rule out an orbiting Earth. Galileo used his telescope to see that the Milky Way is made of countless individual stars: showed that stars must be much farther than Tycho thought. If stars were much farther away, then lack of detectable parallax was no longer so troubling. Astro Lecture 8 31

32 The final nails in the coffin of the geocentric model Two of Galileo s earliest discoveries contributed to the demise of the geocentric model Astro Lecture 8 32

33 1. Galileo s discovery of four moons orbiting Jupiter Galileo thus proved that not all objects orbit Earth. Page from Galileo s notebook written in His sketches show four stars near Jupiter (the circle) but in different positions at different times (and sometimes hidden from view). Galileo soon realized that the stars were actually moons. Astro Lecture 8 33

34 2. Galileo s observations of phases of Venus proved that Venus orbits the Sun and not Earth. In the Ptolemaic model, Venus orbits Earth, moving around a smaller circle on its larger orbital circle; the center of the smaller circle lies on the Earth-Sun line. If this view were correct, Venus phases would range only from new to crescent In reality, Venus orbits the Sun, so from Earth we can see it in many different phases. This is just what Galileo observed, allowing him to prove that Venus orbits the Sun. Astro Lecture 8 34

35 Galileo Galileo observed all of the following. Which observation offered direct proof of a planet orbiting the Sun? 1. Phases of Venus 2. The Milky Way is composed of many individual stars 3. Four moons of Jupiter 4. Patterns of shadow and sunlight near the dividing line between the light and dark portions of the Moon's face Astro Lecture 8 35

36 Galileo Galileo observed all of the following. Which observation offered direct proof of a planet orbiting the Sun? 1. Phases of Venus 2. The Milky Way is composed of many individual stars 3. Four moons of Jupiter 4. Patterns of shadow and sunlight near the dividing line between the light and dark portions of the Moon's face Astro Lecture 8 36

37 In 1633 the Catholic Church ordered Galileo to recant his claim that Earth orbits the Sun. His book on the subject was removed from the Church s index of banned books in Galileo was formally vindicated by the Church in Astro Lecture 8 37

38 What have we learned? How did Copernicus, Tycho, and Kepler challenge the Earth-centered idea? Copernicus created a Sun-centered model; Tycho provided the data needed to improve this model; Kepler found a model that fit Tycho s data. What are Kepler s three laws of planetary motion? 1. The orbit of each planet is an ellipse with the Sun at one focus. 2. As a planet moves around its orbit it sweeps our equal areas in equal times. 3. More distant planets orbit the Sun at slower average speeds: p 2 = a 3. Astro Lecture 8 38

39 What have we learned? What was Galileo s role in solidifying the Copernican revolution? His experiments and observations overcame the remaining objections to the Sun-centered solar system. Astro Lecture 8 39