Astronomy Assignment – Cosmological Models

Reading The student is responsible for material in Chapter 2: The Copernican Revolution

Objectives/HW

The student will be able to:		HW:
1	Describe and illustrate the apparent motion of each of the eight planets as seen from Earth bringing special attention to the similarities and differences.	1 – 5
2	Define, illustrate, and apply the following concepts: direct or prograde motion, retrograde motion, conjunction, opposition, and elongation.	1 – 5
3	Explain and illustrate aspects of ancient geocentric models of the universe including the concepts of deferents, epicycles, and the works of Ptolemy.	6 – 8
4	Explain and illustrate the heliocentric model of the universe proposed by Copernicus including its seven main points and its own inconsistencies.	9 – 11
5	Explain and illustrate how Galileo was able to provide evidence for the validity of the heliocentric model.	12
6	Desribe Tycho Brahe’s contribution to the formation of Kepler’s Laws.	13 – 14
7	Define and apply the characteristics of ellipses: focus, semi-major axis, semi-minor axis, and eccentricity.	15 – 16
8	Define, illustrate, and apply the concepts of aphelion and perihelion.	15 – 16
9	Explain, illustrate, and apply Kepler’s three laws of planetary motion and properties of ellipses to solve problems involving orbits.	17 – 21
10	Explain, illustrate, and apply methods for determining the absolute and relative scale of the solar system.	22 – 25
11	Explain, illustrate, and apply Newton’s Laws of Motion and Universal Gravitation.	26 – 29
12	Compare and contrast Newton’s Laws with Kepler’s Laws.	30 – 32

Problems

1. When Mars is most near to the Earth it appears brightest to us (and largest when viewed through a telescope). (a) Does this occur at opposition or conjunction? Explain. (b) Is Mars at this point moving prograde or retrograde? Explain. (c) Which other planets would behave like this and which would not?

2. (a) Which planet(s) undergo conjunction? (b) Which planet(s) undergo opposition? (c) Which planet has the smallest range of elongation? (d) Which planet has the smallest retrograde “loop”?

3. What are some key differences in the appearance of Venus and Mercury compared to the other planets?

4. For each of the following cases determine the approximate time of day or night when an outer planet is highest in the sky for an observer. (a) Opposition. (b) Conjunction. (c) 90° west elongation (west of the Sun on the celestial sphere – think “right”).

5. Although this is not normally done, let us try to apply the terms conjunction, opposition, and elongation to the Moon. NOTE: Astronomers do not use these words in this way!
(a) “Opposition” would be what phase of the Moon? (b) “Conjunction” would be what phase? (c) 1^st quarter would be what “elongation”?

6. Briefly describe the geocentric model of the universe. What aspect(s) of the celestial sphere’s appearance are most difficult to explain in the geocentric model?

7. The benefit of our current knowledge lets us see flaws in the Ptolemaic model of the universe. What is its most essential flaw?

8. (a) In Ptolemy’s model a planet’s “deferent” was used to explain what aspect(s) of the planet’s motion? (b) The “epicycle” was used to explain what aspect(s)?

9. What was the great contribution of Copernicus to our knowledge of the solar system? What was still a flaw in the Copernican model?

10. What is the “Copernican principle”?

11. Before the mid-1990’s no planet outside our own solar system had ever been detected. Since then astronomers have found evidence for planets orbiting other stars and have confirmed the existence of such. (see pp. 396 – 399 for more information) (a) How does this relate to the “Copernican principle”? (b) It is still unknown if there is another planet outside our own solar system that harbors life the way Earth does. Based on the Copernican principle would you expect that astronomers will ever find a planet teeming with life similar to Earth? Explain.

12. Name three discoveries of Galileo that helped confirm the views of Copernicus, and for each of these explain how it supports our modern views and refutes that of the ancient Greeks.

13. Tycho Brahe’s observations of the stars and planets were accurate to about the nearest 1 arc minute. Suppose Tycho was viewing the Moon at a distance of 380,000 km. (a) At that distance, a change of 1′ in the Moon’s apparent position would correspond to what actual change in the Moon’s position in space. (b) Repeat the calculation for the Sun at a distance of 150 Gm. (c) Repeat the calculation for Saturn at its nearest approach of 1200 Gm.
Hint: use the skinny triangle approximation for all three calculations.
Note: this problem is like asking how accurately you could determine an object’s position in space using Tycho’s observations.

14. How did Tycho contribute to Kepler’s laws?

15. Use the semi-major axis, a, (average distance from the Sun) and eccentricity, e, to determine the perihelion and aphelion distances for:
(a) Earth; a = 1.00 AU, e = 0.017, and (b) Mars; a = 1.524 AU, e = 0.093.

16. The average distances from the Sun for Neptune and Pluto are 30.07 AU and 39.48 AU and yet there are times when Neptune is farther from the Sun than Pluto! Use the eccentricities of the two planets, 0.009 and 0.249 respectively, to calculate how much farther, at most.

17. The asteroid Ceres has an orbital semi-major axis of 2.8 A.U. as it orbits the Sun. Determine the time for it to orbit the Sun once.

18. Suppose a certain minor planet (asteroid) orbits the Sun once every 4.00 years. What would be its average distance from the Sun during its orbit?

19. The asteroid Icarus has a perihelion distance of 0.20 A.U. and an orbital eccentricity of 0.69. (a) Determine its average distance from the Sun. (b) Determine its greatest distance from the Sun. (c) Determine its orbital period.

20. An asteroid has a perihelion distance of 2.0 A.U. and an aphelion distance of 4.0 A.U. Calculate the following orbital properties: (a) semi-major axis, (b) eccentricity, and (c) period.

21. Halley’s comet has a perihelion distance of 0.6 A.U. and an orbital period of 76 years. What is its aphelion distance?

22. Suppose there was a planet Vulcan that orbited the Sun closer than Mercury does. If Vulcan had a maximum elongation of about 10° what would be its approximate distance from the Sun? (Hint: use trigonometry or skinny triangle or construct and measure a scale diagram.)

23. Suppose two alien worlds, Uresophagus and Urradius, orbit an alien sun, Urnotsirius. Uresophagus completes an orbit in exactly one third the time that it takes for Urradius to complete an orbit. Assuming Kepler’s Laws would apply to this alien planetary system, which world is farthest from Urnotsirius and by what factor?

24. Suppose a radar signal is sent from Earth, reflects off of Mars, and returns to a detector on Earth in 704 seconds. How far away is Mars at that point? (Hint: use the speed of light for the radar signal.)

25. Radio waves cannot be reflected from the Sun, but radar can be reflected from the surface of planets. Explain how radar measurements of distance from Earth to planets like Venus or Mars allows us to determine (indirectly) the distance from Earth to the Sun.

26. According to Newton’s 3^rd Law a baseball attracts the Earth just as much as the Earth attracts a baseball. If this is so, why does a baseball fall toward Earth and not Earth toward the baseball?

27. Consider our planet Earth as it moves (at essentially constant speed) in its orbit (essentially circular) about the Sun. (a) According to Newton’s 1^st Law, what would happen to Earth if the Sun’s gravity were suddenly “switched off”? (b) According to Newton’s 2^nd Law the Sun’s gravity should cause the Earth to accelerate toward the Sun and yet we never get closer to the Sun. If this is so, what is the effect of the Sun’s gravity on the Earth?

28. Consider the two planets Venus and Earth. The two planets are about equal in mass and size but Venus has a smaller orbit. Use Newton’s Laws to explain why Venus must move at a greater speed in its orbit than does Earth.

29. According to Newton’s Law of Universal Gravitation, everything in the solar system is attracted toward the Sun. And yet, there are many celestial bodies in the solar system that move away from the Sun – for example a planet moving toward aphelion or a comet heading back out into deep space. Use Newton’s Laws to explain how an object can move like this.

30. According to Kepler’s 2^nd Law a planet must move faster as it gets closer to the Sun so that the area it “sweeps out” will remain constant for a given interval of time. How do Newton’s Laws explain the increase in a planet’s speed as it gets closer to the Sun?

31. What are two modifications that Newton made to Kepler’s laws?

32. What does it mean to say that Kepler’s laws are empirical? Are Newton’s laws also empirical?

Objectives/HW

Problems

Selected Answers