b. So at 12:00 p.m., are the shadows pointing in the direction you predicted? If they are not, you must explain this observation.

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1 Astronomy 100 Name(s): Exercise 2: Timekeeping and astronomy The following exercise illustrates some basic ideas about time, and how our position in the solar system uniquely configures the measurement of time. The hours and the diurnal (daily) cycle 1. a. How many degrees in a circle (such as the ecliptic around the Earth)? The ecliptic is broken up into the twelve Zodiacal constellations (e.g., Aries, Taurus, Gemini, etc.); assuming each constellation is equally large, how many degrees does each Zodiacal constellation subtend? b. How many degrees of the sky does one hour represent? Another way to look at this question: how many degrees of the sky disappear over the western horizon every hour? 2. a. We used to be able to check out the sundial on campus, which was, sadly, stolen. However, we can still observe the shadow of a flagpole or light standard. Local noon is defined as when the Sun is directly overhead (also called the Sun transit ). In what cardinal direction should the shadow be pointing? b. So at 12:00 p.m., are the shadows pointing in the direction you predicted? If they are not, you must explain this observation. The lunar cycle 3. In the figure of the Moon below, indicate the direction of the Sun with an arrow (point the arrow from the Sun to the Moon.

2 4. a. Telling time using the Moon: At what time of day would you expect to observe a full Moon rising? Over which (eastern or western) horizon would it rise? b. At what time of day would you expect to observe a first quarter Moon setting? c. At what time of day would you expect to observe a new Moon overhead? A guide for naming the phases of the Moon: If only a minimal crescent of the Moon is illuminated, it is a new Moon. If less than half of the Moon is lit, it is a crescent Moon. A half-illuminated Moon is a quarter (not half) Moon (first quarter occurs before full Moon, third quarter occurs after). If more than half is illuminated, but the Moon is not full, it is gibbous. If it is fully illuminated, it is a full Moon. When the lighted side s apparent area increases from night to night, the Moon is said to be waxing; when the apparent area decreases, the Moon is waning.

3 5. In the upper portion of the figure on the previous page (which represents looking down from the celestial north pole), eight positions of the Moon (labeled 1 through 8) are given in its orbit around the Earth. The lower part of the figure shows what an Earth observer would see of the Moon at each of the eight positions. a. For each position on both parts of the figure, shade the dark and illuminated portions of the Moon. b. For the lower part of the figure, label each Moon's phase. c. For the upper part of the figure, is the Earth rotating clockwise (CW) or counterclockwise (CCW)? Indicate this with an arrow. 6. Night after night, in what compass direction does the Moon appear to move, relative to the fixed stars? So does the Moon rise earlier or later each night? Does this mean the Moon can rise during daytime as well? 7. From your phase drawings, why isn t every new moon, when the Moon is between the Earth and the Sun, coincident with a solar eclipse? In other words, why isn t the sun eclipsed every 28 or so days when there s a new moon? Hint: a related idea is the 8. If your answer to the previous part is true, then, how do we ever see a fully lit, full Moon? Shouldn t some of its face toward us always be dark?

4 9. a. Does the Moon rotate on its own axis? Note: Rotate, not revolve. Equivalent question: Does the Moon always present the same face to us on Earth? b. So what percent of the whole Moon s surface should we be able to see and map from telescopes on the surface of the Earth? c. In fact, we have been able to map 59% of the Moon s surface from Earth-based telescopes (the rest was done by sending satellites around the back ). How are we able to see that much? Hint: look up libration. The annual cycle For the next figures, pretend that the stars are so bright and the Sun so dim that the stars can be seen during the day and do not wash out the Sun. On December 1 at noon, you are looking toward the south and see the Sun among the stars of the constellation Scorpius (informally, the Sun is in Scorpius ) as shown in the figure below. 10. Two students are discussing their answers to the question At 3 p.m. that afternoon, which constellation will the Sun appear? Student 1 (Goofus): The Sun moves from the east through the southern part of the sky and then to the west. By 3 p.m., it will have moved from being high in the southern sky to the west into the constellation Libra. Student 2 (Gallant): You re forgetting that stars and constellations will rise in the east, move through the southern sky and then set in the west just like the Sun. So the Sun will still be in Scorpius at 3 p.m. a. With whom do you agree? What argument convinced you?

5 b. So is it reasonable to pretend that the Sun is at a fixed position on the celestial sphere from one day to the very next day, and is carried along its path in the sky by the sphere s rotation? By carefully observing the sky night after night (which we will speed up in the planetarium), we find that the celestial sphere rotates slightly more than 360 every 24 hours. The figures below show the same view of the sky as the previous one, except one day later and one month later, respectively (for comparison, the gray constellations show the positions of the constellations on December 1 at the same time). Figure 3 Figure Draw the location of the Sun as accurately as possible on figure Two students are discussing their answers to the question Figure 4 shows the same view of the sky one month later on January 1. Draw the location of the Sun as accurately as possible on figure 4. Student 1 (Davey): The Sun will always lie along the dotted line in the figures when it s noon. Student 2 (Goliath): I don t know, Davey; we saw in question 10 that the Sun s motion can be modelled by assuming it is stuck to the celestial sphere. The Sun must, therefore, stay in Scorpius. Student 1 (Davey): If that were true, then by March the Sun would be setting at noon. The Sun must shift a little along the celestial sphere each day so that in 30 days it has moved to the east in the next constellation. Whom do you believe? What convinced you? And sketch the Sun in its proper location in figure 4.

6 13. Why is it reasonable to think of the Sun as attached to the celestial sphere over the course of a single day as suggested in question 10 even though we know from question 12 that the Sun s position is not truly fixed on the celestial sphere? Now consider the whole celestial sphere; the Sun s position on the celestial sphere on December 1 is shown in figure 5 below, among the stars of the constellation Scorpius. 14. a. Draw where the Sun will be located on the celestial sphere on January 1. Label this position Jan. 1. Then locate and label the Sun s position for February 1, March 1, April 1, May 1, June 1, July 1, August 1, September 1, October 1 and finally, November 1. b. Clearly, the line shown in figure 5 going through the Zodiac is the ecliptic, the pathway of the Sun. And the length of time it takes to complete one cycle is. Figure 5: The celestial sphere

7 From the viewpoint of the Earth, it appears that the Sun revolves around the Earth once per year. The Sun follows a nearly circular orbit around the Earth (from the perspective of the Earth), which the ancients named the ecliptic. The constellations along the ecliptic, called the Zodiac, allowed the ancients to predict what part of the sky the Sun inhabited during various portions of the year. The apparent motion of the Sun, from noon to noon compared to the background stars, is west to east across the sky. 15. a. When viewed from high above the North Pole of the Earth, does the Earth move around the Sun CW or CCW? How does this compare to the direction of the Earth s rotation? b. This is no mere coincidence. What is the deeper reason why the two motions (rotation and revolution) are similar? (Hint: consider the current origin theory of the solar system) 16. Given the diagram on the previous page, then, how come news reports always mention that the autumn equinox (or for that matter the vernal equinox or either of the solstices) occurs at a particular time and day? This year, the autumn equinox was at 2:04 a.m. last Friday. What definition of equinox allows this kind of precision? The seasonal cycle If the Earth s axis of rotation was oriented so that each pole received equal illumination from the Sun, we could not explain why the poles receive no sunlight six months of the year. In June, the North Pole must be tilted toward the Sun to produce daylight over an entire day in the Arctic and darkness over an entire day at the South Pole. In December, the North Pole must be tilted away from the Sun to produce daylight over an entire day in the Antarctic and darkness over an entire day at the North Pole.

8 I will use a globe and tilt its North Pole so that points towards the north wall of the room. As the globe rotates around its axis, note which pole is in complete darkness and which pole is in complete light during a full rotation. Next, I ll revolve the Earth around the Sun until the Earth reaches the opposite side of its orbit from where it started. Remember, the North Pole is always pointed toward the north wall. Again, note which pole is in complete darkness and which pole is in complete light during a full rotation. 17. a. If summer is the term used to describe the season when daylight is longer than night for a given hemisphere, should summer occur when that hemisphere s pole is tilted toward or away from the Sun? b. Look at the shadow of a stick figure on the globe during the course of a day at four different positions around the Sun (the different seasons of the year ). I ll orient the stick figure at each of these positions so that the Sun is overhead. What is the orientation of the North Pole (toward or away from the Sun) when is the shadow of the stick figure is the shortest? Which season does this correspond to? c. Are the changes in the seasons due to the North Pole is oscillating in space, alternately toward and away from the Sun over the course of one year? We know from experience that the Sun is lower in the sky at noon during the winter months than during the summer. The annual variations in the Sun s altitude and the length of the day, resulting from the tilt of the Earth s axis, cause the seasons. In the summer, long days and the high altitude of the Sun give us more daylight hours and more direct sunlight than in the winter, which means we receive a higher concentration of sunlight on the surface of the Earth in the summer than in the winter. 18. a. Consider a planet with no tilt at all (in other words, its rotational axis is exactly perpendicular to the planet-sun plane). Will this planet have seasons? b. Consider a planet that has its rotational axis parallel to the plane on which it orbits. Will this planet have seasons?

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