ASTRONOMY 111 LABORATORY MANUAL

Transcription

1 ASTRONOMY 111 LABORATORY MANUAL DR. TSUNEFUMI TANAKA PHYSICS DEPARTMENT CALIFORNIA POLYTECHNIC STATE UNIVERSITY DR. BRETT TAYLOR DEPARTMENT OF CHEMISTRY AND PHYSICS RADFORD UNIVERSITY FALL 2003 EDITION

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3 Contents A Laboratory Experiments 1 A.1 Celestial Coordinates A.2 Angular Resolution: Seeing Details with the Eye A.3 How Big and How Far is the Moon? A.4 The Solar System Scale Model A.5 The Shape of the Earth s Orbit A.6 Phases of the Moon A.7 The Shape of the Mercury s Orbit A.8 The Orbit of Mars A.9 Obtaining Ages for Martian Surfaces via Cratering A.10 Optics and Spectroscopy B Computer Laboratories (CLEA) 61 B.1 Astrometry of Asteroids B.2 Rotation of Mercury B.3 Jupiter s Moons C Observations 83 C.1 Constellation Quiz: Get To Know Your Night Sky! C.2 The Sun and Its Shadow C.3 Moon Observation C.4 Sunspot and Prominence Observation C.5 Observation With A Telescope C.6 Moon Journal C.7 Observation of a Planet C.8 Observation of Deep Sky Objects iii

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5 2 CHAPTER A. LABORATORY EXPERIMENTS

6 CHAPTER A. LABORATORY EXPERIMENTS 3 Name: Section: Date: A.1 Celestial Coordinates I. Introduction How do you pinpoint the position of your house on the Earth? You can specify the street address or give a pair of coordinates. You can divide the surface of the Earth into grids in the east-west direction and the north-south direction. By measuring coordinates (i.e., distances or angles) from some reference points, you can determine the exact position of your house. For example, the City of Radford is located at a longitude of 80.6 west and a latitude of 37.1 north. In this case the reference points are the meridian through Greenwich, England, the reference point for longitude, and the equator, the reference point for latitude. In astronomy we are interested in specifying the positions of objects in the sky as seen by an observer on the Earth. It is accomplished by giving a pair of coordinates in a similar manner as determining locations on the Earth. It helps to picture the night sky as an immense glass sphere with the Earth (and the observer) at its center and all of the stars and planets projected on the sphere (see Fig. A.1). This sphere is known as the celestial sphere. Observer Horizon Figure A.1: The observable half of the celestial sphere above the horizon. There are various ways to define coordinates on the celestial sphere. In this lab we are going to study two such systems: the alt-azimuth system and the equatorial system. II. Reference The Cosmic Perspective, Supplement 1, pp III. Materials Used planetarium Starry Night Backyard

7 4 CHAPTER A. LABORATORY EXPERIMENTS IV. Activity The Alt-Azimuth System Let us define some terminology. Suppose the observer is located at the center of the celestial sphere in Fig. A.2. The point directly overhead on the celestial sphere is called the zenith, while the point directly opposite of the zenith is the nadir. The horizon is the circle extending around the celestial sphere and located exactly 90 from the zenith and the nadir. Local Celestial Meridian Zenith W S N Observer Horizon E Nadir Figure A.2: The Celestial sphere. The north point (N) is located on the horizon in the direction of geographic north as seen by the observer at the center. The east (E), south (S), and west (W) points are also located along the horizon at 90 intervals. The local celestial meridian is the imaginary circle on the celestial sphere that runs from the north point, through the zenith, to the south point and through the nadir back to the north point. Now let us consider a star on the celestial sphere (see Fig. A.3). The circular arc running from the zenith through the star to the horizon at H is a vertical circle. The azimuth of the star is the angle along the horizon from the north point eastward to H. This is basically the compass direction (SSW for example), but measured in degrees. The altitude of the star is the angle of the star above the horizon along the vertical circle. The altitude is a positive number if the star is above the horizon; it is negative if the star is below the horizon. Altitude combined with azimuth can specify the position of any object in the sky. Find the altitudes and azimuths of some reference points on the celestial sphere and complete the following table (Table A.1). If an item does not have a well-defined value or range of values, then it will be represented by an. The Celestial Sphere In part of this activity, you will use the planetarium software, Starry Night Backyard. This software can be used for many purposes, but its use in this lab will be to show you the sky as it appears from Radford or any other place on the Earth. In this way, it is very much like the planetarium.

8 CHAPTER A. LABORATORY EXPERIMENTS 5 Vertical Circle Zenith Altitude N W E H S Azimuth Nadir Figure A.3: Azimuth and altitude. Table A.1: Azimuth and azimuth of celestial reference points and circles. Point or Circle Azimuth Altitude North point 0 0 South point West point Local celestial meridian 90 0 Horizon 0 to Zenith Southeast point

9 6 CHAPTER A. LABORATORY EXPERIMENTS The Earth itself rotates counterclockwise 15 every hour as seen from above the North Pole. Starry Night will allow you to view the sky rotate at this rate, stand time still, or rotate at a much faster rate so that you can view yearly details (or even changes over centuries). Please remember that even though our idea of the celestial sphere is a useful tool, it is not a real model of the universe. For example, although all of the stars are located at the same distance from the Earth in our model, this is not true in reality. Also, stars do move, albeit slowly, and the constellations will change, but over time scales much much longer than a human lifetime. Finally, the Earth wobbles while it rotates on its axis, much like a top, and the positions of the stars relative to our fixed points on the sphere (the north and south celestial poles and the celestial equator) will change. 1 Start up Starry Night Backyard. The program can be found under Start Programs Radford University Course Software Curie Lab Starry Night Backyard Starry Night Backyard 4. 2 In the upper left hand corner, there should be a Home location noted. Make sure that the location shown there is Radford, Virginia. 3 You will need to choose a time to observe the stars show below. It should start up at the current time as set on the computer clock. Change the date so that it is September 1 at 9:30 PM. Hit the Stop button on the time controls to fix time at this moment - it s the filled in square in the upper left hand corner to the right of the time. 4 You need to now make some adjustments to the program to make things easier. First, right click anywhere in the dark background and a menu will appear. Select Small City Light Pollution. This will decrease the number of visible objects in the field of view. 5 On the left hand side of the window, you will see a number of tabs including Find and View Options. Click on the View Options tab. Inside of that you will see a number of sub-categories. Select Constellations. Turn on Stick Figures and Labels. In the Stars sub-category turn on Labels as well. 6 Record in Table A.2 the azimuth and altitude of the stars listed there. You can move around in the field of view by left clicking anywhere on the field of view and dragging the mouse in the direction you wish to view. To get started, you can find Vega almost directly overhead. Move the field of view so that you are looking directly overhead. Right click on Vega. Choose Show Info from the menu. The tabs should open. You can find the altitude and azimuth under the submenu Position in Sky. 7 If you cannot easily find the star or object you are searching for, open the Find tab. Type in the first few letters of the object and a list of matching items will appear. If the item name is in bold it is up and visible. If not, it is below the horizon. You can still get the information for this item by right clicking on the name of the object. The Equatorial System Before we learn about the equatorial system of coordinates, we need to define a few more reference points and circles in the sky. You are undoubtedly aware of the rising and setting of the stars. However, you may not be aware that the stars appear to be rotating about a fixed point in the sky directly above the north point on the horizon. This fixed point is called the north celestial pole (NCP). The north celestial pole is in the direction of the Earth s rotational axis, and it is the point on the celestial sphere directly above the Earth s geographic north pole. The apparent motion of stars around the north celestial pole is due to the rotation of the Earth. There is a bright star called Polaris approximately at the location of the north celestial pole. The corresponding point in the the sky south of the Earth s equator is the south celestial pole (SCP). The only difference is that the stars appear to rotate counterclockwise about the north celestial pole but clockwise about the south celestial pole. The altitude of the north celestial pole is equal to the latitude of

10 CHAPTER A. LABORATORY EXPERIMENTS 7 Table A.2: Azimuth and altitude of bright stars on the celestial globe. Star Name Azimuth Altitude Vega Fomalhaut Sirius Arcturus Capella Antares Canopus the observer s location. For example, the north celestial pole is located at 37.1 altitude (and obviously 0 azimuth) in Radford. The circle on the celestial sphere which is 90 from both the NCP and the SCP is the celestial equator. The celestial equator is the imaginary circle around the sky directly above the Earth s equator. Figure A.4 illustrates the relationship of the NCP, SCP and celestial equator to the alt-azimuth system discussed earlier. In order to set up a system of coordinates on the celestial sphere, it is necessary to specify both a reference point and a reference circle. In the alt-azimuth system, the north point and the horizon were chosen. For the equatorial system, coordinates are given that are analogous to latitude and longitude on the Earth. In the same way that the Earth s equator is a reference point for latitude, the celestial equator will serve the equivalent role for the equatorial system. On the Earth, longitude is specified by measuring the angle east or west of a single point, Greenwich, England. In the same way, we must choose a refernce point to measure angles from in the east-west direction in the sky. The reference point astronomers have chosen is the vernal equinox, which is the point on the celestial sphere where the Sun crosses the celestial equator moving northward. This occurs on approximately March 21 st. The apparent path of the Sun around the sky is called the ecliptic. The circles on the celestial sphere which pass through both celestial poles and cross the celestial equator at right angles are called hours circles (see Fig. A.5). The hour circle which passes through the vernal equinox is labeled 0 h. Every successive 15 interval measured along the celestial equator constitutes 1 h. The right ascension (RA) of a star is the angular distance measured in hours, minutes, and seconds from the hour circle of the vernal equinox (0 h ) eastward along the celestial equator to the the point of intersection of the star s hour circle with the equator. The star s declination (Dec.) is the angle measured along its hour circle from the celestial equator. The declination is positive for an object north of the celestial equator and negative for an object south of the equator. The declination is 0 everywhere on the celestial equator. Right ascension and declination are analogous to longitude and latitude respectively. The main advantage of the equatorial system is that it is independent of the observer s location because it does not depend on the locally defined horizon. The equatorial coordinates are fixed on the celestial sphere and move with stars. If one expresses the position of a star in the sky in terms of RA and Dec., another observer anywhere else on the Earth, will be able to locate the star. 1 For the data in Table A.3, assume that the vernal equinox is on the local celestial meridian when looking south. Determine the right ascension and declination of points listed in the following table (Table A.3). If an entry does not have a well-defined value, put an in the appropriate blank.

11 8 CHAPTER A. LABORATORY EXPERIMENTS Celestial Equator Zenith NCP W 37.1º S E Observer Horizon N SCP Nadir Figure A.4: Celestial poles and equator. NCP Hour Circles Vernal Equinox 0 h 1 h 2 h SCP Celestial Equator Figure A.5: Hour circles.

12 CHAPTER A. LABORATORY EXPERIMENTS 9 NCP Star's Hour Circles Vernal Equinox RA Dec. Celestial Equator SCP Figure A.6: Right ascension and declination. Table A.3: RA and Dec. of points on the celestial Sphere. Point RA Dec. Zenith 0 h +37 NCP North point East point South point West point SCP Nadir

13 10 CHAPTER A. LABORATORY EXPERIMENTS 2 Using Starry Night Backyard, determine the RA and Dec. of the stars listed in Table A.4. This information is in the same Position in Sky submenu. Record only the J2000 information. Table A.4: RA and Dec. of bright stars on the celestial globe. Star Name RA Dec. Vega Fomalhaut Sirius Arcturus Capella Antares Canopus

14 CHAPTER A. LABORATORY EXPERIMENTS 11 V. Questions 1. Would you say that a star s azimuth and altitude remain fixed throughout the course of an evening? Explain. 2. Would an observer at a different location observe the same azimuth and altitude for a particular star if he were observing at the same time as you? Explain. 3. Is there a point on the celestial sphere at which an object s azimuth and altitude would not change in the course of an evening? If so, describe this point. 4. Some stars never set and are called circumpolar stars because they lie close enough to the NCP (or SCP) that they are always above the horizon. What is the minimum declination a star must have to be circumpolar as seen from Radford? VI. Credit To obtain credit for this lab, you need to turn in appropriate tables of data, observations, calculations, graphs, and a conclusion as well as the answers to the above questions. Do not forget to label axes and give a title to each graph. Show your work in calculations. A final answer in itself is not sufficient. Don t leave out units. In the conclusion part, briefly summarize what you have learned in the lab and possible sources of error in your measurements and how they could have affected the final result. (No, you cannot just say human errors explain what errors you might have made specifically.) You may discuss this with your lab partners, but your conclusion must be in your own words.

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16 CHAPTER A. LABORATORY EXPERIMENTS 13 Name: Section: Date: A.2 Angular Resolution: Seeing Details with the Eye I. Introduction We can see through a telescope that the surface of the Moon is covered with numerous impact craters of various sizes. Some craters are hundreds of kilometers across; some are less than one millimeter. But, what is the diameter of the smallest crater that you can seen on the Moon with your naked eyes? In this activity you are going to determine the smallest object (or separation) that your eyes can see at a given distance. II. Reference 21 st Century Astronomy, Chapter 4, pp. 94, 96; Appendix A5 A6 III. Materials Used fantailed chart blank sheet meter stick IV. Activities One measure of the performance of an optical instrument is its angular resolution. Angular resolution refers to the ability of a telescope to distinguish between two objects located close together in the sky. If someone holds up two pencils 10 cm apart and stands just 2 m away from you, you can tell there are two pencils. As the person moves away from you, the pencils will appear to be closer together to your eye. In other words, their angular separation decreases although their actual separation has not changed. This is the same phenomenon that makes railroad tracks appear to come together in the distance. For telescopes and most other optical instruments, the diameter of the aperture is the factor which determines the angular resolution. The finer (smaller the angle) the resolution, the better the instrument. In this lab, rather than directly measuring the angle, you will measure the spacing between lines in a grating that you can see and compare that to the distance from the grating. In this case, the higher the ratio, the better your eyes angular resolution. 1 Tape the fantailed chart (Fig. A.8) to a wall in a well-lit classroom. 2 Stand 10 m from the chart. 3 Your partner will hold a sheet of paper over the chart, hiding all but the bottom tip. Tell your partner to move the paper very slowly up the chart, keeping the paper horizontal. When you start to see the chart lines clearly separated from each other just below the paper, tell your partner to hold the paper in place. 4 Your partner will read the line spacing printed on the chart nearest to the top of the paper. 5 Repeat the measurement at 5 m.

17 14 CHAPTER A. LABORATORY EXPERIMENTS Table A.5: The distance-to-size ratio for your eye. distance (m) line spacing value (mm) distance-tosize ratio 10 5 Suppose when your classmate stood 10 m (= 10,000 mm) from the chart, she was just able to distinguish the separation of the lines spaced 4.5 mm apart. The distance-to-size ratio for her eyes is 10,000 mm (the distance to the chart) divided by 4.5 mm (the line spacing): 10, 000 mm 4.5 mm 2, 200 =. (A.1) 1 This ratio can be written as 2,200/1, the distance-to-size ratio for her eyes. This ratio is read as 2,200 to 1 and can also be written as 2,200:1. The larger the distance-to-size ratio, the more detail your eyes can see. 5 Calculate the distance-to-size ratio for your eyes. 6 How do the two ratios compare? 7 Find the average of two measurements for your distance-to-size ratio. You will be using this average value for the problems in the Questions section later. The distance-to-size ratio for your eyes determines how much detail you can see. Using the triangle method, you can estimate the sharpness (ability to see detail) of your eyesight. In Fig. A.7, O is the position of your eyes; A and B are two side-by-side lights. The distance of the observer from the lights is OA (or OB); the distance (i.e., size) between the lights is AB. In the previous example, your classmate had the distance-to-size ratio of 2,200/1. This ratio means that if she were closer than 2,200 m away from two lights separated by 1 m, she would see two separate lights. If she were farther away than 2,200 m, she would not be able to distinguish the two lights; she would see only one light. V. Questions 1. What is the farthest distance you could be from two lights, separated by 1.0 cm, and still see them as two lights?

18 CHAPTER A. LABORATORY EXPERIMENTS 15 Light A Eye O Figure A.7: Sharpness of your eyesight. Light B 2. What is the farthest distance you could be from two lights, separated by 30 m, and still see them as two lights? 3. Will you be able to distinguish two lights separated by 50 cm if you were standing 500 m from them? Show your work. 4. An automobile has headlights placed 1.2 m apart. If the car were driving toward you at night, how close to you would it have to be for your to tell it was a car and not a motorcycle? 5. The Moon is about 384,000 km from the Earth. What is the diameter of the smallest crater that you could see on the lunar surface?

19 16 CHAPTER A. LABORATORY EXPERIMENTS VI. Credit To obtain credit for this lab, you need to turn in appropriate tables of data, observations, calculations, graphs, and a conclusion as well as the answers to the above questions. Do not forget to label axes and give a title to each graph. Show your work in calculations. A final answer in itself is not sufficient. Don t leave out units. In the conclusion part, briefly summarize what you have learned in the lab and possible sources of error in your measurements and how they could have affected the final result. (No, you cannot just say human errors explain what errors you might have made specifically.) You may discuss this with your lab partners, but your conclusion must be in your own words.

20 CHAPTER A. LABORATORY EXPERIMENTS mm 5.5 mm 5.0 mm 4.5 mm 4.0 mm 3.5 mm 3.0 mm 2.5 mm 2.0 mm 1.5 mm 1.0 mm 0.5 mm Figure A.8: The fantailed chart for measuring the distance-to-size ratio.

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22 CHAPTER A. LABORATORY EXPERIMENTS 19 Name: Section: Date: A.3 How Big and How Far is the Moon? I. Introduction Have you ever wondered how we know the size of and distance to a far away object in outer space, such as the Moon, without actually getting out there and measuring it directly? Astronomers use the principles of geometry to estimate the size and distance. In this activity we are going to learn techniques used by astronomers to measure the sizes of and distances to distant objects. II. Materials Used 1 meter stick 1 index card 2 protractors 2 pins 1 ruler III. Activity Similar Triangles Many astronomical calculations are based on the geometry of similar triangles. The two triangles in Fig. A.9 are similar because the three angles of the small triangle are the same as the three angles of the large triangle. Angle θ equals angle θ ; α equals α ; and β equals β. Therefore, the lengths of the sides of the small triangle are proportional to the lengths of the sides of the larger triangle. A' O θ α β A B O' θ' α' β' B' Figure A.9: Similar triangles. In the following activities you will be estimating the sizes of and distances to faraway objects, ranging from buildings on campus to the Moon, by using the size of and the distance to a known object. 1 Tape an index card to a wall so that the longer side is oriented vertically. 2 Stretch one arm out to full length and make a fist. Straighten your index finger, keeping the rest of your hand closed in a fist. Position your hand so that your index finger is straight up and down. With your arm stretched in front of you, close one eye.

23 20 CHAPTER A. LABORATORY EXPERIMENTS O A Finger Width A' Card Width Eye Eye-to-Finger Distance B Eye-to-Card Distance B' Index Card Figure A.10: Your eye the width of your finger forms a triangle similar to that triangle formed by your eye and the width of the card. 3 Facing the wall, stand at where the upper joint of your index finger appears to just cover the width of the card (see Fig. A.10). 4 Measure the distance from your finger to your eye with the meter stick. This may call for the help of your partner. Keep your arm straight and horizontal! Be careful when the meter stick is near your eye! Table A.6: The distance and size of your index finger. Eye-to-Finger Finger Width (cm) Eye-to-Finger Distance Distance (cm) Finger Width 5 Measure the width of the upper joint of your index finger. 6 Calculate the ratio of Eye-to-Finger Distance to Finger Width by dividing the former by the latter. Note that this ratio is just the number of fingers need to get from your eye to your outstretched index finger. 7 With the help of your partner, measure the distance from your eye at the location where your finger appears as wide as the index card to the index card on the wall. Table A.7: The distance and size of the index card. Eye-to-Card Card Width (cm) Eye-to-Card Distance Distance (cm) Card Width 8 Measure and record the width of the card and determine how many Card Widths fit between your eye and the card. As before, this is just the ratio of the distance between your and the card compared to the width of the card. 9 What are the similarities between the ratio of Eye-to-Finger Distance to Finger Width and the ratio of Eye-to-Card Distance to Card Width?

24 CHAPTER A. LABORATORY EXPERIMENTS Repeat the procedure so that your partner can determine her or his own Eye-to-Finger relationship. In Fig. A.10 line AB is the width of your index finger and line A B is the width of the card. The distance from your eye to your finger is OB and the distance from your eye to the card is OB. The triangles AOB and A OB are similar triangles: the angles in AOB are equal to those in A OB. This equality means that the ratio of the height to the base of each triangle is the same. Eye-to-Finger Distance Finger Width = Eye-to-Card Distance. (A.2) Card Width Apparent Size and Distance 1 Now stand at a position where the index card is just covered by two index finger widths. Has the apparent size of the card increased, decreased, or stayed the same? 2 What has happened to the distance between you and the card? 3 Measure the distance between you and the card with the meter stick and compare this distance to the distance measured in the previous part. This illustrates the following relationship: Eye-to-Finger Distance 2(Finger Width) = Eye-to-Card Distance. (A.3) Card Width 4 Move to a position where the card is just covered by only half of your finger width. Has the apparent size of the card increase, decreased, or stayed the same? 5 What has happened to the distance between you and the card? 6 Measure the distance between you and the card with the meter stick and compare it with the distance measured in Part A.

25 22 CHAPTER A. LABORATORY EXPERIMENTS This comparison is illustrated by the following ratio: Eye-to-Finger Distance 1 2 ( Finger Width) = Measuring the Size and Distance Using Similar Triangles Eye-to-Card Distance. (A.4) Card Width Now you have a tool to determine the distance from you to some distant object if you can estimate the object s size. You can also determine the size of a distant object if you can estimate its distance. 1 Find an object outside your classroom window (such as a tree or a building) that you can just cover with the width of your index finger when your arm is fully extended. 2 How many of the objects would fit in the distance between you and the object? Use Eq. (A.2) to find the ratio of the Eye-to-Object Distance to the Object Width. 3 Determine the distance to the object by a rough estimate of the width of the object in meters. Measuring the Distances by Triangulation The technique using similar triangles only allows you to find the ratio of the object s distance to its size. You cannot find the distance and the size separately. So how can we find the distance to a distant object? We can use the following fact: a relatively close object appears to move with respect to a more distant background as the location of the observer changes. This is called parallax, and the technique which uses parallax to estimate the distance to the object is called triangulation. We are going to learn triangulation in this part of the activity. 1 Tape two protractors to a meter stick so that the centers of the protractors are 50 cm apart as shown in Fig. A.11. This distance is called base line. Be sure that the straight edges of the protractors lie along the edge of the meter stick. 2 Place a straight pin at the center of each protractor. Sight from one of the vertex pins to the object of interest in the classroom. Place another pin along the curved edge of the protractor so that it also lies along this line of sight. You can now read off the angle to the object. Repeat the procedure at the other end. The two sight angles do not have to be equal to each other. Now you know two angles and one length. This is sufficient information to determine all other lengths in the triangle. One way to obtain the distance to the object is to draw a similar triangle but with a more convenient scale. For example, you could let 1 cm represent 1 m. You can then measure any distance that you want on the scaled triangle and convert your measurement back to the actual size.

26 CHAPTER A. LABORATORY EXPERIMENTS 23 Protractor Left Sight Angle Eye Base Line Meter Stick Distance to Object Right Sight Angle Distant Object Figure A.11: Finding the distance to an object by triangulation. Table A.8: Sight angle to the object. Left Sight Angle Right Sight Angle 3 On a large piece of paper, draw a scaled triangle. First, pick an appropriate scale factor so that the triangle will fit on the paper, then draw the base line. Next, construct the measured angles at each end of the base line. 4 Draw a straight line from the intersection of these two lines to the base line at a right angle. Measure the scaled distance from the base line to the object. Table A.9: Scaled Distance and Actual Distance to the Object. Scale Factor Scaled Distance Estimated Distance Actual Distance 5 Estimate the distance to the object by multiplying the distance on the scaled drawing by the scale factor you have chosen for the scaled drawing. Check your result by actually measuring the distance with the meter stick. 6 Does your accuracy depend on the distance to the object? Explain the reasoning behind your answer.

27 24 CHAPTER A. LABORATORY EXPERIMENTS IV. Questions For the following questions, assume a standard Ratio of Eye-to-Finger Distance to Finger Width of 40 to You are standing in a park on a hill outside Boston, Massachusetts. At this position, the width of an index finger will just cover the height of the John Hancock building found in downtown Boston. The visitor s guide to the city states that the Hancock building is 240 m tall. Estimate your distance from the building. 2. You attend the launch of the space shuttle from Cape Canaveral, Florida. The observing site is 11 km from the launching pad. The shuttle (with fuel tank) appears about one fifth of an index finger width tall. What is the height of the shuttle in meters? V. Credit To obtain credit for this lab, you need to turn in appropriate tables of data, observations, calculations, graphs, and a conclusion as well as the answers to the above questions. Do not forget to label axes and give a title to each graph. Show your work in calculations. A final answer in itself is not sufficient. Don t leave out units. In the conclusion part, briefly summarize what you have learned in the lab and possible sources of error in your measurements and how they could have affected the final result. (No, you cannot just say human errors explain what errors you might have made specifically.) You may discuss this with your lab partners, but your conclusion must be in your own words.

28 CHAPTER A. LABORATORY EXPERIMENTS 25 Name: Section: Date: A.4 The Solar System Scale Model I. Introduction Our solar system is inhabited by a variety of objects, ranging from a small rocky asteroid only a few meters in diameter to the Sun whose diameter is 1,390,000 km. Each object has its own unique characteristics. This lab is a brief tour of the solar system and will help you become familiar with our neighboring planets. II. Reference The Nine Planets ( III. Materials Used calculator geometric compass meter stick large piece of paper IV. Activities How Big Are Other Planets? Planets come in various sizes. How big are other planets, such as Mars, compared to the Earth? Because planets are so much larger than objects we regularly interact with, we will use a scale model to get a more intuitive feel for the sizes of objects in the solar system. Let us shrink the solar system so that the diameter of the Earth becomes 1 cm; i.e., we will use a scale factor of 1 cm equals 13,000 km. 1 To find the scaled size of a planet in cm, divide the actual distance in km by the scale factor 13,000 km/cm. For example, Mercury s diameter is 4,900 km. Then, ( ) 1 cm Mercury s scaled diameter = 4, 900 km = 0.38 cm. (A.5) 13, 000 km 2 Find the scaled diameters for all planets and Sun and complete the following table. 3 What is the largest planet? Smallest? 4 Draw a circle corresponding to the scaled diameter of each planet on a large piece of paper using a compass.

29 26 CHAPTER A. LABORATORY EXPERIMENTS Planet Table A.10: The scaled diameters of planets. Actual Scaled Planet Actual diameter diameter diameter (km) (cm) (km) Mercury 4, Saturn 120,000 Scaled diameter (cm) Venus 12,000 Uranus 51,000 Earth 13, Neptune 50,000 Mars 6,800 Pluto 2,300 Jupiter 140,000 Sun 1,400,000 Venus Sun Mercury 1 AU Earth Mars Figure A.12: The inner solar system.

30 CHAPTER A. LABORATORY EXPERIMENTS 27 How Far Away Is Pluto? Planets do not collide with each other because the solar system is mostly empty and because the planets circle around the Sun at different distances at different rates. The path of a planet around the Sun is called its orbit. All planets orbit the Sun in the same direction as the Earth (counterclockwise as seen from above the north pole). To measure the distance from the Sun to a planet, astronomers use the distance standard called the astronomical unit (AU). One AU is defined as the average distance between the Sun and the Earth, 150 million km. 1 AU = km (A.6) In astronomical units, the distance from the Sun to Mercury can be expressed as 0.39 AU. In this part of the lab you are going to experience the vast size of our solar system. 1 The solar system is a big place. It is too big for us appreciate its size in the classroom. So, let us shrink the entire solar system. This time we are going to pick a scaling factor such that the Earth is 1 mm in diameter, ie. 13,000 km equals 1 mm in our scale model. The distance between the Sun and the planet (or the orbital radius) in the scaled solar system can be found by using this conversion to convert from kilometers into millimeters. For example, ( ) ( ) 1 mm 1 m Earth s orbital radius = km = 12, 000 mm = 12 m. (A.7) 13, 000 km 1000 mm 2 While we could do the above conversion for each planet, there is an easier way. We know that the Earth s actual distance of 1 AU is the same as 12 m in our scaled model. The scaled orbital radius to a particular planet can then be more easily found by multiplying the scaled radius for the Earth (= 12 m) by the actual orbital radius of the planet in astronomical units. Calculate the scaled orbital radii for all planets and record in Table A Next, you are going to express all distances in terms of your average stride size. In a hallway, mark a starting point and casually walk forward 10 strides. Mark the ending point. Using a meter stick, measure the total distance between the starting and ending point. Divide this distance by 10 to determine your average stride size. 4 Divide the distance between the Sun and the Earth in the shrunken solar system by the average stride. Now you have the Earth s distance from the Sun in the unit of your stride. We can find the distance from the Sun to another planet by multiplying the distance in the astronomical unit by the number of strides to the Earth. For example, suppose the distance to the Earth is equal to 18 strides. Then, the distance to Saturn is 5 Complete the third column of Table A strides 9.54 = 172 strides. 6 Go outside and take a piece of chalk with you. Find a straight section of sidewalk. Mark the position of the Sun. 7 Take an appropriate number of strides toward Mercury. Mark the position on the ground with chalk. Keep walking till you are at the Venus position. Keep marking the positions of the planets up to Saturn. While doing this, recall that in this scale model, the Earth is only 1 mm in diameter!

31 28 CHAPTER A. LABORATORY EXPERIMENTS Table A.11: The scaled orbital radii of planets. Planet Actual radius (AU) Scaled radius (m) Scaled radius (strides) Mercury 0.39 Venus 0.72 Earth Mars 1.52 Jupiter 5.20 Saturn 9.54 Uranus Neptune Pluto Table A.12: Average stride. Total Average distance stride size (m) (m)

32 CHAPTER A. LABORATORY EXPERIMENTS 29 8 Describe what happens to the distance between two consecutive planets as you walk away from the Sun. How Old Would I Be On Mercury? Each planet takes a different amount of time to orbit around the Sun. We call that time a year or the orbital period. It takes the Earth days to go around the Sun once. In contrast, it takes only 88.0 days for Mercury. Therefore, one Mercury-year is equal to 88.0 days. Similarly, one Jupiter-year is equal to 11.9 Earth-years. 1 Convert your age in the Earth-years to Mercury-years. your age in Mercury-years = (your age in Earth-years) ( ) 365 days 1 Earth-year ( ) 1 Mercury-year 88.0 days (A.8) 2 Repeat the conversion for other planet-years. Table A.13: Your age on other planets Planet Orbital period (Earth-years) Your age (planet-year) Mercury 88.0 days Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto 225 days 365 days 687 days 11.9 years 29.5 years 84.0 years 165 years 248 years

33 30 CHAPTER A. LABORATORY EXPERIMENTS V. Questions 1. A typical person walks at 4.8 km/h. At this speed, how long does it take you to get to the Moon? The orbital radius of the Moon is 384,000 km. 2. The nearest star to our solar system is Alpha Centauri. It is over 7,000 times the distance between the Sun and Pluto. Find the distance to Alpha Centauri in the scaled model of the solar system in which the Earth s diameter is 1 mm. 3. Now imagine that you are asked to create a model of the solar system that can fit on a single sheet of paper along its long axis. If the paper is 279 mm long, what scaling factor should you use such that Pluto s orbit just fits on the sheet of paper? How big would the Earth be in this scale model? VI. Credit To obtain credit for this lab, you need to turn in appropriate tables of data, observations, calculations, graphs, and a conclusion as well as the answers to the above questions. Do not forget to label axes and give a title to each graph. Show your work in calculations. A final answer in itself is not sufficient. Don t leave out units. In the conclusion part, briefly summarize what you have learned in the lab and possible sources of error in your measurements and how they could have affected the final result. (No, you cannot just say human errors explain what errors you might have made specifically.) You may discuss this with your lab partners, but your conclusion must be in your own words.

34 CHAPTER A. LABORATORY EXPERIMENTS 31 Name: Section: Date: A.5 The Shape of the Earth s Orbit I. Introduction Does the distance between the Sun and Earth stay constant throughout the year? If so, the Earth s orbit is a perfect circle centered at the Sun. If not, when is the Earth closest to and farthest from the Sun? In this activity, you are going to determine the shape of the Earth s orbit around the Sun from the fact that an object appears larger when closer than the identical object much further away. II. Reference 21 st Century Astronomy, Chapter 3, pp (Kepler s 1 st law). III. Materials Used ruler protractor calculator graph paper compass IV. Activities On each day that you observe the Sun, you can determine its direction and its angular diameter. From the observed angular diameter you can find its relative distance from the Earth. Therefore, each date yields one point on the Earth s orbit. Connecting your data points with a smooth curve gives the Earth s orbit around the Sun. The closer an object is to you, the larger it appears. This is because it fills a larger angle as seen by your eye. For small angles, we can obtain a relationship between the size d of an object, its angular size θ, and the distance r to the object by using an approximation that simplifies the calculation. θ r r d a Figure A.13: Small angle approximation. For a very small angle θ the length of the short side of the triangle d is almost equal to the length of the arc a of a circle of radius r subtended by the angle θ. Then, d is approximately the same fraction of the circumference of the circle as θ is of 360. d 2πr a 2πr = θ 360. (A.9)

35 32 CHAPTER A. LABORATORY EXPERIMENTS Therefore, the distance r to the object is inversely proportional to its apparent angular size θ. ( ) d 360 r = 2π θ. (A.10) Table A.14 shows the direction and apparent angular size of the Sun as seen from the Earth on various dates throughout the year. Table A.14: Apparent diameter of the Sun. Date Direction of the Sun Direction of the Earth Apparent angular size (θ) Relative distance (cm) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec According to Table A.14, on January 1, the Sun is in the direction 282 east (counterclockwise) of the vernal equinox as see from the Earth. Then on the same date, the Earth must be at = 102 east of the vernal equinox as seen from the Sun. Find the direction of the Earth for each date in Table A Place a dot at the center of the graph paper to represent the Sun. Let 0 point to the right along one of the grid lines. You will use this as the direction of the Sun as seen from the Earth with respect to the background stars on the vernal equinox, assumed to be March 21 (See Fig. A.14). Notice that the Earth is 180 from the vernal equinox with respect to the Sun. 3 Draw radial lines from the Sun in each of the directions for the Earth in Table A.14 in the counterclockwise direction from 0 and label their dates. 4 The average value of the apparent diameters in Table A.14 is about Since an orbit with a radius of roughly 10 cm will nicely fit on a sheet of paper, let us choose the constant in the parentheses

36 CHAPTER A. LABORATORY EXPERIMENTS 33 Mar Earth 180 Dir. of Sun Vernal Equinox Figure A.14: The Earth s orbit around the Sun. in Eq. (A.9) to be cm. Then, the relative distance r between the Sun and Earth can be found by the following equation: r = ( cm) 360 θ. (A.11) Calculate the relative distance for each date in Table A.14. Then plot the location of the Earth on each of the radial lines. 5 Draw a best circle through your data points with a compass. The circle should pass close to each of the data points. In order to do this, you will need to move the center of the circle away from the Sun (the central dot). This circle gives you the shape of the Earth s orbit around the Sun. Be sure to label where the pointed end of the compass was as that marks the center of the orbit. V. Questions 1. Using your completed diagram, on what date is the Earth furthest from the Sun? The point where the Earth is furthest from the Sun is called the aphelion. Also, mark the aphelion on your diagram of the orbit. 2. Using your completed diagram, on what date is the Earth closest to the Sun? The point where the Earth is closest to the Sun is called the perihelion. Also mark the perihelion in your drawing. 3. Compare the answers to the previous two questions with the answers you would get using the calculated values from Table A.14. If they do not match, give an explanation as to why.

37 34 CHAPTER A. LABORATORY EXPERIMENTS 4. You have been told many times that the orbits of the planets are ellipses instead of circles. How can you explain the fact that we could draw a circle to represent the Earth s orbit? 5. During what season is the Earth the closest to the Sun for observers in the northern hemisphere? 6. The eccentricity e of an orbit is defined as e = c/a where c is the distance between the Sun and the center of the ellipse and a is the semi-major axis which is 10 cm in your case. Find the eccentricity of the Earth s orbit from your drawing. How does your value compare to the actual value of 0.017? Calculate the percent error in your measured value. % error = (measured value) (actual value) (actual value) 100% (A.12) VI. Credit To obtain credit for this lab, you need to turn in appropriate tables of data, observations, calculations, graphs, and a conclusion as well as the answers to the above questions. Do not forget to label axes and give a title to each graph. Show your work in calculations. A final answer in itself is not sufficient. Don t leave out units. In the conclusion part, briefly summarize what you have learned in the lab and possible sources of error in your measurements and how they could have affected the final result. (No, you cannot just say human errors explain what errors you might have made specifically.) You may discuss this with your lab partners, but your conclusion must be in your own words.

38 CHAPTER A. LABORATORY EXPERIMENTS 35 Name: Section: Date: A.6 Phases of the Moon I. Introduction Earth has only one natural satellite, the Moon. It is the one of the largest satellites in the solar system. It takes the Moon 29.5 days to orbit around the Earth, and it always shows the same side towards the Earth. In this activity we are going to study the most noticeable feature of the Moon, the phase. The phase of the Moon is a result of the relative angles between the Moon, Earth, and Sun. First Quarter Full Moon 12 midnight 180 Earth 12 noon To the Sun 0 New Moon Sunlight Third Quarter Figure A.15: The phase of the Moon seen from the Earth depends on the relative positions of the Sun, Earth, and Moon. II. Reference 21 st Century Astronomy, Chapter 2, pp III. Materials Used ball light bulb

39 36 CHAPTER A. LABORATORY EXPERIMENTS IV. Activities Lunar Phases 1 Turn on the light bulb. We are going to pretend the bulb is the Sun. Hold a ball at arm s length. Which side of the ball is illuminated? Which side is in shadow? 2 In Fig. A.15, shade the dark sides of the Moon and the Earth. The side facing away from the Sun is always in the dark. 3 We are going to measure all angles from the direction of the Sun (0 ) in the counterclockwise direction. Find the angle to the Moon at each location on the orbit. 4 Pretend your head is the Earth. The ball is going to represent the Moon. Hold the ball in your hand and stretch your arm. As you spin counterclockwise, the Moon orbits around you. Notice that the Moon is illuminated by the Sun from different angles with respect to the Earth. At 0, your head, the ball, and the bulb are aligned in a straight line. You can see only the dark side of the Moon. It is the new moon. 5 Now rotate counterclockwise by 45. You should be able to see a crescent moon. Sketch the phase and label the phase. Keep rotating by 45, and for each angle, sketch and label the phase. New Moon 0 Full Moon Figure A.16: Lunar phases and corresponding angles between the Sun and Moon. What time does a full moon rise? We can use Fig. A.15 to find what time the Moon of a particular phase rises or sets. Also, we can find the time of the transit. The transit is the time when the Moon, or any celestial body, is exactly on the local celestial meridian (LCM). 1 The local time is defined by the position of the Sun in the sky. When the Sun is on the LCM, it is the local noon. From Fig. A.15, find the local time for the transit for each lunar phase.

40 CHAPTER A. LABORATORY EXPERIMENTS 37 Table A.15: The transit depends on the phase of the Moon. lunar phase rise transit set new moon first quarter 12:00 noon full moon 12:00 midnight third quarter 2 The Moon rises 6 hours before the transit and sets 6 hours after the transit. Find when each lunar phase rises and sets. V. Questions 1. What is the phase of the Moon if the angle between the Sun and Moon is 150 in the counterclockwise direction? 2. What is the phase of the Moon during a solar eclipse? 3. Your younger brother swears that he saw a crescent moon at midnight. Can you trust him? Explain your reasoning. VI. Credit To obtain credit for this lab, you need to turn in appropriate tables of data, observations, calculations, graphs, and a conclusion as well as the answers to the above questions. Do not forget to label axes and give a title to each graph. Show your work in calculations. A final answer in itself is not sufficient. Don t leave out units. In the conclusion part, briefly summarize what you have learned in the lab and possible sources of error in your measurements and how they could have affected the final result. (No, you cannot just say

41 38 CHAPTER A. LABORATORY EXPERIMENTS human errors explain what errors you might have made specifically.) You may discuss this with your lab partners, but your conclusion must be in your own words.

42 CHAPTER A. LABORATORY EXPERIMENTS 39 Name: Section: Date: A.7 The Shape of the Mercury s Orbit I. Introduction Mercury is the innermost planet of the solar system and, therefore, always remains close to the Sun as seen from the Earth. It can be seen only right after sunset or right before sunrise. A simple way to determine the orbit of Mercury is to use pairs of angles measured at different locations. The angle between the Sun and Mercury as seen from the Earth is called the elongation. When the elongation reaches its maximal value as shown in Fig. A.17, the line of sight from the Earth to Mercury is tangent to Mercury s orbit. Earth's Orbit Mercury's Orbit Sun 90 Mercury θ Earth Figure A.17: The greatest elongation for Mercury. If the orbits of Mercury and Earth were both circular, the greatest elongation would be the same for every observation. however, the greatest elongation varies from revolution to revolution because of the elliptic shapes of both orbits. In this activity you are going to construct the orbit of Mercury. II. Reference 21 st Century Astronomy, Chapter 3, pp (Kepler s 1 st law). III. Materials Used protractor compass ruler graph paper