Fall 2001: Angles and Parallax: Scientific Measurements, Data, & Error Analysis

Transcription

1 In this exercise, you will construct and use two simple devices for measuring angles and angular sizes, the cross-staff and quadrant. These instruments probably originated in ancient Greece. Using a cross-staff-type instrument, Ptolemy, the great encyclopedist of astronomy of the first century AD, made measurements of the altitudes of the sun that were accurate to fractions of a degree. Following this lead, the Arabs developed the astrolabe. Except for their work, European astronomy was greatly neglected in the Dark Ages. A renaissance of science began after the invention of the printing press. The fifteenth-century German astronomer Regiomontanus redeveloped several instruments based on his survey of ancient writings. The cross-staff (known by several names, including Jacob s staff or cross-lath) became the most common tool of navigators until the invention of the modern sextant in In using your cross-staff to map the sky, you will obtain accuracy roughly equivalent to that obtained by most of the world s astronomers up until the naked-eye work of Danish astronomer Tycho Brahe ( ) and the subsequent work of Galileo in 1609, when he first observed the sky with a simple telescope. By the beginning of the sixteenth century, new series of observations were begun, most having an error of less than one degree. The use of new and accurate instruments greatly enhanced the navigational efforts of the new world explorers. The quadrant reached its pinnacle of development with Brahe, the last great pre-telescopic observer. He built several fixed quadrants with radii of about two meters and an accuracy of one minute of arc. Brahe s exquisitely accurate naked-eye observations played a crucial role in overthrowing the 2000-year-old Ptolemaic theory of the Universe and substituting the modern Sun-centered solar system. Even today, Brahe s observations cannot be improved upon without a telescope. Later astronomers used quadrants with telescopic sights which helped them map the sky and lay the foundations of basic navigation. You will become familiar with the units of angular measurement, the various types of measuring errors, and how to achieve greater accuracy by averaging repeated measures in other words, you will learn the techniques of measurement as practiced through the centuries. Retain the instruments which you construct for use in other labs in this course. Objectives:! Learn how to make and use a cross-staff and quadrant! Take and analyze data! Distinguish between types of error and do basic error analysis! Measure angular sizes and altitudes! Investigate the relationships between angular sizes, physical sizes, and distances Equipment:! Cross-staff with meterstick (pattern, scissors, tape, stapler, heavy paper)! Quadrant with weight (pattern, scissors, tape, heavy paper, straw or pencil, string)! Cabinets of uniform physical size Set-Up: First, you should divide up into five groups. These are your collaborators for this lab. Get to know them and don t forget to credit them in your notebook! Skim through the exercise before you begin; get familiar with what you ll be doing so you have a better idea of how to arrange you notebook. Clearly mark answers to discussion questions. Page 1 of 8

2 I. ASSEMBLING THE CROSS-STAFF The cross-staff has two pieces a meterstick and sliding crosspiece. Your instructor will provide you with the pattern for the sliding crosspiece. Cut the pieces out and follow the assembly instructions above the pattern. The rectangle marked cutout is where the crosspiece slides onto the meterstick; after this occurs, fasten A to A and B to B. Your instructor will demonstrate the proper way to hold the cross-staff and how it is used to measure an angle. Resting one end of the meterstick lightly against your cheek, you can sight down the stick and line up various objects with the edges of the sliding piece by moving it back and forth. Notice that there are three sets of vertical edges on the sliding crosspiece; they are 4, 2, and 1 inches apart. These edges allow the measurement of a wide range of angles. Suppose that you wanted to measure the angle between two stars and you used the medium edges. You would slide the crosspiece until the two stars were lined up with the edges as in Figure 1. Then read off the number of centimeters from your eye to the front of the sliding piece. Using the nomograph, you can convert the meterstick reading into an angle which Figure 1. Sighting stars with the medium edges. is the angle between the two objects you were sighting. To use the nomograph, lay a straight-edge from the reading you measured (left side of nomograph), through the mark indicating which sized sights you used. Then read the correct angle at the intersection of the straight-edge with the scale on the right side of the nomograph. II. ANGULAR MEASUREMENT Angular measurement is something we do regularly without thinking about it. For example, a car that looks half as big as another one is approximately twice as far away as the other one, assuming they are roughly the same size. Thus, the larger car subtends an angle approximately twice the size of the smaller car. We use the fact that we know the approximate linear sizes of common objects such as cars, buildings, and bicycles to estimate their distances from their apparent angular sizes. The mental conversion of an observed angular size into an estimated distance is done easily by most persons for common objects that are relatively close. However, we would have difficulty in estimating the distance to the Goodyear blimp. We don t see it very often and if it is high in the sky there is no common object at the same distance to compare it to. The eye can measure only the angular separations or angular sizes of objects; it takes the mind to convert to real physical measurements. We want to make this process more systematic and quantitative, and use a simple and practical technique for measuring angles to measure distances and/or linear sizes of various objects. The basic rules describing angular size are as follows: As an object moves further away, its angular size decreases Page 2 of 8

3 If two objects are at different distances but appear to have the same angular size, the smaller object is closer. The most common unit of angular measurement is the degree (written 1 ). A circle is divided into 360 degrees. In astronomy, small angles are quite common so it becomes necessary to subdivide the degree into smaller units, called minutes and seconds of arc. There are 60 arcminutes (written 60 ) in one degree and 60 arcseconds (written 60 ) in one arcminute. Using these relations, we can convert angles into tenths and hundredths of a degree if need be. As an astronomical example, the apparent size of the full moon as seen from Earth is 0.5. By apparent size, we mean that if we measured the angle between the two edges of the moon in the same fashion as we measure the angle between two stars (or edges of an object), the resultant angle would be the angular size, or apparent size, of the moon. A very convenient unit of angular measure is the radian. There are 2π radians in 360 (a circle). Thus, one radian must equal 360 divided by 2π. Look at the diagram in Figure 2. For an object that is very far away (r is very large), you can calculate its physical size (s) given its distance (r) and angular size (θ) and vice versa. We can express this relationship as follows: θ s r 2π s o = o r θ (1) 360 where the angle θ is in degrees. For example, if we are given that the angular size of M31 (the Andromeda Galaxy) is 3 and it is 680 kpc away, we can find its physical size like so: Figure 2. Relationship between angular and physical size. s = (2π) (r) (angle /360 ) s = (2π) (680 kpc) (3 /360 ) s = 35.6 kpc So, M31 is 35.6 kpc (or ly) across. ANSWER THE FOLLOWING QUESTIONS IN YOUR LAB NOTEBOOK. 1. What is the angular size of the moon in arcminutes? 2. How many arcseconds are in one degree of arc? 3. Convert into decimal form, rounding to the nearest hundredth of a degree. 4. We learned above that the apparent size of the moon is half a degree. If its radius is 1738 km, how far away is it? (Do not confuse the moon s radius with the r above!) Page 3 of 8

4 III. SCIENTIFIC MEASUREMENTS: DATA AND ERRORS The first step in learning to be a good scientist is learning how to take good data. No measurement is ever 100% accurate and we often introduce irregularities into our data in unavoidable ways. If you time an event several times with a stopwatch, you may find that sometimes you will be slow to stop the clock while other times you will anticipate it and be too quick. If your stopwatch is faulty in some way, this may cause inconsistencies in the measurements. To get the most out of your data, you must learn how to recognize and correct for, or limit, the effects of errors. Averaging is one way to determine the best value in a series of measurements with random error, which assumes that your measurements are too large as often as they are too small. Say we make 6 measurements (let s call it N measurements where N = 6). We call them x 1, x 2,... x 6 and call the average x. Then to find the average, we sum up all the measurements and divide by how many we took (N; here it s 6). In mathematical notation, we write the average as in Equation (2). N 1 x = x i (2) N i= 1 However, not all error is random. Say your stopwatch has a built-in delay of 0.2 seconds, that is, it begins timing exactly 0.2 seconds after you start it. All of your measurements are now 0.2 seconds too fast. This is a classic case of a systematic error. If you are lucky enough to recognize this type of error, you can calibrate your instrument against a standard source, determine the correction to apply to your data, and correct for it. In the example above, we would need to add 0.2 seconds to every measurement made with that particular stopwatch. Regardless, many factors will affect the quality of your data no matter how careful you are. The errors in individual measurements will always introduce uncertainty into your final result. You know that your result is close to the actual number, but you need to quantify how close it is to the actual number. This is often done using the standard deviation, σ. The standard deviation of a dataset is the typical difference between an individual measurement and the actual value. You must first calculate the average of your dataset in order to find its standard deviation. Equation (3) shows how to calculate the standard deviation of N measurements. Your instructor 2 ( xi x) i= 1 (3) σ = N 1 will go over an example in class. Make sure you can do this by hand. DO NOT SIMPLY USE THE STANDARD DEVIATION FUNCTION ON YOUR CALCULATOR! Most calculators give the population deviation rather than the standard deviation; they are different! You can determine which your calculator gives and you should do this if you plan to use the deviation function in the future. IV. THE EXPERIMENT N Position your group such that you can measure the angular size of one of the grey metal cabinets with doors. Your instructor will let you know what distance you are to measure from (one of the following: 1.5 m, 3.0 m, 4.5 m, 6.0 m, or 7.5 m). Measure off this distance as carefully as Page 4 of 8

5 possible from the base of the cabinet. Set up tables similar to those in Figure 3 in your lab notebook in which to record your measurements. You will make three measurements at each distance and then take the average to arrive at a final measurement for each distance. Once everyone in your group has finished his/her three measurements, rotate to the next group and continue until you have measured at all five distances. When everyone is finished, average your group s numbers for each distance and calculate its standard deviation. Make sure you have converted your linear measurements to angular measurements using the nomograph before you calculate the averages and standard deviations. Record these in your table. Plot the angle in degrees as a function of distance for your group. Use a reasonable scale and include error bars the size of the standard deviations. Sketch a smooth curve that connects the average values. Do not use straight lines, since you have been measuring a quantity that varies smoothly, not in jerks (Figure 4; also see handout on graphing techniques). Distance (m) x 1 x 2 x 3 x avg Distance (m) Person 1 Person 2 Person 3 Person 4 average σ Figure 3. Sample data tables. angle (degrees) distance (m) Figure 4. The correct way to plot data. Page 5 of 8

6 ANSWER THE FOLLOWING QUESTIONS IN YOUR LAB NOTEBOOK 5. Describe the variation of the angle with distance as you determined it experimentally. Is the relationship linear or some other function? How did you determine this? 6. Does the amount of scatter in the data change with distance? Why or why not? 7. What factors contributed to the error in your class? 8. How could you improve the instrument so there is less error in your measurements? 9. If the sights on the cross-staff were closer together than they should be, would the measured angle be smaller or larger than the true value? Hint: check the nomograph! 10. What is the actual linear size of the grey cabinets, as calculated from your data? 11. The actual linear size of the cabinets is 91 cm. If you got a different answer, tell me why and then calculate the percentage error using the formula below. calculated size - actual size % error = 100% actual size V. PARALLAX One of the basic problems in astronomy is the determination of distances to celestial objects. By definition, parallax is one-half of the angle formed at the celestial body by two intersecting lines drawn from the ends of a baseline. More directly, it is the apparent shift in position of a nearby object with respect to more distant objects when viewed from two different positions (see Figure 5). As an example, hold up your thumb at arm s length and look at it first with only one eye, then with the other. Your thumb appears to shift back and forth against the background; it is this apparent shift that we call parallax. Parallax is the result of looking at an object from two different vantage points, and each eye has a different position relative to your thumb. If you move your thumb closer to your face and repeat the Figure 5. Parallax: Two views of a star S as seen from Earth, E1 and E2. experiment, you see that the apparent shift of your thumb is much larger. Therefore, the more distant an object is, the smaller is its shift, or parallax, when viewed from two different positions. In order to deal with such large distances as those we encounter in astronomy, we have defined a distance unit: if the parallax is one second of arc, the distance is said to be one parsec (a coined word consisting of the first three letters of the words parallax and second). Using a Page 6 of 8

7 E1 a 2p d E2 baseline of 2 AU, we find that one parsec equals 206,265 AU and we have the simple relationship in Equation (5). Note the similarity between this relationship (Figures 5 and 6) and that of angular size with distance (Figure 2); they are essentially the same. Make sure you can see the connection here. 1 d = (5) p Figure 6. Parallax illustrated as with angular measurement. ANSWER THE FOLLOWING QUESTIONS IN YOUR LAB NOTEBOOK 12. What is the distance in parsecs of a star with a parallax of one arcsecond? 13. What is the distance in AU of a star with a parallax of one arcsecond? 14. What is the distance to a star with a parallax of 0.1 arcsecond? VI. ASSEMBLING AND USING THE QUADRANT Your instructor will provide you with the pattern for the quadrant. Cut it out and attach it to a drinking straw (or pencil) with tape, being careful that the straight side of the quadrant template is aligned perpendicular to the length of the straw. Your instructor will show you the proper way to construct the cross-staff and how it is used. The quadrant is a device for measuring the altitude (or angular height) of an object, that is, its angular distance above the horizon. To use the quadrant, sight along or through the straw at the object whose altitude you wish to measure. Let the weight hang down freely under the influence of gravity until it stops swinging. With your finger, you can hold the thread against the quadrant and slowly move it away from your eye and read the altitude off the scale. To check that the quadrant is properly mounted on the straw, you should find someplace with a clear horizon and sight the straw toward the distant horizon. The quadrant should then read zero, since 0 is indeed the altitude of the horizon. If it does not, reset the template. The altitude of the zenith, or point directly overhead, is 90. Practice using the quadrant to measure some angles and their errors. You can do this with a friend who is taller than you, the top of a door, a building, or anything that you can stand back from a bit. While there are no formal measurements for you to make here, you should make sure you are comfortable with using the quadrant. You will need it to measure the angular heights of particular celestial objects at a later date. If you need help, ask! VII. REFERENCES Duckett, K.E. & Moore, M.A. A Laboratory Textbook for Introductory Astronomy, 2 nd Edition, Contemporary Publishing Company, 1986 Page 7 of 8

8 Hemenway, M.K. & Robbins, R.R. Modern Astronomy: An Activities Approach, Revised Edition, University of Texas Press, 1991 Hoff, D.B., Kelsey, L.J., & Neff, J.S. Activities in Astronomy, 3 rd Edition, Kendall/Hunt Publishing Company, 1992 Shaw, J.S., Dittman, M., & Magnani, L. Laboratory Textbook for Elementary Astronomy, 7 th Edition, Contemporary Publishing Company, 1996 Page 8 of 8