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name MEASURING ANGLES Fig 1: Islamic Astronomers of the Middle Ages in an Observatory in Istanbul. Fig 2: Tycho Brahe in Denmark. His observations were later used by Kepler. Image can be found in the book “History of the Modern World” by Palmer, Colton & Kramer at http://highered.mcgrawhill.com/sites/0072316551/ or http://www.unf.edu/classes/freshmancore/core1images/muslimastro nomers-MS-1.jpg Image from http://buhlplanetarium2.tripod.com/Buhlexhibits.htm or http://casswww.ucsd.edu/public/tutorial/images/history/brahe_quadrant.gif or http://zebu.uoregon.edu/~imamura/121/lecture-3/brahe.html or http://commons.wikimedia.org/wiki/Image:Tycho_Brahe_in_his_laborator y.png Have you ever thought how we determine the positions of stars, or the sizes of anything? If you want to know the size of Jupiter (or even just the moon), you can hardly fly to it with your ruler in your hand and measure it. However, the moon has a definite size in the sky. And what’s the size of the entire sky? This seems a ridiculous question, however if we think in terms of angles it gets easier. One whole circle has 360 degrees. A huge cloud that occupies about one 10th of the sky then has a diameter of 180o/10, i.e., 18o. Similarly, you can measure the diameter of the moon, which turns out to be about 0.5o. So in Astronomy we always think in angles - i.e., in “angular sizes.” Thousands of years ago we observed and measured angles in the sky, and today we still do the same - and ironically we still use pretty much the same methods. In this Lab you will learn the simplest and most basic method of how to measure angles. Measuring Angles Lab 1 1 PART I There are several methods of measuring angles and rather sophisticated instruments have been designed. Yet it is surprising how few people actually know how to determine angles with their own body parts… Maybe it is not quite as accurate, but at least it provides rather good estimates. This is how you do it – stretch out your arm. Your fist will cover an angle of about 10o, your index finder 1o, and you whole hand about 20o. Let’s check if this hypothesis. What is the angular size of your finger? a) Extend your arm in front of you. Measure the distance between your eye and the tip of your index finger. cm b) Measure the width of your index finger (across your finger nail). cm c) Calculate the angular size of your finger in degrees. Consult the TOOLKIT to figure out which trigonometric function to use. Write the trigonometric function into the box. SHOW your calculation; ZERO points otherwise! Angle width of finger Distance from eye to finger d) Rewrite your answer using the rules of significant figures e) How could you determine the uncertainty in the angular size? Make a suggestion. degrees Your estimated uncertainty (guess this) in the width of your finger is f) Rewrite your answer using significant figures and error estimates (Your answer should have the form of 13.4 ± 0.5 or so.) f) Does the One-Degree-per-Finger Rule apply to you? (In Astronomy we often use estimates. If your value falls within 30% of the expected value, your estimate is considered reasonable.) 2 Lab 1 Measuring Angles degrees ± degrees DETERMINING THE HEIGHT OF THE CAMPUS CENTER USING YOUR FINGERS Fig 3: Woodcut from 1533: surveyors are using a cross-staff to measure angles. See http://www.math.nus.edu.sg/aslaksen/teaching/heavenly.html or http://www.reformation.org/sir-francis-drake.html Whether you use your fingers, or a cross-staff like in the above picture, the basic method is the same. While the measurements with the cross-staff are more accurate, your finger is a hell of a lot ‘handier.’ a) b) Chose a tall landmark – any building will do. How many finger widths is that building? Estimate the accuracy of your measurement. (Your answer should have the form of 13.4 ± 0.5 or so.) ± fingers Go roughly halfway to that building. How many finger widths is the building at half that distance? ± fingers Go roughly quarter the way to that building. ± fingers What is the angular size of the campus center from building? ± degrees What is the angular size at half the distance to that building? ± degrees What is the angular size at quarter the distance to that building? ± degrees c) Explain in words how angular size and distance are related d) Abbreviate what you said (e.g., angular size increases/decreases as distance increases/decreases) Measuring Angles Lab 1 3 e) Insert three points into the plot below. Max # of degrees Angular Size (in degrees) Distance Quarter the Way to the tall building Half-Way to the tall building All the way to the tall building Is this plot consistent with your statement in (d)? f) Now figure out the exact relationship between the angular size, the linear size and the distance. Use your toolkit to choose the right formula. Write it down below. g) To figure out the physical height of the building we need to know how far it is. Either look up the exact distance to that building, measure it (paste it out), or estimate it. distance = _________________meters h) What is the physical height of the building in meters? Show your calculation. e) Estimate the uncertainty in that measurement. f) The height of the building is meters ± h) Do you think this is a good method to determine the height of any building? EXPLAIN. 4 Lab 1 Measuring Angles meters DO EITHER OPTION A OR B PART II — A (Sunset Labs ONLY –or– do as a Homework Assignment) There is a rule of thumb (no pun intended) that the sun moves “one finger every ten minutes.” Your assignment is to check this statement. a) Find a spot where you can watch the Sun. Bring a watch and write down the time from the watch. b) Using your fingers determine the height of the Sun above the horizon. fingers c) Estimate the uncertainty (fractions of fingers are okay) fingers d) This corresponds to ... ±... degrees: e) Predict the time the sun will set according to the 10 min per finger rule. f) Come back at Sunset; record the time g) Was it close to your prediction? (A measurement is generally regarded as “reasonable” if it falls within one third of the predicted value. For example if the predicted value is 100, any measurements within 60 and 130 are considered reasonable.) h) Calculate the theoretical value, i.e., determine how many minutes it takes the sun to move 1o. (HINT: The Sun takes 24 hours to move around the sky. This corresponds to 360o. So you need to figure out how many minutes it takes the Sun to move 1o.) SHOW calculation! i) Is the scheme of “10 minutes per finger” reasonable? (A measurement is reasonable if it falls within one third of the predicted value.) k) If it is not reasonable suggest a different wording of the “rule” ± degrees Measuring Angles Lab 1 5 DO EITHER OPTION A OR B PART II — B There is a rule of thumb (no pun intended) that the Sun and the Stars move “one finger every ten minutes”. Your assignment is to check this statement. a) Take a pencil and piece of chalk, and go outside. Find a tree, a lamppost, or some other distinct mark on the horizon and position yourself so that the Sun (or star) is directly above that landmark. Use the chalk to mark the exact position where you are standing (you will return to that position). T he position of the S un (or star) w hen you co me back some time later Initial position of the Sun (or star) T he Land M ark Us e your fingers to meas ure the dista nce the Sun (star) moved. Chalk Circle Horizon Towards W est Time of first observation. b) Come back about half an hour later to exact same location. The Sun will appear to have moved. Time of second observation. How many fingers did the Sun move?. fingers Estimate the uncertainty (fractions of fingers are okay) fingers Transform this to degrees. ± degrees How many degrees did the sun move in 10 minutes ± degrees c) Calculate the theoretical value, i.e., determine how many minutes it takes the Sun to move 1o. (HINT: The Sun takes 24 hours to move around the sky. This corresponds to 360o. So you need to figure out how many minutes it takes the Sun to move 1o.) SHOW the calculation! d) Then determine how many degrees the sun should have moved in 10 minutes. 6 Lab 1 Measuring Angles DO EITHER OPTION A OR B PART III — A (FALL SEMESTER) Go outside and look at the night sky and try to find the constellations below (look towards the Southwest, all the way up to the Zenith). You’ll see Deneb and Vega above you (Vega in you Zenith), and Altair a little further South. These are the three brightest stars in the summer sky and are often referred to as the Summer Triangle. Use your fist and fingers to determine the size of the Summer Triangle. How far is Vega from Deneb, Deneb from Altair, and Altair from Vega? Use your fist and fingers to determine the distances. ___________ ___________ ___________ Deneb is in the constellation Cygnus (or Swan), which is sometimes called the Northern Cross. The star at the other end of the cross is Alberio, a rather spectacular double star. A little further to the south you’ll see Altair in Aquila, west of that you’ll see Hercules, and further in the Southwest will be Ophichus, the Serpent Holder. Scorpio (below Serpens) will be on the Southwest Horizon, but you might only see Scorpio’s brightest Antares, which will be a little above the Horizon. Cygnus X-1 is an elliptical galaxy with a dust-ring. It has a mass black hold in its center. It is in Cygnus, between Deneb and Altair. How many finger widths is it from Deneb and Alberio? ________ ________ The Ring Nebula, M57, is worth checking. You’ll even see it in a small telescope. It is between the two fainter stars in Lyra. How far are these two stars from Vega and Altair? _______ ________ M13 is one of the larger globular clusters. It is rather spectacular and can be easily seen with a small telescope. M5 – Globular Cluster The M16 challenge M12 (top) & M10 More globular clusters Measuring Angles Lab 1 7 DO EITHER OPTION A OR B PART III — B (SPRING SEMESTER) Go outside, look at the night sky, and try to find the constellations below (look the Sout, all the way up to Zenith). You can recognize Orion by a box of four stars with three diagonal stars in the middle (Orion’s Belt). Rigel is the blue star at the bottom left, and Betelgeuse is the orange star at the top left. The 3 belt stars point down towards Sirius, the brightest star in the sky, and up towards Aldebaran, a red giant. You have now recognized three stars of the Winter Hexagon: Sirius, Rigel, and Aldebaran. Continue to Capella, which should be more or less in your Zenith. Then go East Castor and Pollux (the Heads of Gemini), then to Procyon and back to Sirius. Use your finger and fist to determine the separation between Rigel and Betelgeuse, and between Castor and Pollux: _______ ______ The Rosette Nebula (bottom red dot) and Cone Nebula The Orion Nebula is one of the most photographed nebulae in the sky. It is also the classical example of a star formation region. It is in the Sword ______ degrees down from Orion’s Belt. 8 Lab 1 Measuring Angles The Horsehead nebula is pretty, but might be difficult to see with a small telescope. LAB REPORT 1) Objective of the Lab. 2) Summarize the main results of this Lab. 3) Critically comment on this method of measuring angles. How accurate is this method? When would you use it? [Be original; do not just propose to measure heights of buildings or mountains.] Measuring Angles Lab 1 9 FOR EXTRA CREDIT DETERMINE YOUR DISTANCE BETWEEN YOU AND ANY TALL BUILDING OF YOUR CHOICE a) Find a spot from which you can see the horizon and a tall building. Write down your location. b) Write down the location of the tall building. c) From your location measure the angular height of that tall building. d) Make an educated guess of the number of floors of that building. e) Make an educated guess (in meters) of the height of each floor. f) Calculate the physical height (in meters) of that building. g) Using the measured angular height and the estimated physical height, calculate the distance (in meters) to that building. A n g le h e ig h t d is ta n c e h) Transform this to miles. i) Take a large map1 on which you see both the tall building and your location. Using a ruler measure the distance in inches. j) Convert that distance to miles. k) Compare the measured distance (g) with the calculated distance (j). Is the measured distance close to the distance obtained from the map? l) Finally comment on this method of determining distances. 1 A Google map is fine. 10 Lab 1 Measuring Angles