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Measuring Distance Using Parallax Parallax is the apparent motion of a stationary object due to a change in the observer’s position. In astronomy parallax is the gold standard (best) way to measure distances. That is, until it doesn’t work any more. The accuracy of parallax measurements depends very strongly on the distance itself, as you will find out from your work. In this exercise you will use parallax to determine distances to three different objects chosen to be at VERY different distances. You will relate the accuracy of your measurements to the accuracy of the distance measurement. Angle B Angle C Angle C Baseline, Historical Background: The Greeks knew that if a Distance between points distant object is viewed from two different points, the where observations are made direction to it will change, as is shown in the figure. They (and you) could figure out all the sides of a triangle whose corners are at the observing positions and the nose of the animal given the length of the baseline and two angles. This procedure is used for finding distances and for surveying and measuring land. The Greeks tried to measure the parallax of the Sun, the stars, the planets etc. They did not get accurate parallax measurements for any of these objects. Since the stars did not seem to move AT ALL over the year they thought that the Earth stands still in the middle of a sphere of stars. Why couldn’t the Greeks measure the parallax of the stars? The problem is that as something gets further away, the apparent motion due to parallax gets smaller. Eventually it becomes too small to measure. The distance which is becomes too far to measure depends on the equipment and care used. Materials: Ruler or tape measure, protractor, calculator, paper, tape or chalk to mark the baseline. Procedure: You will measure the distance to three objects and then estimate the accuracy of these measurements. The idea for measuring the distances is to measure one side and two angles of a triangle, then you will use this information (and math) to figure out the distance to an object. The angles and distances to measure are shown in the picture. You may use EITHER type of protractor. Protractor Measure 3 times to nearest 0.1 deg Protractor, different type Because you are exploring the effect of distance on measurement accuracy, choose 3 objects at different distances, one at a distance of about 5 times the baseline, one at about 10 times, and one Much further.. Guess the distances to decide what to measure. Decide on baselines to use to measure your objects. The baseline must be AT LEAST 200 cm. You MAY use the same baseline for all three. You must be able to see the object from both ends of the baseline. Mark the ends of the baseline with tape or chalk and be certain to have something to show the direction from one end to the other (a fence or railing is good). Measuring Distance Using Parallax 1 Measure the length of the baseline to the nearest millimeter (0.1 centimeter). So the number of centimeters in the length that you measure will not normally end with a zero. You must have three different values for the length. (The length does not change, but if you measure with enough precision, your measurement errors will cause the values to be different.) Look at the object from each end of the baseline. Measure the angle between the baseline and the direction to the object by putting the center of a protractor at the end of the baseline, lining up the zero degree line with the baseline, Looking across the center of the protractor to the distant object and marking the spot on the protractor where the line of sight crosses the protractor. Decide whether the angle is acute(<90deg) or obtuse(>90deg) before reading the protractor so that the many scales on a protractor do not confuse you. SKETCH your set ups. Show your baseline, some landmarks in the background (like trees or buildings) and identify the objects which you are measuring. Do NOT try to measure of f the angles and sides but o DO be sure to show which angles are acute (less than 90 ) and which are obtuse. Record your measurements in the following table. Be sure that the baseline is measured in centimeters to the nearest millimeter and that the angles are measured and computed to the nearest 1/10 degree. Object # 1 Object # 2 Object #3 Name of Object Baseline Length Middle Value Largest Value Baseline Length Smallest Value Baseline Length Angle B Middle Value Angle B Largest Value Angle B Smallest Value Angle C Middle Value Angle C Largest Value Angle C Smallest Value Angle A Middle Value o (180 -Middle B-Middle C) Largest A Value o (180 -Smallest BSmallest C) Smallest A Value o (180 -Largest BLargest C) Getting the distance when you know two angles and a side Your measurements for each object amount to two angles and the included side (the baseline) for a triangle. Your observation points are at two of the vertices and the object which you are trying to measure is the third. This is enough information that, there is only one triangle which corresponds to the measurements. When the triangle is completed, the position of the distant object will be determined. But drawing is awkward and introduces its own errors. Measuring Distance Using Parallax 2 You could draw the triangle to scale and measure the distance on the scale picture. This is comparatively easy to think about, but the process of drawing introduces additional errors and the triangles are long and skinny. The “distance” to the object depends on whether the distance is defined to be from one of the observation points, from the baseline or from some other point. In astronomical measurements, the center of the earth is often used for nearby objects and the center of the Earth’s orbit is used for distant objects. For your experiment, you will find side b and call that the distance. As the distance gets large, it Angles B and C are known doesn’t matter much which side you choose for the distance. B Baseline length, a is Your picture may look like Figure 3. Notice the labels on the angles and the sides. All of the angles are known. Two have been measured at the observers. The third is found because: c known Baseline a The distant object is here A C A+B+C= 180 degrees b Angle A = 180 degrees - B - C Figure 3 There is a theorem (called the Law of Sines) which says that a b c = = sine A sine B sine C . The lower case letters represent the lengths of the sides of a triangle; the uppercase letters represent the angles. Side a is across from angle A. Figure 3 shows the situation. The notation “sine A” means to use a function called “sine” (abbreviated sin) and to use the value for the angle A. The sine of an angle is the ratio (answer) you get when the angle is part of a right triangle and you divide the length of the side opposite the angle by the length of the hypotenuse, so the definition is s in (a n g l e ) le n g t h o f s i d e o p p o s it e le n g t h o f h y p o t e n u s e Once you know that a triangle is a right triangle, and you know one of the other angles, you know the shape of the triangle. This occurs because the third angle must make up the 180 degrees total for the triangle. Since all the angles are known, the shape is fixed, and so are the relative sizes of the sides. The trigonometric functions just tabulate the ratios of the sides of triangles. You could think of them as tables which describe the shapes of right triangles. There is a table of the values of sine at the back of this write up. Your scientific calculator almost certainly has the sin function. Check out how it works and use the table to verify, Sine of 90 deg is 1. If you don’t get that from your calculator, check to be certain that the calculator is in degrees mode. To find the distance, we will use a b = sine A sine B Here a is the baseline, which you know. The angle B is one of the measured angles, which you know. The angle A is known because A = 180 degrees - B - C, and you known B and C both. The values of sine A and sine B are found by looking at the table in the back of the write up. Thus you will know the values of all the quantities except b, the distance from one of the observers to the distant object. This is exactly what was wanted. So now you can solve the equation and find the distance Measuring Distance Using Parallax 3 b sinB a sinA distance sinB baseline sinA Example Problem: The measurements are as follows: Baseline = 204.2 cm, 203.7 cm, 205.5 cm Angle B = 87.2 degrees, 88.1 degrees, 86.9 degrees Angle C = 85.4 degrees, 85.9 degrees, 85.1 degrees Let’s just consider the middle values. First compute angle A as 180 degrees= A + B+ C 180 degrees= A + 87.2+85.4 A = 180 degrees – 87.2 degrees – 85.4 degrees A= 7.4 degrees So now you are prepared to find sine A and sine B. Your calculator is the easiest way (or you can interpolate using the table at the back). sine A = sine 7.4 degrees = 0.128796 sine B = sine 87.2 degrees = 0.9988. The sines have no units. The lengths of the two sides had the same units as one another and the units cancel. Using the formula distance sinB and substituting the known values. baseline sinA 204.2cm 0.12876 dis tan ce 1583.565cm dis tan ce 0.9988 So the distance to the object is found. The numerical value 1583.565 cm is not really correct. We do not know the contributing quantities well enough to say the distance to the nearest thousandth of a centimeter. You only measured the baseline, 204.2 cm to four digits, and the variation among the measurements tells us that the uncertainty is about 0.9cm (so more like 3 significant figures). The B values in this example vary by around 0.6 degree(half the difference between largest and smallest). This shouldn’t be expressed as a percentage,since the measurement error probably would have been about the same number of degrees regardless of the value of b or C But the sines don’t change as much. Sine (87.2 degrees) =0.998806 Sine (88.1 degrees) =0.99945 Sine (86.9 degrees) =0.998537 So here the variation is about ±0.0005 in the sine. This is like 3 significant figures. So we might expect that 3 the answer should have only 3 significant figures. We would be better off expressing it as 1.58x10 cm. You will assess extreme values possible with your set of measurements. They may be further off than you would expect from the number of significant figures. Take another look at the formula for the distance Measuring Distance Using Parallax 4 distance sinB baseline sinA The bigger the values of sine B and the baseline, the bigger the distance that will result. The smaller the value of sine A, the bigger the distance (since we divide by sine A). So choosing the largest sine B and baseline and the smallest sine A will result in the largest value of the distance. Conversely, the smallest sine B, smallest baseline and the largest sine A result in the largest distance. The table that follows helps you to organize your work to compute these extreme values of the distances. Use your measurement data to compute the distance to each body. Record your work in the following table. Be sure to write the units for each of your quantities, Distance Formula- choose max or min Di st values to fit formula Middle baseline* Middle(sine B) /Middle(sine A) # 1 Max baseline* Max(sine B) / Min(sine A) 1 Min baseline * Min(sine B) / Max(sine A) 1 Middle baseline* Middle(sine B) Middle(sine A) 2 Max baseline* Max(sine B) / Min(sine A) 2 Min baseline* Min(sine B) /Max(sine A) 2 Middle baseline* Middle(sine B) /Middle(sine A) 3 Max baseline* Max(sine B) / Min(sine A) 3 Min baseline* Min(sine B) /Max(sine A) 3 Baseline Length Angle B Angle A Sine B Sine A Measuring Distance Using Parallax Distance 5 1) Which of your distance measurements has the greatest percentage of uncertainty? (hint, per cent error 100 (value official value) Treat the Middle value for each distance as the official. official value Negative values do matter, since that means a value is smaller than expected. But a negative value is considered comparable to a positive one. E.g. A value with -10% error has larger error than one with +9%) 2) If you tried to measure another object which is even more distant using the same equipment which you were already using, would your answer become more or less accurate? Distance Best Estimate (the middle value from the previous table) Plus (the difference between the best estimate and the largest) or Minus amount (the difference between the best estimate and the smallest) #1 #2 #3 3) How accurately do you think that you could measure the angles in this experiment? (Hint:: Look at the variation among the repeated measurements of the same angle B or C. The variation is due to your personal measurement error. Calculate the difference between largest and smallest measurement of the same quantity. If you find half the difference, that is a measurement of the uncertainty inthe measured value. You have 6 different B and C angles and all of them are measured the same way. That is like having 6 different samples of your measurement process. Look at all these uncertainties and choose value that encompasses the general run of values.) How accurately could you determine the value for the angle A, the angle at the distant object (the one which you did not measure, but which you computed from the other two)? (a number of degrees please!) How accurately could you measure the length of the baseline? (a number of centimeters please!) Which of these factors, the baseline, angle A, or angle B is the largest contributor to the uncertainty in the distance which you found? 4) Suppose you want to measure the distance to a distant light pole. Would you get a more accurate value using a 25 foot baseline or one with a 10 foot baseline? Assume that you can get the baseline laid out with no trouble and that you are using the same equipment that you used for the rest of this experiment. Extra Credit The Greeks wanted to know the distance to the Moon and to the stars. Suppose that they had stationed two persons on opposite sides of the Earth, and both of the persons observed the Moon against the stars at the same time. Draw a sketch of how two people would set up this measurement so that they would be as far apart as possible. Look up how far away the moon is. How large would angle A be (a numerical value please)? Do you think the Greeks could have observed this change? Why or why not? What difficulties would there have been in measuring the distance to the Moon with this method? Measuring Distance Using Parallax 6 Sine Table o What if the angle is MORE than 90 ? The value of the sine starts to decrease from its high of 1. The way o that it works is that sine x = sine (180 -x). For example o o o o sine 95 =sine(180 -95 )=sin 85 . Your scientific calculator will find sines and relieves the problem of what to do with the tenths of degree). Check at least one value to be certain that the calculator is in the correct mode. Cheap calculators have no problem. Some calculators have several modes. Be certain to use DEGREES mode. Angle, Degrees 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Sine A Angle, Sine A Degrees 0 34 0.559192903 0.017452406 35 0.573576436 0.034899497 36 0.587785252 0.052335956 37 0.601815023 0.069756474 38 0.615661475 0.087155743 39 0.629320391 0.104528463 40 0.64278761 0.121869343 41 0.656059029 0.139173101 42 0.669130606 0.156434465 43 0.68199836 0.173648178 44 0.69465837 0.190808995 45 0.707106781 0.207911691 46 0.7193398 0.224951054 47 0.731353702 0.241921896 48 0.743144825 0.258819045 49 0.75470958 0.275637356 50 0.766044443 0.292371705 51 0.777145961 0.309016994 52 0.788010754 0.325568154 53 0.79863551 0.342020143 54 0.809016994 0.35836795 55 0.819152044 0.374606593 56 0.829037573 0.390731128 57 0.838670568 0.406736643 58 0.848048096 0.422618262 59 0.857167301 0.438371147 60 0.866025404 0.4539905 61 0.874619707 0.469471563 62 0.882947593 0.48480962 63 0.891006524 0.5 64 0.898794046 0.515038075 65 0.906307787 0.529919264 66 0.913545458 0.544639035 Measuring Distance Using Parallax Angle, Sine A Degrees 66 0.913545458 67 0.920504853 68 0.927183855 69 0.933580426 70 0.939692621 71 0.945518576 72 0.951056516 73 0.956304756 74 0.961261696 75 0.965925826 76 0.970295726 77 0.974370065 78 0.978147601 79 0.981627183 80 0.984807753 81 0.987688341 82 0.990268069 83 0.992546152 84 0.994521895 85 0.996194698 86 0.99756405 87 0.998629535 88 0.999390827 89 0.999847695 89.1 0.999876632 89.2 0.999902524 89.3 0.99992537 89.4 0.999945169 89.5 0.999961923 89.6 0.999975631 89.7 0.999986292 89.8 0.999993908 89.9 0.999998477 90 1 7