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Optical Powers
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Optical Powers
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Standards (see Appendix A):
Unifying
Concepts: Size and Scale; Evidence, Models and
Explanations.
Objective:
Students create models that demonstrate the angular size of one degree at various distances from the center of concentric circles. Students apply mathematical skills of measurement and calculate distances needed using the formula for circumference of a circle. Students consider the role of angular measurements in describing location in the sky, on the celestial sphere. Note: A circle of dimes has a radius of approximately one meter.
Overview:
Students build a model of angular size by placing 360 objects in a circle, each object representing one degree. Then they add another circle inside or outside the first using an object of a different size to represent a degree. Building on the concept from the previous activity that larger objects at a greater distance may appear the same size as smaller objects which are closer, students now explore angular size. In this activity they learn that angular size can be used to describe both an object’s location and its apparent size. This activity also reinforces the concept that apparent size is dependent upon distance. Understanding angular size is important for calculating the field of view of telescope/CCD systems and making decisions about what objects can be successfully imaged. This will be addressed in Activity 9 Telescope Systems, 9b Field of View.
Background:
Vocabulary
Celestial Sphere - an imaginary sphere centered on the earth on which all of the stars are imagined to be projected.
Circumference - the distance around a circle or sphere. Circumference = 2 p radius.
Concentric - circles that have the same center.
Diameter - the distance across a circle through the center.
Radius – the distance from the center to the edge of a circle (equal to 1/2 the diameter).
Subtended angle – The angle measured from the center of an imaginary circle. The observer is located at the center of this imaginary circle. The observer's is at the vertex of the angle. The angle is formed from by drawing imaginary lines from either side of the observed object to the observer. For example, the subtended angle can be that of the observer’s eye at the telescope, the CCD camera, or even an individual pixel.
Measuring distances in the universe is one of the most important pursuits in all of astronomy and one of the most difficult. Consider a familiar experience such as walking away from an object and seeing its apparent size become smaller. It is obvious that the object’s true size is not changing. Knowing what the object is, a house or tree for example, and the presence of numerous other visual clues helps our minds make sense of what we are seeing. When we move to the less familiar objects in the night sky, and in the absence of visual frames of reference, making judgments about actual size becomes nearly impossible.
In spite of these problems, astronomers frequently need to compare the size of objects relative to each other. Because we think of the night sky as a massive dome in which we, the observer, stand at the center, we commonly use angle measurements to describe distances and sizes. With a 360-degree circle in mind, 180 degrees is a useful measurement from horizon to horizon, as is 90 degrees from horizon to zenith (the point overhead). The smaller portions of the sky occupied by celestial objects are measured in degrees, minutes (60 arcminutes in a degree) and seconds (3600 arcseconds in a degree). Understanding angular measure also gives the astronomer an easy method of finding objects in the night sky and a measurement by which to compare the field of view of her telescope with the size of the target object.
Preparation:
Assign homework several days in advance. Student teams will need time to accumulate 360 ‘somethings’ for each of the circles needed for this activity. Students may either collect 360 objects or make 360 tracings of an object in segments of ten objects each. Using dimes for this activity results in a circle with a 1 meter radius. Since collecting dimes is rather expensive, we have provided a sheet that you can photocopy that contains a “strip of dimes” with a measurement of 15°.
Plan for access to a large space for constructing circles: the gym, an outdoor recreation area or open parking lot, with barriers to exclude car traffic.
Find 2 long pieces of brightly colored yarn, ribbon, or rope; two “horizon” signs; and small figure of a person or simply a drawing of a stick figure.
Extension: Locate a small circle or tiny round object representing the moon and a larger one representing the sun. The sun's diameter is about 400 times larger than the moon.
Time: 45 minutes
Homework:
Several days prior to doing this activity assign groups the task of collecting 360 objects (stickers, coins, empty pop cans, etc.). These objects will represent the 360 degrees in a circle. Taping subsets together makes the project easier to manage. One could tape together fifteen pennies in a line; in this case 24 subsets would be needed to have 360 altogether. If students want to use objects that are difficult to find in large quantities, suggest tracing the circumference of the object onto paper and cutting them out. They could also do a series of rubbings of the object onto strips of paper.
Directions:
View example pictures.
1. Review with students the definitions of radius and circumference.
2. Ask the groups to share what object they collected for their 360 somethings. Say: “Each group has collected something different as their round object. Each of these round objects is a different size. How could we compare the sizes of these objects?” Radius and diameter are convenient ways to compare size.
3. You may want to record the objects and their diameters on the board.
4. Mark a center point on the floor in the area you have chosen for this activity. Tell students that this will be the center for all of the circles. “If each group arranges all 360 objects in a circle around this center point, what order do you think the circles will be in?” The smallest object will form the smallest circle.
5. Have groups construct concentric circles around the center point on the floor. Place the dimes on the floor first as a class to provide a point of reference for constructing the remaining circles. A circle of dimes will have a radius of a meter.
6. When they have finished review what you have just done, and then ask: “If there are 360 objects along the circumference of each circle, what is the angle subtended by each of the objects?” You may need to demonstrate to students what the word subtended means.
7. Tape the ends of two pieces of yarn to the center of the circles and extend them outward so that they follow the 1-degree angle subtended by the diameters of the objects on the floor. Ask students repeatedly to identify the size of the angle along the circumference of each circle to reinforce the fact that each angle is still 1 degree even though the diameters of the objects increase as distance from the center increases. “If you are standing on Earth looking up at a 1-degree angle in the sky, what can you say about the actual size of the object you are looking at?” Nothing. “What else would you need to know in order to determine how big the object is?” You would need to know the distance to that object.
8. You may want to have each group measure the distance from the center to the edge of their circle (radius) and the diameter of their object. Plotting these points on a graph should reveal that the larger the object representing one degree, the larger the radius of the circle. You may wish to use this exercise to explore the geometry of circles, and analyze the data using algebraic methods.
9. Now ask your students to imagine that they are standing at the center of the circles looking up at the night sky. Place the paper stick figure or toy figure at the center of the circles. Extend a piece of yarn across the diameter of all the circles to simulate the horizon line. Ask a student to place the horizon signs at the appropriate spots.
10. Place a small image of the moon on one of the objects of the inner circle. The image should be 1/2 the diameter of the object it is placed on since the moon appears to be 1/2 degree in diameter. Ask:n style="mso-spacerun: yes"> “How can we describe the position of the moon in the sky?” Students should be able to count the number of degrees above the horizon you placed the moon. “How can we describe how large the moon looks in the sky?” They should be able to see that the angle subtended by the moon is approximately 1/2 degree.n style="mso-spacerun: yes">
11. Place a sketch of the image of the Sun on one of the outer circles. (For the size and distance scale to be correct, the size of the model of the Sun should be 400 times larger than the Moon. This would make the distance to the model Sun 400 times as far away. If your Moon is 0.5 cm, then your model Sun would have to be 200 cm. Consider that you may not be able to create the entire outer circle for the Sun using accurate size and distance scales.) The sun should be of a size to match 1/2 degree on its circle.n style="mso-spacerun: yes"> “If you look up at the sun (with appropriate eye protection of course) how large will it appear to the observer on Earth?” Like the moon it will appear to be 1/2 degree in diameter. Ask students to make statements about the Sun and the Moon using the words “apparent size” and “angular size” and “actual size”.n style="mso-spacerun: yes">
12. You may want to demonstrate how an eclipse occurs by placing the two objects in line with one another. Students should be able to see that although the Sun is much larger than the Moon, they appear to be the same size from the point of view of the Earth bound observer. If the Sun is 400 times larger in diameter than the Moon, how much further from Earth must the Sun be than the Moon?
Extension (Comet Model)
13. Now place a picture or sketch of a comet over several of the objects on one of the circles further out from the moon. “You spot this comet in the sky and you call a friend across town. How are you going to tell him to find this comet?” Students should be able to count the number of degrees from the moon the object is located and the number of degrees from the horizon. “How would you describe the size of this comet?” They should be able to describe the APPARENT size of the object by the number of degrees it intercepts on the circle. Students should be comfortable at this point with the difference between apparent and actual size.
Extension: Leading to Activity 9 Will it fit on the Chip?, 9b Field of View.
14. Finally ask students to imagine that they want to take an image of this comet. They have a choice of using a camera or a telescope with a CCD. “How would you decide which instrument to use?” You want to use the instrument that will capture the entire comet while filling as much of the field of view as possible. “How would we need to measure field of view in order to make a decision?” We would have to know the field of view in terms of the angular measurement. You may want to explain to students briefly that the field of view of most CCDs they will be using are much less than one degree and that for large objects such as a comet, cameras are a better choice. In the next activity, they will learn what they need to know in order to determine the field of view of any telescope/CCD system. The angular size of target objects is something they can look find in reference books, planetarium programs, or on the Internet.
Web Resource:
The NGC/IC Project, http://www.ngcic.com/ has a number of observing tools. The list generators will show you the angular size of any object as long as you know its NGC (New General Catalogue) number.
Evaluation/Assessment:
Students create a Learning Log, reflecting on how their ideas about size and distance evolved during these activities. Logs should include words and sketches, describing before and after understandings of the concepts explored. Encourage students to share how creating the physical representations of models and how listening to others’ ideas helped them further develop their own understandings.
