Name: Lab Partner: Instructor's Signature: Lab Title: Tracking Sunspots. Equipment:

Transcription

1 Name: Lab Partner: Instructor's Signature: Lab Title: Tracking Sunspots Equipment: Scientific calculator, Macintosh computer, and the National Institutes of Health computer program ImageJ. Purpose: In this exercise, you will get a chance to observe sunspots on the surface of the Sun, and use your observations to measure the rotation of the Sun s equator. You will use images of the Sun from the Boulder, Colorado, solar observatory of NOAA, the National Oceanic and Atmospheric Administration. This exercise will also introduce you to the way astronomers can take and store images in digital form. It will demonstrate some of the fundamentals of digital image processing the manipulation of digital images with a computer to enhance, magnify, or alter the display of the images. Requirements: This lab is to be performed individually (or in groups of no more than two if there are not enough computers). If you must work in pairs, take turns manipulating the computer and taking the data. You should switch off from time to time so everyone gets a chance to use the computer. Although you may use the computer and the program with your partner to collect data, all calculations, graphing, and any narratives in your lab report must be your own original work! -1-

2 Introduction: Recorded observations of sunspots date back 2000 years to ancient Chinese observers. Indian astronomers also recorded observations of sunspots, and there are references to them by Greek observers from around 400 B.C. In western history, Galileo was among the first observers to look at the Sun through a telescope in 1610 and observe dark spots on its face. In ordinary visible light sunspots appear dark compared to the rest of the solar surface because they are cooler than the surrounding regions. The temperature of sunspots is only about 4000 K, while the average temperature of the photosphere (solar surface) is about 6000 K. The reason why sunspots are cooler than the rest of the Sun s surface has to do with the complicated interaction between the solar magnetic field, the convecting ionized gases that rise to the surface in the outer layers of the Sun, and the fact that the Sun does not rotate as a rigid body. Heat Transport in the Sun The Sun is made mostly of ionized hydrogen. In the core, where the thermonuclear reactions that make the Sun shine are taking place, the temperature is millions of Kelvin. Heat is carried away from the core to the outer layers by the slow process of radiative transport. It takes photons hundreds of thousands of years to travel from the interior of the Sun to the outer layers. Within the upper 100,000 km of the Sun, however, heat is transported to the surface more rapidly by convection. Magnetic Fields in the Sun The strong magnetic fields of sunspots keep the hot, ionized gases from reaching the surface and releasing their heat directly below the sunspots; this causes sunspots to be somewhat cooler than the rest of the Sun s surface and appear dark in comparison. If you could isolate sunspots, however, they would appear far brighter than the full Moon, but somewhat red. The Sun as a whole has a weak magnetic field, similar to that of the Earth, with the magnetic field lines emerging at the north pole and re-entering at the south pole. The overall intensity of the solar magnetic field is about 1 gauss the same as that of the Earth. In sunspots, however, the local magnetic field can be several thousand times stronger! We know this is so, because of the splitting of the spectral lines in the absorption spectra of sunspots. This effect, called the Zeeman effect after the Dutch physicist who discovered it in 1886, was first observed for sunspots by George Ellery Hale in 1908, at the Mount Wilson Observatory. -2-

3 Rotation Rates of the Sun and the Formation of Sunspots The Sun behaves like a fluid. While terrestrial planets such as the Earth are solid, and rotate as rigid bodies, the Sun rotates faster at its equator than at its poles. As a result of the differences in rotation rates at different latitudes, the north-south magnetic field lines of the Sun get stretched out in an east-west direction at lower latitudes (closer to the equator). The field lines eventually pinch off and form intense localized fields, aligned roughly east-west. Sunspots tend to form in pairs, one a north magnetic pole, where the local field lines emerge from the surface, and the other a south pole, where the local field lines plunge back into the solar surface. Sunspots are often accompanied by towering arcs of ionized gases which follow these local magnetic field lines. New sunspots first appear at latitudes around 35 north and south of the equator. Occasionally sunspots are observed at higher latitudes. By observing the motion of sunspots at different latitudes, astronomers have been able to measure the rotation rate of the Sun at different latitudes. Another way to do this is by observing the Doppler shift of spectral lines in the photosphere at various latitudes. Safely Observing the Sun You CANNOT observe the Sun safely by looking directly at it, either with the naked eye or with a telescope! However, you can observe sunspots with a small telescope if you project the image of the Sun onto a screen. If there are any sunspots, they will appear as dark, hairy spots on the solar disk. Other ways to observe the Sun are with special filters or digital cameras that are sensitive at x-ray or other wavelengths. The Images of the Sun Used in this Exercise The images you will use in this exercise were taken with an Hα (pronounced H-alpha ) filter. This is a special, very narrow-band red filter, centered on the 6563-Ångstrom line emitted by excited neutral hydrogen. Since the Sun is made up mostly of hydrogen, this 6563-Å line is one of the strongest absorption lines in the solar spectrum. Most of the photons of that wavelength coming from the surface of the Sun (the photosphere) are absorbed before they reach the next thin layer of the lower atmosphere of the Sun (the chromosphere). Only 6563-Å photons that originate in the -3-

4 chromosphere escape into space and are visible to us. So, when you look at these images, you are mainly looking at features in the chromosphere, or outer part of the solar atmosphere. Since sunspots are about 2000 K cooler than the surrounding photosphere, the ionized hydrogen in the chromosphere directly above the sunspots actually cools down enough to capture a free electron, and become neutral hydrogen (1 proton, 1 electron). This neutral hydrogen is heated and excited, causing the hydrogen to emit photons at its characteristic frequencies in the visible spectrum. The strongest emission is the one at a wavelength of 6563 Ångstroms, corresponding to red light. If you are lucky enough to observe a total solar eclipse, you can see a faint red rim around the Sun at totality, which is the red glow of the Hα photons in the chromosphere! The images used in this exercise were all taken with a digital camera and appear in black & white. The parts that are dark in the picture are Hα absorbers; the features that appear bright are Hα emitters. Locally intense magnetic fields in the photosphere often cause gases in the chromosphere to heat up and glow brighter than their surroundings. Called plages ( beaches in French), these irregular bright patches of hot, disturbed gas usually precede the formation of sunspots and persist even after the sunspots have disappeared. You can see plages on some of the images used in this lab; sunspots show through these plages as dark patches. Solar prominences Figure 1. A portion of an Hα image near the eastern limb of the Sun -4-

5 You will also notice irregular, snaky dark lines on these images; these are filaments of gas that stick up above the surface of the Sun. When these filaments approach the limb of the Sun as it rotates, they can be seen as bright curtains of gas called solar prominences that protrude many hundreds to thousands of kilometers above the surface. They are strong Hα absorbers, and hence appear dark when seen through the 6563 Ångstrom filter with the solar surface in the background. Procedure: Your instructor will inform you about which computers on campus currently have the program installed on their hard drives. Position the mouse cursor arrow over the icon (picture) of the Mac's hard disk and double-click on the mouse button. A list of the contents of the hard disk will appear. Move the mouse cursor over the Astronomy 30 folder's icon and again double-click the mouse button to reveal the contents. Clicking the mouse button twice when the cursor arrow is on the picture of the Image Processing folder will open the folder and reveal a picture (icon) of the program itself called ImageJ in the file list. You can now start the program by positioning the mouse cursor over the ImageJ icon and double-clicking to run the program. ImageJ is a public-domain Java image-processing program inspired by the National Institutes of Health s NIH Image. It is used widely throughout the scientific research community to study and analyze digital images obtained from a wide variety of sources. Downloadable versions of ImageJ are available for Windows, Mac OS, Mac OS X and Linux. When the program has finished loading, a menu bar will appear along the top portion of the screen. Each word in this bar (File, Edit, Image, Process, Analyze, Plugins, Window, Help) is a menu consisting of a list of commands to ImageJ. You can open a menu by positioning the cursor over a word in the menu bar. While pressing the mouse button, move the mouse downward. As you drag the mouse, an item list appears and each item is highlighted with a black bar as the cursor moves over it. Some items under names in the menu bar have a series of three dots (... ) next to them. Releasing the button on these dotted items causes a dialog box to appear on the screen. If an item is a command, releasing the mouse button with the item highlighted executes the command. The Quit command in the File menu lets you exit from the ImageJ program. An item that is lighter than the other items in a menu is inactive and can not be used until the program decides it is appropriate to do so. -5-

6 Measuring Sunspot Positions on the Face of the Sun When the program has finished loading, select File Open... and scroll down in the list in the ImageJ folder to another folder called Solar Images that contains all of the images that you will use in this lab. Once you select and Open the Solar Images folder you will find another file list. You will see a list of all the available images of the Sun, which are labeled by the date on which the image was recorded. For example, the image called Sun was taken on 1992 February 24. In this exercise you will look at four consecutive images of the Sun made on different days. Select the first image you want to study by clicking the mouse button to highlight and Open it. (Don t be boring and simply choose the first image in the list...) Also remember that you will need images for FOUR consecutive dates, so don't choose your first image to be too close to the end of the list! In the Tools window, click once on the tool to select it and notice that a little crosshairs appears that you can move across the image to pick out interesting pixels. As you move the cursor around the image, look at the numbers displayed in the ImageJ window below the tool buttons. The first two numbers indicate the column (X) and row (Y) of the pixel to which the center of the cursor is pointing. The X coordinate increases from left-to-right across the image while the Y coordinate increases from top to bottom down each image. The next number (Value) is the actual number that represents the brightness of that pixel. Make sure the image is displayed in black-and-white tones by choosing Image Lookup Tables Grays, and then choosing Image Lookup Tables Invert LUT. For Grayscale images, a pixel brightness value of 255 represents pure white and a pixel brightness value of 0 represents pure black. Pixel values like 25, 71, and 193 represent various shades of gray between white and black. Notice that there is a wide range of pixel values between the brightest and darkest parts of the image. If the image on your screen seems too dark or too light, try to enhance the picture by changing the brightness and contrast. From the menu bar, choose Image Adjust Brightness/Contrast. To change the settings, drag the small black squares to the left or right in the brightness and contrast controls in the B&C window. -6-

7 It is also possible to magnify or to shrink the image and to change the centering of the display so you can examine parts of it better. Click once on the tool to activate it and note that the cursor has now changed to a little magnifying glass with a plus sign. You can repeatedly click on parts of the image using the tool to zoom in for a closer look. Each click of the mouse zooms you closer and simultaneously centers on the particular pixel you click (unless you are too close to the edge of the image). The magnification factor is shown in the title bar of the image window. Try magnifying the image by 200%, 400%, 800%, 1600%, and 3200%. Notice that the pixels get bigger each time. Double-click the tool to return the image to its original size. On the Mac: You can change the size of the window by dragging the bottom right-hand corner of the window. On the PC: The window will resize itself as you magnify the image. You may zoom out by right-clicking the mouse. You should see one or more sunspots in each image. To get a better idea of the position of any suspected sunspot, you should repeatedly zoom in on each image of the Sun by using the tool. On this first image you re looking at, locate a sunspot. If you can, choose a sunspot on the LEFT-HAND SIDE of the image. You re going to track the position of this sunspot over several days, and the sunspot will move from left to right across the face of the Sun, so if you choose a sunspot that s too close to the right-hand edge of the image, over a few days it may move to the back side of the Sun and be lost from view! Move the mouse to center the crosshairs on the sunspot you ve chosen and use the tool to ZOOM IN and get a close look at a particular part of the sunspot you've chosen to track. Record the X and Y coordinates of the particular part of the sunspot in the table below, along with the date of the image and the time shown on the image. The time shown is UT or Universal Time, formerly known as Greenwich Mean Time because it was measured at the Royal Observatory in Greenwich, England. Universal Time is given using a 24-hour clock and is used by astronomers because it is a standard that makes no reference to the time zone of the observatory at which the image was recorded. Then use File Open... to open up the next day s image, and record the coordinates of the same sunspot (which will have moved to the right). Each sunspot is larger than one pixel across, so try to record the coordinates -7-

8 of the same point on the sunspot for each image say, the center or one edge of the sunspot. In addition, record the diameter of the Sun in pixels from the image. To measure the diameter, select the tool and then use the mouse to point the crosshairs to the extreme left-hand edge of the Sun. Then, holding the mouse button down, move the crosshairs to the extreme right-hand edge of the Sun moving across the widest part of the Sun. Then release the mouse button. This process draws a horizontal line on the image, connecting the points where the crosshairs began and ended its journey. To determine the distance between the beginning and the end of the line, select Analyze Measure. The line Length will be reported in the Results window. You can get rid of the line you drew on the image by first selecting the tool and then clicking the mouse anywhere else inside the image frame. REPEAT THE ABOVE STEPS for each solar image until you have measurements of the Sun for FOUR consecutive dates. 1. Record your raw data in the table below: Image 1 Image 2 Image 3 Image 4 File Name: Date: Time (UT): X coordinate: Y coordinate: Sun's Diameter: (pixels) IMPORTANT NOTE: You may find that after opening a few images, ImageJ gives you an error message saying that it doesn t have enough memory to continue. If this happens, just close one of the images from which you ve already taken measurements. To do this, click the mouse on the window you want to close, then choose File Close. -8-

9 2. In each image, the north pole of the Sun is at the top of the image. As time increases, does the sunspot move across the face of the Sun from east to west, or from west to east? Explain your choice. 3. Since sunspots are fairly large, they appear as more than one pixel across in your image. Estimate your uncertainty in picking exactly the same spot on the sunspot from one image to the next. That is, by how many pixels (+ or ) do you think your measurements could have potentially been in error each time? Why did you choose this number of pixels? Animating the Hα Images of the Sun You can make a short movie by assembling all of these separate solar images into something called a stack and then by animating the stack. To do so, first Close any images you may have open. Then select File Open... each one of the six images of the Sun in chronological order and leave them all open. Next, select Image Stacks Convert Images to Stack and then Image Stacks Start Animation. Select Image Stacks Animation Options... to change the speed of the blinking frames. Using the < and > keys will allow you to step through the sequence one at a time in either direction. The original names of the images are lost in this process, but they can still be identified: Stack(l/6) is the image Sun Stack(2/6) is the image Sun Stack(3/6) is the image Sun Stack(4/6) is the image Sun Stack(5/6) is the image Sun Stack(6/6) is the image Sun

10 Note that this is just the same chronological order in which you opened the six files. The movie may be a bit jumpy since the time differences between each of the frames are not the same. 4. Which image in the Stack shows the largest solar prominence near the eastern (right-hand) limb of the Sun's image close to the solar equator? 5. Let's now determine the scale of these images of the Sun that is, how many kilometers are represented by each pixel. We'll assume that the Sun is spherical and use the average of your FOUR measurements in Question 1 for the diameter of the Sun. Average diameter of the Sun = pixels Consult your astronomy textbook to find the actual diameter of the Sun in kilometers: Actual diameter of the Sun = km Now you can find the scale of the images of the Sun in kilometers per pixel by dividing: Scale = Actual diameter of the Sun in km Average diameter of the Sun in pixels = Scale = km/pixel 6. Again consult your astronomy textbook to find the actual diameter of the Earth in kilometers: Actual diameter of the Earth = km To find the corresponding size of the Earth in pixels on the scale of these solar images, just divide the actual diameter of the Earth in km by the scale factor in km/pixel you just found above: Diameter of the Earth = pixels To better appreciate the size of even a relatively modest solar prominence, first Close the Stack of images and then select File Open

11 the single image Sun Use the tool to zoom in on the prominence on the right-hand edge of the Sun in this image until you reach a magnification of 8:1. Now select the tool and hold down the shift key while you drag this tool to draw a nice circle the size of the diameter of the Earth in pixels on this image. The ImageJ window will show you the diameter (Width or Height) in pixels of the circle as you draw it. You can point at the edge of the Earth-size circle you just drew and drag it around to compare it to the size of the solar prominence. 7. About how many Earth diameters above the edge of the Sun does the solar prominence rise? When you are finished using ImageJ, select the command Quit in the File menu. Reduction of the Data: (Computer NOT needed from here on.) Changing Pixel Coordinates The sunspot positions you ve recorded are measured from the upper left corner of the image. However, for the next step in the analysis, we need to measure coordinates from the center of the image of the Sun. Here s how to convert the coordinates. The pixel numbers in each solar image run from (0,0) at the upper left corner to (511,511) at the lower right corner. Therefore, the center of the image, which coincides with the center of the image of the Sun, is at (255.5,255.5). We ll use the coordinates a and b to refer to the position of the sunspot relative to the center of the image of the Sun. In terms of the X and Y that you recorded in your answer to Question 1, the coordinates a and b are: a = X b = Y -11-

12 8. Using the raw data that you recorded in Question 1, calculate the a and b coordinates of the sunspot in each image. Record these in the table below: Image 1 Image 2 Image 3 Image 4 File Name: Date: Time (UT): a coordinate: b coordinate: Sun's Diameter: (pixels) Translating from Flat Images to the Spherical Sun Now comes the (only slightly) tricky part! The images you have been looking at are two-dimensional: they are projections of a round object like the Sun onto a flat plane. When we observe the motion of a feature going around the sphere of the Sun on such a flat image, we may get the wrong idea about how fast the feature is moving. In particular, when a sunspot is near the edge of the Sun, it will appear in our two-dimensional images to be moving more slowly than when it is near the center of the Sun s disk. For this reason, we have to translate the motion we observe on the flat image to what is really happening on the spherical surface of the Sun. On a flat plane, like our images, it s most convenient to express the position of a point in terms of its a and b coordinates. But on the surface of a sphere, it s most convenient to express the position of a point in terms of two angles. One angle describes how far north or south of the equator the point lies, and the other angle describes how far east or west the point lies. This is how geographers and navigators describe positions on the Earth s surface: the first (north-south) angle is called latitude, and the second (east-west) angle is called longitude. As the Sun rotates, it carries sunspots with it, so that the eastwest angle of a sunspot changes with time. So to study the motion of a sunspot across the face of the Sun, we need only keep track of this east-west angle. Sunspots move very little relative to the surface of the Sun, so a -12-

13 sunspot won t move appreciably to the north or south. We ll call the eastwest angle φ (the Greek letter phi ). The definition of φ is shown in the figure below. View of Sun as seen from Earth N View of Sun as seen from above the Sun's north pole Sunspot Equator φ = 0 line a b φ = 0 line N +φ Sunspot Negative φ in this hemisphere Positive φ in this hemisphere To Earth Figure 2. Translation to a spherical coordinate system Note that φ is positive for positions in the right-hand hemisphere of the Sun (as seen from Earth), and φ is negative for positions in the left-hand hemisphere. A position along the north-south centerline of an image of the Sun is at φ = 0. In order to translate the sunspot coordinates a and b that we observe in our two-dimensional images to the angle φ, we first need to know the radius of the Sun (in pixels). -13-

14 9. For each image from which you made measurements, find the radius of the Sun (in pixels) by dividing the Sun's diameter by 2. Record your answers in the table below: Image 1 Image 2 Image 3 Image 4 File Name: Date: Time (UT): R=Sun's Radius: (pixels) To find the longitude angle φ of a sunspot, use the following formula: φ = tan -1 a R 2 a 2 b 2 The latitude angle θ (the Greek letter theta ) of a sunspot is positive for positions north of the solar equator and is negative for positions to the south of the equator. The angle θ is given by the formula: θ = tan -1 b R 2 b 2 In these formulae, R is the radius of the Sun (in pixels) that you found in Question 9, and a and b are the coordinates of the sunspot that you found in Question 8. The function is the square root, and the function tan 1 is the inverse tangent function. On some calculators, it is also called atan. When using this function, make sure that your calculator is configured to give you an answer in degrees. If you re not sure how to do this, look in the manual for your calculator, or check with your instructor. -14-

15 10. Enter the data you recorded in Question 8, and the radius R you found in Question 9, use the above expressions for φ and θ to find the coordinates of the sunspot on the Sun s spherical surface for each image. Record this information in the table below: Image 1 Image 2 Image 3 Image 4 File Name: Date: Time (UT): a (pixels): b (pixels): R (pixels): R 2 a 2 b 2 pixels : a : R 2 a 2 2 b φ (degrees): R 2 b 2 pixels : b R 2 b 2 : θ (degrees): -15-

16 11. The relevant quantity for measuring the rotation rate of the Sun is the change in the east-west coordinate φ. Using your calculations in Question 10, find the change in φ between each image and the one that precedes it, and also find the elapsed time between images. Note that you must properly account for the fact that times on a given day are 24 hours later than the same times on the previous day. When calculating the elapsed time, remember that the times are given using a 24-hour clock, and express your answer in hours. For instance, if the elapsed time between consecutive images is 19 hours and 25 minutes, you would first express this as 19+25/60 hours = hours. Record your results in the table below: φ 2 φ 1 = degrees T 2 T 1 = hours φ 3 φ 2 = degrees T 3 T 2 = hours φ 4 φ 3 = degrees T 4 T 3 = hours 12. Convert the elapsed times in Question 11 from hours to days: T 2 T 1 = T 2 T 1 (hours) T 3 T 2 = T 3 T 2 (hours) T 4 T 3 = T 4 T 3 (hours) 1 day 24 hours 1 day 24 hours 1 day 24 hours = days = days = days 13. Using the results of Questions 11 and 12, calculate the rotation rate of the sunspot using the formula: -16-

17 Rotation rate = Change in angle φ (degrees) Elapsed Time (days) Between the 1st image and the 2nd image: φ 2 φ 1 T 2 T 1 = = degrees/day Between the 2nd image and the 3rd image: φ 3 φ 2 T 3 T 2 = = degrees/day Between the 3rd image and the 4th image: φ 4 φ 3 T 4 T 3 = = degrees/day Now take the average of these three rotation rates to obtain: AVERAGE rotation rate = degrees/day 14. Now convert your average rotation rate from Question 13, which is in degrees per day, into rotations per day: Rotation Rate = (Average number of degrees/day) 1 rotation 360 degrees = rotations/day 15. Finally, take the reciprocal of the result of Question 14 to find the number of days per rotation. This is the rotation period of the Sun! -17-

18 1 Number of rotations per day = days/rotation 16. According to your astronomy textbook, what is the rotation period of the Sun s equator? How does this compare to the answer you found in Question 15? (If your answer agrees to within a couple of days, you re doing just fine!) 17. For instance, your answer may be different because your sunspot was not precisely at the equator (θ = 0 ). What was the average value of the four values of the latitude angle θ of your chosen sunspot? Was it above or below the solar equator? Uncertainty Analysis 18. On the average, about how many pixels (change in X coordinate) did you find that a sunspot moved across an image in a day? 1 3 X 2 X 1 T 2 T 1 + X 3 X 2 T 3 T 2 + X 4 X 3 T 4 T 3 = pixels/day 19. In Question 3, you estimated your uncertainty in picking exactly the same spot on the sunspot from one image to the next. What percent of the average daily change in position of a sunspot is your error in picking the same point on that sunspot? To get this, divide your answer for Question 3 by your answer for Question 18 and then convert to a percentage: -18-

19 20. Considering your answer to Question 19, how do you think that your uncertainty in measuring the coordinates of the sunspot in each image affected your calculation of the true solar rotation rate? For Further Investigation The Sun is the closest star to the Earth; the next closest star, Alpha Centauri, is about 4.3 light-years away. (One light year is around 9 trillion kilometers.) If the images you ve seen of the Sun have intrigued you, you may wish to study more about the Sun. Here are just a very few ideas for you to investigate: The relation between sunspots, flares, and magnetic storms on Earth; The solar corona, solar wind, and the aurorae that are seen at high latitudes on Earth; The production of energy in the Sun by thermonuclear reactions; Heat transport in the Sun from the core to the surface, by radiation and convection; Studying the interior structure of the Sun with acoustic waves (helioseismology) that travel throughout the Sun like earthquake waves do on Earth; The mission of the space probe Ulysses, which traveled beyond Jupiter and is returning to the Sun in a polar orbit; What we can learn about the Sun by observing it in the light of x-ray photons; The life cycle of the Sun: What is its ultimate fate? How long will it last? Will it ever become a black hole? -19-

20 Conclusions and Comments -20-

21 Name: Lab Partner: Pre-lab Exercises: Tracking Sunspots 1. Use the information given in the Introduction of the lab write-up and in your astronomy textbook for the properties of the solar surface. (a) According to Stefan's Law, how much more radiation (per square centimeter per second) is emitted by the brighter surrounding surface of the Sun than is emitted by a darker sunspot? Show all of your work below. (b) Why should a sunspot appear slightly redder than the rest of the Sun's surface? (c) If only 10% of the entire surface area of the Sun were covered with sunspots, by what factor would the luminosity of the Sun decrease? Show all of your work below. -21-

22 2. The first sketch below shows the pixel numbers (X,Y) for each corner of a square image and for the center of the image. (0,0) (511,0) X (255.5,255.5) (0,511) (511,511) Y For each of the five labeled points above, convert the X and Y pixel coordinates using the formulas given in the lab write-up into a and b coordinates. On the second sketch shown below, label each of the corners and the center with its a and b coordinates (instead of X and Y). b (, ) (, ) (, ) a (, ) (, ) -22-

23 3. (a) The speed of light is 300,000 km/sec. At that speed, how long should it take light to travel in a straight line directly from the center of the Sun out to the surface (the Sun's radius is about 700,000 km). Show all of your work. (b) Yet your lab write-up says that light generated in the center of our Sun can take up to 100,000 years or more to finally work its way to the surface. Explain this apparent contradiction in the space below. 4. Carefully define each of the following astronomical terms. Use your astronomy textbook or any other sources if necessary. (a) Plages (b) Chromosphere (c) Photosphere -23-

24 (d) Zeeman effect (e) Hα filter (f) Universal Time (g) Solar prominence 5. Using your scientific calculator, complete the table of arctangents shown below: tan -1 (0.0) = tan -1 ( 0.577) = tan -1 (1.0) = tan -1 ( 1.732) = tan -1 (11.0) = -24-