THE SUN. LENGTH: Course of the semester

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1 ASTR 1030 Astronomy Lab 139 The Sun THE SUN SYNOPSIS: This exercise involves making measurements of the Sun every week throughout the semester, and then analyzing your results at semester s end. You ll learn first-hand where the Sun is in the solar cycle, its rotation rate, and what factors are important in producing the seasonal changes in temperature. EQUIPMENT: Gnomon, sunlight meter, heliostat, sunspot record forms, Astronomical Almanac, Stonyhurst disk overlays, protractor, ruler, calculator, and a pencil. LENGTH: Course of the semester WARNING: The intense solar light from the heliostat can cause instant eye damage! Do NOT look back up the beam of sunlight! REVIEW: Angles and trigonometry (page 19), using your calculator (page 22). Today you ll learn how to take the solar measurements. Then, every week during the semester, you or your classmates will collect additional observations. At semester s end, you ll return to this exercise to analyze your findings. You will find out just what factors are responsible for the Earth s heating and cooling. You will also use your sunspot charts to find where the sun is in its solar cycle and determine its rotation rate. Part I. Start of the Semester: Learning to Make Solar Measurements Everybody knows that it s colder in December than in July - but why? Is it because of a change in the number of daylight hours? The height of the Sun above the horizon? The intensity of the sunlight? Or are we simply closer to the Sun in summer than in winter? We can measure each of these factors relatively easily, and will show you how to do so. But seasonal changes occur rather slowly, so we ll need to monitor the Sun over a long period of time before the important factors become apparent. We ll also need to collect a considerable amount of data from all of the lab classes - to gather information at different times of the day, and to make up for missing data on cloudy days. TIME OF DAY As the Sun moves daily across the sky, the direction of the shadows cast by the Sun moves as well. By noting the direction of the shadow cast by a vertical object (called a gnomon), we can determine the time-of-day as defined by the position of the Sun. Such a device, of course, is called a sundial.

2 ASTR 1030 Astronomy Lab 140 The Sun I.1 Note the time shown by the sundial on the deck of the Observatory. Officially, this is known as local apparent solar time, but we ll just refer to it as sundial time. Determine the time indicated by the shadow to the nearest quarter-hour: Sundial Time =. (Don t be surprised if your watch and the sundial disagree - this is normal and is to be expected due to a number of factors which we won t go into here!) ALTITUDE OF THE SUN When the Sun first appears on the horizon at sunrise, shadows are extremely long; as it rises higher in the sky, the lengths of shadows become shorter. Hence, the length of the shadow cast by a gnomon can also be used to measure the altitude of the Sun (the angle, in degrees, between the Sun and the point on the horizon directly below it). The figure below shows that the shadow cast by a gnomon forms a right-triangle. The Sun s altitude is the angle from the horizontal ground to the top of the gnomon, as seen from the tip of the shadow. We ve prepared a gnomon (of height H) on a stand to help you make the measurement. I.2 Carry the gnomon and a metric tape-measure to the Observatory deck. Measure the height of the gnomon: H = mm. I.3 Carefully measure the shadow length S from the base of the gnomon to the tip of the shadow: S = mm. I.4 Now use your measurements to find the solar altitude angle in degrees: Solar altitude = THE SUNLIGHT METER The sunlight meter is a device that enables you to deduce the relative intensity of the sunlight striking the flat ground here at the latitude of Boulder, compared to some other place on the Earth s surface where the Sun is, at this moment, directly overhead at the zenith. I.5 On the observing deck, aim the meter by rotating the base and tilting the upper plate until its gnomon (the perpendicular stick) casts no shadow. When properly aligned, the upper surface of the apparatus will directly face the Sun.

3 ASTR 1030 Astronomy Lab 141 The Sun The opening in the upper plate is a square 10 cm x 10 cm on a side, so that a total area of 100 cm 2 of sunlight passes through it. The beam passing through the opening and striking the horizontal base covers a larger, rectangular area. This is the area on the ground here at Boulder that receives solar energy from 100-square-centimeter s worth of sunlight. I.6 Place a piece of white paper on the horizontal base, and draw an outline of the patch of sunlight that falls onto it. I.7 Measure the width of the rectangular region; is it still 10 cm? Measure the length, and compute the area of the patch of sunlight: Area = cm 2. I.8 Now calculate the relative solar intensity - the fraction of sunlight we re receiving here in Boulder compared to how much we would receive if the Sun were directly overhead: Relative Solar Intensity = 100 cm 2 Area = 100cm 2 2 cm = THE RELATIVE SIZE OF THE SUN AS SEEN FROM EARTH In the lab room your instructor will have an image of the Sun projected onto the wall using the Observatory s heliostat, or solar telescope. As you know, objects appear bigger (we say they subtend a larger angle ) when they are close, and appear smaller at a distance. By measuring the projected size of the Sun using the heliostat throughout the semester, you ll be able to determine whether or not the distance to the Sun is changing - and if so, whether the Earth is getting closer to it, or farther away, and by how much. I.9 Use a meter stick to measure the diameter, to the nearest millimeter, of the solar image that is projected onto the wall. (Note: since the wall isn t perfectly perpendicular to the beam of light, a horizontal measurement will be slightly distorted; so always measure vertically, between the top and bottom of the image). Solar Diameter = mm. MAPPING SUNSPOTS Mapping and counting sunspots is a principal method for studying solar activity. I.10 Your instructor will use the heliostat (solar telescope) to focus an image of the Sun on the wall chart holder. Position your sunspot record form in the holder so that the Sun's image is centered on the circle. The diameter of the circle on your sunspot record corresponds to the average apparent size of the Sun as seen with our heliostat. Carefully trace with a pencil all of the sunspots that are visible. I.11 North is not necessarily at the top of the image. You can determine direction by noting which way the image shifts when the heliostat is driven in a known direction. Without disturbing your drawing, drive the heliostat briefly in the west direction, using the direction toggle on the heliostat control box. Since the field of view is now westward of its original position, the solar image appears to have shifted to the east. Follow the motion of a single sunspot. When the spot clears the disk circle, make an "X" at its new position and label it "earth east" (see the attached example).

4 ASTR 1030 Astronomy Lab 142 The Sun I.12 Remove the form from the holder. Record the date, time, and the name(s) of the observer(s) on the form. I.13 Draw a line from the original position of the selected sunspot to its final location. This line is parallel to the Earth's equator. Use a protractor to draw a second line perpendicular (90 ) to the equatorial line and through the center of the solar disk. The new line marks the projection of the Earth's axis of rotation onto the disk of the Sun. Label the upper end of the line N earth (for "Earth north") and the lower end S earth (for "Earth south"). The Earth's and the Sun's axes of rotation are not aligned with each other: the Earth's north pole is aimed approximately towards the star Polaris in Ursa Minor, while the Sun's north pole is oriented about 26 away towards the star Delta Draconis. As a result, when we view the Sun from the Earth at different times during the year, the Sun's north pole may appear tilted eastward or westward of the Earth's north pole, and may be tipped either towards or away from us as well. The number of angular degrees of tilt and tip are defined by two angles, P and Bo, as described below. N sun P Earth North + - Looking towards the Sun from the Earth The angle P describes how much the north pole of the Sun is tilted, in the plane of the sky (or the plane of your paper), towards the east (or west) from Earth north. A positive angle P means the Sun's north pole lies to the east of N earth. Earth East Plane of sky and drawing paper Sun Earth West Earth South S sun The angle Bo describes how much the north pole of the Sun is tipped towards (or away from) you, the observer on Earth. A positive angle Bo means that the north pole of the Sun is tipped towards the Earth (so that the north pole of the Sun would, at least theoretically, appear on your drawing). B o - + N sun Earth North Sun Looking sideways at the Sun and Earth Earth S sun Plane of sky and drawing paper Earth South

5 APS 1030 Astronomy Lab 143 The Sun I.14 On your form, record the orientation angles P and Bo for the day of your observation. The angles are listed in Section C of the Astronomical Almanac for each day of the year. I.15 Use a protractor to draw a line through the center of the solar image at an angle P from the N earth -S earth line; remember, the line should lie to the east (left) of Earth north if P is positive, and to the west (right) if negative. This line marks the solar meridian, the north-south line dividing our view of the Sun into eastern and western hemispheres. Mark the upper end of the solar meridian N sun (solar north) and the lower end S sun (solar south). The Stonyhurst disk overlay will help you find the latitude and longitude of the sunspots. The grid is marked every 10 in latitude north and south of the solar equator, and every 10 east or west of the solar meridian line. I.16 Choose the Stonyhurst overlay with a Bo closest to the actual value corresponding to your observation. Center the circle of the Stonyhurst disk over your circular drawing, with the axis aligned with the solar meridian. Be sure that the correct sign (+ or -) for Bo appears at the top of the overlay. I.17 Number the prominent sunspots (see the attached example). Estimate, to the nearest degree, the solar latitude (N or S of the solar equator) and solar longitude (E or W of the solar meridian) of every numbered spot. I.18 Analyze the distribution of today's sunspots with latitude. Estimate the average latitude of all sunspots, regardless of northern or southern hemisphere. Average latitude of sunspots =. PLOTTING YOUR RESULTS You ll use four weekly group charts to record your measurements, which will always be posted on the bulletin board at the front of the lab room: the Solar Altitude Chart, the Solar Intensity Chart, the Solar Diameter Chart, and the Sunspot Latitude Chart. This first week, your instructor will take a representative average of everyone s measurements and show you how to plot a datum point on each graph. After this week, it will be the responsibility of assigned individuals to measure and plot new data each class period. You yourself will be called upon at least once during the semester to perform these measurements, so it s important for you to understand what you re doing, and to make your measurements and plots as accurately and as neatly as possible! I.19 On the weekly Solar Altitude Chart, carefully plot a symbol showing the altitude (I.4) of the Sun in the vertical direction and the sundial time (I.1) along the horizontal direction, showing when the measurement was made. Use a pencil (to make it easy to correct a mistake) and use the symbol appropriate for your day-of-the-week (M-F) as indicated on the chart. Other classes will have added, or will be adding, their own measurements to the chart as well. I.20 On the weekly Solar Intensity Chart, carefully plot a point that shows the relative solar intensity (I.8) that was measured at the corresponding sundial time (I.1). Again, use the appropriate symbol.

6 APS 1030 Astronomy Lab 144 The Sun I.21 On the weekly Solar Diameter Chart, plot your measurement of the diameter in mm (I.9) vertically for the current date (horizontal axis). (There may be as many as three points plotted in a single day from three different classes.) I.22 On the weekly Sunspot Latitude Chart, plot your estimate of the average sunspot latitude (I.18) vertically for the current date (horizontal axis). ASTR 1030 students using a primitive sundial. Part II. During the Semester: Graphing the Behavior of the Sun At the end of each week, the Observatory staff will collect the four weekly charts and construct a best-fit curve through the set of datum points. In addition, the curves will be extrapolated to earlier and later times of the day so that the entire motion of the Sun, from sunrise to sunset, will be represented (adding a few additional early-morning and late-day measurements of our own if necessary). Although the data represent readings over a 5-day period, the curve will represent the best fit for the mid-point of the week, Wednesday. These summaries of everyone s measurements will be available for analysis the following week and throughout the remainder of the semester. Every week take a few moments to analyze the previous week s graphs, and record in the table below: II.1 (a) The date of the mid-point of the week (Wednesday). (b) The greatest altitude above the horizon the Sun reached that week. (c) The number of hours the Sun was above the horizon, to the nearest quarter-hour. (d) The maximum value of the intensity of sunlight received here in Boulder, relative to (on a scale of 0 to 1) the intensity of the Sun if it had been directly overhead.

7 APS 1030 Astronomy Lab 145 The Sun (e) The average measured value of the apparent diameter of the Sun as determined using the heliostat. (f) The mean latitude of sunspots. (a) (b) (c) (d) (e) (f) Week # 1 Date Maximum Solar Altitude (deg) Number of Daylight Hours Maximum Solar Intensity Solar Diameter (mm) Mean Lat. of Sunspots II.2 Also each week, transfer your new data from columns (a) through (e) in the table above to your own personal semester summary graphs on the next two pages: Maximum Solar Altitude, Hours of Daylight, Maximum Solar Intensity, and Solar Diameter. Be sure to include the appropriate date at the bottom of each chart.

8 APS 1030 Astronomy Lab 146 The Sun Week 90 Maximum Solar Altitude (Semester Summary) Maximum Solar Altitude (degrees) Date:

9 APS 1030 Astronomy Lab 147 The Sun

10 APS 1030 Astronomy Lab 148 The Sun Part III. End of the Semester: Analyzing Your Results By now, at the end of the semester, your collected data should provide ample evidence to explain the cause of the seasonal change in temperatures. III.1 Draw best-fit curves through your graphed datum points on the previous two pages. The lines should reflect the actual trend of the data, but should smooth out the effects of random errors or bad measurements. Do you expect these downward or upward trends to continue indefinitely, or will they eventually flatten out and then reverse direction? Explain your reasoning. Now, use your graphs to review the trends you've observed: III.2 III.3 III.4 III.5 III.6 III.7 Measured at noontime, did the Sun s altitude become higher or lower in the sky during the course of the semester? On average, how many degrees per week did the Sun's altitude change? Did the number of hours of daylight get greater or less? On average, by how many minutes did daylight increase or decrease each week? What was the maximum altitude of the Sun on the date of the equinox this semester? (Consult the Celestial Calendar at the beginning of this manual for the exact date.) On that date, how many hours of daylight did we have? Explain or sketch how we could deduce our own latitude on the Earth, using the observed maximum altitude of the Sun on the date of the equinox. (Hint: see page 33) Based on this reasoning and your measurement, what is the latitude of Boulder? Did the sunlight meter indicate that we in Boulder received more or less solar intensity at noon as the semester progressed? Did the Sun s apparent size grow bigger or smaller? Does this mean that we are now closer to, or farther from, the Sun as compared to the beginning of the semester? Has the weather, in general, become warmer or colder as the semester progressed? Which factor or factors that you've been plotting (solar altitude, solar intensity, number of daylight hours, distance from the Sun, mean sunspot latitude) appear to be correlated with the change in temperature? Which factor or factors seems to be contrary to an explanation of the seasonal change in temperature? If you receive a bill from the power company, you re aware that each kilowatt-hour of electricity you use costs money. One kilowatt-hour (KWH) is the amount of electricity used by a 1000-watt appliance (1 kilowatt) in operation for a full hour. For example, four 100-watt light bulbs left on for 5 hours will consume 2.0 KWH, and will cost you about 15 cents (at a rate of $0.075/KWH). Every day, the Sun delivers energy to the ground, free of charge, and the amount (and value) of that energy can be measured in the same units that power companies use. The amount of energy received by one square meter on the Earth directly facing the Sun is a quantity known as the solar constant, which has a carefully-measured value of 1,388 watts/m 2. That is, a one-square-meter solar panel, if aimed constantly towards the Sun, will collect and convert to electricity KWH of energy every hour (worth slightly more than a dime).

11 APS 1030 Astronomy Lab 149 The Sun Each weekly Solar Intensity Chart contains all the information you need to find out how much energy was delivered by the Sun, in KWH, on a typical day that week. Note that one solarconstant hour is equivalent to the rectangular area on the chart 1.0 intensity units high (the full height of the graph, corresponding to a solar panel that always directly faces the Sun) and one hour wide. The actual number of solar-constant hours delivered in a day to flat ground in Boulder is likewise the total area under the plotted intensity curve, from sunrise to sunset. To find the area under a curve, we would normally integrate. Since we do not have an equation for the intensity curve, we will use an approximation method to find the integral. Divide the curve into a series of trapezoids. The base of each trapezoid should be along the x-axis. You can use the vertical lines of the boxes for the sides of your trapezoids. The top of each trapezoid should pass through the curve at the midpoint between the two sides. Now add the areas of each trapezoid to approximate the total area under the curve. This is the number of solar-constant hours. Finally, multiply this number by KWH to obtain the total energy contribution of the Sun to each square meter of ground during the course of a day. III.8 What is the ratio of the amount of energy received at the end of the semester compared to that at the beginning of the semester? (This ignores any effect due to a change in the distance to the Sun.) Final Week' skwh First Week' skwh = kwh kwh = Fractional Change in Solar Energy A ratio greater than 1 implies than an increase in energy was received over the semester, while a ratio smaller than 1 implies that less solar energy was delivered as the semester progressed. Now let s find out just how important was the change in distance from the Sun: III.9 What is the ratio of the apparent diameter of the Sun between the end and the start of the semester? Diameter ratio = Final Week' s Diameter First Week' s Diameter = mm mm = The energy delivered to the Earth by the Sun varies inversely as the square of its distance from us. The diameter ratio calculated above is already an inverse relationship - so all we have to do is square that ratio to determine the change in energy from the Sun caused by its changing distance from us. For example, if the ratio is 1.10 (10% closer), the Sun will deliver 21% more energy ( = 1.21). If the ratio is 0.90 (Sun 10% further away), it will deliver = 0.81 = 81% as much energy (or, if you prefer, 19% less energy). III.10 How much more or less energy (expressed as a percent change) does the Sun deliver to us now, compared to the start of the semester, solely as a result of its changed distance? If only the distance from the Sun caused the seasonal changes in temperature, would we be warmer or colder in the wintertime? Explain your reasoning.

12 APS 1030 Astronomy Lab 150 The Sun III.11 III.12 Compare the relative importance of the sunlight intensity-duration effect (III.8) with the solar-distance effect (III.10). Which of the two factors is, by far, the most important in causing seasonal changes? Explain clearly how you arrived at your conclusion. How would things be different if we were at a higher (more northerly) latitude in the northern hemisphere? If we were at a lower (more southerly) latitude in the northern hemisphere? If we were at the same latitude as Boulder, but in the southern hemisphere? Astronomers have been mapping sunspots for several hundred years. From their data we know that the number of sunspots visible to a viewer on Earth changes in a cycle with a period of about 11 years. The time when the most sunspots are visible is called solar maximum and likewise, when few or no sunspots are visible, we call it solar minimum. In addition to the total number of sunspots changing during the sunspot cycle, the latitude of sunspots also changes. At the start of the cycle, sunspots form mostly at higher latitudes (30-40 o ). At the end of the cycle, most sunspots tend to form closer to the Sun s equator at about 10 o. The chart below shows the latitudes at which sunspots have occurred over the past 80 years. Although spots can appear at nearly any latitude, note the trend from high to low latitudes in each cycle. The pattern of the distribution has given the chart its name: the butterfly diagram. The lower chart shows the annual average sunspot number; it clearly illustrates the cyclical nature of solar activity. SUNSPOT NUMBER SOLAR LATITUDE DATE

13 APS 1030 Astronomy Lab 151 The Sun III.13 Describe the change in the mean latitude of sunspots over the course of the semester. Is this what you would expect? Explain. Estimate where you think the Sun is in its cycle from the average latitude of sunspots. Over the course of several days, sunspots are observed to move across the disk of the Sun. observing the day-to-day motion of the sunspots, we can determine the solar rotation rate. III.14 III.15 III.16 III.17 III.18 Obtain a copy of a Solar Record Form from your lab instructor that contains sunspot observations of the Sun made during the course of one week. Each group will look at a different weeks worth of data. The maps must have several recognizable sunspots in common. Before you do any calculations, what can you say about the period of the Sun s rotation compared to that of the sun spot cycle? Identify sunspots that appear on both maps, and assign them the same spot label; use the appearance, relative location, and latitudes of the spots to aid in the identification. In order to determine the rotation period of the Sun, we must assume that individual sunspots do not change in latitude but only in longitude. Record the latitude and longitude of each of the sunspots you will use to determine the rotation period. Is our assumption correct? If not, by how much do the latitudes of individual sunspots differ between diagrams? Is it significant (i.e. is our assumption wrong)? Explain. For each sunspot, determine the angle through which it appears to have rotated (treat E longitudes as negative, W longitudes as positive, and subtract the first longitude measurement from the second). Average your measurements from the different spots to produce your best estimate of the observed rotation angle. Determine the number of hours between the two observations, and convert the time interval into fractional days. Divide the observed rotation angle by the elapsed time in days to determine the apparent solar rotation rate in degrees per day. By The Earth orbits around the Sun at a rate of 360 in one year (365 days), or an average motion of almost exactly 1 per day (this is probably not a coincidence; it is generally assumed that ancient astronomers/mathematicians divided the circle into 360 parts for just this reason!). Since we orbit the Sun in the same direction that it rotates, our motion "chases after" the sunspots; thus, the apparent movement of spots is less than their actual rotation by about 1 per day. III.19 III.20 III.21 Compensate for the orbital motion of the Earth by adding 1 to your computed apparent daily rotation. Use your daily solar rotation rate to compute the average rotational period of the sunspots: the number of days that it takes for a sunspot to travel 360 around the Sun. Compare your measurement of the rotational period of the Sun with the textbook. Discuss known or suspected sources of error in your measurement. Share your value for solar rotation period with the rest of the class. Average the values from each group and calculate error from the spread. Does your value now agree better with that given in your textbook? Explain.

14 APS 1030 Astronomy Lab 152 The Sun SUNSPOT RECORD FORM Observers Names: EXAMPLE (Finished Drawing) N sun N earth Date Time P B o August 18th, 19 11:45 am MDT o +6.3 o P Earth East #1 #4 #3 #2 Major Sunspots S earth S sun # Latitude (N/S) Longitude (E/W) o 25 N o 12 W o 12 S o 13 S o 16 S o 38 E o 11 W o 41 W # Latitude (N/S) Longitude (E/W)

15 APS 1030 Astronomy Lab 153 The Sun SUNSPOT RECORD FORM Observers Names: Date Time P B o Major Sunspots # Latitude (N/S) Longitude (E/W) # Latitude (N/S) Longitude (E/W)

16 APS 1030 Astronomy Lab 154 The Sun SUNSPOT RECORD FORM Observers Names: Date Time P B o Major Sunspots # Latitude (N/S) Longitude (E/W) # Latitude (N/S) Longitude (E/W)

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