G7 Estimating Star Brightness Activity G7 Grade Level: 6 12 Source: Adapted by permission from Hands-On Astrophysics: Variable Stars in Science, Math, and Computer Education, copyright 1998 by The American Association of Variable Star Observers, 49 Bay State Rd., Cambridge, MA 02138. Tel: 617-354-0484. Web: http://www.aavso.org (Note that the full Hands-on Astrophysics kit includes slides and prints that can convert this to an indoor activity.) What s This Activity About? A key characteristic of a star that astronomers need to measure is how bright it looks to us in the sky. An old system of brightness measurement called magnitudes is still in use today by both professional and amateur astronomers. Such magnitude estimates are especially important in following how variable stars change in brightness. What Will Students Do? Using the estimating activity provided, each student or team of students will learn about interpolation, a technique of estimating an unknown brightness of a star by comparing it with two or more stars whose brightness does not vary. Then, by familiarizing themselves with a sky chart, students will learn how to recognize the pattern of stars making up the constellation Cepheus in the night sky, and learn the location of the naked-eye variable star Delta Cephei and its comparison stars. Students will observe Delta Cephei on every clear night for at least two weeks and estimate its brightness by comparing it to nearby constant stars. They will keep track of their nightly observations by making a graph, called a light curve, which shows Delta Cephei s pattern of brightness change over the course of time, and allows them to estimate the period of its variations. Tips and Suggestions We strongly recommend that this activity be done with groups of students working together. Activity G13 is another, more concise, version of this activity. Activity 2 in this sequence requires students to be able to go outside at night and observe some relatively faint stars. In many urban areas, this may be a serious obstacle, either because, for safety reasons, students parents will not let them go outside when it is dark, or because the lights of the city makes it difficult to see fainter stars. You need to decide, in consultation with the students, and perhaps the parents, whether students can do this part of the activity. An excellent way to begin the observations in Activity 2 would be to hold a class star party, an evening observing session, attended by all the students (and members of their families), where everyone finds Cepheus together, and gains confidence in estimating magnitudes in the real sky. As noted in the activity, observing the constellation of Cepheus is best done in the fall, when it is high in the evening sky. Note that the star chart shown in the activity is for the fall evening sky; the stars patterns will look different in other seasons. (Good star maps for each month are published in two popular magazines called Sky & Telescope and Astronomy, which can be found in many libraries.) Or see: www.skymaps.com Teachers and students can learn more about variable stars and variable star observing by visiting the Hands-On Astrophysics pages of the AAVSO website: www.aavso.org What Will Students Learn? Concepts Star Brightness Magnitudes Variable Stars Interpolation Light Curves Inquiry Skills Measuring Estimating Classifying Organizing Inferring Graphing Big Ideas Patterns of change Interactions Models and simulations Page 1
Estimating Star Brightness Janet Mattei (AAVSO) John Percy (University of Toronto) Introduction Variable stars are stars that change their brightness. Sometimes stars change in brightness because they are actually not single stars, but pairs of stars that orbit one another, eclipsing each other from our point of view on Earth. Other stars often very young stars and very old stars change in brightness because of internal physical processes, some growing larger and smaller and brighter and dimmer. Some of these stars erupt dramatically, and some shine steadily for a time before starting to pulsate again. The star that students will be studying in this activity, Delta Cephei, was discovered to be variable in the late 1700s by a deaf teenager named John Goodricke. Delta Cephei is the prototype of a very important type of variable star called Cepheid variables. These stars pulsate on a very regular, or periodic, basis. (A well-known example of a Cepheid variable is Polaris, the North Star, which takes about 4 days to go through a complete cycle of variations.) It turns out that the rate at which such stars vary is connected with how much light they put out on average (their intrinsic brightness). Comparing the intrinsic brightness of the star derived from this connection to how bright the star looks from Earth, allows astronomers to get a measure of how far away they are. We have even been able to use Cepheid variables to measure the distances between galaxies, which has helped improve our understanding of the size and age of the Universe. There are far too many variable stars for professional astronomers to keep track of all of them. Luckily many variable stars can easily be measured with small telescopes, and scores of dedicated amateur astronomers monitor variable stars as hobby. The American Association of Variable Star Observers (AAVSO), the world s largest amateur astronomy organization devoted to variable star observing, collects measurements from amateur astronomers and sends the data to professional astronomers to assist them in their studies. This activity will introduce students to variable star research by enabling them to observe Delta Cephei, estimate its brightness, and record their estimates using some of the same techniques that professional and skilled amateur astronomers use. Activity 1: Estimating Magnitudes Using Interpolation For reasons that have to do with the history of their field, astronomers use a system to mark the brightness of stars called magnitudes. The approach comes from the ancient Greeks, who called the brightest appearing stars stars of the first magnitude and somewhat dimmer stars second magnitude, and so forth. Much to the regret of modern astronomers (and many generations of students), this cumbersome system, based on human vision, has been widely adopted among those who study the sky. The magnitude system thus works backwards the smaller the number, the brighter the star. A star whose magnitude is one unit greater than another turns out to be about a factor of 2.5 dimmer. A sixth magnitude star is about 100 times fainter than a first magnitude star. One of the first things that an astronomer (or amateur astronomer) who wants to get to know the stars has to learn to do is to get good at gauging the magnitudes of stars. To estimate magnitudes of the variable stars in our activity, students will need to interpolate. Interpolation is the process of estimating a value between two known values. Students will be observing two or more comparison stars of known magnitude that lie near the variable star of interest in the sky. These stars do not change in brightness and are used to compare to the brightness of the variable star. Knowing the values of the magnitudes of the comparison stars and the magnitude range of the variable star itself, students can interpolate or estimate Page 2
the magnitude of the variable star as it changes over time. For example, if the comparison stars were magnitude 4.0 and 4.6 and the variable star had a brightness halfway between them, it would be classified as magnitude 4.3. After you discuss this technique with your students, have them look at and fill out the exercise on Worksheet 1. Activity 2: Observing Your First Variable Star Delta Cephei From most latitudes in the northern hemisphere in the autumn, Delta Cephei is bright and high in the sky away from the horizon and local light pollution. It is in a fairly dark and uncluttered region of the sky. If the circumstances of your students (and your location) allow them to do evening observing, spend some time in the class discussing how to find the constellation Cepheus in the sky (see the instructions on Worksheet 2). Be sure you emphasize that everyone finds a star map like the one on the worksheet confusing at first, and that it takes some getting used to. On such flat star maps, the dome of the sky over our heads is reduced to a circle. The dark outer line of the circle represents the horizon all around us; the point in the middle of the map is the zenith, the point in the sky directly over our heads. To use the map to find things in the real sky, students must hold the maps over their heads, lining up the compass directions shown around the circle with the real world. In other words, they should hold the map over their heads so that north on the map lines up with the direction of north in their location or hold the map in front of them, so that the direction they are facing is down. When students make observations outside, having just come out from a brightly-lit house or car, it is important that they allow about 10 to 15 minutes to pass, so that their eyes can acclimate to the dark. (Just as you can see more spilled popcorn and soda on the floor of a movie theater after a few minutes in the dark, so you can see more faint stars if you wait a while.) By the way, if students have binoculars available, their use will make the fainter stars even easier to spot, although they can take a bit of getting used to. If students need to refer to the star map when they do their observations, they will need a flashlight with them. The bright white light of a flashlight can often interfere with the eye s ability to see faint objects in the sky. An old astronomer s trick is to cover your flashlight with red cellophane, holding it to the barrel of the flashlight with a rubber band; red light is much kinder to your night vision. If no red light is available, then a faint flashlight will be better than a brilliant one. To make sense of the jumble of stars we can see on a clear, dark night, we rely on the patterns that stars make, such as the Big Dipper or the W-shaped star grouping called Cassiopeia. In the activity, students will find the Big Dipper (easily identified toward the northern horizon) and star-hop among the star patterns to find fainter Cepheus. The Big Dipper looks like a pot with a long handle, or a square ladle for soup. Students should begin by finding the pointer stars of the Dipper the two stars that make the side of the pot furthest from the handle. If you connect the two pointer stars (called Merak and Dubhe), the line connecting them (moving away from the horizon) points toward the North Star, Polaris, which is the tip of the handle of the Little Dipper. If students keep going following the direction of the Big Dipper pointer stars in the sky, they should be able to find the distinctive W shape of Cassiopeia. (See Sky Map on Worksheet 2). The fainter star pattern Cepheus is found right next to Cassiopeia, between its W shape and the bright star Deneb (part of a pattern of three bright stars called the Summer Triangle). Cepheus looks a bit like a square house and a triangular roof (see star map) and Delta Cephei is in the lower left corner of the house. Practice doing the star hops with the students in the classroom, before they go out for their evening observing. You may want to assure them that identifying star patterns and star hopping is a skill that takes a while to acquire, and that they should not be hard on themselves if they can t orient themselves in the sky in the first ten seconds of their observations. Also, you may want to discuss the technique of using averted vision for the observations before the students go out under the stars. This will be a new technique for many of them, and may take some practice. The idea is that the edge of one s field of vision is more sensitive to black and white contrasts, so the difference in magnitude will be easier to discern if the students look at the star whose brightness they are estimating from the Page 3
corner of their eye. On the other hand, this is perhaps best saved for a time when the students are past the first anxieties of just finding their way around the sky and learning to make magnitude estimates. Students should observe Delta Cephei on each clear night for at least two weeks one month would be even better, but this may not be practical for most students and classes. If students are not able to do outside observing, or if you want to give them some practice in making light curves (see Activity 3), you can use the table of observations above; these measurements were made by an AAV- SO member in 1990. Activity 3: Plotting a Light Curve Observations of variable stars are plotted on a graph called a light curve; the apparent brightness (magnitude) is shown on the vertical axis, and the time of the measurement on the horizontal axis. (Astronomers usually measure the date and time in a special system called the Julian Date; for this exercise, we will simply record the regular date and time. For a Julian date conversion calculator on the Web, see: aa.usno.navy.mil/aa/data/ docs/juliandate.html.) The light curve is the single most important graph in variable star astronomy. It allows astronomers to unlock some of the secrets of stars and decode the messages hidden within the starlight. Information about the periodic behavior of variable stars, the orbital period of star systems where the two stars actually eclipse each other, or the regularity of stellar eruptions can be directly determined from the light curve. More detailed analysis of the light curve allows astronomers to calculate such information as the masses or sizes of stars. Several years worth of observational data can reveal the changes in the period of a star (the time for one cycle of variation), which is a signal of a change in the star s internal structure or its overall size. Students will use Worksheet 3 to construct a light curve for Delta Cephei from either their own data, or the data given in the table in Activity 2. They will be asked to draw a smooth curve to find the best fit for their data. Drawing such a curve (where the curve itself may not actually fall on many of the points) may be a new experience for the students, who may be more accustomed to drawing graphs where the data points are connected with a line. Remind the students that the estimates they made (and any visual observer may make) of a variable star are necessarily limited in accuracy, and so there will be a scatter in their points. Scientists working at the forefront of their observational capabilities face similar issues; there is always an uncertainty in the data, and they will often average many data points over long periods of time to get the best value for a number they are seeking. Note that the period of Delta Cephei (the time to go through one cycle of variations), is about 5.4 days (5 days and 9 hours). If at all possible, let the students discover this value before you give it to them. Once they have measured the period, you can remind them that this would allow astronomers to calculate the intrinsic brightness of the star. The intrinsic ( close-up ) brightness of a star is typically greater than how bright it looks to us in the sky. This is because the light of a star Measurements of the Magnitude of Delta Cephei in 1990 Date/Time Oct. 6 at 3:07 4.0 Oct. 9 at 1:12 4.4 Oct. 11 at 1:55 4.0 Oct. 12 at 1:55 3.5 Oct. 13 at 1:11 3.8 Oct. 14 at 1:41 4.0 Oct. 15 at 2:09 4.4 Oct. 17 at 0:58 3.7 Oct. 18 at 1:12 3.8 Oct. 19 at 23:46 4.0 Oct. 21 at 0:14 4.2 Oct. 22 at 1:12 4.0 Oct. 24 at 1:12 4.0 Oct. 28 at 2:10 3.8 Oct. 29 at 0:00 4.0 Oct. 29 at 23:46 4.1 Oct. 31 at 1:41 4.3 Nov. 2 at 0:14 4.0 Nov. 6 at 0:00 4.5 Nov. 7 at 0:14 4.1 Magnitude Note that times are given as Universal Time, roughly the time at Greenwich, England. Page 4
gets dimmer with distance. By comparing the intrinsic brightness and the (average) apparent brightness, astronomers can then find the star s distance, something that is typically very hard to do for individual stars that do not vary. Page 5
ESTIMATING MAGNITUDES WORKSHEET 1 Name: Date: 1. Six star fields are shown below. The magnitudes of the comparison stars are given in each field. As you can see, the bigger the dot representing the star, the brighter the stars is (and the lower the magnitude number is). Estimate to the nearest tenth the magnitude of the star that is not labeled with a number (the star we are interested in is marked by four short lines around it). **NOTE: In star fields, the decimals are not indicated. A magnitude of 6.4 is written as 64, so that the fields are not as cluttered and the decimal points are not mistaken for stars. Also record the estimates made by two of your classmates. Do your estimates differ from theirs? A D B E C F Page 6
ESTIMATING MAGNITUDES WORKSHEET 1 Data Table Your Magnitude Classmate 1 Classmate 2 A B C D E F 2. Compare your estimated magnitudes with those of the rest of the class. Does everyone have the same answers? If answers differ, what do you think is the reason? Page 7
MEASURING DELTA CEPHEI WORKSHEET 2 Name: Date: 1. Enter in your logbook the name of the star (Delta Cephei), the date of the observing session, and the time. 2. Using the constellation chart on the next page to find the Big Dipper and Polaris (the North Star), star hop to Cassiopeia and then to Cepheus. Cepheus lies among Polaris, the W shape of Cassiopeia, and the bright star Deneb. The whole Cepheus pattern is about the size of your clenched fist, held at arm s length. 3. Using the chart below (showing the constellations Cassiopeia and Cepheus), find the four stars that make a rough rectangle in Cepheus. Now find the group of three stars near one corner of the rectangular portion of Cepheus. These are Delta Cephei and its comparison stars Zeta Cephei (magnitude 3.6) and Epsilon Cephei (magnitude 4.2). This is the most difficult part and may take you several attempts, as you alternately look at the chart and the sky. Page 8
MEASURING DELTA CEPHEI WORKSHEET 2 Page 9
MEASURING DELTA CEPHEI WORKSHEET 2 (continued) 4. Using averted vision if necessary, observe the variable star and its comparison stars at the center of your field of view. (Averted vision is a technique in which you orient the star at the center of your field of view, and then gaze at the edge of the field.) 5. Estimate your variable star's magnitude to the nearest tenth by using the nearby comparison stars. Look quickly back and forth and ask yourself: Is it dimmer or brighter than this comparison star? Is it dimmer or brighter than the second comparison star? If it is brighter, by how many tenths? Make a note of it. Then estimate the magnitude of your variable star again, and do it a third time. Enter the three numbers and average them in your logbook; then record your result in the data table. (Remember that the brighter the star, the lower the magnitude.) Do not be discouraged if you initially cannot tell the difference between a star of 3.0 and 3.5 in magnitude. Remember, your observations are a valuable "part of the whole" even if you are not yet an expert observer. With experience, you will be able to make your observations much more accurately and quickly. 6. Record the names and magnitudes of the comparison stars used in your log book. 7. Record the time of your observation to the nearest quarter hour. 8. Place a colon [:] after your observation if you are unsure of your observations due to a bright Moon or faint clouds high in the sky. 9. Use a less-than symbol < to indicate that a variable star appears to be fainter than the faintest companion star on your star map. Page 10
PLOTTING A LIGHT CURVE WORKSHEET 3 Name: Date: 1. Using graph paper, mark the brightness (or magnitude) intervals on the y (or vertical) axis, indicating each tenth of a magnitude (Delta Cephei's brightness ranges from approximately 3.4 at its brightest to approximately 4.4 at its dimmest, so begin your graph with 3.0 at the top and go down to 5.0). Then mark your x (or horizontal) axis with the number of planned observation days (using the actual dates, even including days you could not make any observations), as shown below: Dimmest Magnitude Brightest 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10 Day 11 Day 12 Day 13 Day 14 Date 2. Mark each student's observations for each day. Each team or student can plot their data on a separate graph, or the whole class can make one combined graph, using different colors or symbols for each team. Page 11
PLOTTING A LIGHT CURVE WORKSHEET 3 (continued) 3. Take the average of each night's estimates (averaging the observations for the whole class) and plot these average estimates, marking these average data points using a bright color. Then connect the averaged points with a line. How do the average estimates compare with each individual's or team's observations? Is the average estimate a better measurement of Delta Cephei's brightness than any one individual's or team's estimate? Why or why not? 4. Try to draw a smooth average curve through the points, showing how the brightness of the star rises and falls. Such an average curve doesn t necessarily touch all the points, but fits around them so as many points are above as below. 5. Estimate the day of maximum and minimum brightness from the curve you drew. Maximum: Minimum: 6. Now estimate the period of Delta Cephei (the time it takes to make one complete cycle of variations from brightest, let s say, back to brightest again). Estimate of Period: After you have finished, your teacher can tell you how close you came. Don t worry if you didn t hit the value right on the head. Estimating magnitudes is a tricky business, and making such a rough graph (without keeping track of the exact time of day for each observation) can t give you a very precise result. Page 12