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Laboratory Exercise #7 – Geologic Features of Mars

Purpose: Study the geologic features of Mars using remote sensing images taken of the surface of Mars by the Viking Orbiter. Interpret the features within the images and compare them to similar features on Earth to hypothesize the forces that led to the development of these features.

IntroductionMars, as seen through the the camera onboard the Viking Orbiters (there were two), displays geologic features. These features can be compared to geologic features on Earth. Some geologists categorize the processes that shape the crust of the Earth into four categories. These four categories are erosion, volcanism, impact cratering, and tectonism. These categories may also be applied to Mars.

Erosion is the dominant geologic process acting on Mars today. There are different types of erosion. The most common erosional process is mass movement. Mass movement refers to the displacement of material by landslides or the slumping (i.e. the collapsing or sinking) of material through the action of gravity. Aeolian (i.e., wind) activity is also a continuing process of erosion. Sand and dust particles carried by the wind form dunes and windstreaks. Erosional processes involving running water were once important on Mars. Current temperatures and pressures simply do not allow liquid water to exist on the surface of Mars today. Valley systems cut through many of the cratered terrains of Mars and some have characteristics analogous to water-cut valleys on Earth. Although there is evidence that water once flowed on the surface of Mars, there is no surface water flowing today. What happened to the water that was once present? Some water probably seeped into the ground and exists there today in the form of ice, similar to the permafrost of our Arctic region on Earth today. Some escaped into space over time. If water evaporates, then gets to a level where UV radiation can break the bonds of the water molecule, the hydrogen escapes and the oxygen is reabsorbed by the surface of the planet. We do know that there is some water contained in the Martian polar ice caps.

Volcanism on Mars produced vast lava flows, broad shield volcanoes, and plains of volcanic material. In fact, Mars has some of the largest volcanoes in the solar system, including Olympus Mons, a massive volcano many times larger than the Island of Hawaii. Olympus Mons is only one of four huge volcanoes in a 3000 km-wide region called Tharsis. These volcanoes erupted repeatedly over many millions of years, growing higher with each lava flow. Enormous collapsed calderas are found on the summits of volcanoes, now long extinct.

Impact craters are sometimes considered another type of erosional process. They are produced by the impact of asteroids and comets that have penetrated the Martian atmosphere and may produce remnant meteorites. Impact craters can be used to determine the relative ages of different areas of the Martian surface. Generally speaking, older surfaces have more craters per unit area than younger surfaces. Older craters are generally also larger and more eroded. The geology principles of superposition and crosscutting relations state that a feature which at least

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partly covers another feature is the younger feature. This is important in determining the relative ages of all surface features. For example, if a valley cuts through a crater, the crater must be older than the valley. Individual craters are degraded or even destroyed over time by erosional processes and further cratering. Therefore, crisp craters that have upraised rims, a central peak, and steep sides are relatively young, while less distinct, eroded craters having only partial rims are relatively older.

Tectonic features visible on the Martian surface indicate that Mars has been deformed by some type of tectonic process. It must be emphasized that Mars does not have tectonic plates like the Earth. Stress (i.e. a pulling apart or a pushing together) can lead to subsurface uplift. Extensional (as opposed to compressional) stress led to the formation of great valleys on Mars. One of these is the Valles Marineris, the longest canyon in the solar system.

By examining individual craters and applying the principles of superposition and crosscutting relations to the features of any planetary surface, the relative ages of the geologic features on a planet can be determined, along with a sequence of geologic events.

Questions and ProceduresAnswer all of the following questions and enter your response on the answer sheets provided at the end of this write-up. The images required for this lab can be accessed at the following website:

http://physics.gmu.edu/~hgeller/marsgeolab.html

Regarding Figure 1 - Olympus Mons http://physics.gmu.edu/~hgeller/MarsLabFig1.jpgOlympus Mons is a shield volcano that rises some 25 km above the surrounding plain. Around its base is a steep cliff wall, known as a scarp, that is as high as 6 km. It also has a summit caldera likely formed when the top, or cone, of the volcano collapsed.

1) Examine the caldera (labeled A) and describe its shape.

2) Measure the diameter of the caldera structure from east to west.

3) Measure the diameter of Olympus Mons. Give both the east-west and north-south measurements.

4) Suggest some ways in which the scarp around the base of Olympus Mons might have formed.

5) Do you think the surface of Olympus Mons is geologically young or old compared to the surface of the Moon? Explain your answer. The following website has several images of the surface of the Moon. Use these images to help you answer this question.

http://rst.gsfc.nasa.gov/Sect19/Sect19_3.html

Regarding Figure 2 – Ius Chasma http://physics.gmu.edu/~hgeller/MarsLabFig2.jpg

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Ius Chasma is part of the western end of Valles Marineris. Note the smaller tributary valleys that join the main east-west chasm.

6) Using pictures from a textbook or accessing one of the websites given below, draw a rough sketch of the outline of the Valles Marineris. On your sketch, draw a box around Ius Chasma.

http://www.exploringmars.com/science/vm.htmlhttp://www.the-planet-mars.com/map-valles-marineris.html

Measure the width of Ius Chasma, from the outer edge of the tributaries on its left boundary to the outer edge of the tributaries on its right boundary.

7) Compare the size of Ius Chasma and its tributaries to the size of the Grand Canyon. Utilize the world wide web to discover the size of the Grand Canyon and note it on your answer sheet. Which is larger the Grand Canyon or Ius Chasma, and by what amount?

8) Which of the four geologic processes might be responsible for the formation of Ius Chasma?

Regarding Figure 3 – Chryse Planitia http://physics.gmu.edu/~hgeller/MarsLabFig3.jpgThe Martian channels of the Chryse Planitia exhibit a branching pattern, similar to some river systems on Earth.

9) In what direction does the water appear to have flowed in the distant past?

10.) Based on the size and number of craters, and their appearance, would you judge this to be a relatively young or old region of Mars? Explain.

11.) Are most of the craters that you see younger or older than the valleys? Justify your answer.

12.) What is the diameter of the large crater near the bottom of the image?

Regarding Figure 4 – Hesperia Planum http://physics.gmu.edu/~hgeller/MarsLabFig4.jpgThe Hesperia Planum region in the southern hemisphere of Mars consists of cratered plains which have been modified by aeolian (wind) processes. Bright, wind-produced features, called windstreaks, are associated with many of the craters.

13.) Describe the appearance of the windstreaks, and comment on their orientation. Be specific!

14.) If windstreaks are dust deposits formed downwind from these craters, what wind direction is indicated here? Remember that wind direction refers to the direction from which the wind blows.

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Regarding Figure 5 – Apollinaris Patera http://physics.gmu.edu/~hgeller/MarsLabFig5.jpgAll four geologic processes can act to shape a planetary landscape, as depicted in the following image of Apollinaris Patera and its surrounding region. Use what you have learned from previous questions to describe the geologic processes that created the Martian landforms seen here.

15.) Compare Apollinaris Patera (marked A in the figure online) to Olympus Mons (in Figure 1 online). Describe the similarities and differences.

16.) What is the approximate diameter of the base of Apollinaris Patera?

17.) What geologic process may have formed Apollinaris Patera (labeled A)? Justify your answer.

18.) What geologic process do you think formed Reuyl (labeled B)? Justify your answer.

19.) Ma'adim Vallis (labeled C) is the channel in the southeastern part of the image. Which of the four geologic processes do you think formed this channel? Justify your answer.

20.) Consider the relationship between Ma'adim Vallis and the 160 km-diameter crater known as Gusev (labeled D). Speculate as to the origin of the material that comprises the floor of Gusev? Hint: this region slopes down to the north.

21.) Based on your observations, what is the probable order of occurrence of the features labeled A, B, C, and D (i.e. which came first, second, third, last)? Justify your answer.

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Answer SheetsTo be submitted to your instructor.

Questions Regarding Figure 1:1.) Examine the caldera (labeled A) and describe its shape.

2.) Measure the diameter of the caldera structure from east to west.

3.) Measure the diameter of Olympus Mons. Give both the east-west and north-south measurements.

4.) Suggest some ways in which the scarp around the base of Olympus Mons might have formed.

5.) Do you think the surface of Olympus Mons is geologically young or old compared to the sur-face of the Moon? Explain your answer.

Questions Regarding Figure 2:6.) Draw a rough sketch of the outline of the Valles Marineris. On your sketch, draw a box around Ius Chasma.

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7.) Measure the width of Ius Chasma, from the outer edge of the tributaries on the left to the outer edge of the tributaries on the right. Compare the size of Ius Chasma and its tributaries to the size of the Grand Canyon. Which is larger, and by what amount?

8.) Which of the four geologic processes might be responsible for the formation of Ius Chasma?

Questions Regarding Figure 3:9.) In what direction does the water appear to have flowed in the distant past?

10.) Based on the size and number of craters, and their appearance, would you judge this to be a relatively young or old region of Mars? Explain.

11.) Are most of the craters that you see younger or older than the valleys? Justify your answer.

12.) What is the diameter of the large crater near the bottom of the image?

Questions Regarding Figure 4:13.) Describe the appearance of the windstreaks, and comment on their orientation. Be specific!

14.) If windstreaks are dust deposits formed downwind from these craters, what wind direction is indicated here? (Remember that wind direction refers to the direction from which the wind blows.)

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Questions Regarding Figure 5:15.) Compare Apollinaris Patera (marked A in the figure online) to Olympus Mons (in Figure

1 online). Describe the similarities and differences.

16.) What is the approximate diameter of the base of Apollinaris Patera?

17.) What geologic process may have formed Apollinaris Patera (labeled A)? Justify your answer.

18.) What geologic process do you think formed Reuyl (labeled B)? Justify your answer.

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19.) Ma'adim Vallis (labeled C) is the channel in the southeastern part of the image. Which of the four geologic processes do you think formed this channel? Justify your answer.

20.) Consider the relationship between Ma'adim Vallis and the 160 km-diameter crater known as Gusev (labeled D). Speculate as to the origin of the material that comprises the floor of Gusev? (Hint: this region slopes down to the north.)

21.) Based on your observations, what is the probable order of occurrence of the features labeled A, B, C, and D (i.e. which came first, second, third, last)? Justify your answer.

Conclusion:

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Laboratory Exercise #8 – Cratering in the Solar System

Purpose: Learn how to estimate the age of the surface of a terrestrial-type planet or moon by using satellite images to quantify the crater densities (number of craters per unit area).

IntroductionCraters are produced by the impact of asteroids and comets that have penetrated the Martian atmosphere and may produce remnant meteorites. One rule of thumb is that an impactor (i.e., anything impacting the surface causing a crater to form) will carve out a crater about 20 times its own diameter. After a crater has been formed, there are several factors, such as erosion by water and wind, that gradually obliterate a crater. On Mars, it is estimated that wind-carried dust can obliterate a small crater in about a million years and a large crater in about 100 million years.

Craters are also effected by a thick lava flow which can totally alter surface features in a very short period of time. A careful study of the number and size of craters gives us a way to estimate the age of the rocks that make up the surface of a planet.

In this lab you will measure crater diameters, plot this data and generate the same type of graph (i.e. curve) for each of four objects in the solar system: Mercury, Mars, the Moon and the Earth. You will use computer digitized images and a special tool designed to facilitate the taking of these measurements. You will also have access to an Excel file which contains a worksheet, and a separate graph on which to plot your data, for each of the four objects.

The graph on which you will plot your data shows the relationship between the crater density (i.e. the number of craters per square kilometer) of an object’s surface and the size of the craters. The curves on the graph are called isochrons (meaning “same age”) and describe the age of the surface on which the craters formed. This graph not only shows that there are more small craters than large ones, no matter the age of the surface, but also that the greater the number of craters there are per square kilometer of surface, the older the surface.

Note: The proportion of small craters to large is about the same throughout the solar system, regardless of the age of the surface.

Procedure1.) To begin, click on the “Scion Image” icon on the desktop. When the program starts, go

to File-Open. Open the subdirectory “Images,” and then open the first file for analysis, “MERCURY.BMP.”

Note: This file is a read-only file. You should not save this file when you have finished with your measurements.

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2.) Before any measurements can be made, a scale must be set. (The scale tells the program how to translate the pixels of the computer image into distance.) Go to Analyze-Set Scale..., and select ‘kilometers’ from the Units drop-down box. Enter 0.725 into the Scale box (see note below) and click “OK.”

Note: Do not leave a zero in the Scale box or it may crash the software.

3.) Now that the scale is set, measurements can be taken. Go to the Tools window and select the line drawing icon,

When you move the cursor over the image, it will change to a black-and-white cross. Position this cross at the edge of a crater whose diameter you want to measure, hold down the left mouse button and drag the cross to the opposite edge of the crater. Release the mouse button. You should have drawn a line across the crater. The program will measure your line, not the size of the crater, so draw it as accurately as possible.

4.) Now, tell the program the information you are interested in logging. Since we need the program to measure length, go to Analyze-Options…. Uncheck Area and Mean Density, select Perimeter/Length, and click “OK.”

5.) Go to Analyze-Measure on the drop-down menu and the program will measure your line. To view the measurement, go to Analyze-Show Results. A list of all the measurements you take will be shown. (Since this is your first measurement, there will be only one value in the Results Table.)

6.) Keep track of the craters you measured. Go to Analyze-Label Selection, and the crater line you just drew will be numbered.

Note: Do not click the mouse button outside the boundary of the object image or any of the Scion Image windows. This may erase your results and you may have to begin again from Step 1.

7.) Repeat steps 3 to 5—you have already selected the appropriate measuring tool—until you have measured 60 craters on the Mercury image. (You should measure 60 craters on each of the first three images; there will be only 45 crater distances for you to measure on the image of the Earth.)

Note: If you leave the Results window on the screen, you won’t need to repeat that part of Step 5 requiring you to “go to Analyze-Show Results.”

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8.) When you have finished taking measurements for Mercury, you can copy them to Excel for analysis. Go to Edit-Copy Measurements to save the work you have done. Open up the Excel file Craters.xls and paste the list into the worksheet labeled “Mercury,” at cell location A2. (Excel will automatically place each measurement into the next A-cell location, so you need only click on cell A2.) The program will group your measurements into 5 different size categories, such as 8km – 16km (see the column labeled Crater Diameter in the worksheet table).

9.) The “Craters.xls” file is a read-only file, so make sure to save the file as craters-your name.xls. Notice that within the workbook each worksheet is labeled. Be careful to paste your data into the appropriate worksheet.

10) Next to each data worksheet is a graph worksheet. (In the case of Mercury, the worksheet is labeled “Mercury Graph.”) Click on this worksheet and examine the graph. You will plot the data you have taken onto this pre-existing graph.

11) Go back to the “Mercury” worksheet and examine the last column in the table, Crater Density (Counts/km2). You will calculate the values in this column by taking the values in the Total Counts column and dividing them by the area of the image. (All the images in this lab, except the image of the Earth, are 600 km × 600 km. Therefore, the area of the image is 360,000 km2. The area of the Earth image is 1.3 × 107 km2.)

12.) The values in the average crater diameter column are the medians between the low and high values for each size category. For example, in the case of the 8km – 16km group, the average crater diameter would be 12 km. Since it is the average (i.e. the median) crater diameter, and not each individual crater diameter, that is being plotted, error bars must be used so that the first data point will cover all values from 8km – 16km. (The procedure for adding error bars is discussed in Step 18, below.)

13.) To add your data to the pre-existing graph, click on the Mercury Graph worksheet. Place the mouse in the graph, right click and the following options will appear:

14.) Choose Source Data and click on the “Series” tab. To add your data, click on the “Add” button. Type “Mercury Data” in the name box.

15.) Click on the red arrow on the right-hand side of the “X-Values” window. Go back to the worksheet (Mercury, in this case), highlight the x-values, and then re-click on the red

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arrow. To add the y-values, repeat this procedure using the “Y-Values” window, and then click “OK” at the bottom, right.

16.) Ideally, you should get a line (i.e. a curve) that is similar in shape to the pre-plotted ones. However, your line may look a bit different, especially for small values of x (i.e. on the left side of the graph).

17.) Now that your data is on the graph, you may want to change the color or the width of your curve. To do this, right click directly on the curve and select “Format Data Series…” from the options that appear. Click on the “Patterns” tab and proceed from there.

18.) To add error bars, right click on the curve, and select the “X Error Bars” tab. Select “Both” under “Display.” Set the percentage to 33%. Excel will now draw a horizontal line to the left and the right of each data point. Thus, the total length of this line will be 8 km for the first data point, 16 km for the second data point, and so on.

19.) Now that the graph has been completed, you can determine the age of the cratered surface by examining the relationship between your curve and the pre-existing isochron curves. Each isochron represents the crater density for the age indicated. Estimate the age of Mercury by analyzing the position of your curve relative to the positions of the pre-existing isochrons. (Observe that the isochrons increase in age in the positive-y direction.)

Click here andenter 33.33%

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How does the shape of your curve differ from the shapes of the curves already on the graph?

Did your data give the results (i.e. an age) you expected? If not, why?

20.) With Mercury completed, move onto Mars, then the Moon and finally, the Earth. When you return to the Scion Image program, Reset the Show Results window to omit the crater measurements taken for a previous image. To do this, go to Analyze-Reset, and the measurements will disappear from the Show Results window. (Check this before proceeding.)

When working with each planet or the Moon, follow the procedures outlined above and create a graph for each celestial body. Estimate the age of each as described before, using the isochrons as a reference. For each planet and the Moon, answer the questions posed in Step 19.

21.) As Earth’s surface has been, and continues to be, modified by weathering, erosion, tectonics, and volcanism to a far greater extent than any of the other planets, most of its craters are very difficult to find. As such, we have taken a map of the contiguous United States and Canada and marked the position of each known crater with a red circle. (Take note of the bottom-most circle, at the tip of the Yucatán Peninsula, bordering the Gulf of Mexico. This crater was made about 65 million years ago, by an impactor 10 km in diameter. It is this impact that is believed to have precipitated a sequence of events that eventually lead to the extinction of the dinosaurs.) Beside each circle is a horizontal yellow line the length of which is proportional to the diameter of the crater. The scale of these lines is one pixel = 1.0 km. Therefore, before taking measurements of the crater lines in the EARTH.BMP map, make sure to change the scale value to 1. To do this, repeat step 2 above, but enter 1.0 in the Scale box. (This scale does not apply to the map shown, but only to the crater lines. The area of this map is about 1.3 × 107 km2. It does not include northern Canada and Alaska because their geology hasn't been studied as thoroughly as the rest of the North American continent.)

That which will be submitted to your instructor:

Four Excel spreadsheets, four graphs, and your answers to the questions posed in Step 19 for each planet and the Moon. Also, rank the four celestial bodies in terms of age, from oldest to youngest. Write your estimate of each body’s age after its name. Provide concluding remarks below, and comment on your ability to estimate the age of the solar system by analyzing the results you obtained. Answer any specific questions asked by your instructor as well.

Conclusion:

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Laboratory Exercise #9 – Radar and the Rotation Rate of Mercury

Purpose: Learn how scientists can calculate the rotation rate and orbital velocity of a planet using radar measurements.

IntroductionSince Mercury is a small planet whose surface features show little contrast, and because it is so close to the Sun that it is not often visible against a dark sky, it is difficult to determine how fast it is rotating merely by observing the planet from the Earth. However, radar techniques developed in recent years have proven very effective in measuring its speed of rotation. (These techniques have wider application than simply measuring the rotation of Mercury. On a large scale they are also used to study various motions of the other planets, and on a smaller scale, they are used to investigate the motions of asteroids and even the particles that comprise the rings of the Jovian planets.

The procedure that you will follow in this lab is a very realistic simulation of the techniques employed by astronomers. The basic idea is to use a radio telescope to send a short pulse of electromagnetic radiation of known frequency towards the planet Mercury, and then to record the spectrum (i.e. the frequency vs. the intensity) of the returning echo. Depending on the relative positions of the Earth and Mercury, the pulse can take anywhere from just under 10 minutes to almost half an hour to make the round trip.

By the time the pulse reaches Mercury, it will have spread out to cover the entire planet. As the planet’s surface is spherical, the pulse will hit different parts of the planet at different times. The pulse will first hit the surface at a point that lies directly on a line between the center of the Earth and the center of Mercury, known as the sub-radar point. Over a time interval of a few hundred microseconds the pulse will hit points farther back along the surface, toward the edges of the planet. Thus, the first echo will come from the sub-radar point. By examining the returning echoes, each of which will arrive about 100 microseconds later than the previous one, we can obtain information about different parts of Mercury’s surface.

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fleftfecho

fright

Figure 9.1: The Spectrum of an Echo

Figure 9.2: Illustration of Doppler Shift due to the Rotation of Mercury

The frequencies of the returning echoes are different from the frequency of the pulse sent out, because the echoes have bounced off a moving surface (i.e. Mercury). Any time a source of radiation is moving radially with respect to an observer (i.e. toward or away from an observer), there will be a Doppler shift in the frequency of the observed signal that is proportional to the velocity of the radiation source along the line of sight of the observer. (Simply stated, the faster an object is moving toward or away from us, the greater the Doppler shift of the signal that we observe.)

There are two motions of Mercury that can produce such a shift: its orbital velocity around the Sun and its rotation on its axis. The first echo, from the sub-radar point, is shifted in frequency only by the orbital velocity of the planet. We can calculate how fast the planet is moving with respect to the Earth from the amount of this shift, but we can’t determine how fast the planet is rotating. This is because the component of Mercury’s rotational velocity is perpendicular to our line of sight at the sub-radar point (see Figure 9.2), and so, there is no additional frequency shift (i.e. no shift due to the rotation of the planet). However, the echoes that are received after the sub-radar echo show additional shifts. They come from farther back along the planet’s surface,

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to the left and to the right of the sub-radar point, where the rotational velocity is more directly along our line of sight.

The rotational motion that is most directly along our line of sight is near the eastern and western edges of the planet. The edge of the planet that is rotating toward us has an orbital velocity that is a little faster than the orbital velocity of the planet as a whole. The edge of the planet that is rotating away from us has an orbital velocity that is a little slower than the orbital velocity of the planet as a whole (again, see Figure 9.2). Therefore, as explained by the Doppler effect, the right-hand part of the returning echo, reflected from the edge of the planet rotating toward us, will have a slightly higher frequency, and the left-hand part of the echo, reflected from the edge of the planet rotating away from us, will have a slightly lower frequency.  We can measure the amount of these frequency shifts, and by applying our knowledge of the Doppler effect we can calculate the rotational velocity of the surface of Mercury. Once we know this velocity, we can determine Mercury’s period of rotation.

ProcedureAccess the CLEA software as you did in a past laboratory exercise (see Lab #4). Click on the Mercury Rotation icon. Click on Log In…at the top left of the screen and proceed as you did previously.

When you get to the screen having the title of the program, click on Start at the top left of the screen. To continue, click on Tracking. The program will indicate that the Radio Telescope is ON. Click on Ephemeris, and then click on “Ok” to see the coordinates (i.e. the R.A. and the Dec.) of Mercury for the present date and time. Click on Set Coordinates and then answer “Yes” to the question, “Use Computed Values?”

The screen also gives you the distance to Mercury from the Earth, and the time it will take for a signal sent from the Earth to reach the surface of Mercury (in Light Minutes). Click on Send Pulse. In about 20 seconds or so you will see a “green” radio pulse leave the Earth, headed in the direction of Mercury. The screen will indicate the time remaining until the expected return of the first echo.

Bouncing a radar pulse off the surface of Mercury (and receiving the return echo here on Earth) will take a minimum of almost 10 minutes, even though the radar signal travels at the speed of light, because the distance between the two planets is considerable. This distance (and therefore the time) will vary, of course, depending upon the relative positions of Mercury and the Earth in their respective orbits. Mercury can be positioned as close to the Earth as possible—this is the planetary configuration known as inferior conjunction—or as far away from the Earth as possible—this is the planetary configuration known as superior conjunction—or anywhere in between these two extremes.

The pulse you have sent is on its way toward Mercury. After reaching the planet it will bounce off the surface and head back toward the Earth. The screen will visually alert you when just under half a minute remains until the arrival of the first echo. You may start on your calculation for Table 9.1 while waiting for the signal echo to return. Watch the screen as the echoes arrive and the Doppler-shifted spectra of five echoes are displayed, each in its own window.

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Measuring and Collecting the DataExamine the first echo (at t = 0). The signal peaks and then falls off to the baseline in a similar way on either side of the peak. Each of the following echo signals will have more than one peak. The leftmost and rightmost peaks are known as the shoulders of the echo, where the intensity on either side of the signal just begins to fall off toward the baseline. Look for the shoulders of the echoes in the second through fifth data windows.

Complete Table 9.1 on the answer sheet as you follow the steps below.

1.) Begin with the second echo (at t = 120 microseconds).

2.) Calculate the distance that the delayed pulse has traveled beyond the sub-radar point, by multiplying half the delay time by the speed of the radar pulse. Enter the result in the first row of the table.

d 12---ct=

Here t is the time delay in microseconds (1 microsecond = 10-6 seconds), c is, of course, the speed of light (c = 3 x 108 m/s), and d is the distance, in meters.

3.) Calculate the lengths x and y (refer to Figure 9.3 to understand the context).

Figure 9.3

x R d–=

y R2 x2–=

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Here R is the radius of Mercury (R = 2.440 × 106 meters). Record your results in the second and third rows of the table.

4.) Calculate favg, the shift in frequency due to the rotational velocity alone. You need simply note that one side of Mercury is rotating toward the Earth as fast as the other side is rotating away from the Earth. Therefore, the difference between the frequency shifts from the two sides, fright and fleft, is twice the shift due to the rotational velocity. So we divide by 2:

|fright - fleft|favg = ----------------------

2

Note: Negative frequency shift values simply refers to the direction of the motion.

fcfavg

2------------=

5.) Since the signal we are measuring is an echo (i.e. a reflection), the Doppler shift due to the orbital velocity of Mercury is only one-half the average value.

6.) Using the signal in the data window we wish to find Vo, the observed line-of-sight component of the rotational velocity. Rewrite the Doppler equation in the following form:

Vo cfc

f-------

=

Here f is the frequency of the transmitted signal (i.e. the radar pulse; f = 430 MHz or 4.3 × 108 Hz).

VVo------ R

y---=

7.) Knowing the line of sight component of the velocity, Vo, the actual tangential velocity, V, can be determined. Using similar triangles (see Figure 9.3), we find that:

Therefore,V Vo

Ry---=

8.) Calculate V. Your answer should be in units of meters/second.

9.) Calculate the rotation period of Mercury, Prot. Divide the circumference of Mercury

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(CMercury = 1.530 × 107 meters) by V. This will give the rotation period of Mercury in seconds.

ProtCMercury

V---------------------=

10.) Convert your answer from Step 9 into (time) units of days, using the fact that 86,400 seconds equals 1 day.

11.) Repeat Steps 1 through 9 for the remaining three echoes: at t = 210, 300, and 390 microseconds.

Calculating the Orbital Velocity of MercuryDetermine the orbital velocity of Mercury from the frequency shift of the echo from the sub-radar point. (Remember that this value must be divided by two because you are dealing with an echo.) Use the Doppler formula to calculate the orbital velocity of the planet. (The mean orbital velocity of Mercury is 47.9 km/sec.) Negative speeds (i.e. velocities) are speeds of recession, and positive speeds are speeds of approach. Show your work below. Express your answer in km/sec (1 km = 103 meters).

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Answer SheetTo be submitted to your lab instructor.

Table 9.1t (sec) 120 210 300 390

d (meters)

x (meters)

y (meters)

fleft (Hz)

fright (Hz)

favg (Hz)

fc (Hz)

Vo (meters/sec)

V (meters/sec)

Prot (days)

Find the average of the values in the bottom row of Table 9.1: Prot average = ________________

Determine the percentage error. (Use 59 days as the accepted value of Prot.)

%age error = (|Experimental value – Accepted value| ÷ Accepted value) × 100 = ___________%

Conclusion:

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Calculating the Orbital Velocity of MercuryDetermine the orbital velocity of Mercury from the frequency shift of the echo from the sub-radar point. (Remember that this value must be divided by two because you are dealing with an echo.) Use the Doppler formula to calculate the orbital velocity of the planet. (The mean orbital velocity of Mercury is 47.9 km/sec.) Negative speeds (i.e. velocities) are speeds of recession, and positive speeds are speeds of approach. Show your work below. Express your answer in km/sec (1 km = 103 meters).

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Laboratory Exercise #10 – Astrometry of Asteroids

Purpose: Learn how astronomers apply the concepts of parallax and astrometry to discover asteroids and determine their distance and velocity using images from a telescope.

Finding an AsteroidThe Technique: Finding the Coordinates of Unknown ObjectsThe lines of right ascension and declination are imaginary lines of course. If there's an object in the sky whose right ascension and declination aren't known (because it isn't in a catalog, or because it's moving from night to night like a planet), how do we find its coordinates? The answer is that we take a picture of the unknown object, U, and surrounding stars, and then interpolate (a way of finding a position between positions) its position using nearby stars A, and B, whose equatorial coordinates are known. The stars of known position are called reference stars or standard stars.

Figure 1: Unknown Star and Reference Stars

Suppose, for instance, that our unknown object lies exactly halfway between star A and star B. Star A is listed in the catalog at right ascension 5 hours 0 minutes 0 seconds, declination 10 degrees 0 minutes 0 seconds. Star B is listed in the catalog at right ascension 6 hours 0 minutes 0 seconds, declination 25 degrees 0 minutes 0 seconds. We measure the pixel positions of stars A and star B and U on the screen and find that U is exactly halfway between A and B in both right ascension (the x direction) and declination (the y direction). (See Figure 1).

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We can then conclude that the right ascension of the unknown object is halfway between that of A and B, or 5 hours 30 minutes 0 seconds; and the declination of the unknown object is halfway between that of A and B, or 17 degrees 30 minutes 0 seconds. This is outlined in Table 1 below

Table 1: Interpolating Position of Unknown Star

Star Right Ascension Declination X Position on Image

Y Position on Image

Star A 5h0m0s 10o0’0” 20 20Star B 6h0m0s 25o0’0” 10 30Unknown Star U ? ? 15 25

If the unknown object isn't exactly halfway between the two known stars, the interpolation is a bit more complicated. In fact, in practice it's a bit more involved largely because the image of the sky appears flat and, in actuality, the real sky is curved. But it is not difficult to write easy to use software that will do the mathematics for you and the program we supply does just that. To measure an unknown object's position from an image, the software will instruct you to choose at least three stars of known coordinates (for best results choose more). Then indicate the location of the unknown object by clicking on it. The computer performs what is called a coordinate transformation from the images on the screen to the equatorial coordinates, and prints out a solution showing you the coordinates of your unknown object. The software we provide can, in principle, calculate coordinates with a precision of about 0. 1 seconds of arc. That's approximately the angular diameter of a dime seem at distance of about 20 miles, a very small angle indeed.

The Problem of Finding AsteroidsIn this exercise, you will be using images of the sky to find asteroids and measure their positions. Asteroids are small rocky objects that orbit the sun just like planets. They are located predominately between the orbit of Mars and Jupiter, about 2.8 Astronomical Units from the sun. Some asteroids do orbit closer to the sun, even crossing the earth's orbit. In the past earth-crossing orbit asteroids have collided with the Earth. Hollywood movie producers have frequently used an asteroid collision as a plot for a disaster movie. The danger is real, but dangerous collisions are very infrequent. Most asteroids are only a few kilometers in size, often even smaller. Like the planets, they reflect sunlight, but because they are so small, they appear as points of light on images of the sky. How then can we tell which point of light on an image is an asteroid, and which points are stars?

The key to recognizing asteroids is to note that asteroids move noticeably against the background of the stars because an asteroid is orbiting the sun, i.e. within our solar system. If you take two pictures of the sky a few minutes apart, the stars will not have moved with respect to one another, but an asteroid will have moved. (See Figure 2).

It's hard to see the forest for the trees, however. Often there are so many stars on a picture that you can't easily remember the pattern when you look at another image, and therefore you can't easily tell which dot of light has moved. Computer software comes to the rescue again! You can

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load and display simultaneously two images of the sky that were previously taken with a telescope. You then instruct the computer to switch the display quickly back and forth from one image to another, a technique called blinking. If you are careful to line up the stars on the first image with the stars on the second image before you blink the two images, the only object that will appear to change will be the asteroid, which will appear to jump, making it easy to spot. Our computer program enables you to easily align the stars on two images and then blink back and forth, making asteroids jump to your attention.

Figure 2: Moving Asteroid

Sometimes the asteroids will be faint; other times there will be spots or defects that appear on one image and not on another. These spots can mislead you into thinking that something has moved into the second image that was not there in the first. So even with the ease of blinking, you should carefully inspect the images, in order to pick out the object (or objects) that really move from a position on the first image to a new position on the second.

Once you've identified an asteroid on a picture, you can then get the computer to calculate its coordinates by measuring its position with respect to reference stars (stars of known coordinates) on the screen. Comparing an asteroid's position at particular time with its position at some other time enables you to calculate the velocity of the asteroid, as we will see later in this exercise.

The Principle of ParallaxMeasurements of precise positions are key to one of the most powerful methods astronomers have of measuring the distances to objects in the sky, a method known as parallax. Parallax is the most direct way astronomers have of measuring the distances to stars.

The parallax of an object is its apparent shift in position when you view the object from two dif-ferent vantage points. It's commonly used on the earth to measure such things as the distance across a wide river (see Figure 3). You view a tree from two vantage points on the opposite

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shore. You carefully measure the distance between the two vantage points, called the baseline, B, and the parallax angle, Θ, between the two lines of sight to the tree. Using trigonometry, you can find the perpendicular distance across the river. The distance, D, makes a right triangle where it intersectis with the baseline B. The tangent of the angle Θ/2 is B/2 divided by D, so the distance across the river is D B 2

2 tan------------------------=

In general, if you can measure the parallax, Θ, of an object as measured from two points sepa-rated by a baseline, B, you can measure its distance.

Figure 3: Parallax Image

In astronomy, a commonly used baseline is the diameter of the earth's orbit. The parallax we measure is due to the shift in vantage point as the earth orbits the sun. If we view a nearby star from opposite sides of the earth's orbit, and measure its position against a background of distant stars, we will see that it appears change position with respect to the background stars. The further the star, the smaller the parallax. Even the nearest stars have parallax shifts of less than a second of an arc, hard to see and hard to measure. However, astronomers have been able to measure the parallaxes of over a hundred thousand stars using sophisticated techniques and a special satellite called Hipparchos.

It is possible to measure the distance to asteroids using parallax. Because asteroids are closer than stars, their parallax is larger, and we can even use a short baseline, such as the diameter of the Earth. Viewed from two points on the earth, asteroids show a very noticeable parallax, as you will see in Part IV of this exercise. Measuring the parallax of asteroids using astrometry is a useful exercise for showing us how the much smaller parallax of stars is measured in practice.

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PART IFinding Asteroids by blinking ImagesLog on to your computer and open the CLEA program Astrometry of Asteriods. On your com-puter you will find a series of images of a region of the sky, about 4 arcminutes squared, in which astronomers were searching for a faint earth-approaching asteroid designated 1992JB. The problem is to recognize the asteroid, which, as the name suggests (asteroid star-like), looks just like a star. How can we distinguish between one type of object and another?

The answer is that asteroids move, while stars do not appear to move. In the first part of this exercise you will examine the images to find a moving object, the asteroid 1992JB. The technique you will use is called blinking. Using 2 stars on each image as reference points, you can align the images electronically in the computer. After alignment, whenever you turn on the blink feature, the computer will quickly switch its display from one image to another. Stationary objects like stars will not appear to change position, but objects that have moved from one image to the next will appear to jump, drawing your attention to them. You can thus identify which starlike object is, in fact, the asteroid.

Procedure for Part ILet's look at one of the images. From the menu bar, choose File ... Load ... Image 1. A directory listing showing you a list of files appears. From this list, select 92JBO5.fts, and click Open to load it.

The computer will show you the progress of the loading and creating a screen for display and then write the name of the file on the screen to let you know it has successfully loaded Image 1. The program can load two files, Image I and Image 2, for blinking purposes.

The image has been loaded, but is not yet displayed. To display it, select Images on the menu bar, then choose Display ... Image 1 from the pull-down menu. A window showing the image 92JB05 will appear on your screen. (The display window containing the image uses the same window control symbols found in any Windows program. Using the "-" button at the upper right minimizes the window, and using the “x” button at the upper right removes the window.)

The image is oriented with west to the right and north to the top. All of the dots you see on the image are distant stars except one, which is the asteroid. We'll make a chart of the image and mark the asteroid when we discover it. In the blank space located on the following page, make a freehand sketch of the image 92JB05. Try to pay attention to the details. Draw it to scale and try to make it fit in the space.

Now let's find the asteroid. Load as before, but this time choose File ... Load ... Image 2 from the main program window (which appears blue, not black as in the Image 1 window). From the list of images, select 92JB07.fts, and click Open to load it.

You can display this image in its own window by using the Images ... Display ... Image 2 selec-tion on the menu bar. Since 92JB07 was taken 10 minutes after 92JB05, the asteroid will have

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moved. But it may not be immediately obvious which starlike object is out of place, even when you compare the images side by side.

So we blink the images, aligning them so they coincide, and use the computer to flip from one image to the other. Here are the steps we use to blink the images.

We must first align the images. From the main menu bar located on the blue window, choose Images ... Blink. You will see one window now, displaying just Image 1.

At the bottom right, a small instruction box asks you to click on an alignment star, which the computer can use to align the two images. If possible, you should try to choose two stars that are on opposite sides of the picture to achieve best results. The best combination is a pair of diagonally opposite stars. Choose one of the brightest ones, click on it, and note it for your own reference by writing the number 1 next to it on your freehand drawn chart.

Click on Continue in the instruction box, and it will ask you for a second alignment star. Two stars are needed to account for possible rotation of the images. Click on this star, record it on the chart above as 2, and click again on Continue.

Now you will see Image 2, and you will be asked to identify star number 1 on Image 2. Check your chart, then click on the same star 1 that you chose in Image 1. If you are satisfied, click Continue in the instruction box. You are now asked to identify star 2 and again you click on the same star 2 that you chose in Image 1. Actually, based on your choice of stars in Image 1, the computer will make a good guess as to where the star should be on Image 2. It will draw a box around where it thinks it should be, and if, in fact, it comes close, you need merely click Continue to accept its choice. You'll find that the computer does this guesswork in many parts of the program, making it easier for you to make measurements quickly.

The computer now has all the information it needs to align the images and blink them. All you need to do is click on Blink from the menu bar at the top of the blinking window, and you will see the computer flip back and forth between Image 1 and Image 2 about once a second. The stars will not move, but you should easily be able to pick out the asteroid as the one object that does jump. Be careful when choosing! For example, occasionally a white spot will appear on one image, and not the other. It is not an asteroid, however, but a defect in the picture itself. This is usually caused by radiation - a cosmic ray most often-that exposes a single pixel in the camera during one exposure. You may also notice all the stars brighten and dim, since one image is a longer exposure and therefore brighter. But the asteroid should appear clearly as a smudge of light that changes position from one image to the other.

To stop the blinking select Stop from the menu bar. To adjust the rate of blinking slower or faster at your convenience, select Adjust ... Blink Rate. The Blink Rate Interval dialog box appears. The blink rate can be adjusted using the slide bar to increase or decrease the rate per msec that the images blink. The range is from 10 to 1000 msec with the default set at 500 msec.

The brightness and contrast of either image can also be adjusted. Select Adjust ... Image Display ... Image 1 (or Image 2) from the menu bar. The Adjust Image 1 (or 2) Display

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Parameter Dialog box appears. Separate slide bar controls are available for the brightness and contrast features. Use the Initialize ... Field Alignment menu option to choose the alignment stars again.

When you have identified the asteroid on Image 1 (92JB05) and Image 2 (92JB07) mark the position of the asteroid with a dot on your chart. On the freehand drawing on your own paper, neatly label the asteroid's position on Image 92JB05 with a small 05 and its position on Image 92JB07 with an 07.

Now continue to find the asteroid on images 92JB08, 92JB09, 92JB10, 92JB12, and 92JB14 by blinking them using Image 92JB05 as Image 1.

To do this, simply select the Load ... Image2 menu item, and then Image ... Blink to blink the images. The computer program has a learning feature that will speed up successive measure-ments. Since you have already blinked one set of images, the next time you blink images, the program assumes that the alignment stars are in the same position. So when you choose Image ... Blink, you will be shown the blink window. You can realign the images using the Initialize ... Field Alignment menu choice in the blink window menu box. You'll be prompted for alignment stars, but the computer will show you its guesses first. If the computer's choices are close to the stars chosen for alignment, you need simply to click Continue to accept the computer's choices. You can also make your own selection by clicking on the star you want to choose and then clicking Continue.

Repeat this procedure until you've identified a pair of stars on both images.

The procedure will then be the same as you followed with image 92JB07 as Image 2. Again, mark the successive positions of asteroid 1992JB by dots labeled 08, 09, 10, 12, and 14 on the chart you made in step 1.

You should note the asteroid is moving in a straight line. Draw an arrow in the space on the answer sheet to show the direction of motion, keeping coordinates the same as the blinking images. What direction is this? (North, Northeast, Southeast, etc.)?

PART IIMeasuring the Equatorial Coordinates of an Asteroid by Comparing it to Positions of Known Stars in the Hubble Guide Star Catalog

Now that you have identified which object on the image is the asteroid, the next step is to deter-mine the location of the object in the sky by measuring its equatorial coordinates, i.e. the object's Right Ascension and Declination. The computer can do this by a type of interpolation or "fitting" process, which compares the position of the asteroid on an image with the positions of stars whose coordinates have been previously tabulated in a catalog. The most widely used catalog today is an electronic file called the Hubble Guide Star Catalog (GSC), which was originally created to aid in pointing the Hubble Space Telescope. The GSC, as it is called, was created by measuring star positions on photographs of the sky. It contains coordinates and magnitudes of about 20 million stars brighter than apparent magnitude 16. The GSC is available on two CD-

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ROMS, but to make it unnecessary for you to insert a CDROM into your computer, we have selected part of the GSC and stored it on your computer's hard drive, where the Astrometry program can access it easily.

For each image of 1992JB, you will determine the coordinates of the asteroid by telling the GSC the approximate coordinates of the center of the image, so that it can draw a map of the stars with known coordinates in the vicinity.

Identify at least three GSC stars that are also on the image, as reference stars for calculating coordinates on the image.

Point and click on each star on your image that is a reference star, so that the computer knows which star on your image corresponds to a particular reference star in the catalog.

Point and click on a target object with unknown coordinates: the asteroid. The computer then calculates its coordinates with reference to the GSC reference stars you have chosen.

Store the results of the computer calculation for later printout.

Write the coordinates of the asteroid in the table at the end of this section.

Procedure for Part IILet's measure the position of the asteroid in Image 92JB05. If it is not already loaded into Image 1, use the File ... Load ... Image1 menu choice from the main window to load the file. Then choose the Images ... Measure ... Image1 menu choice. A window will open asking you for the approximate coordinates of the center of the image, so that the computer can retrieve data from the proper files in the Guide Star Catalog. You can find the coordinates of the centers of these images in the following table.

Table 2: Image InformationFile Name RA (2000) of Image

Center (h m s)DEC (2000) of Image Center (o ‘ “)

Exposure Length (s)

92JB05 15h 30m 44.30s 11o 15‘ 10.4“ 30s92JB07 15h 30m 44.30s 11o 15‘ 10.4“ 120s92JB08 15h 30m 44.30s 11o 15‘ 10.4“ 30s92JB09 15h 30m 44.30s 11o 15‘ 10.4“ 180s92JB10 15h 30m 44.30s 11o 15‘ 10.4“ 30s92JB12 15h 30m 44.30s 11o 15‘ 10.4“ 120s92JB14 15h 30m 44.30s 11o 15‘ 10.4“ 30s

Type them into the proper boxes for Right Ascension and Declination. Set the Field size at 8 arcminutes. The images are about 4 arcminutes on a side, but this leaves you some margin for error in finding your reference stars if the field center was a little off. The default value for the magnitude limit is 20. When you have entered the field center information, click OK.

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The computer will now search for stars in this region in the GSC, and will draw a star chart, based on the GSC coordinates in a window on the left-hand side of the screen. It will display the Image 1, 92JB05, in a window on the right-hand side of the screen. The image will show more stars than the GSC map, partly because one of the "stars" is the asteroid, and also because the GSC only includes stars brighter than 16th magnitude, and some of the stars in your images are fainter than that. However, you should see a distinctive pattern of at least three stars in the GSC chart that you can match to the brighter stars on your image.

When you see the match, you will want to identify it. Sketch the reference stars you have chosen on your own paper, and label them 1, 2, 3, etc. Now you will tell the computer which stars you've chosen.

Using the mouse, click on reference star number one in the left window (the GSC map). A dialog box will open at the bottom of the page, identifying the GSC data on that star; simultaneously the computer will draw a colored box around the reference star on the chart.

Click select in the dialog box at the bottom of the page to set your first reference star. Repeat the process for reference star 2, etc. until you have marked as many reference stars as you can see on both the GSC chart and image 92JB05. (You need to identify and mark at least 3, well spaced over as wide an area of the image as possible, for best results.). Record the ID#, RA, and DEC of each reference star in the table on the answer sheet.

When you have selected at least 3 reference stars and marked them, go to the Select Reference Stars dialog box that is in the bottom center of the screen and click OK to tell the computer you're finished selecting stars. If you've only chosen 3, the computer will warn you that more reference stars would give you better results, but if 3 is all you could choose, assure the computer by clicking NO, you don't want to choose any more.

You will now be asked to point to reference star 1, then star 2, etc. in the right window-Image 92JB05. Click on each star in turn, making sure to match it with the corresponding reference star you've selected on the GSC chart, and click OK on each to signal you've picked the right star. The computer will start to second-guess you after 2 stars have been identified, it will be able to tell approximately where star 3 is, and if the computer has done a good job (Star 3 is anywhere inside the colored box the computer draws on the screen), you need only click OK to accept its choice.

When all the reference stars have been identified on your image, the computer will ask you to point and click on the "unknown" or "target" star-the asteroid. If you click OK after choosing this star, the computer will have enough information to calculate the coordinates of the target star with respect to the reference stars. A small text window will open on the left side of the screen displaying the right ascension and declination coordinates of your target object. Write these down in the proper place in the table on the answer sheet.Click on OK in the center box at the bottom to accept the solution.

When the computer asks if you want to record the data, click Yes, and write the image name (92JB05) in the form labeled OBJECT NAME.

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Finally click OK to record your measurement on a data file. You can view this report form from the Report menu item on the main window at any time for your own reference, and you may want to print it out later when you have added all the other measurements to it.

Now use the File ... Load ... Image1 menu choice on the main window, along with the Image ... Measure.. Image1 menu choice, to measure the asteroid position in images 92JI307, 92JB08, 92JB09, 92JB10, 92JB12, and 92JB14. You will proceed just as you did above, recording your data both on this worksheet and on the computer. You will note that the computer quickly learns the setup you are using-it will know what reference stars you chose the first time, and outline them on the GSC window the next time through. You will still have to match the stars on your image with reference stars on the GSC, and you will, of course, have to show the computer where the asteroid is, since it moves from one image to the next.

The measuring process should be rapid. If you need to change any of the field settings, you can change them in the measuring window using the Image ... Adjust ... Reference Stars menu, or in the main window by using the File ... New ... Target Object menu choice, which clears all the images from computer memory and starts over.

When you have recorded the positions of the asteroid on all the images, you are ready to proceed to the Part III. Print out the data using the Report ... Print menu option. Look at the data you have taken to make sure it makes sense.

The asteroid appears to move in a straight line (actually it moves in an elliptical orbit around the sun, but the short portion of the path seen on the image appears straight). Therefore you should note a smooth change in the numbers from beginning to end - a regular increase in right ascension (if it's moving eastward) or decrease (if its moving westward) and a regular increase or decrease in declination (depending on whether it's moving northward or southward). Re-measure any images that seem to be in error.

PART IIIThe Angular Velocity of Asteroid 1992JB

How fast is 1992JB moving? We can calculate its angular velocity in arcseconds per second of time using data taken in Part II of this exercise. The procedure we follow is to subtract the asteroid's starting position on image 92JB05 from its ending position on image 92JB14, and divide by the number of seconds between the starting image and the ending image. We express this mathematically as

t--------=

where is the angular velocity of the asteroid, ΔΘ is the angular distance it moved, and ΔΘ is the time that elapsed.

Procedure for Part III

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Measuring the Elapsed Time.

Use the following worksheet to calculate the angular velocity of the asteroid. Record selected values on the answer sheet.

Use values recorded on the answer sheet in Part II to find the time when image 92JB14 and 92JB05 were taken.

Time of image 92JB14: ____hours _____minutes _____seconds

Time of image 92JB05: ____hours _____minutes _____seconds

Convert to hours and a decimal to make subtraction easier. (Note: Use standard conversion factors for changing hours to minutes and seconds.)

Add the decimal fractions to the hours to get the final value:

Time of image 92JB14 as a decimal: _________hours.

Time of image 92JB05 as a decimal: _________hours.

Express to at least five decimal places!

Time elapsed between 92JB14 and 92JB05 _______________________hours.

Subtract the time when image 92JB05 was taken from the time image when image 92JB14 was taken.

Convert to seconds by multiplying hours by 3600s/1hr.

Time elapsed between 92JB14 and 92JB05 _____________seconds.

Measuring the Angular Distance Travelled by 1992JB

In order to calculate the angular distance traveled, ΔΘ, we use the Pythagorean theorem, which states that:

c a2 b2+=

As illustrated in Figure on the following page, because right ascension and declination are perpendicular coordinates, we can find the total angle moved using this mathematical relationship derived from the Pythagorean theorem. Figure 4: The Mathematical Illustration of the Motion of an Asteroid

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Using Figure as a guide, if we let RA represent the change in the number of seconds of right ascension, and Dec represent the change in the number of arc seconds moved in declination, then using the relationship expressed by the Pythagorean theorem, we can construct the fol-lowing equation to determine the total angle moved.

RA 2 Dec 2+=

Using your table from Part II of the answer sheet record the values for the declination of images 92JB 14 and 92JB05 below.

Declination of asteroid on 92JB14 _____° _____' _____"

Declination of asteroid on 92JB05 _____° _____' _____"Subtract to find the change in declination

Δ Dec ____________” (units of seconds of arc)

Repeat the previous steps to calculate Δ RA.Right Ascension of asteroid on 92JB14 _____ h _____ m _____ sec

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Right Ascension of asteroid on 92JB05 _____ h _____ m _____ sec

Subtract to find the change in right ascension:

Δ RA __________” (sec of RA)

BUT WAIT! We're not done yet- 1 second of RA is 15 arcseconds times the cosine of the dec-lination. (Remember the RA lines come together at the poles, and so there are smaller angles between them at high declination. Multiplying by the cosine of the declination adjusts for this physical change). For declination you can use the value of the declination you recorded for either image (in decimal degrees) in the previous steps:

RA 15 ine DEC cos =_________________ RA = __________” (units of arc seconds)

Using the Angular Distance Traveled Formula and the compatible units we can calculate the value:

RA 2 Dec 2+= = ________________” (units of arc seconds). Record this value on your answer sheet.

Calculating the Angular Velocity of Asteroid 1992JB on May 23, 1992

t-------- ________________= =

Note: We've only calculated the angular velocity of the asteroid. We need to know its distance from us to calculate how fast it's traveling in km/second. We will calculate the distance of 1992JB in the next section, using the method of parallax.

Part IVMeasuring the Distance of Asteroid 1992JB on May 23, 1992

In this section we'll use our skill in measuring coordinates to determine the parallax of asteroid 1992JB. For this purpose, we took two images of 1992JB simultaneously from observatories at opposite sides of the United States. One image, you have already seen: Image 92JB12, which we have also stored under the name ASTWEST. It was taken by Dr. Laurence Marschall using a 0. 8m diameter telescope at the National Undergraduate Research Observatory in Flagstaff, Arizona. The other image ASTEAST was taken by Dr. Thomas Balonek using an 0.4 m telescope at the Foggy Bottom Observatory of Colgate University in Hamilton, New York.

Table 3: Time of Observation at Both Sites 06 57 00 UT 23 May, 1992.

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Site Latitude Longitude Image File ExposureFoggy Bottom Observa-tory, Colgate UniversityHamilton, NY

42o 48’ 59.1” W 75o 31’ 59.2” ASTEAST 120

National Undergraduate Research ObservatoryFlagstaff, AZ(Telescope Operated by Lowell Observatory)

35o 05’ 48.6” W 111o 32’ 09.3” ASTWEST 120

Because the asteroid is much closer than the stars, it appears in a different position on the two images with respect to the background stars. This is called its parallax. By measuring the position with respect to reference stars on both pictures, we can determine the parallax, which is just the angular difference, ΔΘ in arcseconds, between its position on the image taken from the eastern site and from the western site. Using simple trigonometry (see Figure 10.3 as the example), if B, the baseline, is the distance between the two telescope sites in kilometers, then : Dist. of Asteroid =

206 265 B--------

Using our program, then, the measurements can be done rather quickly.

Procedure for Part IV

Loading the Images

Load ASTEAST as Image 1. Load ASTWEST as Image 2. Display the two images side by side for comparison, note that the two cameras had different sensitivities (the east-coast telescope was smaller), and had CCD chips of different dimensions, so the images don't look quite the same. But the same reference stars are visible on both images. Find asteroid 1992JB again on each image. You can refer to your chart in part I; remember, ASTWEST is 92JB12.

Now using the methods you learned in part 2, measure the coordinates of the asteroid in ASTEAST and ASTWEST. You can use the Images ... Measure ... Image 1, and Images ... Measure ... Image2 menu options on the main window. Tabulate your results on the answer sheet.

Calculating the parallax of 1992JB

The parallax of 1992JB is just the difference between the two positions. We can follow the methods of calculating the angular difference between two positions that we used Part 3. We separately calculate the difference in the declinations and the difference in the right ascensions in arcseconds ( " ) and then find the total angular difference as the square root of the sum of the squares of the declination and right ascension differences.

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Express the coordinates of 1992JB on both images in decimal form to make subtraction easier:

File RA (h.xxxxx) of 1992JB Dec (o.xxxxx) of 1992JBASTEASTASTWEST

Express the difference ΔDec in decimal degrees: ____________________°

Convert to arcseconds by multiplying by (3600 arcsec/1 degree).ΔDec =____________”

Express the difference ΔRA in decimal hours:_______________ h of RA

Convert to seconds by multiplying by 3600 RA: ______________sec of RAConvert to arcseconds by using the equation

RA in arc seconds = RA 15 ine Dec cos

ΔRA (in arcseconds) = ______________________.

Calculate the total parallax in arcseconds:

Parallax = RA 2 Dec 2+

Parallax = __________________. Record on answer sheet.

Calculating the distance of Asteroid 1992JB:

Knowing the parallax of Asteroid 1992JB when seen from the Flagstaff, AZ as compared to Hamilton, NY, and knowing baseline, i.e. the separation of the two telescopes (3172 kilometers). We can use a simple trigonometric formula to calculate the distance of the asteroid.Dist. of Asteroid =

206 265 BaselineParallax-----------------------

Where the baseline and the distance are both expressed in kilometers and the parallax in arcseconds.

Using this formula, calculate the distance of 1992JB on May 23, 1992 at 06 57 UT

Distance of 1992JB _____________________km.

Distance of 1992JB _____________________Astronomical Units.

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Compare this with the distance of the moon. How many times further or closer is it than the moon?

Asteroids are classified by their average distance from the sun. Belt Asteroids orbit in the asteroid belt; Trojan Asteroids orbit at the same distance as Jupiter. Near-Earth or Earth Approaching asteroids have orbits that bring them near the earth. What kind of an asteroid do you think this is? Why?

Part VThe Tangential Velocity of Asteroid 1992JB

The tangential velocity, Vt, of an asteroid is the component of its velocity perpendicular to our line of sight. Again, a simple trigonometric formula lets you calculate its velocity in kilometers/ second if you know its angular velocity ( in "/sec) and its distance (km). We determined the angular velocity in Part 3 and the distance in Part 4 of this exercise. Then it follows that the tangential velocity is represented as:

Vtangularvelocity dist

206 265----------------------------------------------------------=

So using our results, calculate the tangential velocity of the asteroid.

Vt = ______________________km/sec.

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Answer Sheet for Astrometry of Asteroids

Part I: Show your drawing of the path of the asteroid as it moves through the reference stars in the images. Indicate direction of motion.

Drawing of Asteroid Direction

Part II

Reference Star Coordinates

Reference Star ID# RA DEC#1#2#3

Measured Equatorial Coordinates of Asteroid 1992JB, May 23, 1992File Name Time (UT) RA DEC92JB0592JB0792JB0892JB0992JB1092JB1292JB14

Part III

Time elapsed between 92JB14 and 92JB05 in seconds ________________.ΔΘ (Angular distance traveled)_________________________.

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μ (Angular velocity of asteroid)__________________________.

PART IV

Measurement Coordinates for ASTEAST and ASTWEST

File RA (hms) of 1992JB Dec (o ‘ “) of 1992JBASTEASTASTWEST

Look at image ASTWEST. Compared to its position on ASTEAST, does 1992 JB look further to the east or further to the west with respect to the background stars?Why is this what you'd expect? Explain using a diagram in the space below.

Parallax = ________________________________

Distance of 1992 JB in km _________________________Distance of 1992 in AU____________________________

Explain what type of asteroid you have been observing, Belt, Trojan or Near-Earth. Give reasons for your choice.

Part V

Vt (Tangential velocity) of 1992 JB ________________________km/sec

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Make-up Laboratory Exercise – The Planisphere

Purpose: Learn how to use a planisphere to find constellations and stars in the sky for any night of any year.

PART A: Class discussion questions:

True or false:1. Polaris is not the brightest star in the sky.

2. Polaris rises in the east and sets in the west just as the Sun does.

3. Polaris is always found at the observer's zenith.

PART B: Group discussion questions:

About how many stars do you think you can see at night with the un-aided eye?

Are all the stars the same brightness? If not, what do you think is causing the differences?

Can you tell if all the stars are the same size? Why / why not?

Explain why you can't see the stars in the daytime.

Explain what is special about the North Star.

Has anyone seen any planets at night? If so, which planets?

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How do you know whether a point of light is a star or planet?

What is the fundamental difference between a star and planet?

PART C: Constructing the Star Finder

1. Cut out the star map along the outer edge of the calendar ring. Be sure to cut smoothly along the curve. If the edge of the map is not cut properly, the star finder will be hard to use.

2. Cut out the star finder envelope. There are two separate pieces of the envelope. Cut out each piece. You will be taping the two pieces together as demonstrated by the instructor. Note that you must cut out the window section of the envelope, but do not cut off the tabs as they are needed to assemble the star finder.

3. Tape the tabs in place after placing the star map in between the two envelope pieces.

4. Slip the star map wheel into the envelope so that the printed side of the map faces out of the envelope's window. The map wheel should fit snugly inside the makeshift envelope.

5. The portion of the star map inside the window displays the stars visible in the sky for the date and time selected. This star map is a model of the sky. The stars are represented by star symbols on the map. The brighter a star appears to be compared to other stars, the larger is the star symbol that represents it on the map. Stars in the center of the window correspond to the stars overhead. Stars near the edge of the window are stars that are near the horizon. Stars on the map near North on the envelope are visible in the northern sky. Stars on the map near East on the envelope are visible in the eastern sky. Stars in the southern and western skies can be located near South and West, respectively, on the envelope.

6. To simulate the movement of the stars across the sky, turn the star wheel in a counterclockwise direction. Note that when you do this, stars move into the window near East on the envelope, these stars are rising. At the same time, other stars move out of the window near West on the envelope. These stars are setting.

7. Adjust the star wheel until Orion is just inside the window near East on the envelope. This represents rising above the east point on the horizon. Turn the wheel counterclockwise until Orion is above South on the envelope. Note that Orion is about halfway between the South

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edge of the envelope and the center of the window. Orion's location in the sky would therefore be about 45 degrees above the south point on the horizon. This is equivalent to a altitude of 45 degrees and an azimuth of 180 degrees. If you continue to turn the wheel counterclockwise you will see Orion reaching the edge of the window near West which represents Orion setting.

8. To set the star finder, turn the star map wheel until the desired date on the calendar ring lines up with the arrow on the envelope for the time desired. If we were in the time when Daylight Savings time was in effect, we would have to subtract one hour from the time selected and use that as the time setting.

PART D: Now set the star finder for 7:00 PM on February 1 and answer the following questions:

1. Name two constellations that would be visible near the overhead location.

2. Name the constellation that has just risen above the east point on the horizon.

3. Name the constellation that is setting at the west point on the horizon.

4. Estimate Orion's position in altitude and azimuth notation.

Altitude: Azimuth:

5. Estimate the altitude and azimuth of the Big Dipper.

Altitude: Azimuth:

6. At about what time will Orion set on February 1?

7. At about what time will Virgo rise on February 1?

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8. Can you find the Square of Pegasus on February 1? If yes, describe its location in the sky at a specific time.

9. Can you find Sagittarius on February 1? If yes, describe its location in the sky at a specific time.

PART E: Now set the star finder for 10:00 PM for today's date.

1. Name the constellation nearest the EASTERN horizon.

2. Name the constellation that is about to set (but none of which is below the horizon).

3. Estimate altitude and azimuth for Taurus.

Altitude: Azimuth:

4. Estimate altitude and azimuth for Cetus.

Altitude: Azimuth:

5. Estimate time for the setting of Andromeda.

6. Estimate time for the rising of Corvus.