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 Astronomy 1010L – The Solar System Laboratory Manual Dr. Kristin B. Whitson Experiments in this manual were last updated by K. B. Whitson January 2011. The author thanks S.R. Whitson for help in assembling the manual. Credits: 1. Hallo Northern Sky: Exercise partially based on previous versions of this experiment developed by R. L. Marlowe and J. Pitkin. Hallo Northern Sky software written and distributed by Han Kleijn is available for download at www.hnsky.org. 2. Parallax Measurement: Manuscript slightly modified from a previous manual by R. L. Marlowe and J. Pitkin. 3. Telescopes and Introduction to Imaging: Telescopes exercise was originally developed by R. L. Marlowe. Imaging exercise developed by J. Pitkin and K. B. Whitson. Remote telescopes for imaging project are operated by LightBuckets Online Telescopes, Steve Cullen, at www.lightbuckets.com. 4. Jones Observatory Field Trip: Exercise developed by R. L. Marlowe. The observatory is managed by J. Pitkin and operated by UTC. http://www.utc.edu/Academic/JonesObservatory/index.php. 5. Inverse Square Laws: manuscript prepared by K. B. Whitson. Exercise adapted from Physical Science with Vernier by Volz and Sapatka. 6. Energy Flow Out of the Sun: Exercise and software developed by the Contemporary Laboratory Experiences in the Astronomy (CLEA) project at the Department of Physics, Gettysburg College, Gettysburg, PA (http://public.gettysburg.edu/~marschal/clea/CLEAhome.html).This exercise is a modified version of that work prepared by K. B. Whitson. 7. Astrometry of Asteroids: Exercise and software developed by the CLEA project at the Department of Physics, Gettysburg College. This exercise is a modified version of that work prepared by K. B. Whitson. 8. Deep Sky Observing Field Trip: Exercise introduced by R. L. Marlowe, B. Thompson, and J. Pitkin. Manuscript and lab report developed by K. B. Whitson. 9. Mass of the Earth: Portions of the exercise were from a previous manual by R. L. Marlowe and J. Pitkin. 10. Rotation Rate of Mercury: Exercise and software developed by the CLEA project at the Department of Physics, Gettysburg College. This exercise is a modified version of that work prepared by K. B. Whitson. 11. Image Processing: Exercise developed by K. B. Whitson in conjunction with J. Pitkin. 12. Moons of Jupiter: Exercise and software developed by the CLEA project at the Department of Physics, Gettysburg College. This exercise is a modified version of that work prepared by K. B. Whitson. Table of Contents Hallo Northern Sky ……………………………………………….…………………………………………………………..…………………… 1 Parallax Measurement ….………………………………….………………………………….…………………………………………..…… 6 Telescopes and Introduction to Imaging …………………………………………………………………………………...………… 12 Jones Observatory Field Trip .………………………………………………………………………………………………………………. 16 Inverse‐Square Laws ………………………………………………………………………………………………………………….……….. 18 Energy Flow Out of the Sun ….……………………………………………………………………………………………………………… 24 Astrometry of Asteroids ….………………….………………………………………………………………………………………………. 33 Deep Sky Observing Field Trip …………………………………………………………………………………………….………………. 43 The Mass of the Earth …………………………………………………………………………………………………………..…………….. 50 Rotation Rate of Mercury .…………………………………………………………..………………………………………………………. 56 Image Processing …….…………………………………………………………………………………………………………………..……… 64 Moons of Jupiter …………………………………………………………………………………………………………………….…………… 68 Hallo Northern Sky UTC Astronomy 1010L Hallo Northern Sky Objective: This lab will acquaint you with methods to explore the night sky. In particular, we will discuss celestial coordinate systems, rising and setting of celestial objects, magnitudes, and eclipses. To acquaint ourselves with the basics, we will be using a celestial sphere and Hallo Northern Sky, which is a free planetarium program that is similar to other commercially‐available software that provides the same service to varying degrees. Procedure: 1. From the computer desktop, open the “Hallo northern sky planetarium” program. 2. Under the Date menu, enter the current date and specify the time as 18 hrs 0 min (6:00 p.m.). 3. By clicking at the top and/or bottom edges of the screen, rotate the celestial sphere so that you find the observer’s horizon (heavy green/yellow line) and the ecliptic (lighter green/yellow dashed line). You may want to zoom in or out (clicking on IN or OUT at the top) to produce a usable screen image. If you need to return to your original view, click RESET at the top. 4. Set the correct location for your observations. a. Under the File menu, choose Settings. Make sure that the longitude is set to 85.0° West and the latitude to 35.0° North (the approximate coordinates for Chattanooga). Make sure that the boxes which allow the program to automatically correct for parallax error and atmospheric refraction are checked. b. In the Time Zone box, enter ‐5.018 (the time difference in hours between Chattanooga and Greenwich). Make sure that the box for Daylight Saving Time is unchecked. c. Click on OK to close the Settings Window. 5. Hallo Northern Sky allows the observer to view the sky in a normal fashion by default, looking from the inside of the celestial sphere out (toward the sky). To center the screen on an object, position the crosshairs on the object and right‐click once. A single left click on any object displays information about the object provided that it is in the program data bank. Try doing this for one of the named objects on the screen. On your report page, list the object you clicked on and the coordinate information displayed about it. Definitions and abbreviations can be found by using the Help menu. 6. RESET your view and zoom OUT. In the software, the RA and DEC coordinates of the cursor change with its position. Right ascension and declination are shown as the gridlines on the screen and provide similar information for celestial objects as latitude and longitude, respectively, for a location on the surface of the Earth. Move the cursor along the ecliptic to determine whether it maintains a constant declination or right ascension. Record your observations on your lab report sheet. 7. Find Mercury, Venus, Mars, Saturn, Jupiter, and the Sun. Notice how they are arranged relative to each another and the ecliptic. On your report page, sketch the orientation of these for today’s date. 8. To conduct a search for an object, you can use the Search menu. Type in the name of an object and initiate the search. If the object exists in the program databank, it will be placed in the center of the screen. Search for Jupiter. On your data sheet, report (A) the rise and set times for today (B) the portion of the sky where it will be located at 6:00 PM and (C) the location of Jupiter in the sky so that a friend on the telephone could find it. In order to describe the location, you can describe where it is relative to the horizon, other celestial objects nearby (you can view the location of constellations under the Screen menu by clicking on Constellations), and its brightness relative to other objects in the same region of the sky (the lower or more negative the value for magnitude, the brighter the object). 9. Search for Rigel (Orion’s left foot). Note the time that Rigel rises today. Now go to the Date menu to change the date to tomorrow. You can either click on Enter Date/Time and change it there, or you can click on +Day. What 1 Hallo Northern Sky UTC Astronomy 1010L time does it rise tomorrow? Calculate and report the difference (in hours and minutes) between the successive risings of Rigel. 10. Return the date to today’s date and search for the Sun. Click on it to find the time it rose today. Advance one day and see what time it will rise tomorrow. Report the difference (in hours and minutes) between the successive risings. Explain the discrepancy between your answers for Rigel and the Sun. 11. A commonly‐used ground‐based coordinate system for Earth observers specifies the altitude and azimuth of a celestial object. The altitude is the height of the object (in degrees) above the observer’s horizon. The azimuth is the compass direction of the object. (Due north is 0°, East is 90°, South is 180°, and West is 270°). Click on RESET to reset your view and zoom OUT. Change the date to tonight at 11:00 PM (23:00) on the Date menu. Locate the Moon and click on it. What is its altitude and position relative to the horizon line? Report the rising and setting civil times (AM and PM) that the Moon rose and set today and the percentage of the Moon’s surface that is illuminated tonight. As closely as possible, what is its phase today (full, first quarter, etc.)? As of 11:00 PM tonight, is it waxing or waning? Crescent or gibbous? (You can use the +Hour or –Hour features on the Date menu to travel forward or backwards). 12. Reset your time to 12:00 (noon) today. Now search for Venus. Check the rising and setting civil times (AM and PM) as seen from Chattanooga today. Consider what times of day Venus will be visible and decide whether Venus is the “morning star” or “evening star” today. You may want to compare to the times for the rise and set of the Sun. Check the percentage of Venus’s surface that will be illuminated tonight from Earth’s viewpoint. What phase is this (not the percent illuminated)? 13. Compare the brightness (magnitude) of Jupiter with that of the brightest star in our sky, Sirius, which is in the constellation Canis Major (you can find Sirius by using the Search feature). Report the magnitudes of each on your report sheet. Astronomers measure the difference in brightness between objects in terms of magnitude. Each unit difference of magnitude between objects correlates with a brightness ratio of about 2.5 (object A, with a magnitude of 0 is 2.5 times brighter than object B, with a magnitude of 1). For example, if the magnitude of object A is mA = 0.2 and the magnitude of object B is mB = ‐2.0, the brightness ratio would be calculated by: BA
= 2.5(mB −mA ) = 2.5−2.0−0.2 = 2.5−2.2 = 0.13 BB
This tells us that A is about 0.13 (or 13%) as bright as B. Equivalently, B is (1/0.13), or 7.7 times brighter than A. Calculate the brightness ratio of Jupiter to Sirius and report explicitly which is brighter and by how much. 14. Search for Mars to compare its size (angular diameter) with that of the Moon. Unless stated explicitly, the sizes are given in terms of arcminutes ('). The " symbol denotes arcseconds (recall that there are 60” in 1’). Determine the ratio of the angular diameter of Mars to the angular diameter of the Moon. Give your answer as a decimal, rounded off appropriately. This ratio tells you how many times larger the Moon is than Mars from our viewpoint. 15. Search for the Sun and determine its angular size relative to the Moon. 16. Go to the Date menu and set the date and time for 12:15 AM (0:15) on December 21, 2010. Notice that Earth’s shadow is now labeled. Zoom in enough to separate the Moon from the outline of the Earth’s shadow. Use the +Hour and +Minute Functions on the Date menu to track the position of the Moon relative to the Earth’s shadow. You may have to use the search feature to re‐center the Moon in the screen occasionally. Determine between what times of the night we were able to detect an eclipse. Note that unless at least half of the Moon enters the penumbra (the yellow outer dotted circle), the eclipse may be undetectable. Totality is the darkest part of the eclipse, where scattered light does not reflect part of the sunlight back toward the Earth as it does when Earth is in the penumbra. Totality would correspond to where the Moon is fully within the shadow of the Earth (the middle blue circle, or umbra). Between which times did this occur? How long did totality last? 2 Hallo Northern Sky UTC Astronomy 1010L Hallo Northern Sky Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. Describe what the ecliptic represents. 2. a) How are the celestial coordinates of right ascension and declination related to lines of latitude and longitude on Earth? b) How are altitude and azimuth defined? 3. How many arc‐minutes are in one hour? How many arc‐seconds are in one arc‐minute? 4. Describe the differences in the following terms in relation to the phases of the Moon: a) Crescent vs. gibbous b) Waxing vs. waning 5. If object X is 0.2 times as bright as object Y, how many times brighter is object Y? (Show your reasoning) Pre‐Lab Page Hallo Northern Sky UTC Astronomy 1010L Hallo Northern Sky Lab Report Name: ________________________________ Lab Partner: _________________________ Date: ______________ 1. List the star or planet that you clicked on in step 5 of the procedure. Give any information displayed for Az, Alt, RA and DEC. Spell out what each abbreviation means. 2. Does the ecliptic maintain a constant declination or right ascension? Circle your answers: Constant DEC? Yes No Constant RA? Yes No 3. Sketch the relative orientations of the Sun, Mercury, Venus, Mars, Jupiter, and Saturn relative to the ecliptic on the celestial sphere. 4. Time for rise of Jupiter today: ______________. Time for set of Jupiter today: _____________. In which portion of the sky will Jupiter be located at 6:00 PM? (Check the horizon line and specify NE, NW, SE, etc.) ____________ Describe the location of Jupiter in the sky so that a friend on the telephone could find it. 5a. Time for rise of Rigel today: ______________. Time for rise of Rigel tomorrow: ______________. Time (in hours and minutes) between successive risings of Rigel: b. Time for sunrise today: ______________. Time for sunrise tomorrow: ______________. Time (in hours and minutes) between successive sunrises: 5c. Explain the difference between your answers in 5a and 5b. Report Page 1 Hallo Northern Sky UTC Astronomy 1010L 6. Altitude of the Moon ________. Position relative to the horizon line: ____________________. Civil time for rise of the Moon today: __________. Civil time for set of the Moon today: _________. Percentage of Moon illuminated: __________. Phase of the Moon: ____________. At 11:00 PM, the Moon is (circle your answers): Waxing / Waning and Crescent / Gibbous 7. Venus is the morning / evening star today (circle one and give your reasoning in the space below). Phase of Venus: ____________. 8. Magnitude of Jupiter: _________ Magnitude of Sirius: _______. Show your calculation for the brightness of Jupiter to Sirius. Which object is brighter? By how much? 9. Angular size of the Moon: __________' Angular size of Mars: ________" = __________' Show your calculations for the angular size of Mars and the ratio of angular diameters here: Angular diameter of Moon
= Angular diameter of Mars
Which is larger, the Moon or Mars? Angular size of the Sun: __________' Ratio of the angular diameters of the Moon and Sun = ___________ 10. Between what times was the Moon located within Earth’s shadow on the morning of December 21, 2010? Between what times should observers in Chattanooga have detected the eclipse? What are the times of totality and how long did totality last (in hours and minutes)? Report Page 2 Parallax Measurement UTC Astronomy 1010L Parallax Measurement Objective: In this experiment, you will learn about a technique used by astronomers to measure distances as far away as ~500 light‐years (nearby stars). The technique is known as the determination of stellar parallax. A closely‐
related technique (triangulation) is used in surveying measurements on Earth. For this lab, we will apply this method not to stars, but to objects more easily accessible here on the Earth. Background: As an illustration of parallax, first line up an object across the room from you with your thumb held at arm’s length. Close one eye. Keeping your arm, thumb, and head motionless, now close the other eye and open the first. Notice that the position of your thumb with respect to the background object moves. This effect is called a parallax shift, and it results from the fact that the small distance between your eyes causes you to view your “thumb‐object alignment” from slightly different angles. When a nearby star, N, is viewed from Earth against the background of stars which are much further away, the near star appears to shift position by a small amount as the Earth’s location changes due to its orbital motion around the sun (see figure 1). The diameter of the Earth’s orbit provides a baseline, AB, from which the nearby star’s position appears to change with respect to the background. If the parallax angle, p, can be accurately measured we can calculate the star’s distance from us. To a distant star Earth in January A
θA
R
θ=2p
Sun N
p
θB
To a distant star B
Earth in July Figure 1. By geometry, the angle θ, is defined as the ratio of the arc length s (where s is the curved path along the baseline AB) to the radius of the circle, R, when θ is measured in radians, or θ (in radians) = s
R
(eq. 1) Since the circumference of the entire circle from which the arc s is created equals 2πR in length and sweeps out 360° in angular measure, it follows that for an entire circle, the angle in radians is: 2πR
= 2π radians = 360° , or θ=
1 radian = R
360 o
2π
≈ 57.3 o (eq. 2) Referring back to figure 1 and using the definition of a radian defined in equation 2, then the parallax AB
. Since R >>> AB, the chord (straight line) AB is an excellent approximation to the true arc length s. angle p ≈ 2R
We know that AB is 2 AU (2 times the distance from the Earth to the Sun), so if we can measure the angle p, then we can calculate the distance R to the nearby star N. 6
Parallax Measurement UTC Astronomy 1010L Procedure: 1. Find a “near” and “far” object. For example, if the near object distance is about the length of the classroom, then the far object should be some point far in the distance like a point on a city building or a mountain. The far object distance should be at least 100 times the near object distance. On your lab report sheet, describe in detail the position from which you are taking your measurement, the near object you have chosen and its position, and the far object you have chosen and its position. Include a comparison of their placement relative to one another. 2. Determine the baseline AB that you are going to use – it should be approximately 2 m. Try to arrange your baseline so that the near object lies approximately on the perpendicular bisector of the baseline (the near object should be in the middle of the baseline). Estimate the uncertainty in your length measurements by estimating how closely you can position your eye over the baseline. In other words, can you position your eye directly over the edge of the baseline so that it is within 5 cm of the baseline? 3 cm? 1 cm? Record this value on your data sheet to the nearest centimeter (1/100th of a meter). 3. Measure the angle θA between the near and far object using your cross‐staff (this is the angle between the bisector line and the line R to point A on your baseline). Repeat the measurement five times and record your data in the table on your data sheet. 4. Repeat step 3 for θB (the angle between the bisector and the radius R to point B on your baseline). 5. Find the average value for each angle, showing your calculations for one of the two angles (either θA or θB) on your report sheet. 6. Find an estimate for the uncertainty in each angle using the following approximation: largest value − smallest
value
uncertainty =
Example: If your measurements were 6.4°, 6.0°, 5.8°, 6.3°, and 6.0°, the approximate uncertainty in the angle would be: N− 1
, where N is the total number of measurements ±
6.4° − 5.8°
(5 − 1)
=±
0. 6 °
= ± 0. 3 ° 2
7. After completing the angle measurements, step off and record the distance between the center (midpoint) of AB and the near object for three of the five trials. Start with your toes on the line AB and use normal‐sized walking steps for this. 8. Calculate the total angle θ. If the near and far objects reverse their right‐left orientation when you move from A to B on the 2‐meterstick, then θ = θA + θB. If the near and far objects do not reverse orientation, then θ = |θA‐θB|. The uncertainty in θ will be the sum of the uncertainties in θA and θB. 9. Calculate the parallax angle p, knowing that p = ½ θ. The uncertainty in p will be ½ that of θ. 10. Using the definition of an angle in radians (equation 2), we can relate the angle in degrees to the ratio of an object’s true size and distance R from us: AB
, which can be solved for the distance, R to give: 2R
AB
R (in meters) ≈ 57 .3° ×
. 2p
p ≈ 57.3° ×
Using this expression, calculate the distance between your baseline and the near object. 7
Parallax Measurement UTC Astronomy 1010L 11. Find the maximum possible distance to the near object by using the largest value for AB (AB + uncertainty in AB) and the smallest value for p (p – uncertainty in p): Maximum calculated distance = 57 .3° × (AB + uncertaint y in AB )
. 2 × (p − uncertaint y in p )
Find the minimum possible distance to the near object N by: Minimum calculated distance = 57 .3° × (AB − uncertaint y in AB )
. 2 × (p + uncertaint y in p )
The maximum and minimum distances are the upper and lower bounds on your calculation of the distance by parallax. 12. Compare your calculated parallax distance to the value determined from your stepped‐off distance. Carefully measure the distance you cover for six normal walking steps (let the toe‐to‐toe distance be called one step). Provide an estimate of the uncertainty in the measurement of this distance. Divide both the 6‐step length and its uncertainty by 6 to find your average “single step length” and its uncertainty. Also find the uncertainty range for these measurements by: Maximum distance = (avg. # of steps + 0.5) x (avg. step size + uncertainty) Minimum distance = (avg. # of steps – 0.5) x (avg. step size – uncertainty) Find the average of the maximum and minimum values. 13. Does your range of values for the calculated parallax distance (R) overlap with the range of values for the stepped off distance? If so, then your calculated and stepped‐off distances agree within experimental uncertainty. If the range of values did not overlap, what could be possible sources of experimental error/uncertainty or limitations to the experiment that were not accounted for in the experimental design and execution of the experiment? 14. To obtain a quantitative measure of the agreement between your parallax‐determined distance and stepped off distance, find the percentage difference between them: % difference = (parallax determined distance ‐ average stepped off distance)
× 100% average stepped off distance
This calculation assumes that the stepped off distance is more accurate than the parallax determined distance, which may or may not be true, depending on your ability to measure with steps. Note that percent differences of 10‐20% are not unusual in this experiment. 8
Parallax Measurement UTC Astronomy 1010L Parallax Measurement Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. Line up an object across the room from you with your thumb held at arm’s length. Close one eye. Keeping your arm, thumb, and head motionless, now close the other eye and open the first. Describe what happens to the position of your thumb’s image relative to the object across the room. 2. What is meant by “the baseline” that you are going to use in the lab experiment? To what physical measurement does it correspond? 3. With reference to figure 1, which variables (distances, angles, etc.) do you actually measure (not calculate) in the experiment? 4. With reference to figure 1, which variables (distances, angles, etc.) do you calculate in the cross‐staff portion of the experiment? 5. Besides the cross‐staff method, what is the other method you will use to determine the distance between the baseline and the near object? Pre‐Lab Page Parallax Measurement UTC Astronomy 1010L Parallax Measurement Lab Report Name: ________________________________ Lab Partner: _________________________ Date: ______________ 1. In terms of relative position, describe in detail (a) the place where you are taking your measurement: (b) the near object: (c) the far object: 2. Length of the baseline AB = ________ ± _______ m. 3, 4, and 7. Trial Number # steps to near object θA (degrees) θB (degrees) 1 2 3 4 5 Average Uncertainty ± ± ± 0.5 For either θA or θB, show your calculations below for the average and the uncertainty. 5. Average: 6. Uncertainty: 8. Calculation for θ and its uncertainty (show your work): θ = ___________° ± _________°. Report Page 1 Parallax Measurement UTC Astronomy 1010L 9. Parallax angle, p, and its uncertainty: p = ___________° ± _________°. 10. Calculation for R (show your work): R = ___________ (meters) 11. Calculations for maximum and minimum distances to the near object (show your work): Maximum R = ___________ (meters) Minimum R = ___________ (meters) 12. Measured distance for 6 walking steps = ______________ meters ± ____________ meters. Average step size = ____________ meters ± ____________ meters. Calculations for maximum and minimum stepped distance (show your work for the maximum and minimum): Maximum stepped distance = ___________ (meters) Minimum stepped distance = ___________ (meters) Average stepped distance = ___________ (meters) 13. Do the calculated parallax distance and stepped‐off distance agree within their uncertainties? Possible limitations of or sources of error in this experiment (in complete sentences): 14. Calculation for % difference between the two determinations of distance (show your work): Report Page 2 Telescopes and Introduction to Imaging UTC Astronomy 1010L Telescopes and Introduction to Imaging Objective: This goals of this lab are to familiarize you with the basic parts of various types of telescopes, how to assemble and use a telescope like we will be using for deep‐sky observing, and to review necessary pre‐requisites for setting up imaging experiments in our group projects. Equipment needs: Optical bench with 75 and 150 mm lenses. Concave demonstration mirror. Reflecting telescopes with different mounts. Background: Telescopes are used to obtain magnified images of distant objects and focus light in some way to form an image of the object under study. Generically, telescopes can be broken down into two types: refractors (which use lenses) and reflectors (which use mirrors). First, we will be viewing objects through a single lens, in order to understand how light rays are focused, then through a double lens system like in a refracting telescope, which will create a magnified, inverted, virtual image of a distant object. We will also examine the concave mirror, which is used in reflecting telescopes, to see how light from a distant object is focused to create and image. We will examine two different types of mounts on telescopes, an equatorial mount and an altitude‐
azimuth mount, which use different coordinate systems to locate objects. Both will be used in our planned observing sessions. For all coordinate systems, reference points are necessary and we will discuss how to locate and determine appropriate points. We will further inspect the telescopes to study other critical components to the system and the purposes behind each. Finally, we will learn to assemble these telescopes and use them. The second part of this laboratory deals with the collection of astronomical data and images. Our group projects will be done through the use of remote telescopes in the LightBuckets network. Use of these will allow us to image astronomical objects from New Mexico, which has far less light pollution than Chattanooga or most of the entire Eastern U.S; thus, we can produce a much more meaningful and enhanced image to analyze than those that can be collected locally. Depending on the object you choose to image, you may select different exposure times, spectroscopic filters, color mode, etc. for your study that will produce the best image. For instance, a black and white image is taken using the luminance filter on the telescope. Color images are created by combining frames taken through luminance, red, green, and blue filters. Spectroscopic filters (such as hydrogen‐alpha) can enhance regions of nebula that contain molecular or atomic gas. We will need to find coordinates for our objects and determine if they are suitable for study at this time of year. Exposure times are important to consider. The longer your exposure, the more light you can collect, and the fainter the object you can see. However, overexposures cause blooming effects that can be difficult to remove. It is best to take multiple short exposures and combine them. Following the introduction to this exercise in lab, your instructor will assign you and your group members “points” that have already been purchased, and you will select an astronomical object to image. Consultations will be had with individual groups about the appropriateness of planned studies and to schedule imaging runs. Later in the semester after data has been collected and returned, we will learn to process the collection of data frames in order to produce your final image and project. Other Reporting Notes: The report for this laboratory covers our telescopes discussion. For the imaging discussion, you should make good notes elsewhere in your notebook on parameters that you will need to determine and decide upon for your chosen astronomical object. Keep these notes and any parameters you decide upon for future use in your report on the imaging project. 12
Telescopes and Introduction to Imaging UTC Astronomy 1010L Introduction to Imaging Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ Go to www.lightbuckets.com. Click on the “Register” tab and sign up for an account. You will have to choose a username and password and provide a valid email address during the registration process that will allow you to complete the registration. Once you have registered, your instructor will be able to add you to the group for our class. Username:_______________________________ Associated email address: _____________________________ After you have registered, you may want to browse the website to view the albums of other users to get an idea of the types of objects you may want to image and the specifications, settings, filters, etc. required to produce different types of images. Pre‐Lab Page Telescopes and Introduction to Imaging UTC Astronomy 1010L Telescopes Lab Report Name: __________________________________________________________ Date: ________________________ Describe in some detail the questions below, in your own words, using complete sentences. Clearly labeled sketches can often help greatly help with your explanations. Use additional paper if needed. 1. Draw rays that show the path of light taken through a refracting telescope (with two lenses). Show the location, orientation, and relative size of the object and the image. 2. Draw rays that show the path of light taken in a reflecting telescope (with a single concave mirror). Show the location, orientation, and relative size of the object and image. 3. Describe the main features of the Schmidt‐Cassegrain telescope used in this lab. Where is the primary mirror located? What is its diameter? Where is the secondary mirror? What is the corrector plate and what is its purpose? 4. How does the alt‐azimuth mount differ from the equatorial mount? Report Page 1 Telescopes and Introduction to Imaging UTC Astronomy 1010L 5. Why must the polar axis of an equatorial telescope be directed towards Polaris? Will this axis be directed towards Polaris 5,000 years from now? Why or why not? 6. Give a brief explanation of the purpose of the setting circles found on the Celestron telescopes. Which one corresponds to which coordinate? 7. What is meant by the term “local celestial meridian?” For what type of telescope might it be more important to know this? 8. What is a star diagonal, and what is it used for? 9. What is the purpose of a “finder ‘scope”? Compare the field of view from the finder scope to that seen through the eyepiece. 10. Why is the magnification ability of a telescope of little interest to stellar astronomers but of greater interest to planetary astronomers? Report Page 2 Jones Observatory Field Trip UTC Astronomy 1010L UTC Clarence T. Jones Observatory Field Trip Meeting Time: You need to be there by 6:00 p.m. in order to begin your scavenger hunt. We will begin the lecture and planetarium show at 6:30 p.m. Directions: From UTC, take McCallie Ave. east (away from downtown) 2 to 3 miles through the Missionary Ridge Tunnel. McCallie becomes Brainerd Rd. as you exit the tunnel. Continue on for about a mile; go past the large intersection of Brainerd Rd. with Belvoir Rd. (see Grace Episcopal Church on the right). Go approximately 2 blocks further, and take the left turn onto N. Tuxedo Lane. (If you pass Brainerd United Methodist Church with its tall steeple on your left, you’ve gone too far). Tuxedo Lane is a very wide street with parking possible in the middle of it. About one block after you turn onto Tuxedo, see the large sign for the Jones Observatory on the right hand side. Park, lock your car, and walk up the stone steps which lead to the Observatory at the top. Allow roughly 15 minutes for travel time from UTC. If special accommodations are needed because of the walk up the steps, please let your instructor know so that you can be provided alternate directions to a back entrance. 16 Jones Observatory Field Trip UTC Astronomy 1010L UTC Clarence T. Jones Observatory Field Trip Scavenger Hunt Name: ______________________________________________________ Date of Visit: _____________________ Instructions: Answer all questions below; you should attempt to answer the questions without help from others! 1. What is the altitude (to the nearest foot) of the Observatory on the U.S. Geological Survey marker? __________ Where is the marker located? ____________________________________________________________________ _____________________________________________________________________________________________ 2. Where is the Biblical quote located? _____________________________________________________________ What does it say? ______________________________________________________________________________ _____________________________________________________________________________________________ _____________________________________________________________________________________________ 3. When was the building completed (opened to the public)? ___________ Where did you find this information? (This is located on the building somewhere…) _______________________________________________________ ____________________________________________________________________________________________ 4. Where is the dodecahedron inside the Observatory? _______________________________________________ What is its function? ___________________________________________________________________________ What is the geometrical shape of each of its sides? ___________________________________________________ 5. Who was the main architect of the Observatory? ___________________________________________________ 6. How many telescopes are actually mounted in the dome room? ________ How many are refractors? ________ 7. What type of telescope is the observatory’s main telescope? _________________________________________ What is the size of the primary mirror of the main telescope? ____________ inches 8. From a telescope in what other famous observatory was UTC’s original telescope modeled? ________________________________ Where is this observatory located? ________________________________ What display in our observatory references this other famous observatory (what does the display say?) _____________________________________________________________________________________________
_____________________________________________________________________________________________ 9. Where is Lookout Mountain in the planetarium? ___________________________________________________ 10. What astronomical object / constellation did you view tonight through a telescope at the observatory? (If poor weather obstructs this viewing, give a constellation discussed in the planetarium). _____________________________________________________________________________________________ Report Page 1 Inverse Square Laws UTC Astronomy 1010L Inverse Square Laws Objective: In this experiment, you will use a computer‐interfaced sensor to measure light intensity in order to examine the relationship between the intensity of a light source as a function of the distance from it. Through studying this correlation, you will be able to understand fundamental relationships described by inverse square laws for many other natural phenomena, including the light intensity radiating from stars and the gravitational force that binds all objects in the universe together. With respect to light intensity, knowledge of the relationship between intensity and position will also allow us to gain an appreciation for the luminosity of our Sun and its brightness on Earth compared to other planets in the solar system. Background: Inverse square laws are some of the most fundamental and universal relationships in physical science, relating the intensity of an effect to the distance from the cause of that effect. Inverse square laws can be used to describe the loudness of a sound, the strength of electromagnetic and gravitational forces, the depth of a radiation field, the potency of gas molecules from an open perfume bottle, and the luminosity of light. They apply in all cases where something from a localized source spreads uniformly throughout a surrounding space; thus, the intensity measured in an inverse square relationship is the rate that a force, molecules, or energy is transferred though an area. The intensity of any effect described by an inverse square law decreases with distance from the source. Mathematically, the cause and effect are inversely proportional to one another. For example, a light appears to be brighter when you are close to it, but as you move away, it seems to become dimmer. Consider a point of light at the center of a set of nested spheres (as in figure 1). Light travels away from the point in all directions in straight lines, and as it spreads out, it becomes more diffuse. Because the surface area of Figure 1. Inverse square relationships depend a sphere is given by 4πr2, a sphere centered at the same position on the surface area of a sphere. having a radius of 1 m would have a surface area of 4πr2 = 4π(1)2 = 4π. A sphere centered at the same location but having a radius of 2 m would have surface area equal to 4πr2 = 4π(2)2 = 16π; therefore, it is (16π/4π = 4) four times as large, and the light radiating from the point source is now spread over four times more area than it was when it was two times closer to the source. Thus, intensity is inversely proportional to the square of the distance, or I
I = 02 , r
Figure 2. Light intensity plotted as a function of distance from the source yields an inverse square relationship to the curve fitting the data. where I is intensity, r is the distance from the source, and I0 is the intrinsic luminosity of the light at the source itself. A plot showing light intensity as a function of distance from the source is shown in figure 2. Notice that intensity decreases rapidly as the distance from the source increases. Examination of inverse square equations such as that given above also reveals that no matter how far the distance, the effect of a source is never eliminated; for the effect to disappear entirely, the distance would have to be infinite. The curve on the plot would have the same relative shape if instead of intensity, gravitational force were plotted on the y‐axis and distance on the x‐axis, where the relationship between gravity and distance is described by Newton’s Law of Universal Gravitation. For example, if the radius of the Earth is 6,400 km, a satellite that 18
Inverse Square Laws UTC Astronomy 1010L orbits Earth at 4 times the distance from Earth’s center (19,200 km above Earth’s surface) would weigh just 1/16th as much as it did when it was on the surface of the Earth. The force of gravitational attraction quickly gets weaker as objects get farther apart (by the square of distance), but all masses in the universe exert some kind of gravitational influence over every other mass, because all are at some finite distance. Mathematically, the inverse square laws for intensity and gravitational force are given by I=
I0
r
2
and F=
Gm1 m2
r2
, respectively. Both equations can be written in the comparable form y = Ax‐2, where y represents either the intensity of light or gravitational attraction, x represents the distance from the source (equivalent to r), and A is a constant that multiplies the curve; in the case of the intensity equation, A = I0, and in the case of gravitational attraction, A = Gm1m2. Procedure: 1. Set up the apparatus as shown below. Be sure that the light sensor is directly lined up on a horizontal plane with the light source. This is absolutely vital to the success of this experiment and the most important aspect of the setup. The base of the ring stand must be able to slide smoothly with one edge flush against the 2‐meter stick, while the light sensor is always in a single direct line with the light source. Light Source Light sensor To Vernier box 2‐meter stick 2. On the computer, open the Logger Pro program. Go to the File menu, and then select the appropriate file to open (given to you by your instructor). The vertical (y) axis is preset to show illumination (intensity) data. The horizontal (x) axis will plot distance scaled from 0 to 50 cm. Follow the instructions that the computer gives you on how to connect your light sensor. 3. You first need to take a reading of the background light intensity (with your light source off) in order to be able to account for ambient light hitting the detector that comes from other sources, for instance from windows in the room. Define the light level as zero by clicking 0 Zero. The intensity reading should now be near zero. Click Auto Scale if the reading of zero is not visible. 4. Now you are ready to collect data from the light source. Turn on the light source and make sure that it has warmed up for at least 30 seconds so that it has had time to equilibrate to its maximum output power. Position the detector element of the light sensor 10 cm away from the light source and watch for the reading to move up and down. Try to make sure that the computer reading is at least 100 lux when the sensor is 10 cm from the light source. If it is not, you may need to re‐align your apparatus. 5. When the meter reading appears to have reached its highest point, click Keep. Type “10” in the edit box in order to correlate your intensity reading with the 10 cm distance you used and press Enter to save the data into the computer. 19
Inverse Square Laws UTC Astronomy 1010L 6. Move the detector element of the light sensor to 15 cm away from the source and repeat step 5, correlating the intensity reading with “15” cm. Repeat step 5 again for distances of 20 cm, 25 cm, 30 cm, 35 cm, 40 cm, and 45 cm away from the light source. Once you have finished all readings, click Stop to end the data collection. 7. Analyze your intensity data to determine if it fits an inverse square relationship. Click on Analyze > Curve Fit. Choose the Power fit on your measured illumination data. An equation A*xn fitting your data should appear. Record the equation on your data sheet. 8. Print a copy of your data and the graph with the fit. 9. If your intensity data exactly fit an inverse square relationship when viewed as a function of distance from the light source, the exponent in your power function would be ‐2.00. How closely does your adjusted intensity fit exponent compare? To answer this, find the percent difference with the formula below. Record your answer and show your calculations on your data sheet. % error = your value − ( − 2.00 )
× 100 % − 2.00
10. On your report sheet, discuss sources of uncertainty or limitations in the assumptions of the experiment that may have influenced the outcome of this experiment. 11. Now that you understand the inverse square relationship, let’s calculate the intrinsic intensity (the luminosity) of the Sun. Assume that your light sensor placed just outside Earth’s atmosphere measured an intensity of 1400 Watts/m2. First, find the total surface area of the sphere surrounding the Sun at the radius of 1.5×1011 m, which is the distance between the Earth and the Sun. Then, multiply the total surface area of the sphere (4πr2) by the intensity of the Sun’s light at this distance. Show all your calculations on your report sheet. 12. Using your value for the luminosity of the Sun, calculate the intensity of the sunlight that would be measured at the surface of Mercury with the same supposed light sensor you used on Earth. The distance between the Sun and Mercury is 5.80×1010 m (which is 0.387 AU). Show your calculations on your report sheet. 13. Find the ratio of the intensity that you calculated for the sunlight at Mercury to the intensity of the sunlight received at Earth by dividing the two values. This will tell you how much brighter the Sun is on Mercury. 14. Calculate the intensity of the sunlight that would be measured at the surface of Neptune with the same light sensor. The distance between the Sun and Neptune is 4.50×1012 m (which is 30.07 AU). Show your calculations on your report sheet. 15. Find the ratio of the intensity that you calculated for the sunlight at Neptune to the intensity of the sunlight received at Earth. This will tell you how bright the Sun appears on Neptune relative to how bright the Sun appears from Earth. 20
Inverse Square Laws UTC Astronomy 1010L Inverse Square Laws Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. What are some examples of physical properties that follow inverse square laws? 2. What is the most important aspect to ensure in setting up the apparatus? 3. What is the purpose of defining the ambient light level as zero in the experiment? 4. The power function used to fit your data has the form y=A*xn. To what physical property does the quantity “A” correspond? What should the numerical value of n be? 5. Find the change in the force of gravity between two planets if the masses of the planets don’t change but the distance between them is decreased to one‐third of the original distance. Pre‐Lab Page Inverse Square Laws UTC Astronomy 1010L Inverse Square Laws Lab Report Name: ________________________________ Lab Partner: _________________________ Date: ______________ 7. Equation for fit to intensity data: 9. Calculation of the % difference between intensity fit exponent and theoretical exponent of ‐2.00. 10. Sources of experimental uncertainty or limitations in the assumptions of the experiment that may have influenced the outcome: 11. Calculation of the Sun’s luminosity: Report Page 1 Inverse Square Laws UTC Astronomy 1010L 12. Calculation for sunlight intensity at Mercury: 13. How much brighter is the Sun on Mercury than on Earth? (Show your calculations.) 14. Calculation for sunlight intensity at Neptune: 15. How bright does the Sun appear on Neptune relative to how bright the Sun appears from Earth? (Show your calculations.) **Attach your computer‐generated graph and data from this experiment to this report sheet. Report Page 2 Energy Flow Out of the Sun UTC Astronomy 1010L Energy Flow Out of the Sun Objectives: This experiment focuses on analysis of computer simulations that will allow you to explore the interaction of photons with gas atoms. Specifically, we will examine the effects of photon absorption and re‐
emission by individual atoms, the effect that a photon’s energy has on the rate of absorption, and the effects of diffusion and its dependence on the number of gas layers in the star. Through these analyses, you will achieve an understanding of how absorption and emission lines in stellar spectra are produced and why the energy flow from the Sun’s core to its surface is such a seemingly slow process. Introduction and Background: The energy emanated by our Sun is the ultimate source of energy on Earth and throughout our solar system, traveling across the vast distances of space by way of electromagnetic radiation. Solar energy originates at the core of the Sun when photons are generated during the processes of nuclear fusion that convert mass to energy. The software used in this exercise examines the different interactions of photons with matter in two regions of the Sun, the solar atmosphere and the solar interior, each having different effects on the ultimate travels of an individual photon into space. The solar atmosphere is a thin layer of gas that makes up the outermost skin of the Sun. It is largely transparent, so it has little effect on most photons. However, the absorption and re‐emission of a few photons by atoms in the solar atmosphere results in dark absorption lines in the continuous spectrum of the Sun. These absorption lines are called Fraunhofer lines. The “Interaction” module of the program will allow us to study how two classes of photons, line radiation and continuum photons, interact with atoms in the solar atmosphere. The “Line Formation” simulation demonstrates how line radiation photons, or photons that have just the right amount of energy to kick an electron of a gas atom to a higher energy state, are absorbed by the atom. The “Continuum” simulation shows how continuum photons, or photons that do not have the precise energy required to be absorbed, pass though a cloud of gas easily. A final gas cloud simulation, “Experiment”, will allow you to match a photon’s energy with various gas clouds, a process that will permit you to plot a line spectrum similar to what you might observe using a spectrograph attached to a real telescope. Before they reach the atmosphere, photons generated in the core travel through the main body of the sun, called its interior, in a zigzag path as they are scattered back and forth by particles (mostly electrons). In fact, so many interactions occur that it literally takes hundreds of thousands of years for a typical photon to travel from the center of the Sun to its surface. In the “Flow” simulation of the program, a two‐dimensional slice of the interior of a star is used to study how a photon diffuses outward from the core and how the number of layers of atoms in the model affects the amount of time it takes for a photon to escape. Part I: Photon Interaction in the Solar Atmosphere 1. Start the Solar Energy program under the CLEA exercises folder on the Start / Programs menu and select Log in from the menu bar. Enter your name, those of your lab partners, and the laboratory table number where you are seated for this experiment. When all the information has been entered, click OK to continue. After confirming, the opening screen will appear. 2. On the Simulation menu, select Interaction. The display portrays many atoms in a gas. In the model of each atom, an electron cloud surrounds a nucleus that is invisible and buried deep in the center. By clicking on the View menu, you can change the scale to “close‐up” or “large scale”. 3. To choose the type of photons to be sent through the gassy region, click on the Photon Type menu and select Line. Line photons have precisely the right amount of energy to be absorbed by the atoms. Absorbing a photon adds energy to the electrons in the atom, and they are kicked into higher orbits. After a brief moment, they fall to a lower energy state and release a photon in some random direction. These photons are said to be undergoing bound‐bound transitions, since the electrons remain attached to the atom even when excited. When you click on 24 Energy Flow Out of the Sun UTC Astronomy 1010L the Run button, photons are sent continuously from the left side of the screen to pass through or interact with the atoms. (Alternatively, click on Step to send a photon one at a time through the field.) 4. Click on the Stop button after 20 photons have been sent. Record the number of photons that were scattered on your lab report sheet. 5. Change to Continuum photons on the Photon Type menu and repeat steps 3 and 4. Continuum photons are the wrong energy to interact easily with these gas atoms and often pass through them without incident. Occasionally, a continuum photon will scatter off of an electron just because it happens to make a direct hit, but not often. Record the number of continuum photons that were scattered on your lab report sheet after 20 have passed through the gas. Figure 1. Photon/Atom Interaction Simulation
6. Close the Photon/Atom Interaction Simulation Window so that you are back on the main screen of the software. From the Simulation menu, select Line Formation. Line photons are responsible for the lines in a spectrum (hence the name). If they are observed against a bright background (like the surface of a star), they cause dark lines to appear because most are scattered away. This is called an absorption spectrum. On the other hand, if line photons are observed against an otherwise dark background, they cause bright lines in a spectrum. This would be the case if you observed from a place which does not receive direct photons, and would be called an emission spectrum. 7. On the Parameters menu, select # of Photons (for Figure 2. Line Photon and Continuum Simulations “Run”) and enter 20. As the simulation proceeds, photons will be sent through a container of gas (red square in the middle) having come from a very bright but off‐
screen object such as the Sun to the left. Your detector is located on the right side of the screen so that it views the Sun through the gas. Thus, a spectrum with the detector located directly opposite the Sun should appear as an absorption spectrum if the photons are scattered away from horizontal. In the simulation, if a photon makes it through the gas cloud and is picked up by the detector, the “Detected” counter increases. Alternatively, the photon may interact with the cloud and get redirected, missing the detector. This situation is scored “Not detected.” Click on the Run button to send photons through the cloud. Record the number of photons that were detected of the 20 sent on your lab report sheet. 8. Select Return from the menu bar to exit then on the main screen, choose Simulation > Continuum. Continuum photons give rise to the solid continuous rainbow of colors in a spectrum. They are photons of various energies (and therefore colors) that cannot interact with electrons of a given atom because their energies don’t match the energies needed to boost electrons to another level – they provide either too much or too little energy. The only way that continuum photons interact with electrons in an atom is occasional scattering, if they happen to exactly coincide in position in space. We observe a star’s light through its atmosphere, so pure continuous spectra are not 25 Energy Flow Out of the Sun UTC Astronomy 1010L normally observed – nearly all spectra show tell‐tale absorption lines characteristic of the cooler, less dense regions of the star’s upper atmosphere. 9. In the Continuum simulation, the configuration with gas container, photon source, and detector is the same as in the Line Formation simulation, but this demonstration uses continuum photons instead of line photons. Therefore, you should see that most photons pass through the gas without interacting. Change the number of photons for the simulation to 20 by choosing Parameters > # of Photons (for “Run”) and Run the simulation. Record the number of photons detected on your lab report sheet. 10. Select Return from the menu bar, then choose Experiment from the Simulation menu on the main screen. In the experiment mode, you are challenged to determine the energy level of a photon necessary to excite the atoms of various gases. You will plot the number of photons that pass easily through the gas at different wavelengths to see where the dark absorption lines appear. The Line Formation and Continuum simulations demonstrated that the photon must have just the right amount of energy to accomplish this. You have a number of atoms available for study. They include thin gaseous clouds of Calcium (Ca), Hydrogen (H), Magnesium (Mg), Oxygen (O), and Sodium (Na). 11. Choose a gas by clicking on Parameters > Select Gas Atoms. Record which gas you chose on your report sheet. 12. On the Parameters menu, select Change Photon Energy and set the photon energy to 1.5 eV. As you change the photon energy, the wavelength (color) of light changes automatically since the two are related by the relationship E = hc / λ, where E is the energy of the photon, h is Planck’s constant, c is the speed of light, and λ is the wavelength of light. On your data sheet in Table 1, record the wavelength of light corresponding to the 1.5 eV energy level. 13. From the Parameters menu, select # of Photons (for “Run”) and enter 20. Click on the Run button to send them through the gas cloud. On your data sheet, fill in the number of detected photons for this energy level in Table 1. 14. Repeat steps 12 and 13 for 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, and 3.2 eV. 15. Construct a graph of your results using Excel. a. Label column A “Wavelength (nm)” and type in your values. Label column B “Number of photons detected” and type in your values. b. In order to make a graph, highlight the data in columns A and B. c. On the Insert tab and the Charts toolbar, click on Scatter and select the Scatter with Smooth Lines and Markers option. 26 Energy Flow Out of the Sun UTC Astronomy 1010L d. The chart will be inserted on your spreadsheet. To move it to its own sheet, make sure the chart is selected, then click on Move Chart on the Design tab of the Chart Tools and select New Sheet. e. On the Layout tab of the Chart Tools, click on Chart Title > Above Chart and enter an appropriate title for your graph (e.g. Na absorption spectrum). Enter a title for the x axis by clicking on Axis Titles > Primary Horizontal Axis Title > Title Below Axis and for the y‐axis by clicking on Axis Titles > Primary Vertical Axis Title > Rotated Title. Enter the same column headings for the axis labels (with units) as you entered previously. Click on the Legend and press the delete key. f.
In order to scale the graph appropriately, on the Layout tab, click on Axes > Primary Horizontal Axis > More Primary Horizontal Axis Options … A dialog box will appear. Beside Minimum, click on Fixed, then enter the value 350. Adjust the maximum x‐value to 900. g. Although crude, you should be able to see a pronounced dip or several dips in the number of photons detected. The wavelength where the dip(s) appear identifies photon energies that match electron energy levels within that specific element. Once the electrons in the gas atoms are in an excited state, you know from the Line Formation simulation that most of the photons will be scattered away from an observer viewing the atom head on. 16. Print a copy of the graph for each lab partner to include with the report, then select Return from the menu bar on the Energy Flow program to exit the simulation. Part II: Flow of Photons in the Solar Interior Most atoms in the Sun and other stars are said to be ionized because the intense temperatures have stripped off most of their electrons. In the interior of the Sun, three primary mechanisms affect the travel of a photon toward the surface: electron scattering, bound‐free absorption, and free‐free absorption. Electron scattering occurs when a photon encounters an electron and gives it some of its energy, causing it to vibrate or oscillate. The energy stolen from the photon in this process is re‐radiated by the electron in some random new direction as a new photon. In bound‐free absorption, a photon can be totally absorbed by an atom, causing the atom to ionize and eject an electron that was bound to it; this free electron can recombine with another ionized atom giving rise to the release of a new photon in some random direction. Finally, in free‐free absorption, a photon transfers all of its energy to an already free electron to make it more energetic; subsequently, the electron may give up this extra energy in the form of a new photon, again to be radiated in some random direction. Though all three processes play a role in affecting a photon’s travel through the interior of the Sun, electron scattering is most hindering to its travel from the core to the surface. The result of these processes is that every time a photon interacts with matter, it is redirected so that it travels in a new and completely random direction. The resulting zigzag path is called a random walk. This is graphically demonstrated in the Flow simulation. 27 Energy Flow Out of the Sun UTC Astronomy 1010L 1. From the Simulation menu, select Flow > 1 photon. This simulation will allow you to explore the number of interactions required for a photon to exit the surface of a simulated star. 2. Go to Parameters > # of Layers to set the number of layers in the Sun to 5. Click on Run Simulation and record the number of interactions on your lab report sheet in Table 2. Repeat this three times for 5 layers, and then take the average of the three data points. 3. Repeat step 2 for stars with 10, 15, 20, 25, and 30 layers. 4. It can be mathematically shown that the number of interactions needed to escape is Figure 3. Solar Energy Flow Simulation very close to n2, where n is the number of layers in the model. This would hold true for a statistical average of all photons, which is many more than the three determinations that you made for each number of layers. Calculate n2 (the theoretical value of interactions needed to escape) for all of your layers and record it in Table 2 on your lab report sheet. 5. Using Excel, make a plot of the average number of interactions as a function of the number of layers for both the theoretical number of interactions you calculated and the “experimental” data that you collected from the simulation. a. Type in the values for the number of layers in the star into column A, your “experimentally‐determined” average number of interactions into column B, and your theoretical calculations for the number of interactions needed into column C. b. Highlight the data in columns A, B, and C. On the Insert tab and the Charts toolbar, click on Scatter and select the Scatter with only Markers option. c. As described above, move the chart to a new sheet, add a chart title, and add x and y axis titles (e.g. “Number of layers in the star” and “average number of interactions per photon”, respectively). d. While selected on the Chart, go to the Design tab and click on Select Data. The Select Data Source Dialog Box appears. Highlight Series 1, click on Edit, then the Edit Series dialog box appears. In Series name, type “Experimental”. Repeat this process for Series 2, naming it “Theoretical” (see figure on next page). 28 Energy Flow Out of the Sun UTC Astronomy 1010L e. Once the chart has been made, click once on a “theoretical” data point so that the entire series is highlighted. On the Layout tab, select Trendline > More Trendline Options... On the dialog box that appears, choose a Power function and click on the box for Display Equation on Chart, then click Close. f.
Repeat step e for your experimental data set. 6. Print a copy of the graph for each lab partner to hand in with the lab report, and record the equations for each data series on your lab report sheet. 7. On your lab report sheet, discuss the differences between your average results from the simulation and the theoretical plot. Are they lower or higher than theory would predict? Give a possible reason for the differences. 8. In the Sun, it takes several hundred thousand years for a photon released in the core to reach the surface. If photons did not interact with matter in the interior of the Sun at all, it would take only a tiny fraction of this time to escape the Sun. By relating the speed at which a photon travels (the speed of light, c = 3.00 × 108 m/s) to the distance it traveled divided by the time it takes to get there: d
d
c = ⇒ t = , c
t
determine what the escape time for a photon from the Sun would be if there were no interactions. Consider the distance that the photon has to travel to be the radius of the Sun, 6.96 × 108 m. Show your calculations on your lab report sheet. 29 Energy Flow Out of the Sun UTC Astronomy 1010L Energy Flow Out of the Sun Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. Which is more likely to be absorbed by an atom, a line photon or a continuum photon? 2. Where is the detector located in the Line Formation and Continuum simulations? 3. How are the energy and wavelength of a photon related to each other? 4. In the Flow simulation, what is the theoretical number of interactions a photon goes through before escaping a star? 5. How much time does it normally take for a photon generated in the Sun’s core to reach its surface? Pre‐Lab Page Energy Flow Out of the Sun UTC Astronomy 1010L Energy Flow Out of the Sun Lab Report
Name: ________________________________ Lab Partner: _________________________ Date: ______________ Part I, 4. In the Interaction simulation, how many line photons were scattered out of 20 sent through the gas? Part I, 5. In the Interaction simulation, how many continuum photons were scattered out of 20? Part I, 7. How many line photons were detected out of 20 sent through the gas in the Line Formation simulation? Part I, 9. How many continuum photons were detected out of 20 in the Continuum simulation? Part I, 11. Gas chosen for the Experiment simulation: Part I, 12‐14. Table 1. Number of photons detected at different wavelengths of light. Photons Photons Energy (eV) λ (nm) Detected Energy (eV) λ (nm) Detected 1.5 2.4 1.6 2.5 1.7 2.6 1.8 2.7 1.9 2.8 2.0 2.9 2.1 3.0 2.2 3.1 2.3 3.2 Part II, 2‐4. Table 2. Number of interactions required for photon to escape. Layers 5 10 15 20 25 Trial 1 30 Trial 2 Trial 3 Average Theoretical (n2) Report Page 1 Energy Flow Out of the Sun UTC Astronomy 1010L Part II, 6. Equation for theoretical number of interactions: Equation for average number of interactions from the simulation: Part II, 7. Discussion of the differences in simulated and theoretical data from the plot, and possible reason for any discrepancies: Part II, 8. Calculation of the escape time for a photon if there were no interactions in the Sun: ***Attach the graph you made of the number of photons detected as a function of photon wavelength in Part I and the graph for number of interactions as a function of layers in the solar interior in Part II to this lab report. Both graphs should have an appropriate title and labels for the axes with units. The graph for Part II should have a best‐fit power function line with equation displayed for both the theoretical and simulated data. Report Page 2 Astrometry of Asteroids UTC Astronomy 1010L Astrometry of Asteroids Objectives: The goals of this exercise are to understand how asteroids are discovered as moving objects on images of the sky; and to be able to measure their celestial coordinates as they orbit the Sun. Introduction and Background: Astrometry is the technique of precise measurement of the positions of stars, a fundamental tool of astronomers which allows construction of sky charts through assignment of right ascension and declination values to astronomical objects. By knowing the coordinates of certain positions in the sky, we can measure distances to some objects by applying the method of parallax. Using computers to measure positions of stars on digital images of the sky, astronomers can use astrometry to determine the coordinates of objects to very high precision; even the relatively simple program you will be using in this exercise can pinpoint objects to better than 0.1 arcseconds. Astrometric measurements can also pinpoint the locations of asteroids, yielding calculation of the speed at which they move through space. Lines of declination are like lines of latitude on the Earth – they specify a given celestial object’s position in the sky with respect to the celestial equator, an imaginary line in the sky that runs above the earth’s equator. Lines of declination are designated by angular distance north or south of the celestial equator measured in degrees (°), arcminutes ('), and arcseconds ("). There are 360 degrees in a circle, 60 minutes in a degree, and 60 seconds in a minute. A star with a declination of +45° 30' lies 45 degrees, 30 minutes north of the celestial equator; negative declinations are used for an object south of the equator. Right ascension lines are like lines of longitude on Earth, running through the north and south celestial poles Figure 1. The Celestial Coordinate System.
perpendicular to the lines of declination; they designate angular distance east of a line through the vernal equinox, the position of the Sun when it crosses the celestial equator on the first day of spring. Right ascension is measured in hours (H), minutes (m) and seconds (s). 1 hour of right ascension is equal to 15 degrees. There are 60 minutes in an hour and 60 seconds in a minute of right ascension. A star with a right ascension of 5 hours would be 5 hours, or 75 degrees, east of the line of right ascension (0 H) that runs through vernal equinox. There are many catalogs of objects in the heavens which list their right ascensions and declinations. One such catalog used in this exercise is the Hubble Space Telescope Guide StarCatalog, (GSC), which lists the coordinates of almost 20 million stars. These can be used as reference objects with known coordinates. To find the coordinates of an object in the sky whose right ascension and declination are not known (either because it isn’t in a catalog, or because it moves from night to night, like a planet or asteroid does), a photo is taken of the unknown object and surrounding reference stars whose coordinates are known; the position of the “unknown” object can then be interpolated from its location relative to the nearby reference stars. For example, suppose an unknown object “U” lies exactly halfway between reference stars A and B as shown in figure 2. Star A is listed in the catalog at right ascension 5h, 0m, 0s, declination 10°, 0’, 0”. Star B Figure 2. Determining the coordinates of an unknown object. is listed in the catalog at right ascension 6h, 0m, 0s, declination 33
Astrometry of Asteroids UTC Astronomy 1010L 25°, 0’, 0”. On the picture we measure the pixel positions of star A (20, 20), star B (10, 30) and U (15, 25) and find that U is exactly halfway between A and B in both right ascension (the x‐direction) and declination (the y‐
direction). We can then conclude that the right ascension of U is halfway between that of A and B, or 5h, 30m, 0s, and the declination of U is halfway between that of A and B, or 17°, 30’, 0”. If the unknown object isn’t exactly half‐way between the reference stars, interpolation of its position is more complicated. Moreover, because images of the sky appear flat when the sky is actually curved, astrometry is more involved in practice; however, the software used in this exercise and in real‐world astrometry measurements accounts for this in internal calculations. In this exercise, the unknown object’s position can be determined by choosing at least three stars of known coordinates, then indicating the location of the unknown object. The computer performs a coordinate transformation from the images on the screen to celestial coordinates in order to determine the coordinates of the unknown object. Overall Strategy: In this exercise, you will be using images of the sky to find asteroids and measure their celestial coordinates. Asteroids are small rocky objects that are usually only a few kilometers in size, often even less; most are located in the asteroid belt between Mars and Jupiter about 2.8 AU from the Sun. Like planets, they reflect sunlight, but because they are so small, they appear only as points of light on images of the sky, like dim stars. The key to recognizing asteroids is to note that over time they move noticeably against the background of stars due to their orbital motion around the Sun. If two pictures of the sky are taken a few minutes apart, stars will not have moved with respect to each other, but asteroids will have moved Figure 3. Finding the Asteroid
(see figure 3). Using the computer, we can simultaneously display two images of the sky 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 reference stars on the first and second images before you blink the two images, the only object that will change will be the asteroid, which will appear to jump. Sometimes asteroids will be faint; other times there are spots or defects that appear on one image and not another. These spots can mislead you into thinking that something has moved into the second image that was not there in the first. Thus, even with the ease of blinking, images should be carefully inspected 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. Seeing the asteroid continue its trend of motion in a third image can confirm your identification. Once an asteroid has been identified, its coordinates can be calculated by measuring its position with respect to reference stars. Finally, comparing an asteroid’s position over time enables us to calculate its velocity in space. Part I: Finding Asteroids by Blinking Images 1. Start the Astrometry of Asteroids program under the CLEA exercises folder on the Programs menu and select File > Log in. Enter your name, those of your lab partners, and the laboratory table number where you are seated for this experiment. When all information has been entered, click OK to continue. After confirming, the opening screen will appear. ** Note: At any time you can select Help from the menu. Within the topics, you should find information about how to load images, modify, print, blink, and measure them; the reports option shows you how to review your data and compute baselines. 34
Astrometry of Asteroids UTC Astronomy 1010L 2. We will be working with images of a region of the sky that is about 4 arcminutes square in which astronomers were searching for a faint Earth‐approaching asteroid designated 1992JB. From the menu bar, choose File > Load Image Files > Image 1. A directory listing showing you a list of files appears. From this list, select 92jb05.fts, and click Open to load it. 3. To display the image, select Images on the menu bar, then choose View/Adjust > Image 1 from the pull‐down menu. A window showing the image 92JB05 will appear on your screen. 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. On your lab report data sheet, make a sketch of the image, paying attention to details, drawing it to scale, and making it fit in the space. 4. Load the next image as before, but this time choose File > Load Image Files > Image 2 from the main (blue) program window. Select 92jb07.fts from the list and click Open. Display this image in its own window by using the Figure 4. Image Display Window Images > View/Adjust > Image 2 selection on the menu bar. Since this image was taken 10 minutes after the first image, the asteroid will have moved, but it may not be immediately obvious which star‐like object is out of place, even when you compare the images side by side. 5. To begin to locate the asteroid, we need to first align the images. Two stars will be needed for alignment in order to account for possible rotation between the images. On the main (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 a star that the computer will use to align the two images. If possible, you should try to choose two stars that are on diagonally opposite sides of the picture to achieve best results. When you click on one of the brightest stars, a yellow square surrounding the star should appear to mark your selection. Note which star you chose by writing #1 next to it on your freehand chart drawn on your lab report page. 6. When you click on Continue in the instruction box, it will ask for a second alignment star. Click on this star and record your selection on your chart as #2 on your lab report sheet. Click Continue again. 7. Now Image 2 appears and you will be asked to identify the same reference stars on Image 2 as you did on Image 1. Click on the same star #1. Hit Continue, click on the same #2, and Continue again. 8. On the menu bar at the top of the image window, click on Blink. The computer will flip back and forth between Image 1 and Image 2 about once per second. The stars will not move, but you should be able to pick out the asteroid as the one object that does jump. Be careful – occasionally, a white spot appears on one image and not the other; this is not an asteroid, but a defect in the picture itself caused by a cosmic ray exposing a single pixel in the camera during one exposure. The asteroid should appear clearly as a smudge of light that changes position from one image to the other. ** Notes: To stop the blinking, select Stop from the menu bar. Also, if you mis‐aligned and need to start over, select the Adjust > Field Alignment menu option to choose the alignment stars again. 9. 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 lab report sheet. On your drawing, label the asteroid’s position in Image 92JB05 (Image 1) with a small 05 and its position in Image 92JB07 (Image 2) with 07. 35
Astrometry of Asteroids UTC Astronomy 1010L 10. Repeat steps 4‐9 for the other images. Continue to find the asteroid on images 92JB08, 92JB09, 92JB10, 92JB12, and 92JB14 by blinking them. You will continue to use Image 92JB05 as Image 1. To do this, simply select the Load > Image2 menu item, and then Image > Blink to blink the images. You will be prompted to identify the alignment stars 1 and 2 on the new image. Mark the successive positions of asteroid 1992JB by dots labeled 08, 09, 10, 12, and 14 on the chart you made on your lab report sheet in step 3. 11. You should see that the asteroid is moving in a straight line. Draw an arrow in the space to the right of your drawing on your lab report sheet to show the direction of motion. Also record on your lab report sheet what direction this corresponds to (i.e., North, Northeast, Southeast, etc.). Part II: Determining the asteroid’s coordinates 1. To measure the coordinates of the asteroid, the computer will use a fitting process that compares the position of the asteroid with positions of stars whose coordinates have been previously tabulated in the Hubble GSC. We will start with the position of the asteroid in Image 92JB05. Make sure this image is loaded into Image 1 (File > Load Image Files > Image1). Choose Images > Measure > Image1. A window will open asking you to confirm the Observation Date and Time. Click OK. A second window will open asking you to confirm the coordinates for the center of the field. For Image 1, these should display correctly. Set the Field Size to 8 arcminutes. Click OK. Figure 5. Selecting Reference Stars 2. 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. Image 1, 92JB05, will be displayed 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 partly because the GSC only includes stars brighter than a 16th magnitude and some of the stars in the 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. **Note that the scale of the chart will not match the scale of the image. On your lab report sheet, sketch all stars in the reference star window on the left, then label the three you will use as #1, #2, and #3. 3. Now tell the computer which stars you have chosen as references. Click on reference star #1 in the GSC map. The Reference Star dialog box will open at the bottom of the page identifying the GSC data on that star (see figure 36
Astrometry of Asteroids UTC Astronomy 1010L 5); simultaneously the computer will draw a colored box around the reference star on the chart. Record the ID#, RA, and DEC of the star in Table 1 on your lab report sheet, then click OK to set the first reference star. 4. Repeat step 3 for reference stars 2 and 3. Make sure to record the data for each star in Table 1 on your lab report sheet. 5. When you have finished selecting at least three reference stars and marked them, go to the Select Reference Stars dialog box that opened when the star chart appeared and click OK. The computer will warn you that more reference stars would give you better results. Click No, you don’t need to choose any more right now. 6. You will now be asked to point to reference star #1, then star #2 and #3 in the image window. Click on each star in turn, making sure to match it with the reference star on the GSC chart, and click OK on the Locate Reference Stars dialog box for each. When all reference stars have been identified on your image, the computer will then ask you to click on the “unknown” or “target” star, which is the asteroid you identified. Hit OK after choosing it. If you are prompted, make sure that there are no other stars within the dotted outline, then click Yes. The computer now has enough information to calculate the location of the asteroid, and a text window will open displaying the right ascension and declination coordinates. Record these in Table 2 on your lab report sheet. 7. Click OK on the dialog box at the bottom to accept the solution. When the computer asks if you want to record the data, click OK. You can view this from the Report menu item on the main window at any time. Finally, press OK to return to the main window. 8. Now use the File > Load Image Files > Image1 menu choice on the main window along with the Image > Measure > Image1 menu choice, to measure the asteroid position in image 92JB14. You will proceed as in steps 3‐
7, recording the final data for the asteroid’s coordinates in Table 2 on your lab report sheet. Part III: Calculating the velocity of asteroid 1992JB To determine how fast 1992JB is moving, we first calculate its angular velocity, the speed at which it traveled across sky as seen in the images from Earth. Mathematically, this can be expressed: μ=
Δθ
, Δt
where μ is the angular velocity and Δθ is the angular distance it moved in the time elapsed, Δt. Δθ = ΔRA 2 + ΔDec 2
Figure 6. Mathematical illustration of the asteroid’s motion. 1. In order to calculate the angular distance traveled between when images 92JB05 and 92JB14 were taken, you can use the Pythagorean Theorem, as shown in figure 6. Because right ascension and declination are perpendicular coordinates, we can find the total angle θ moved across the sky by the asteroid, letting ΔRA represent the change in the number of arcseconds in right ascension and ΔDec represent the change in the number of seconds moved in declination. First, convert the declination for each coordinate that you recorded in Table 2 to arcseconds by multiplying ' by 60 and ° by 3600, then add all the values together. Show a sample calculation for one of these conversions on your lab report sheet. 2. Subtract the two values to find the change in declination, ΔDec, in arcseconds. Then convert this back to degrees by dividing your answer by 3600. Record both values on your lab report sheet. 3. Now determine the number of seconds of right ascension through which the asteroid traveled. Similar to declination, convert the hours, minutes, and seconds recorded in Table 2 to seconds by multiplying minutes by 60 37
Astrometry of Asteroids UTC Astronomy 1010L and hours by 3600, then adding all values together. Show a sample calculation for one of these conversions on your lab report sheet. 4. Subtract the two values to find the change in right ascension, ΔRA, in seconds and record the answer on your lab report sheet. In order to convert seconds into arcseconds of right ascension, several considerations are needed. First, remember that 1 second of RA is equal to 15” in angular measure. Also, we have to consider that right ascension lines merge at the poles, so there are smaller angles between them at higher declinations. To account and adjust for this physical change then, we must multiply by the cosine of the declination. Thus: ΔRA (") = ΔRA (seconds )×
15"
× cos(ΔDec ) , 1 second
where ΔDec is in degrees (calculated in the second part of step 2). Determine ΔRA in arcseconds, showing your calculation on your lab report sheet. **Note: Make sure your calculator is in degrees mode for this calculation. 5. Determine the angular distance that the asteroid moved, Δθ, using the equation given in figure 6 and the values for ΔRA and ΔDec in arcseconds (determined in the second part of step 4 and the first part of step 2, respectively). Show your calculations on your lab report sheet. 6. Calculate the angular velocity, μ, by dividing the distance traveled by the time it took to travel this distance. From the image data given in Table 2 on your lab report sheet, we can find that Δt is 8580 seconds. Use your value for Δθ determined in step 5. Show your calculations on your lab report sheet. 7. Finally, calculate the component of the asteroid’s velocity perpendicular to our line of sight, called the tangential velocity. Because the apparent speed of an object across the sky (what we measured as μ) is highly dependent on how far it is away, we have to know the distance to asteroid 1992JB in order to calculate its tangential velocity in way that is more meaningful to us. An object that is closer to us would appear to move across the sky faster than one that is farther away, even though the velocities of the two objects may be the same. By using parallax measurements of 1992JB from two different locations, the distance on the day that these images were taken was about 50,000,000 km. The equation for tangential velocity is: Vt =
μ (" /s )× distance (km)
, 206,265"
where 206,265 is from the parallax conversion of distance to angular measure, and is equal to the more familiar 57.3° in parallax calculations (see Parallax lab exercise for further detail). Calculate the tangential velocity of 1992JB, showing your calculations on your lab report sheet. 38
Astrometry of Asteroids UTC Astronomy 1010L Astrometry of Asteroids Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. Describe what is meant by “blinking” the images. 2. How many reference stars are chosen in each image to align them? What is the purpose of doing this? 3. How many reference stars are needed for the computer to calculate the asteroid’s coordinates? 4. Why is the angular measure of right ascension multiplied by the cosine of the declination in determining the change in right ascension of the asteroid? 5. What is the difference between the angular velocity of the asteroid and its tangential velocity? Pre‐Lab Page Astrometry of Asteroids UTC Astronomy 1010L Astrometry of Asteroids Lab Report Name: ________________________________ Lab Partner: _________________________ Date: ______________ Part I. 3, 5‐6, 9‐11. Sketch of 92JB05, labeling stars used for alignment (#1 and #2) and the asteroid’s position in images 05, 07, 08, 09, 10, 12, and 14. Indicate the direction of motion of the asteroid with an arrow to the side of the image. NORTH
EAST
Direction of asteroid’s travel: Part II. 2. Sketch of reference star map from the Hubble GSC, labeling reference stars #1, #2, and #3. Report Page 1 Astrometry of Asteroids UTC Astronomy 1010L Part II. 3 and 4. Table 1. Reference Star Coordinates Reference Star ID # RA (h, m, s) DEC (°, ′, ″) # 1 # 2 # 3 Part II. 6, 8 and 9. Table 2. Measured celestial coordinates of asteroid 1992JB on May 23, 1992. File Name Time (UT) RA (h, m, s) DEC (°, ′, ″) 92JB05 04 53 00 92JB14 07 16 00 **Note: The values for RA should be approximately the same. Part III. 1. Declination in 92JB05 in arcseconds: Declination in 92JB14 in arcseconds: Sample calculation for conversion: Part III. 2. ΔDec = ________________” = ________________ °. Part III. 3. Right Ascension in 92JB05 in seconds: Right Ascension in 92JB14 in seconds: Sample calculation for conversion: Report Page 2 Astrometry of Asteroids UTC Astronomy 1010L Part III. 4. ΔRA = ________________ seconds = ________________ ”. Calculation for conversion to arcseconds: Part III. 5. Calculation of Δθ: Part III. 6. Calculation of the angular velocity, μ: Part III. 7. Calculation of the tangential velocity, vt: Report Page 3 Deep Sky Observing Field Trip UTC Astronomy 1010L Deep Sky Observing Field Trip We will be meeting at the Chickamauga National Battlefield site on Thursday evening from 6:30 ‐ 9:30 PM, where we will be setting up the telescopes and observing various astronomical objects. Directions to the observing site are given below. Allow at least 30 minutes driving time from UTC. You will need to bring this handout as well as a writing implement with you, as those will not be available at the site. As you will be assembling the telescopes, review your notes from the lab that we performed on telescopes earlier this semester. Background: There are several ways in which to locate an object in the sky, but in this exercise, we will be using right ascension and declination, which are convenient to use with the setting circles on the telescopes. Right ascension values are imaginary lines that run in the north/south direction along the celestial sphere, increasing eastward from the position of the Sun at vernal equinox. Right ascension is measured in units of hours, minutes, and seconds, and can be thought of as analogous to longitude on Earth. Declination values are lines that run east/west along the celestial sphere, and can be thought of as analogous to lines of latitude on Earth. Declination is measured in degrees (°), arc‐minutes ('), and arc‐seconds (″) north or south of celestial equator. For reference, the celestial equator has a declination of 0°, the north celestial pole, +90°, and the south celestial pole, ‐90°. Right ascension and declination values do not change depending on the time of night. Constellations are another convenient way in which to locate celestial objects, if you know in which constellation the object is located. The IAU has designated 88 constellations, which are actually bounded regions of the sky, not merely the stars which have traditionally been connected to form an object. Twelve constellations are part of the zodiac, which are the constellations that fall along the ecliptic, but only six to seven are visible at the same time, depending on what time of the night and what time of year you are observing. As the ecliptic is the plane in space of Earth’s orbit around the Sun, the Sun and all planets also fall along the ecliptic in the night sky. Because the Moon’s orbit around Earth is tilted at 5° to the ecliptic, its path across the sky is near, but not on, the ecliptic. We will be viewing several different types of astronomical objects on our trip. Table 1 gives a list of possible objects to view, noting the spectral type of each star. As mapped on a Hertzsprung‐Russell (H‐R) diagram, the spectral sequence from left to right is O B A F G K M, with the hottest stars appearing on the left and cooler stars on the right. For example, M type stars have a surface temperature as low as 3000 K, A type stars around 10,000 K, and O type stars up to 30,000 K. Using a blackbody diagram and Wien’s law, hot stars emit shorter wavelengths of light and appear blue in color, while cooler ones emit longer wavelengths of light, and appear redder. The magnitude of each celestial object is also given. The smaller the magnitude (even negative), the brighter the star will be in the sky. In the Chattanooga area, we are restricted by light pollution to viewing objects with magnitudes of around 6 or less, but it may be possible to see objects with slightly larger magnitudes, dependent on the particular weather and light conditions. Star clusters are formed from collapsing and fragmenting interstellar clouds of gas and dust called nebulae. Open clusters are loose and irregular, generally containing 10‐1000 young, bright, hot, blue stars. By the theory of stellar evolution, a star’s lifetime can be tracked in a specific pattern along the H‐R diagram. A star resides on the main sequence for the majority of its life. If it has enough mass, it moves to the red giant stage when it starts burning hydrogen and/or helium in its outer shells. When carbon fusion begins in the core of the star, the envelope is ejected into space. The remaining carbon core of the star is a white dwarf. Globular clusters are roughly spherical and contain millions of cooler, old stars which are in the last stages of their lifetimes and have left the main sequence. Generally, open clusters are in the main plane or spiral arms of the Milky Way, and globular clusters are found in the surrounding halo of a galaxy. In addition to the Milky Way, our Local Group of galaxies includes Andromeda and the Triangulum, both of the spiral type. 43
Deep Sky Observing Field Trip UTC Astronomy 1010L Table 1. Astronomical Objects for Observation Catalog Number RA Type / Star (hr:min) Declination (Magnitude) Constellation Notes Stars Mizar/Alcor 13:23 +54° 55’ A2/A5 (2.4/4.0) Ursa Major Polaris 2:31 +89° 15’ F8 (2.1) Ursa Minor Rigel Aldebaran 5:14 4:35 ‐8° 12’ +16° 30’ B8 (0.3) K5 (1.1) Orion Taurus Sirius 6:45 ‐16° 42’ A0 (‐1.6) Canis Major “Horse and Rider” vision test in the Big Dipper. Mizar is the bend of the handle, Alcor just a little above. Actually a triple star; the North star. Has brightened about 15% in past 100 years. The left foot of Orion. The eye of the bull. One of the brightest stars in the sky. Orion’s dog. Double Stars γ Andromeda 2:03 +42° 19’ K0/A0 (2.3) Andromeda Blue one is a triple star system Galaxies M31 0:42 +41° 16’ (3.4) Andromeda M33 1:33 +30° 39’ (5.5) Triangulum Spiral; in the Local Group (fuzzy patch) Spiral; in the Local Group Open Clusters M42 5:35 ‐5° 23’ (4.0) Orion NGC 869 NGC 884 2:19 2:22 +57° 09’ +57° 07’ (5.3) (6.1) Perseus M45 3:47 +24° 07’ (1.2) Taurus The Orion Nebula; a cluster with nebulosity. Together, these are the famous “double cluster” of Perseus. Also known as the Pleiades; a cluster with nebulosity. Globular Clusters M3 13:42 +28° 22’ (6.3) Coma Berenices Fuzzy ellipse appearance Solar System Objects Moon Mercury Jupiter Saturn 00:08 00:38 13:01 0° 47’ 2° 55’ ‐3° 36’ Planet (‐1.2) Planet (‐2.0) Planet (0.1) Pisces Pisces Virgo Titan Moon Virgo Rises 9:40AM, sets 12:20 PM Rises 7:30 AM, sets 7:40 PM Rises 7:55 AM, sets 8:15 PM Rises 8:30 PM, sets 8:30 AM Saturn’s largest moon. About 3‐4 ring
lengths east of Saturn. Constellations Cassiopeia Ursa Major Ursa Minor Leo Cancer Orion Taurus Aries Gemini Canis Major Canis Minor Lepus 44
Deep Sky Observing Field Trip UTC Astronomy 1010L Figure 1. Ideal view of the night sky on March 10, 2011 at 8:00 pm. In order to get your bearings, hold this figure above your head with the N direction on the diagram facing north. 45
Deep Sky Observing Field Trip UTC Astronomy 1010L Figure 2. Sky view at 8:00 pm on March 10, 2011, showing the positions of various constellations and planets. Constellation boundaries are denoted by the blue dotted lines. Use this figure by holding it above your head with the N direction on this chart facing north. 46
Deep Sky Observing Field Trip UTC Astronomy 1010L Directions to the observing site: From UTC: Take McCallie Ave. east about 3 blocks to Central Ave. Turn right at the traffic light onto Central, and go about 1.3 miles to the traffic light at Rossville Blvd. Turn left onto Rossville Blvd. and follow it for about 3 miles until you enter Georgia. When you see the Family Dollar store on the right, get in the middle lane – the road you should follow (Rossville Blvd., which turns into Chickamauga Ave.) veers to the left, while McFarland Ave. turns to the right. Follow Chickamauga Ave. (which becomes Lafayette Rd.) about 3.7 miles through Fort Oglethorpe into the battlefield. Alternate Route to Chickamauga Battlefield: Take I‐24 East to I‐75 South towards Atlanta. Go about 6 miles on I‐75 and take exit 350 onto Battlefield Parkway (GA Hwy 2). Go about 6.3 miles, then turn left onto Lafayette Rd. Travel less than a mile to enter the battlefield. Inside Chickamauga National Battlefield: Note: Be careful to obey the speed limits and one‐way signs! From Lafayette Road, turn right onto Dyer Rd (at site 4 on the map), just before the Brotherton cabin on the right. Follow Dyer until it hits Chickamauga‐Vittetoe Road, and turn left. Follow Chickamauga‐Vittetoe Road for about ½ a mile until it meets Glenn‐Kelly Road. There will be a parking area here. We will be just off the parking lot in the Glenn Field Recreation Area. 47
Deep Sky Observing Field Trip UTC Astronomy 1010L Deep Sky Observing Lab Report Name: ______________________________________________________ Date of Visit: _____________________ Instructions: Answer all questions below; you should attempt to answer the questions without help from others! 1. (a) What types of telescopes were set up tonight for viewing? (b) What was the star you used to calibrate the setting circles on your telescope? (c) For the telescope you were using, what was the focal length of the eyepiece? 2. View the stars listed in Table 1. For Mizar/Alcor, are you able to distinguish Mizar from Alcor without a telescope? Are you able to resolve the two stars with a telescope? 3. Describe the color of Aldebaran as compared to Rigel. 4. For γ‐Andromeda, describe the colors of the two components of the system. 5. Which constellations of the Zodiac did you see tonight? In what area of the sky are these? Draw and describe two you saw tonight. 6. (a) View and sketch an open cluster of stars (in the Orion nebula or the double cluster of Perseus) and the globular cluster of stars. (b) Describe how open and globular clusters are different from one another. Report Page 1 Deep Sky Observing Field Trip UTC Astronomy 1010L 7. (a) View the Pleiades, otherwise known as the seven sisters. How many bright stars can you see when you look directly at this open cluster? (b) How many stars can you see in the Pleiades if you look at stars beside it? (c) What automobile manufacturer uses the Pleiades as their logo? 8. (a) To what galaxy do all the individual stars and clusters viewed tonight belong? (b) Draw and/or describe another galaxy we viewed tonight. 9. What was the phase of the moon tonight? How do you know? (The weather almanac told me is not an acceptable answer. Think about the percentage illuminated, rise/set times, or the side illuminated.) 10. (a) Draw and describe the planets (at least Jupiter or Saturn) you observed. (b) Can you distinguish features on these planets or any moons? 11. (a) What constellation is the Big Dipper part of? (b) In what constellation is Polaris located? (c) Is Cassiopeia found on the ecliptic? What letter in our alphabet does Cassiopeia resemble? 12. What set of “animals” surrounds Orion? Report Page 2 The Mass of the Earth UTC Astronomy 1010L The Mass of the Earth Objective: In this experiment, you will measure the value for the acceleration due to gravity, g, near the Earth’s surface by timing the oscillations of a simple pendulum. Knowing the value of g will allow you to determine the mass of the Earth given its radius. Further, from these quantities, you will also be able to determine the average density of the Earth. Background: The period of a pendulum’s oscillation, T, is the time required for one complete back‐and‐forth cycle of its swing. Using laws of classical physics, it can be shown that for small angle oscillations (less than ~5°), the relationship between the period, T, and the length, L, of a pendulum is given by: T ≈ 2π
L
or g
T2 ≈
4π 2
L g
(1) where L is measured in meters, T is measured in seconds, and the acceleration due to gravity is in m/s2. Notice that in equation 1, T2 is linearly dependent on L. By relating this equation to the standard form of a line, y = mx + b, this means that if a graph were constructed where T2 was plotted on the y‐axis and L on the x‐
axis, the slope of the graph (m) should be approximately equal to 4π2/g. Theoretically, the y‐intercept of this plot (b) would be zero. Thus, the slope of the line which best fits experimentally‐measured values of T2 and L can be used to find the value of g by: 4π 2
g=
(2) slope
Because Earth is very close to spherical and the acceleration due to gravity depends on your position relative to the center of Earth, the value of g varies little dependent on your location on the surface, averaging about 9.81 m/s2. Very slight deviations (ranging ~0.25% around the average) do occur based on latitude (g is slightly less at the poles because Earth bulges at its equator) and altitude on (g is slightly less on mountaintops and greater toward sea level). Newton’s 2nd law of motion tells us that the force acting on an object is equal to its mass times its acceleration, or F = ma. Thus, in the case of an object falling toward the Earth, the force of gravity (FG) attracting an object toward the Earth is simply FG = mg, where m is the mass of the object. Newton’s Law of Universal Gravitation describes the force of gravity in different terms, that is, that gravitational force (FG) is directly proportional to the product of the interacting masses (m1 and m2) and inversely proportional to the square of the distance between them (r2), or FG = (Gm1m2)/r2. In this case, G is the universal gravitational constant, and is not the same as g, the acceleration due to gravity. However, note that FG is the same in both equations, and so if we examine the case of an object being attracted to the center of the Earth, we can use Newton’s two relationships to find the mass of the Earth. Equating the right sides of each relationship in order to eliminate FG, we have: mg =
Gm1 m2
r2
On the right side of the above equation, m1 and m2 are the masses of the object and the Earth, respectively. Since m1 is m (the mass of the object), we can eliminate this variable from the equation. Also, we can substitute Me for m2 (the mass of the Earth) and Re (the radius of the Earth) for r, since in the case of an object on Earth’s surface being gravitationally attracted to it, the distance between the interacting masses is simply the radius of the Earth. g=
GMe
Re2
50
The Mass of the Earth UTC Astronomy 1010L Finally, we can re‐arrange to solve for the mass of the Earth Me =
gR e 2
G
(3) where g was experimentally determined from the pendulum oscillation measurements, G has a value of 6.672 × 10‐11 Nm2/kg2, and the average value for the radius of the Earth (Re) is 6.378 × 106 m. By knowing the density of a planet, we can make good speculations about what materials may compose it. The density of an object is equal to its mass divided by its volume, and typical units are kg/m3. If a large amount of matter is packed into a small volume, then the density will be high. Conversely, if a small amount of matter occupies a large volume, the density will be low. If the density of a planet is high, it suggests that its interior contains heavy elements like iron and it likely has a solid, rocky surface. On the other extreme, if a planet’s density is low, it may be gaseous or liquid with no solid surface and composed of lighter elements like hydrogen and helium. Largely, conclusions such as these are drawn on the basis of comparisons with Earth, since it is the planet of which we have the most knowledge. Since we determined the mass of the Earth we can now determine its average density. The volume of the Earth can be determined from its radius using the relationship for the volume of a sphere, 4/3 πr3 (an assumption which is fairly close, as previously discussed). ρ=
Me
3M e
m
=
=
V ⎛ 4 3 ⎞ 4πRe3
⎜ πRe ⎟
⎝3
⎠
(
)
(4) where ρ is the density of the Earth, Me is the mass of the Earth, and Re is its radius, 6.378 × 106 m. Procedure: 1. Set up the apparatus as shown in figure 1. Clamp the large C‐clamp to the end of the table so that the groove in the clamp is vertical, place the aluminum pole in the groove, and tighten the screw. Clamp the pendulum clamp to the top of the aluminum pole, loosen the screws on the clamp, place the strings under two metal fingers and tighten the knobs back, so that the pendulum bob hangs in the middle and can freely swing back and forth. 2. Adjust the string in the pendulum clamp so that its length (refer to figure 1) is about 1 meter. Measure the length to the nearest millimeter and record it as trial 1 on your report sheet. 3. Displace the pendulum bob so that the string makes an angle of no more than 5° relative to vertical. For a length of 1 m, this would mean you pull it back no more than 8 cm. (**Note: for 0.88 m, this is < 7.5 cm; for 0.76 m, this is < 6.5 cm; for 0.64 m, this is < 5.5 cm; for 0.52 m, this is < 4.5 cm; and for 0.40 m, this is < 3 cm). The length of the pendulum is the distance from the position of the fixed point of the string to the position of the center of mass of the bob. Lab Table Figure 1. Experimental Set‐up
51
The Mass of the Earth UTC Astronomy 1010L 4. Release the bob and let it swing several times to come to steady‐state. With your stopwatch, measure the time required for 20 complete oscillations of the pendulum. Record the time for 20 oscillations on your lab report page. 5. Determine the period, T, for a single oscillation by dividing the time for 20 oscillations by 20. 6. Shorten the length of the pendulum and repeat steps 2‐5 for lengths of about 88 cm, 76 cm, 64 cm, 52 cm, and 40 cm, making a total of 6 independent measurements for different lengths in this fashion for Trials 2‐6. 7. Calculate the value of the period squared, T2, for all lengths of the pendulum. 8. Using your data, make a plot of T2 as a function of L (that is, the length of the pendulum is on the x‐axis and the corresponding values of T2 are on the y‐axis). Graphs should have a title, all axes labeled (including units), and a best‐fit curve to the data. You can construct the graph using Excel and the instructions provided below: (a) Label column A as your independent variable (whatever should be on the x‐axis) and type in your values. Label column B as your dependent variable (the values plotted on the y‐axis) and type in your values. (b) In order to make a graph of the dependent vs. independent variable, highlight the data. On the Insert tab and the Charts toolbar, click on Scatter and select the Scatter with only Markers option. (c) The chart will be inserted on your spreadsheet. To move it to its own sheet, make sure the chart is selected, then click on Move Chart on the Design tab of the Chart Tools and select New Sheet. (d) On the Layout tab of the Chart Tools, click on Chart Title > Above Chart and enter an appropriate title for your graph (like Period Squared vs. Length of the Pendulum). Enter a title for the x axis by clicking on Axis Titles > Primary Horizontal Axis Title > Title Below Axis and for the y‐axis by clicking on Axis Titles > Primary Vertical Axis Title > Rotated Title. Enter the same column headings for the axis labels (with units) as you entered previously. Click on the Legend and press the delete key. 52
The Mass of the Earth UTC Astronomy 1010L (e) Once the chart has been made, go to the Layout tab and select Trendline > More Trendline Options... On the dialog box that appears, choose a Linear function and put a check mark in the boxes for Display Equation on Chart, and Display R‐squared value on Chart. Although the theoretical intercept of the line should be zero, you should not force the line through zero by checking the “Set intercept = 0” box, since you did not experimentally measure that point. (f) The equation of the line and the estimate of the goodness of its fit (R2) should now appear on your chart. A value of R2 that is equal to 1 would indicate a perfect correlation. (g) Print your graph to hand in with your lab report. 9. Using equation 2 and the slope of the best‐fit line to your data, determine your value for g, showing your calculations on your lab report sheet. 10. Since the acceleration due to gravity does fluctuate depending on your location on Earth, your value may not be precisely equal to the accepted average value for g of 9.81 m/s2. In order to compare your experimentally‐
determined value of g with the accepted average value, find the percentage difference between the two using the formula below. Show your calculations on your lab report sheet. % difference = your value ‐ 9.81 m/s 2
9.81 m/s 2
× 100% 11. Calculate the mass of the Earth using equation 3 and your determined value for the acceleration due to gravity, g, showing your calculation on your lab report sheet. 12. Quantitatively compare your value for the mass of the Earth to the published value for the mass of the Earth (5.97 × 1024 kg) by calculating a percent difference calculation similar to the one above. Show your calculation on your lab report sheet. 13. Using equation 4 and your determined value for Me, calculate the volume of the Earth, then its density. Show your calculations on your lab report sheet. 14. Compare your value for the average density of the Earth with the published value of 5500 kg/m3 by doing a percent difference calculation, showing your work on your lab report sheet. 53
The Mass of the Earth UTC Astronomy 1010L The Mass of the Earth Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. What is the physical meaning of one period of a pendulum oscillation? 2. What quantities are plotted your graph in order to determine g? State which is the data for the y‐axis and which is the data for the x‐axis. Also give the units for each. 3. How will you determine the mass of the Earth (what equation will you use)? 4. If a planet had a very low density, what might this indicate about its composition? 5. What assumption is made when using the calculation in the lab manual for determining the volume of the Earth? Pre‐Lab Page The Mass of the Earth UTC Astronomy 1010L The Mass of the Earth Lab Report Name: ________________________________ Lab Partner: _________________________ Date: ______________ 2, 4‐7. Trial Length (meters) Time for 20 cycles (s) Period, T (s) T2 (s2) 1 2 3 4 5 6 **Attach your computer‐generated graph to this report sheet. 9. Calculation for the experimentally‐determined value of g (including units): 10. Calculation of the percent difference between your value of g and the accepted average value: 11. Calculation for Me (including units): 12. Calculation of the percent difference in your value of Me from the published value: 13. Calculation for Earth’s volume (including units): Calculation for Earth’s density (including units): 14. Calculation of the percent difference in your value for the Earth’s density from the published value: Report Page Rotation Rate o
of Mercury UTC Astron
nomy 1010L Rotation Rate of Meercury Objectives: The goal of thiss exercise is to
O
o measure the broadeningg of the refleccted signal fro
om a rotating body (M
Mercury) in order to determine the ratee at which it rrotates. Baackground: Becausse Mercury iss a small planet whose surface featuress have low contrast, and beecause it is so
o close to th
he Sun that it is rarely visib
ble against a d
dark sky, it is difficult to deetermine how
w fast it is rotating merely by looking att it from Earth
h. In recent years radar techniques havve proven mo
ost effective in
n measuring iits speed of ro
otation. Th
he method yo
ou will emplo
oy here has wider applicatiion than just tthe measurin
ng the rotation of Mercuryy – it can bee used to stud
dy cloud‐cove
ered Venus, rrings of the m
major planets, and rotation
n of the smalleest asteroids.. The baasic method o
of this experim
ment is to usee a radio telesscope to send
d a short pulsse of electrom
magnetic raadiation with a known freq
quency towarrd the object, then to reco
ord the spectrrum (frequency versus inteensity) of th
he returning eecho. Depend
ding on the reelative positio
ons of Earth aand Mercury, the pulse will take betweeen 10 m
minutes and a half‐hour to travel to Merrcury, bouncee off, and retu
urn. By the tim
me the pulse has reached Mercury, it has spread o
out to cover th
he entire plan
net. Howeverr, because thee planet’s surrface is a spheere, the pulsee hits diifferent partss of the planet at different times. The pulse first hits the surface aat a point direectly on a linee between th
he centers of Earth and Me
ercury (the “ssub‐radar point”). A few m
microseconds later, the pullse hits pointss farther baack, toward tthe edges of tthe planet. Th
hus we wait fo
or the first eccho (from thee sub‐radar po
oint), and theen by lo
ooking at the returning ech
hoes at succeeding times, each a few m
microseconds later than thee next, we geet in
nformation ab
bout differentt parts of Mercury’s surfacce. Figure 1
1. The Doppler Shift
The freequencies of the returningg echoes are d
different from
m the frequen
ncy of the pullse sent out b
because th
he echoes havve bounced o
off the movingg surface of M
Mercury. Any time a source of radiation
n is moving raadially (towards or aw
way from the observer), th
here will be a Doppler shiftt in the receivved frequencyy that is proportional o the velocity along the line of sight. Th
here are two m
motions of M
Mercury that ccan produce ssuch a shift: itts orbital to
veelocity around the Sun and
d its rotation on its axis. Th
he first echo (from the sub
b‐radar point) is shifted in frequency on
nly by the orb
bital velocity of the planett around the SSun. We can ccalculate how
w fast the plan
net is movingg with reespect to Eartth from the amount of thee Doppler shifft, but we can
n’t tell how faast Mercury iss spinning. Th
his is 56
Rotation Rate o
of Mercury UTC Astron
nomy 1010L beecause the co
omponent of the rotationaal velocity of tthe surface o
of Mercury is p
perpendicular to our line o
of sight at th
he sub‐radar point (see Figgure 1), so theere is no addiitional frequeency shift. However, the ecchoes that arrive after th
he sub‐radar eecho show ad
dditional Dop
ppler shifts beecause they co
ome from farrther back on the planet w
where the ro
otational velo
ocity is more d
directly alongg our line of sight. Becausee of the rotatiion of Mercurry, one edge o
of the pllanet is movin
ng toward us a little fasterr than the cen
nter of the plaanet, and thee other edge is moving tow
ward us a litttle slower than the center of the planeet (see Figure 1). Due to th
he Doppler efffect, part of tthe returning echo (ffrom the fasteer moving edgge of Mercuryy) is at a sligh
htly higher freequency, and part of the reeturning echo
o (from th
he slower moving edge) is at a slightly lower frequen
ncy. Figure 2 d
depicts the frrequency spectrum that is obtained. L
Lower frequen
ncy echo from sside moving to
owards Earth a little Higheer frequency echo from side moving towards Earth a
s
slower (rotatio
onal velocity co
omponent awaay from Earth)
little faster (rotational velocity coomponent towaards Earth) Figuree 2. Frequency of returning eecho (Hz differe
ence from transsmitted frequeency) W
We measure th
he amount off the frequency shift and aapply our kno
owledge of the Doppler efffect to calculaate the veelocity of the surface of M
Mercury, and ffrom this, its p
period of rotaation. Prrocedure: 1.. Open the M
Mercury Rotattion program
m from the CLEEA Exercises menu. Selectt Log In on thee main menu, and fill in the requ
uested inform
mation. After completing th
he log‐in, preess Start at the top of the w
window. The control panel that appears has displays for tthe frequencyy the telescop
pe is tuned to
o and the coordinates it is pointed to in the sky. Press the Traacking button
n to turn on th
he sidereal drrive so that th
he telescope will track thee planets as Earth tu
urns. This will activate the other controls menu. 2.. Select Ephe
emeris at the top of the wiindow; this acctivates a pro
ogram that caalculates the p
position of a p
planet for any date and time. Use
e December 1
1, 1999 for the launch datee, 0 (hours, m
minutes, and seconds), and
d Universal TTime. Press O
OK and recorrd the distancce between Eaarth and Merrcury (in AU) aand the expected length of time for the p
pulse to return
n on your datta sheet. 3.. Leave the eephemeris win
ndow on the screen or min
nimize it (do not close it), and press Sett Coordinatess. Respond Y
Yes to Use Computed Valu
ues and the teelescope will begin slewingg (moving rap
pidly) to pointt to the planet. (If yyou left the EEphemeris Co
omputation w
window on thee screen it will minimize w
when the slew
w starts. If you minim
mized it, it rem
mains minimizzed and you ccan reopen it for referencee.) When thee telescope move is complete, the red light will go out an
nd the telesco
ope’s pointing coordinatess will appear on the contro
ol panel. 4.. Press Send Pulse to tran
nsmit a radar pulse toward
d Mercury. The frequency o
of the pulse w
will be shown
n on the 6
lower left. Record it on your data sheet, and convvert f to unitss of Hz from M
Megahertz (1 MHz = 10 Hzz). 5.. A message w
will appear on the screen with the estim
mated time u
until reception of the returrn pulse. A wiindow showing th
he initial specctrum (frequeency versus in
ntensity) of th
he pulse also appears on th
he screen when the pulse is traansmitted. TThis window w
will minimize itself after 10
0 seconds. W
When this occu
urs you will seee an 57
Rotation Rate of Mercury UTC Astronomy 1010L animated display on the telescope screen. This display shows the correct positions of Mercury, Venus, the Earth, and the Sun on the date of the coordinates you have used. The bright radar pulse moves toward Mercury and then the fainter echo returns. (The pulses move at the speed of light relative to the scale of the display. The distances on the display are correctly scaled, but the images of the disks of Sun, Earth, and Mercury are not to scale.) Sketch the relative positions of the planets on your data sheet for reference. Figure 3. Pulse sent and echo. 6. While you are waiting for the return pulse, you can calculate some geometrical terms needed to convert the measured velocities from points not on Mercury’s equator into velocities at Mercury’s equator (see figure 4). The delay distance (d) is the distance the delayed beam has traveled beyond the sub‐radar point. x is the distance parallel to our line of sight from the center of Mercury to the point from which the echo comes back y is the distance perpendicular to our line of sight to the extreme outer edge of the region of Mercury from which the echo comes back. R is Mercury’s radius, V is its rotational velocity, and VO is the measured component of the rotational velocity parallel to the line of sight at that point. Figure 4. Geometry of Mercury’s rotation.
For the time intervals 120×10‐6, 300×10‐6, and 390×10‐6 seconds, calculate d, x, and y, with the following formulas and record them in your data table. The values for 210×10‐6 seconds are calculated for you. a.) To calculate d (in meters): use distance = rate x time, but since we are measuring an echo, which has to travel over the same path twice (down and back) we take half this value: d=
cΔt
2
where c is the speed of light (3 x 108 m/s) and Δt is the time delay for the particular pulse in seconds. b.) To calculate x: by fig. 4, this is Mercury’s radius minus d (calculated above), where R = 2.42 x 106 meters. x = R – d 58
Rotation Rate of Mercury UTC Astronomy 1010L Since the echoes we measure come back from points only a few kilometers back from the sub‐radar point, x will be only slightly smaller than R. c.) To calculate y: Note that y is one side of a right triangle whose hypotenuse is R and whose other side is x. y = R2 − x2 Make sure you have shown the calculation for one of your times in the column on your data sheet. 7. The return pulse is spread out over a few hundred microseconds due to the curved surface of the planet. A series of five windows will appear on your screen when the pulse is received. These windows show snapshots of the spectrum of the returning echo beginning at the instant of reception (labeled Reference), followed by another 120×10‐6 seconds later, and three more at successive 90×10‐6 second intervals. These pulse spectra show a certain amount of noise which increases with the later‐arriving pulses, since they are weaker. Compare the appearance of the received pulse with the initial pulse. You will note that the initial pulse, which is stronger, appears much smoother and sharper. The frequency of the peak on the reference spectrum is the change in frequency between the transmitted pulse and the returned sub‐radar echo pulse at time equal to zero seconds, Δft=0. In order to find this value (in Hz), click on the central peak of the spectrum. Record the value displayed for the frequency shift from the zero position (with units) on your data sheet. 8. For all the other delayed echoes, you must measure the positions of the peaks at the left and right “shoulders” of the plots. These represent echoes from the parts of the planet that are turning toward and away from us the fastest. When you have positioned the cursor on the left “shoulder” of the plot, double click the left mouse button. A red arrow will appear on the screen, along with the measured position in Hz. To measure the right “shoulder”, follow the same procedure. A blue arrow appears at the measured position. Record the results of all measurements for Δfright andΔ fleft in your data table. Figure 5. Measurement of the return echoes. 9. In order to determine the shift in frequency due to the rotational velocity of Mercury (Δftotal), we need to note that one side of Mercury is rotating toward us as fast as the other side is rotating away from us. Thus, the difference in the frequency shifts from the two edges, Δfright and Δfleft is twice the shift due to rotational velocity, shown by the equation below. Calculate these values and record them on your data sheet. Δftotal =
Δfright − Δfleft
2
59
Rotation Rate of Mercury UTC Astronomy 1010L 10. We need to correct for the fact that our measurement is an echo (that is, the shift is twice what would be produced by a source which is simply emitting at a known frequency). The pulse arrives at Mercury and appears shifted as seen from the surface, and then it is shifted again because the surface of Mercury is moving as seen from Earth. Calculate the corrected frequency Δfc as below and record it on your data sheet. ΔfC =
Δftotal
2
11. To calculate VO, the component of the rotational velocity of the edge of Mercury along the line of sight at the point from which the echo returns, apply the Doppler equation to the observed frequency shift: ⎛f ⎞
VO = c⎜⎜ C ⎟⎟ ⎝ f ⎠
where c is the speed of light and f is the transmitted frequency of the pulse (recorded in step 4 – use the one in units of Hz). 12. Finally, we can find the rotation rate of Mercury. This velocity, V, is your observed velocity multiplied by a factor from the geometry shown in figure 4. In doing this, you are correcting for the fact that the velocity we measured is only the component of the rotational velocity directed along our line of sight, and that the component perpendicular to the line of sight produces no measurable Doppler shift. In the equation below, R is the radius of Mercury, and y is calculated in step 6c. ⎛R⎞
V = VO ⎜⎜ ⎟⎟ ⎝y⎠
13. For each of the delayed echoes, we can calculate the rotational period for the planet by dividing the circumference of Mercury by its rotation rate. We can convert our answer (which will be in seconds) to Earth days by dividing the result by the number of seconds in an Earth day: Prot (sec) =
2πR
V
Prot (days) =
Prot (sec)
86,400
Note that Prot should decrease as Δftotal increases. 14. Determine the average period of rotation in days for Mercury from your values of Prot (days). 15. Calculate a percent difference between your value and the accepted value of 59 days using: % difference =
your average − 59
59
× 100% 16. You can use your value of frequency shift for the echo from the sub‐radar point to calculate the orbital velocity of Mercury around the Sun. Note that the shift you get must be divided by two to account for the doubling due to an echo. If you get a negative value, it indicates Mercury is receding; positive speeds are speeds of approach. Calculate the orbital velocity as below, using Δft=0 as recorded in step 7. The transmitted frequency of the pulse is f (recorded in step 4 – use the one in units of Hz). ⎛ Δf
Vorbital = c⎜⎜ t = 0
⎝ 2f
17. Complete the remainder of questions on your data sheet. 60
⎞
⎟⎟ ⎠
Rotation Rate of Mercury UTC Astronomy 1010L Rotation Rate of Mercury Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. About how long does it take for a radar pulse to travel from Earth to Mercury and back? 2. How does the frequency of the return echo change if the pulse reflects off the edge of Mercury that is rotating away from Earth? Toward Earth? 3. What is meant by the sub‐radar point? From where on Mercury is this signal reflected? In relation to the returned echoes, when is this echo received? 4. What is the difference between VO and V? 5. How will we calculate the rotation period for Mercury? What equation will we use? What does each variable in the equation stand for? Pre‐Lab Page Rotation Rate of Mercury UTC Astronomy 1010L Rotation Rate of Mercury Lab Report Name: ________________________________ Lab Partner: _________________________ Date: ______________ 2. Distance between Earth and Mercury (AU) ______________________ Time for Pulse to Return (light‐minutes) ____________________________________ 4. Transmit frequency, f = __________________MHz = __________________ Hz 5. Sketch of the Sun, Mercury, Venus, and Earth as seen on your display screen for December 1, 1999. Label each planet and the Sun. 6, 8‐10. Data table Δt (seconds) 120×10‐6 210×10‐6 300×10‐6 390×10‐6 SAMPLE CALCULATION d (meters) 3.15×104 x (meters) 2.39×106 y (meters) 3.89×105 Δfright (Hz) 107429 Not Applicable Δfleft (Hz) 107431 Not Applicable Δftotal (Hz) ΔfC (Hz) 1.38 0.69 Report Page 1 Rotation Rate of Mercury UTC Astronomy 1010L 7. Transmit frequency, Δft=0 ____________________ 11‐13. Calculation of Mercury’s rotation velocity and rotation period. Δt (seconds) 120×10‐6 210×10‐6 300×10‐6 390×10‐6 VO (m/s) 0.48 V (m/s) 2.99 Prot(sec) 5.09×106 Prot(days) 58.9 SAMPLE CALCULATION 14. The Rotation Period of Mercury = _________________________ days 15. Percent difference (show your calculation below) = ________________% 16. Calculation of the orbital velocity of Mercury (show your work below and report your units): ______________ 17. The relative sizes of the orbits of the planets were known from Kepler’s laws long before the actual number of kilometers in an astronomical unit (AU) was measured. The delay in the return time of a radar signal provides a neat and accurate way of measuring the AU. Your ephemeris calculations, which you recorded above, gave you the distance of Mercury from Earth in AU and the time for the return pulse in light minutes. Use these values to calculate the number of kilometers in an AU. Show your work below. 3 × 10 5 km/s (time in light ‐ minutes )(60 sec/ min)
km in 1 AU =
distance between Earth and Mercury in AU
(
)
Report Page 2 Image Processing UTC Astronomy 1010L Image Processing Most modern telescopes collect astronomical images using CCD technology. Charge‐coupled devices (CCDs) are silicon plates that hold a two‐dimensional array of individual photocells sensitive to light that work by the photoelectric effect. When a photon of light strikes a photocell, electrons are freed from the material of the cell, generating an electric charge. The larger the amount of light, the larger the number of electrons ejected, and the larger the charge. The charge on each cell (or pixel) can be read out by applying a voltage, and in this way, the CCD transforms the optical image into electrical signals that are translated into digital language. CCDs are rated by the number of pixels (which relates to the size of the array) and by quantum efficiency (which relates how many photons are captured relative to the number of photons that strike a cell). For good quality astronomical images of deep‐space objects, the efficiency should be as high as possible at the desired wavelength of light. It is possible with most CCDs to combine multiple individual pixels into one “super pixel” in a process known as binning. 1x1 binning means that the individual pixel is used as is; 2x2 binning means that an area of 4 adjacent pixels are combined into one larger pixel. Binning has the effect of increasing the sensitivity to light for faint or out of focus objects. Conversely, the resolution of the image is decreased in this process. The level of binning is thus an important consideration when processing images to maximize both sensitivity and resolution. A CCD only renders an image in black and white. In order to construct a colored image, various filters have to be used. All colors can be generated by mixing red, green, and blue light in various ratios. Thus, a full color image requires that frames taken with red, green, and blue filters be combined in the image processing stage. Each filter permits certain wavelengths of light to pass through. A luminance filter allows most of the visible portion of the electromagnetic spectrum to pass (white light), and is used for black and white as well as color images. A blue filter only allows all blue light to pass, etc. For more advanced imaging, narrow‐band filters that only let through very specific wavelengths of light to the CCD chip corresponding to spectral lines of hydrogen, sulfur, and oxygen may be used. These can be especially useful for studies of nebulae. The raw data files collected are the scientific measurements with no calibration applied. One of our main goals in processing the image is to increase the signal to noise ratio of the data, because the light we are collecting is coming from very distant objects. Several processes help us achieve this goal. In “calibrating” our images, we are trying to reduce the noise. This is achieved by applying dark frames, flats, and bias frames. Dark frames are subtracted to remove electronic “noise” from thermal fluctuations on the CCD chip. Light flats remove optical imperfections in the camera lenses and the telescope mirrors such as dust. In applying the calibration frames, the same binning should be used on all exposures to be combined. Shorter duration times of individual images helps to prevent overexposure of pixels which can cause “blooming” of stars; stacking these multiple exposures together will increase signal in the data such that the merged image is equivalent in intensity to a single long exposure. Another benefit of shorter duration times is that if a satellite, cosmic ray, or anything else crosses the field of view, only a small period of imaging time is lost as opposed to an entire run. In image processing, you can remove a transient object that does not appear in multiple exposures. In order to stack images properly, they must be aligned using reference stars to account for possible rotation and translation between frames. For the most effective alignment, stars that are far apart on the frame should be chosen. In addition, it is important that the reference stars chosen are relatively small and tight, but are not saturated, necessary conditions for good alignment by the computer algorithm. Multiple frames are usually taken with each filter, be it luminance, red, green, blue, or other spectral filters. When processing images, the set of subframes corresponding to each filter is usually processed into a master frame before combining various filters into a final color image. Further post‐processing should be done on your stacked and aligned color image to enhance details, color correct or balance, or remove imperfections due to noise. Blurring is often done to remove small bits of noise from data on the RGB master. The luminance layer contains most of the detail in the image, so blurring color images does not reduce the resolution. There may be other settings to try on your image depending on the software used or your particular target. 64
Image Processing UTC Astronomy 1010L Project Outline and Pre‐Lab Work for Image Processing January February March April Assign groups, select astronomical targets for imaging, and determine appropriate coordinates. Action items: You should be using your group area at UTC Online to discuss and share ideas for objects and verify the prospects. Determine parameters for imaging runs: You need to decide on which frames (i.e., luminance, color, spectroscopic filters) to take, the telescope to use, and the number and length of exposures necessary. Based on your plans, your instructor will allot you a certain number of points to use. Action items: You should be discussing your imaging plans in your group area. When your plan is approved, the group leader will schedule the run at LightBuckets. This needs to be done as early in February as possible to allow time for re‐runs if an unforeseen problem occurs. Run imaging projects on remote telescopes and imaging systems. It will be best to let LightBuckets select the date and time for your run, so that factors such as weather and phase of the moon can be accounted for. Action items: Your group leader will receive an email when your imaging run is complete, at which time he/she can log in to the Account section on LightBuckets and retrieve your images from the completed run. Your group leader will need to download all the image data and then upload it to the group’s section at UTC Online so that all group members can access the files for processing. Process the images to the final project. Action items: Based on your previous meetings and discussions, your group has collected several different images or frames which you will now process. Although LightBuckets has automatically generated one image from your data by an automated process, we can often do better by processing the raw data files ourselves. Since the raw data files are the scientific data, we can decide not only how to calibrate the data, but which features of the object we’d like to emphasize, what level of noise is acceptable in our images, the final balance of the image, etc. In class, we will work together to process the images with software designed to work with astronomical images, that is, Nebulosity or Maxim DL (although other programs, including Adobe Photoshop, are used by astroimagers). You will complete the report page, including writing a summary of the project and a possible caption for your image. Finally, you own the copyright to your image data. It is your intellectual property, although you may choose to let the UTC Astronomy program use it for future exercises and publicity of the program. Pre‐Lab Page Image Processing UTC Astronomy 1010L Image Processing Report Name: __________________________________________________________ Date: ________________________ Describe in some detail your answers to the questions below, in your own words, using complete sentences. Use additional paper if needed. 1. Describe your astronomical target. 2. Describe the telescope you used to make this image. (What were the specs?) 3. List all the images (including luminance, RGB frames, spectroscopic filters, calibration frames, etc.) you collected and the details of each: 4. What is binning? Which bin setting did you use? 5. What are you doing when you align/stack the images? Report Page 1 Image Processing UTC Astronomy 1010L 6. What is the purpose of each calibration frame you used (e.g., the dark frame)? 7. What color settings did you use? 8. List any other digital steps you took to process your data. (Example: did you blur your image? Why/why not?) 9. What, if anything, would you do differently for this specific target/image next time? 10. Submit your final image electronically to your instructor with all group members’ names. A summary of the outcome of your image and a possible caption for it should be included for each member of the group. Write your own below. Report Page 2 M
Moons of Jupite
er UTC Astron
nomy 1010L Moon
ns of Jupiter
U
O
Objectives: Th
e goal of thiss exercise is to
o determine tthe mass of Ju
upiter using K
Kepler’s third law, applied to the orrbit of the Gaalilean moonss. Baackground: Johann
nes Kepler de
educed three empirical maathematical laaws governingg the orbit off one object aaround an
nother. Kepleer’s Third Law
w states that ffor a moon orrbiting a much
h more massiive parent bo
ody: M=
a3
p2
Here, M is the mass of the p
parent body iin solar massees, and a is th
he length of the semi‐majo
or axis in units of AU. If th
he orbit is circcular, a is equ
ual to the radius of the orb
bit, which is also called amplitude. The period of orb
bit (in Eaarth years), p
p, is the amou
unt of time required for thee moon to orrbit the paren
nt body once. In this lab, we will o
observe the fo
our moons off Jupiter that Galileo saw tthrough his teelescope, kno
own today n moons. In o
order of distance from Jupiter, they aree Io, Europa, G
Ganymede, an
nd Callisto. If you ass the Galilean
lo
ooked at Jupitter through a small telesco
ope, you migh
ht see someth
hing similar to
o figure 1. The moons app
pear to be lin
ned up becau
use we are loo
oking edge‐on
n at the orbital plane of th
he moons of Jupiter. If we watched, as Galileo diid, over a succcession of cle
ear nights, wee would see the moons shuttle back and forth, moree or less in a line. W
While the moo
ons actually m
move in rough
hly circular orrbits, you can only see the perpendiculaar distance off the m
moon to the li
ne of sight be
etween Jupiteer and Earth. If you could view Jupiter ffrom “above”” (figure 2), th
he moons w
would travel in
n apparent cirrcles. As sho
own in figure 3, the perpen
ndicular distance of the mo
oon should be a sinusoidal curve when plotted veersus time. B
By taking enou
ugh measurements of the moon’s posittion, a sine cu
urve can be fiit to the data to deetermine the radius of orb
bit (the amplittude of the curve) and thee period of orrbit (the perio
od of the curvve). Once th
he radius and period are known, the maass of Jupiter can be found
d by Kepler’s Third Law. YYou will determine Ju
upiter’s mass using measurements of each of the fou
ur moons; theere will be errors of measu
urement asso
ociated w
with each moo
on, and therefore your Jup
piter masses m
may not be exxactly the sam
me. Figure 1. Jup
piter and its moons through a sm
mall telescope.. Figure 2 (left). View from abo
ove the plane o
of orbit. Rapparent shows the ap
pparent at would be seen from Earth.. distance betweeen the moon and Jupiter tha
Figure 3 (botto
om). The apparrent position of
of a moon variees sinusoidally with the changing angle from the linee of sight, θ, ass it orbits Jupiteer. Apparent po
osition is measured in units of the radiius of the moon’s orbit, R. θ iis measured in degrees.
68
Moons of Jupiter UTC Astronomy 1010L This program simulates the operation of an automatically controlled telescope with a charge‐coupled device (CCD) camera that provides a video image to a computer screen. It also allows convenient measurements to be made at a computer console, as well as adjustment of the telescope’s magnification. The computer simulation is realistic in all important ways, and using it will give you a good understanding of how astronomers collect data and control their telescopes. Instead of using a telescope and actually observing the moons for many days, the computer simulation shows the moons to you as they would appear if you were to look through a telescope at the specified time. Procedure: 1. Start the Revolution of Jupiter Moons program from the CLEA Exercises folder. Select Log In from the File menu, and fill in the requested information. Now select File > Run; when the Start Date & Time dialog appears, click OK to accept the defaults; you will change these later. 2. You should now see the observation screen (like in figure 4), showing Jupiter much as it would appear in a telescope. The small, point‐like moons are on either side of Jupiter. A moon may be difficult to see if it is in front of Jupiter or hidden behind it. Notice that you can change the level of magnification by clicking on the 100X, 200X, 300X, and 400X buttons. The screen also displays the date, Universal Time (the time at Greenwich, England), the Julian Date (a running count of the date and time used by astronomers in decimal format that begins at Jan. 1, 4713 B.C.), and the interval between observations (or animation step interval if Animation is selected). Figure 4. Observation screen.
3. To do something you can’t do with the real sky, select File > Features > Animation > OK, then click on the Cont. (Continuous) button on the main screen. Watch the moons zip back and forth as the time and date scroll by. With this animation, it’s fairly easy to see that what the moons are really doing is circling the planet while you view their orbits edge‐on. To reinforce this, stop the motion by selecting Cont. again, select File > Features > Show Top View > OK. A new window appears. Start the motion again (Cont.) on the main observation screen. Note that under the Features menu you can also choose ID Color to avoid confusing the four moons. 4. Turn off the Animation feature before going on. Click on Cont. to stop the motions of the moons, then go to File > Features > Animation > OK. The Cont. button should not be active. 5. Select File > Observation Date > Set Date/Time. The Set Date & Time window will appear, and now you will change the defaults. Use today’s date for the Day, Month, and Year, and the current time for UTC. Click OK. To enter the Observation Interval, select File > Timing, and enter 12 for the Observation Step (Hrs). Click OK. 6. In order to measure the position of a moon, move the pointer to a moon and left‐click the mouse. The lower right‐hand corner of the screen will display the name of the moon (for example, II. Europa), the X and Y coordinates of its position in pixels on your screen, and its X coordinate expressed in diameters of Jupiter (Jup. Diam.) to the east or west of the planet’s center. This is the crucial figure for our purposes. Note that if the name of the moon does not appear, you may not have clicked exactly on the moon, so try again. To measure 69
M
Moons of Jupite
er UTC Astron
nomy 1010L the moon’s position acccurately, swittch to the higghest magnifiication that w
will keep the m
moon on the screen and then cclick on the m
moon again. Note: Sometimes, the p
program will d
display a pictu
ure of clouds on the observving screen. TThis is to simu
ulate the real‐life prroblem of a clloudy observin
ng night where it may nott be possible tto take data. If this occurs, write cloudy in yyour data tablle for the resp
pective Jupiteer Diameter m
measurementss and do not record the da
ata on your comp
puter. In anotther scenario,, a moon mayy be hidden behind Jupiter;
r; in this case, write “behind” in your data table in the space ffor that moon and do not record anyth
hing on the co
omputer for th
hat moon. 7.. After clickiing on the Mo
oon, press Re
ecord and you
ur data will au
utomatically b
be entered in
nto a dialog box for you. Verifyy that the vallue is correct and click OK.. You can go back and add
d to or edit th
his data later using File > Data > Reeview. 8.. Zoom backk out and clickk on the remaaining moons, recording th
he measurem
ments on the ccomputer as yyou go. Before cliccking OK on th
he dialog boxx for the last m
moon, make ssure you reco
ord the Jupiter Diameter (w
with E or W) for eacch of the four moons on yo
our lab reportt sheet. Day 1.0 is your firrst observing session; when you move to th
he next obserrving session 12 hours lateer, use the row
w for day 1.5,, etc. Also, record the Datee and Universal TTime (in hourrs, minutes, and seconds). Click OK on tthe dialog boxx. 9.. Click on Ne
ext on the maain observingg window to m
move to the o
observation tiime 12 hours later. Repeatt steps 6‐8 for each m
moon to comp
plete the dataa table. Data Analysis: You no
ow need to an
nalyze yo
our data. By plotting the p
position of th
he moon verssus time, you can obtain 1 Period
a graph similarr that in figure 5. We kn
now the following: (1) the
e orbits of th
he moons aree regular, thatt is, they do
o not speed u
up or slow do
own from on
ne period to tthe next and (2) the raadius of each orbit does no
ot change frrom one perio
od to the nextt. The sine cu
urve should therefore also
o be reegular. It sho
ould go througgh all of th
he points, and
d not have a vvarying m
maximum heig
ght or a varyin
ng width Figure 5
5. Sample graph for an imagin
nary moon witth p = 14 days ((0.0383 frrom peak to p
peak. years) a
and a = 3 Jupiteer Diameters (0
0.00286 AU). Takingg as an examp
ple the im
maginary moo
on in figure 5,, we can deteermine the radius and period of the orb
bit. The perio
od will be the time beetween two ssuccessive maaxima, minim
ma, or any two
o equivalent p
points on the curve. The ttime between
n crossings att 0 Jupiter diaameters (J.D.)) is half the peeriod; this is tthe time it takes the moon
n to get half w
way around Ju
upiter. For so
ome moons, yyou may not get data from
m your observvations for a ffull period, an
nd in this casee, the half‐period will bee useful. The radius of the
e moon’s orbitt is the maxim
mum position
n eastward orr westward (the amplitudee of the cu
urve, which iss half the totaal height). 10
0. To create tthe appropriaate sine curvee: Select File > Data > Anaalyze. On the window thatt appears, go to the Data menu
u, then Selectt moon. Choo
ose Ganymed
de III. 11
1. When the graph of dataa points appeears, try to fin
nd a pattern b
by eye, and taake note of an
ny data pointts that seem to bee out of place
e. To better ssee the patterrn, select Datta > Plot > Plo
ot Type > Con
nnect Points. This will display a “connect‐the‐dots” version
n of your grap
ph. Click on aa point at which the line co
onnecting thee points 70
Moons of Jupiter UTC Astronomy 1010L crosses the x‐axis and goes from negative to positive. If this happens more than once, choose the left‐most point on the screen. A Julian date (Mod. JD) and a value close to zero for Jp. Diam. should appear in the box marked Cursor Position for X and Y, respectively. Record the X cursor position as T‐zero on your data sheet. 12. Estimate the period of orbit by clicking on a point on the curve, recording its x‐position, and clicking on a subsequent equivalent point on the curve if a full period is displayed (see figure 5). Record these two dates as 1st and 2nd dates on your data sheet. Subtract the smaller date from the larger date to estimate the period, recording it on your data sheet. If the curve does not include a full period (e.g., for Callisto), you can use a half‐period estimation and then double it. In the case of a fast moving moon (e.g., Io), you can measure the time to complete several periods and divide it by the number of periods for a more accurate estimate. 13. Finally, estimate the amplitude of the sine curve by clicking on the maximum or minimum peak or valley of your graph and reading the y‐value in the cursor position box. The amplitude is equal to the absolute value (i.e., ignore a – sign) of the y‐position. Record this as the estimated amplitude on your data sheet. 14. Now select Data > Plot > Fit Sine Curve > Set Initial Parameters and enter the data you just estimated for T‐Zero, Period, and Amplitude. Click OK and a blue sine curve will appear. It should be a rough fit for all your data points. If it does not look near to a good fit, you may need to repeat the Set Initial Parameters step with new estimates. If only one or two points are significantly off the blue line, they may represent inaccurate measurements. Figure 6. Jupiter satellite orbit analysis. 15. Now you will begin adjusting the fit with the three scroll bars, as shown in figure 6. First adjust the T‐Zero point. As you adjust this value, the entire curve will slide to the left or right. Try to achieve the best fit for the data points closest to the T‐zero value you selected above. After you’ve adjusted the T‐zero value, you may notice that the points farther away from your T‐zero point no longer fit as well. Now use the Period scroll bar to stretch or shrink the curve and achieve a better fit. If at any time you cannot scroll far enough to get a good fit, click on the green circular arrow button to reset the scroll bar to the center. Next adjust the Amplitude bar to better fit the points near the peaks and valleys. You may wish to return to the other scroll bars for further adjustment, but the goal at this point is a good, though not necessarily perfect, fit. 16. In order to fine tune the curve, adjust the Slider Sensitivity Bar from Course to Fine. Now adjust the T‐zero scroll bar, this time observing the RMS Residual. The smaller the value of this number is (note that it is expressed in scientific notation), the better the fit. The RMS Residual value will turn from red to green when it is more accurate. Move the scroll bar with the arrows until this number is smallest, that is, when one click in either direction would make it larger. Continue this process with the Period and Amplitude bars. Because the three adjustments affect each other, you should then return to the T‐zero bar and repeat the scroll bar adjustments until all three yield the lowest RMS Residual. 17. Write down the RMS residual, period (in days), and the amplitude (the semi‐major axis in Jupiter diameters) for your final fit in the table on your data sheet. Then select Data > Print > Current Display to print the graph. 18. Choose a different moon from the Select menu and repeat steps 10‐17 for the other three moons. 71
Moons of Jupiter UTC Astronomy 1010L 19. You now have all the information you need to use Kepler’s Third Law to find the mass of Jupiter. First, we need to convert the values we obtained for the period (in Earth days) to Earth years. To do this conversion, divide the period (in days) by the number of days in a year (365). Show a sample calculation on your data sheet and fill in the values to the appropriate spaces in your data table. 20. We will also need to convert the length of the semi‐major axis we found (in Jupiter Diameters) into AU. To do this, divide your value by the number of Jupiter Diameters in an AU (1050). Show a sample calculation on your data sheet and fill in the values to the appropriate spaces in your data table. 21. With the period, p, and the radius of orbit, a, in the correct units, we can calculate the mass of Jupiter using data from each of the four moons and Kepler’s Third Law, M = a3/p2. Show a sample calculation and record these on your data sheet. If one value differs significantly from the other three, look for a source of error. If no error is found, the data may not be adequate for a better result, in which case you should leave the data as you found it. 22. Find the average value of your masses and record it. NOTE: All values for the mass of Jupiter, MJ, should be approximately 0.0009 solar masses. This is an approximate value; you should record your own calculated value. If your calculated value seems far from this approximate value, try to give some reason for the discrepancy. 23. Answer the remaining questions on your lab report sheet. 72
Moons of Jupiter UTC Astronomy 1010L Moons of Jupiter Pre‐Lab Exercise Name: __________________________________________________________ Date: ________________________ 1. At the same time you record your data by hand for each moon’s position in the data table, where else are you also recording it? 2. What is the difference between Universal Time and the Julian Date? 3. What two things can prevent data‐taking for a particular moon on a particular day? 4. Label the amplitude and period on the sine curve shown below. Denote the span of each. 5. In order to find the mass of Jupiter in solar masses using the equation M = a3/p2, what must be the units of a and p? Pre‐Lab Page Moons of Jupiter UTC Astronomy 1010L Moons of Jupiter Lab Report Name: ________________________________ Lab Partner: _________________________ Date: ______________ 6‐9. Data Table. Example data is shaded in gray. Position in Jupiter Diameters Date 7/24 7/24 UT (hr:min) 0:00 12:00 Observing Day 1.0 1.5 Io 2.95 W Europa Ganymede 2.75 W 7.43 E Cloudy – cannot observe Callisto 13.15 W Position in Jupiter Diameters Date UT (hr:min) Observing Day Io Europa Ganymede Callisto 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 Report Page 1 Moons of Jupiter UTC Astronomy 1010L 11‐13. Estimated fitting parameters. T‐Zero 1st date 2nd date Estimated Period Estimated Amplitude Moon (Julian Date) (Julian Date) (Julian Date) (days) (Jup. Diam.) Io I Europa II Ganymede III Callisto IV 17, 19‐20. Determined fitting parameters. Show your sample calculations below. Period Period Semi‐major axis Moon RMS Residual (days) (years) (Jup. Diam.) Io I Semi‐major axis (AU) Europa II Ganymede III Callisto IV Sample calculation for period in years: Sample calculation for length of the semi‐major axis in AU: 21. Mass of Jupiter in solar masses calculated from Kepler’s third law. Using Io, M = ____________ (show sample calculation to right): Using Europa, M = ____________ Using Ganymede, M =____________ Using Callisto, M = ____________ 22. Average determined mass of Jupiter in solar masses = _____________ Possible reasons for discrepancies (if any): Report Page 2 Moons of Jupiter UTC Astronomy 1010L 23. Additional questions: a. Express the mass of Jupiter in earth units by dividing it by 3.00 x 10‐6, which is the mass of Earth in solar mass units, showing your work below. b. There are moons beyond the orbit of Callisto. Will they have larger or smaller periods than Callisto? Why? c. Which do you think would cause the larger error in M: a 10% error in p or a 10% error in a? Why? **Attach all 4 graphs generated to this report. Report Page 3