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name: NI WNQNY 2 Ecipses & Transts small and distant it is from us, the transit of Mercury is difficult to observe. Venus transits only 4 times every 243 years but is easier to observe with a telescope. The last Venus transit occurred July 5-6, 2012 and won't happen again until the year 2117. In this lab you are going to investigate circumstances when the positioning of celestial bodies aligns and what we observe when that happens. You will also model (on a simplified level) observations made by astronomers over 100 years ago to calculate the distance of Earth from the Sun. Finally, you will begin to consider how astronomers might be able to use transit events in other solar system to determine information about the planets found there. LAB EXERCISE PART I: TOTAL ECLIPSES In the Retrograde Motion and Phases of Venus labs, we looked at the apparent diameters of Mars and Venus, and how they change as the planets move around the Sun. Let’s investigate this further. Your lab instructor has a number of Styrofoam spheres for you to investigate. Pick two that are noticeably different in size. Measure the diameter of each sphere and record it in the table below. Hold the larger sphere at arm’s length and measure the distance from the center of the sphere to your eye. Record this below. Now, while holding the larger sphere at arm’s length, position the smaller sphere such that it “appears” the same size to your eye. 1. Will the smaller sphere be Iocated r farther than the larger sphere? URS AW smoVlesh §pnive 1S Closer Measure the distance from the center of the smaller sphere to your eye and record it below. Diameter (d) Distance (D) Angular Diameter (3) Larger Sphere 1[.5W 2\ a2y, b Smaller Sphere qQ.6m W\ 7% ¥ \37,4 © 53 Scanned with CamScanner

NameN\C/\(/\ N\\)YYV\\\ ‘Lb Eclipses & Transits 2. How many times bigger is the larger sphere to the smaller one? L s e - 3. How many times farther is the larger sphere than the smaller one? 150 Hwes Bowxrer Use the following formula to calculate the angular diameter of each sphere: 8 = 206,265 Ly = : 2655 This yields an angular diameter in units of arcseconds (the angular diameter measurements you've observed in previous labs have used this angular unit). MAKE SURE THAT THE UNITS FOR DIAMETER AND DISTANCE ARE THE SAME! Angular diameter of larger sphere: 1.6 é\: ZO(Q,’L(OS "Z'z J=91,2%2.590L Angular diameter of smaller sphere: S’.Z()(alwg ?—6/ W\ $-\1$ \31.964 4. How do the two angular diameters compare? e diamerey 6 YR Symoney Spreve iS OWBSY Lunice Mg Yhon Ang | Y Yty SM\ It turns out that the same effect occurs with the Sun and Moon. When viewed in the sky, the Moon appears to be as big in the sky as the Sun. 54 Scanned with CamScanner

name: INTCAA Mg Iy Uer PART I1: PLANETARY TRANSITS As Mercury and Venus orbit within Earth’s orbit, they periodically pass directly between Earth and the Sun, at a point called inferior conjunction (refer to the Planetary Configurations section in the Appendix for more details.) We can use observations of these transits to determine the distance from Earth to the Sun. Eclipses & Transits With the Sun as a backdrop, observers at different locations on Earth will see a transit occur across different portions of the Sun. By measuring the angular shift of the transiting planet when viewed from one location versus another, astronomers can use basic trigonometry to calculate the distance from Earth to the transiting planet. We will investigate the distance to Venus in the following example. ’ ’ Earth Not to Scale Figure 1: The transit of Venus is o'bserved to cross different parts of the Sun depending from where on Earth you observe. Looking at Figure 1, you can see that the angular separation (indicated by the Greek letter 6) of the two transiting paths L 4 depends on the distance between the two different i observation points on Earth (known as the baseline). If we H can measure this value, along with the baseline distance (B), "' then we can measure the distance (D) between Earth and fo] & =FA Venus. The trigonometric equation that applies to this situation is the tangent: i I (9) .82 AN3L5°D i Earth Figure 2: The distance to Venus from Earth can be calculated using basic trigonometry. 56 Scanned with CamScanner

Name: N\(’}Z{ V\AW\\)j ’L@ Eclipses & Transité We can solve the previous equation for D, the distance to Venus: 2tan(§ In general, if you can measure the angle shift, 8, and the baseline, B, then you can determine the distance using the equation above. Luckily, the angles we are dealing with are small. This allows us to get rid of the trigonometric functions all together. To do this we use something called the small angle approximation. For small angles: tanf = 6. However, if we want to use the equation, we need to make sure our angle is in units of radians. We can take care of this by making a small adjustment to our equation: p- 87138 0 Where D is the distance to the object, B is the size of the baseline, and 6 is the angular separation measured in degrees. This will be the equation you will use to calculate distances in this lab. PART III: MEASURING THE DISTANCE TO VENUS To experience how this measurement works, we're going to measure the distance to a much closer object, and to take those measurements, we’re going to use our cellphones. First, you need to estimate the Field of View of the mobile camera. This is the angle between the left-hand side of the photograph and the right-hand side of the photograph. You can work this out easily by placing the camera lens just above the center point of a protractor and seeing what the angle is at which a pencil placed on the angle part of the protractor just falls out of the field of view. This is shown in Figure 3, and you may use the printed protractor. 57 Scanned with CamScanner

Name: N \w W\\N? 1o Eclipses & Transits Venus seems to have moved! We can use properties of triangles to determine the distance to Venus. A precise expression would involve trigonometry, but because the distance is much larger than the baseline, we can get a really good estimate of the distance is from: p. 5738 0 where B is the width of the baseline (in this case 0.2 m) and 0 is the angle in degrees measured using the camera. Now we need some way to measure the angle through which Venus has moved. Earlier, we calculated the Field of View of the camera using the protractor. We can now measure how much Venus has moved in the two images taken by the camera at the two different positions. First, determine the width of the camera screen in centimeters: Width of camera screen = Y'l M Now you need to measure the distance in cm that Venus has appeared to move in the two images. Measure the distance from the left edge to Venus in each image. Distance from left edge to star in image 1= ?) o Distance from left edge to starinimage2=_ 7 | Sora Now simply subtract to find the distance Venus appeared to shift between the two images. Distance Venus “moved” =, o 6 v Finally, you can estimate the angle by determining the fraction of the Field of View the star shifted through, and then hence the angle Venus moved and multiplying this by the Field of View. Distance Venus “moved” = Angle Venus “moved” Width of camera screen total field of view (degrees) Solve the ratio to determine the angle that Venus “moved” -3 Angle Venus “moved” = "‘ .2 Cl +0) @ Scanned with CamScanner

Name:N&QX'\' ‘(\[\\N\{)W}\ /L((? Eclipses & Transits This is the angle through which Venus has appeared to shift relative to the background as measured from two observation points on Earth, or 0. Recall that our baseline is 20 cm (which equals 0.2 m) across, or B = 0.2 m. Use the equation D = (57.3 * B)/0 in the space below to solve for the DISTANCE (D) to your paper Venus. Distance to paper Venus = O f \ Cl \W\ Measure the actual distance to the paper Venus from Earth using a ruler: ‘L\ ' q) .d-) '\‘5 G g“’\ It’s highly unlikely that your measured distance and your computed distance to Venus are the same, SO NOW you can compute your error: % error = [(Your value — actual value) + actual value] x 100. (5.1 - *36%) 1. 5L ¢ 100 % error = = TUR\w, 8. What do you think you could do to improve your results? ale Qv ko howe ol e Uik Fhe Shve R o< Cagu\e-W\Pr This method allows us to measure the distance to either Mercury or Venus at inferior conjunction, but not the distance to the Sun. BUT, as we demonstrated in last week’s lab (Mercury’s Orbit), we can determine how big the orbit of Mercury (or Venus) is relative to Earth’s orbit. Combined with our ability to measure the distance to either planet during a transit, we can now determine the distance from Earth to the Sun. Scanned with CamScanner

Name:N\rQ« \ \N\\j\l\l\\,\\ Q(/ Eclipses & Transits -9 S 3 U0 ywa- 361 Let’s observe our solar system from the nearest star syste outside of our, Proxima Centauri. PART 1V: EXOPLANET TRANSITS Proxima Centauri is 4.25 light-years from our solar system. 11. How far is this in meters? SO A 12, What is the angular size of the Sun at this disti'm(_fln}_m/wuhflwa\from part |) \O J= 100105 WvioT, 4000M8335%2 b0y \0 13. What is the angular size of Earth at this distance? 600 266 208 ol LM Y.0Lno 14. What percentage of the Sun's light would Earth block from our view? (To determine this, take the square of the angular size of the Earth and divide it by the square of the angular T 00 13221 A 1T L WO \(,}L\o\pdfié If we were to witness Earth transiting the Sun, this would occur when Earth passes between us and the Sun. 15. Given that Earth would be “1 AU closer” to us during a transit, what would its angular size be? \ b 1 0oyl (e - LHAMo = .3 %10 200105 (g ) - 16. Does the distance of a'p anet from its parent star make any difference in its angular size? NG, Yhe aNgic size &WSXWS(AM 62 Scanned with CamScanner

Name: N\\Q)G,\\ \N\\}WM /LU Eclipses & Transits PART V: FURTHER INVESTIGATIONS In today’s lab you observed how precise alignments between celestial objects can be used to determine information, such as the scale distance of our solar system (the Astronomical Unit) and basic information about the presence of extrasolar planets in other solar systems. Following procedures similar to the ones you conducted in the lab you performed today, what other related research questions could you pursue? Come up with at least TWO questions related to today’s lab that could be answered by gathering and analyzing data similar to what you did in lab. Kow wond 4oV £NA e angu\ay §ZL 0t O Meewp\b Vidwer 0“ How da m’d\V\%\f\@w-' INY 4N CN\NPG\V‘-“'\. Scanned with CamScanner