# Astronomy calculations

Name: Astronomy 1301 Fall 2018

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PROJECT 1: ORBIT OF MERCURY • You may collaborate with your classmates but your submission should be your own work. • Please answer questions in complete sentences and/or equations. Type or write legibly. • The project is due in hard copy by 6 p.m. on Tuesday, September 25. You may hand it in in

class or in my drop box outside B-233. Late projects will not be accepted.

How can we tell, from Earth, what the orbit of a planet is like? It seems like a difficult question but it can actually be surprisingly easy to find the orbit from basic observation. In this project you will use a simple set of observation – given to you for the sake of time but that you could have made yourself – to find the size, shape and period of the orbit of Mercury. You will be reproducing the work that Johannes Kepler did in the 17th century to formulate his laws of planetary motion.

“Making” the observations Ancient Greeks were familiar with Mercury but did not recognize it as the same object when it appeared in the morning sky (Apollo) and evening sky (Mercury). Almost two thousand years later the Danish astronomer Tycho Brahe made detailed observations of the motions of the planets. On his death in 1601, these record were taken over by one of his assistants, Johannes Kepler. From Brahe’s records, Kepler was able to find the maximum angular distance from the Sun that Mercury reached during each one of its cycles. You could make the same observations yourself, but to save time here is a website where you can find these distances, called maximum elongations: http://www.fourmilab.ch/images/3planets/elongation.html.

Name: Astronomy 1301 Fall 2018

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• Go to the above website and find Mercury’s maximum elongations for 2016, 2017 and 2018. Enter the data in the table below.

# date elongation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Name: Astronomy 1301 Fall 2018

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Drawing the orbit Figure 21-1 shows Earth’s orbit, with the dates indicating Earth’s position throughout the year. The scale is marked in astronomical units to allow you to convert measurements made on the diagram into those units. The position of the Sun is indicated near the centre of Earth’s orbit. To draw the orbit you will need a sharp pencil (the width of even a sharp pencil line on the diagram is about 100,000 km), a millimetre ruler and a protractor. Now plot each elongation of Mercury on the diagram as follows: • Locate the date of the maximum elongation on Earth’s orbit. Draw a light pencil line from

Earth’s position to the Sun. • Centre the protractor on Earth’s position and construct a line, so that the angle from this

constructed line to the Earth-Sun line is equal to the maximum elongation. Extend this line well past the Sun. It represents the direction toward Mercury. Remember that, facing the Sun from Earth, eastern elongations are to the left and western ones to the right, as shown in Figure 21-2. On the date in question Mercury was somewhere along the sight line you have drawn.

• Number the lines in chronological order. • After you have plotted all the data, sketch in the orbit of Mercury. The orbit must be a

smooth, closed curve that just touches each of the sight lines you have drawn. It may not cross any of those lines.

Name: Astronomy 1301 Fall 2018

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Name: Astronomy 1301 Fall 2018

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Questions

• Does it appear that the curve you have drawn is a circle? How can you tell? Be specific.

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Kepler had the same problem: his orbit was not a circle either. Eventually he hit on the idea of using ellipses for orbits.

• On Figure 21-1 draw the longest diameter possible of the orbit of Mercury. Note that this diameter must pass through the Sun. This is the major axis of the ellipse.

• Measure the major axis and bisect it to find the centre of the ellipse. Draw the minor axis through the centre of the ellipse and perpendicular to the major axis.

• What is your “observed” value of the semimajor axis of Mercury’s orbit in cm as measured on Figure 21-1?

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• What is your “observed” value of the semimajor axis of Mercury’s orbit in AU? Show the conversion from cm to AU obtained using the scale at the bottom of Figure 21-1.

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• What is the accepted value of the semimajor axis of Mercury’s orbit in AU?

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Combining the information from the orbit you drew with the chronologically numbered sight lines, you now know where Mercury was in its orbit on those dates.

• Find two orbital positions on different dates that are as close to one another along Mercury’s orbit as possible. Mark these two positions on Figure 21-1 and write the corresponding dates on the line below.

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Name: Astronomy 1301 Fall 2018

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• How many days elapsed between those two dates?

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• How many orbits did Mercury complete between those two dates? (Hint: To determine the number of orbits completed, you may find it helpful to number the positions where the sight lines are tangent to Mercury’s orbit in chronological order.)

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• From the time elapsed and the number of orbits completed between the two dates, calculate the orbital (i.e. sidereal) period of Mercury in days. This is your “observed” value of the period. Show your calculation.

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• What is the accepted value of the orbital period of Mercury in days?

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• Kepler’s Third Law states that the orbital period 𝑃𝑃 squared equals the semimajor axis 𝑎𝑎 cubed, 𝑃𝑃2 = 𝑎𝑎3. Use your “observed” value of 𝑎𝑎 and Kepler’s Third Law to calculate 𝑃𝑃 in days. Show your calculation. (Hint: In the above formula 𝑃𝑃 is in years.)

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• Compare the calculated and “observed” values of 𝑃𝑃. Are the orbital parameters you found for Mercury consistent with Kepler’s Third Law? Explain.

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Name: Astronomy 1301 Fall 2018

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The Sun lies at one of the foci of the elliptical orbit. Measure the distance from the Sun to the centre of the ellipse and call it 𝑐𝑐. The ratio 𝑒𝑒 = 𝑐𝑐

𝑎𝑎 is the eccentricity of the ellipse.

• What is your “observed” value of the eccentricity of Mercury’s orbit? Show your calculation.

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• What is the accepted value?

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• Use http://www.fourmilab.ch/images/3planets/elongation.html to find the first maximum elongation for Mercury in 2020. Write down the date and the maximum elongation below and plot it on Figure 21-1. Mark the line from Earth to Mercury in red. Is it tangent to Mercury’s orbit?

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