A.5 The Shape of the Earth s Orbit

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1 CHAPTER A. LABORATORY EXPERIMENTS 31 Name: Section: Date: A.5 The Shape of the Earth s Orbit I. Introduction Does the distance between the Sun and Earth stay constant throughout the year? If so, the Earth s orbit is a perfect circle centered at the Sun. If not, when is the Earth closest to and farthest from the Sun? In this activity, you are going to determine the shape of the Earth s orbit around the Sun from the fact that an object appears larger when closer than the identical object much further away. II. Reference 21 st Century Astronomy, Chapter 3, pp (Kepler s 1 st law). III. Materials Used ruler protractor calculator graph paper compass IV. Activities On each day that you observe the Sun, you can determine its direction and its angular diameter. From the observed angular diameter you can find its relative distance from the Earth. Therefore, each date yields one point on the Earth s orbit. Connecting your data points with a smooth curve gives the Earth s orbit around the Sun. The closer an object is to you, the larger it appears. This is because it fills a larger angle as seen by your eye. For small angles, we can obtain a relationship between the size d of an object, its angular size θ, and the distance r to the object by using an approximation that simplifies the calculation. θ r r d a Figure A.13: Small angle approximation. For a very small angle θ the length of the short side of the triangle d is almost equal to the length of the arc a of a circle of radius r subtended by the angle θ. Then, d is approximately the same fraction of the circumference of the circle as θ is of 360. d 2πr a 2πr = θ 360. (A.9)

2 32 CHAPTER A. LABORATORY EXPERIMENTS Therefore, the distance r to the object is inversely proportional to its apparent angular size θ. ( ) d 360 r = 2π θ. (A.10) Table A.14 shows the direction and apparent angular size of the Sun as seen from the Earth on various dates throughout the year. Table A.14: Apparent diameter of the Sun. Date Direction of the Sun Direction of the Earth Apparent angular size (θ) Relative distance (cm) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec According to Table A.14, on January 1, the Sun is in the direction 282 east (counterclockwise) of the vernal equinox as see from the Earth. Then on the same date, the Earth must be at = 102 east of the vernal equinox as seen from the Sun. Find the direction of the Earth for each date in Table A Place a dot at the center of the graph paper to represent the Sun. Let 0 point to the right along one of the grid lines. You will use this as the direction of the Sun as seen from the Earth with respect to the background stars on the vernal equinox, assumed to be March 21 (See Fig. A.14). Notice that the Earth is 180 from the vernal equinox with respect to the Sun. 3 Draw radial lines from the Sun in each of the directions for the Earth in Table A.14 in the counterclockwise direction from 0 and label their dates. 4 The average value of the apparent diameters in Table A.14 is about Since an orbit with a radius of roughly 10 cm will nicely fit on a sheet of paper, let us choose the constant in the parentheses

3 CHAPTER A. LABORATORY EXPERIMENTS 33 Mar Earth 180 Dir. of Sun Vernal Equinox Figure A.14: The Earth s orbit around the Sun. in Eq. (A.9) to be cm. Then, the relative distance r between the Sun and Earth can be found by the following equation: r = ( cm) 360 θ. (A.11) Calculate the relative distance for each date in Table A.14. Then plot the location of the Earth on each of the radial lines. 5 Draw a best circle through your data points with a compass. The circle should pass close to each of the data points. In order to do this, you will need to move the center of the circle away from the Sun (the central dot). This circle gives you the shape of the Earth s orbit around the Sun. Be sure to label where the pointed end of the compass was as that marks the center of the orbit. V. Questions 1. Using your completed diagram, on what date is the Earth furthest from the Sun? The point where the Earth is furthest from the Sun is called the aphelion. Also, mark the aphelion on your diagram of the orbit. 2. Using your completed diagram, on what date is the Earth closest to the Sun? The point where the Earth is closest to the Sun is called the perihelion. Also mark the perihelion in your drawing. 3. Compare the answers to the previous two questions with the answers you would get using the calculated values from Table A.14. If they do not match, give an explanation as to why.

4 34 CHAPTER A. LABORATORY EXPERIMENTS 4. You have been told many times that the orbits of the planets are ellipses instead of circles. How can you explain the fact that we could draw a circle to represent the Earth s orbit? 5. During what season is the Earth the closest to the Sun for observers in the northern hemisphere? 6. The eccentricity e of an orbit is defined as e = c/a where c is the distance between the Sun and the center of the ellipse and a is the semi-major axis which is 10 cm in your case. Find the eccentricity of the Earth s orbit from your drawing. How does your value compare to the actual value of 0.017? Calculate the percent error in your measured value. % error = (measured value) (actual value) (actual value) 100% (A.12) VI. Credit To obtain credit for this lab, you need to turn in appropriate tables of data, observations, calculations, graphs, and a conclusion as well as the answers to the above questions. Do not forget to label axes and give a title to each graph. Show your work in calculations. A final answer in itself is not sufficient. Don t leave out units. In the conclusion part, briefly summarize what you have learned in the lab and possible sources of error in your measurements and how they could have affected the final result. (No, you cannot just say human errors explain what errors you might have made specifically.) You may discuss this with your lab partners, but your conclusion must be in your own words.

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