Craddock_Lab6

Rowan Introduction to Astronomy Lab 6 / Plotting the Orbit of the Moon Name: ________________________________________________ Score: __________________________  Objectives After completing this lab, the student will be able to  use a series of lunar photographs to make a scale drawing of the Moon’s orbit.  be able to confirm the elliptical nature of the Moon’s orbit by determining the major axis, minor axis, and eccentricity of t he orbit.  Materials Needed  pencil  mm ruler  compass (get cheap ones in dollar stores, Wal-Mart)  calculator  Procedure The Moon’s Orbit As applied to the Moon’s orbit, Kepler’s first law would state that the Moon’s orbit is an ellipse with the Earth at one focu s. If the Moon’s orbit is an ellipse, its distance from the Earth should change during one complete orbit. This means the Moon should appear larger at perigee (closest approach to Earth) than it does at apogee (farthest from Earth). Figure 1 shows a series of lunar photographs. These were obtained as the Moon passed several different positions in its orbit. It is possible to use these photographs to determine a few orbital properties of the Moon. Fill in Table columns in this order: 6, 7, 3, 4, 5 and before plotting your data points. 1. Measure the diameter, d , of each lunar image shown in millimeters. Be careful not to simply measure the illuminated portion. Make all measurements vertically through the center of each image. Estimate each value to ½ millimeter (0.5 mm). Record your measurements in column 6 in Table 1. (Note: These values should range between 40 mm and 60 mm.) 2. From each diameter measurement, the relative distance to the Moon can be determined. This distance will be scaled down to fit on a piece of graph paper. Calculate the scale distance, D, (in mm) to the Moon using the relation D = 4000/ d, where d is the diameter recorded in column 6 of Table 1, and 4000 is a scale factor. Record your values of D, to its nearest tenth of a millimeter, in the last column of Table 1 (column 7). Round to 1 decimal place. 3. Calculate the average scale distance, D ave , to the Moon and record your answer in the space provided at the bottom of column 7. 4. Calculate the difference in days from one image to the next by subtracting the prior day count from the next day count (column 3). 5. Calculate the number of degrees moved from image to image (column 4). Note: The Moon moves in a roughly circular orbit around the Earth. A circle contains 360°. The Moon completes its orbit around the Earth in about 27.3 days. Thus, each day, the Moon moves about 13.2° around the 360° orbit. Multiply (13.2)  (number of days from col. 3) to get degrees moved from prior image. 6. Calculate the number of degrees of longitude for each image (column 5) starting at 270º. Add the degrees moved for the next image (from column 4) to the longitude determined for the prior image. When you reach 360º, start over at 0º. Round to 1 decimal place. 7. Plot the Moon’s orbital position on the polar coordinate graph paper given as Figure 2 . You will have to plot longitude vs. scale distance, D. On the graph paper, longitude starts at the bottom, labeled 0°, and increases counterclockwise. This will correspond to the true direction of the Moon’s orbital motion as viewed from above the Earth’s north pole. To plot the first data point, place the edge of your millimeter ruler along the line labeled 270° (longitude). Be sure that 0 mm is at the origin of the graph paper, which corresponds to the Earth’s position on this diagram. Now measure o ut along the 270° line and place a pencil dot at your value of D for a longitude of 270°. Continue in this manner for the remaining data in Table 1 (columns 5 and 7). You may have to plot some points in the margin outside of the grid lines. This is OK. When all the data points are plotted, connect the points by drawing a smooth line through them (this is your orbital path). Draw freehand (do not use the compass). 8. If you examine the plotted data on Figure 2, you can see that they are NOT centered on t he Earth. In order to draw the Moon’s orbit, you must first establish the orbit’s center. Open your compass equal to the D ave calculated in step 3. Remember to use the edge of your millimeter ruler to properly set the compass. Locate the center by placing the compass point on four or five (or more) of the plotted data points and draw a small arc near the center of the graph. Ideally, these should all cross at one place, the center. In reality, they will show you only approximately where the center should be. Put a dot where you estimate the orbit’s center is. Label the center of the Moon’s orbit “CMO” where you placed your dot. (Note that these steps count for points in the grading.)

Lab 6 / Plotting the Orbit of the Moon 2 Table 1. Moon’s Orbit Data Image Age (days) Difference in Days from Previous Image Degrees Moved from Previous Image Position Plot at Longitude (°) Moon Diameter d (mm) Scale Distance D (mm) 1 4.0 ---------- ---------- 270 2 5.0 3 7.1 4 9.1 5 11.1 6 15.0 7 17.0 8 19.0 9 19.5 10 21.4 11 23.5 12 26.6 D ave = 9. Though the Moon’s orbit looks nearly circular, it is an ellipse with the Earth at one of the foci. Draw and label the major axis of the orbit. The line for the major axis is determined by two points: the Earth (a focus) and the center of the Moon’s orbit (determined through the compass arcs). Label the perigee (closest point of approach) and apogee (farthest point of approach) through visual inspection. Measure the length of the semi-major axis, a (to an accuracy of 0.5 mm). [ Hint: Draw the major axis, measure it in mm, and divide that measurement by 2 to get the length of a or the semi-major axis. Remember that the major axis must go through two points: the center of the Moon’s orbit and the Earth, which is one of the two foci of the ellipse.] a = _______________ mm [ Round to 1 decimal place.] 10. Remember, from Lab 4 you learned that c is the distance between the center of the ellipse (i.e., the Moon’s orbit) and one of the foci (here, the Earth). Measure c to an accuracy of 0.5 mm. (See the attached sheet on the ellipse if you need a refresher.) c = _______________ mm [ Round to 1 decimal place.] 11. The eccentricity of an ellipse is given by the equation e = c/a . Calculate the e ccentricity of the Moon’s orbit from your measurements of a and c . Express your answer using 4 decimal places. e = c/a = _______________ [ Round to 4 decimal places.] 12. The published eccentricity of the Moon’s orbit is 0.0549 . What percentage of the published eccentricity is your computed value? _________________% [ Round to 1 decimal place. ] If your value of e is less than the published value (0.0549), then the percentage must be less than 100%. If your value of e is more than the published value (0.0549), then the percentage must be more than 100%. To determine the correct percentage, use the following algorithm: e your / 0.0549 = x / 100 Scoring for Lab 6 Completing Table 1: 30 pts Plotting orbital points: 20 pts Drawing smooth orbital curve thru plotted points: 0 5 pts Finding (drawing compass arcs) & labeling center of Moon’s orbit: 17 pts Drawing major axis through Earth and center o f Moon’s orbit: 0 5 pts Labeling Major Axis , Perigee , Apogee : 0 3 pts Measuring a : 0 5 pts Measuring c : 0 5 pts Calculating e : 0 5 pts Calculating % of actual e your value of e is: 0 5 pts TOTAL 100 pts  Hint: Use this as a checklist to make sure you have completed all of the tasks required. Round all values in this table to 1 decimal place. Note : Answers for questions 9 – 12 MUST be derived from measurements made from the graph. Answers that are guesses or copied will NOT be accepted.

Lab 6 / Plotting the Orbit of the Moon 4 Figure 2. Polar coordinate graph paper. Note : It is permissible to plot points “off” of the grid if necessary. [ Hint: Your first plotted point along the 270° line will be in the margin.] Be sure to check the sketch on page 5. If your graph does not look like the sketch, then you have not correctly prepared your graph.  Ignore these answer blanks to the left. NOTE: Read the important information at the bottom of this page and in the box at the top of page 5.

Lab 6 / Plotting the Orbit of the Moon 5 Rough sketch of how your graph should appear: c is the distance between the center of the ellipse (orbit) and one of the foci, F, which typically is the Sun or a planet (like Earth) that is being orbited. c = e  a a = semi-major axis AB = major axis The sketch shows the following elements that must appear on your completed graph: ▪ Correctly plotted points. ▪ Points are connected with as smooth a curve as possible. ▪ The “+” represents the Earth, which is at a focus (already provided). ▪ The CMO or Center of Moon’s Orbit. Must show with a dot at intersection of compass arcs. ▪ The compass arcs, which are necessary to determine the CMO. ▪ The distance “c” between the Earth and the CMO must come from your graph. ▪ The major axis, 2a. You can only draw the major axis AFTER the CMO is determined. It takes two points — CMO and Earth — to determine a line. ▪ Note that the distance “a” is the length of the major axis (2a) divided by 2. You need to do ALL steps described in the box below. Ignoring these steps will render your answers to questions 9, 10, 11, and 12 as UNACCEPTABLE . Your answers to questions 9 – 12 can ONLY come from a successfully completed graph. Compass arcs