The Orbit of Mars Materials: protractor, compass, centimeter ruler, engineering paper Object: To reconstruct the approximate orbit of Mars by using some of Tycho Brahe's data. During the latter part of the 16 th century, Tycho Brahe collected a very large number of precise observations of the motion of Mars through the sky. He recorded the position of Mars against the background stars to within an arcminute, and the time of the observation to the minute. Below is a list of some of Tycho's data Date Heliocentric Longitude of Earth Geocentric Longitude of Mars 1585 Feb 17 159 135 1587 Jan 05 115 182 1591 Sep 19 6 284 1593 Aug 06 323 347 1593 Dec 07 86 3 1595 Oct. 25 42 50 1587 Mar 28 197 168 1589 Feb 12 154 219 1585 Mar 10 180 132 1587 Jan 26 136 185 Once every martian year (687 days) Mars returns to the same point in its orbit around the Sun. If we plot the lines of sight from the earth to Mars at this interval we can triangulate a point on the orbit of Mars. 1. Turn the graph paper so that the side without the lines is facing up and the long edge is horizontal. Place a small dot near the center of the page. Label the dot "Sun". 2. Using the ruler, draw a straight line to the right, starting at the Sun and ending roughly two centimeters from the right side of the page. Label this line the vernal equinox. This is the direction an observer on the earth would look to see the Sun on March 21. All angles will be measured counter-clockwise from this line. 3. Using a compass or circle guide, draw a 5.0 cm. radius circle centered on the Sun. Label this circle "Earths Orbit (figure 1) We know from our studies that the earth's orbit is actually an ellipse. You will see the effect that this difference has on our model when you finish constructing the orbit of Mars. 4. This sets the scale of out drawing at 5.0 cm. = 1 A.U. Where 1 A.U. is the average distance between the Earth and Sun (93 million miles or 1.5 x 10 8 km). 5. Note that the data table is divided into pairs of dates. Each pair represents an interval of one martian year. Starting with February 17, 1585, use a protractor to plot the heliocentric longitude of the Earth (159 o ) given in the table as a point on the earth's orbit (figure 2) 6. Next center the protractor on the point you just marked and plot the geocentric position of Mars. (135 degrees) Orbit of Mars 1/7
7. Now repeat for the Jan. 5th 1587 date. Label the positions of the Earth with the date (you can omit the year). The point of intersection is the position that Mars had occupied on these two dates. Draw a dot there to represent Mars. Label this as position 1 FIGURE 1 FIGURE 2 8. Repeat this procedure for the next four pairs of data, numbering each point in sequence. You should have 5 positions for Mars. 9. Kepler chose the first two sets of data to represent aphelion and perihelion for Mars, respectively. Draw a line from the first position for Mars to the second position for Mars. This line should pass close to the Sun. This line is the major axis of the orbit. 10. Measure to the middle of the major axis Mark the center of the major axis and label it midpoint. 11. Using the compass draw a circle representing Mars' orbit by placing the point of the compass on the midpoint (make sure something is under the point to protect the desktop) and draw a circle which passes through both perihelion and aphelion points (the first two points you plotted). 12. The circle should come quite close to the other three points you plotted, but will not intersect any of them. 13. Once more, measure the distance from the midpoint of the major axis to either end (aphelion or perihelion) in centimeters. Label this length "semimajor axis" and record its value on your data sheet FIGURE 3 Orbit of Mars 2/7
FIGURE 4 14. Calculate the length of semimajor axis of the orbit of Mars in A.U. Remember that on our drawing that 5 cm = I A.U. = 1.5 x 10 8 km. To convert from cm to AU, divide the value in cm above by 5 cm/a.u. 15. Compare this to the actual value of 1.52 A.U by calculating the percent error (your instructor will show you this computation) 16. To convert this distance to kilometers, multiply the value recorded in step 14 by 1.5 x 10 8 km/a.u. 17. What is the closest that Mars can get to Earth? Measure the shortest distance between two orbits on your diagram, then follow the procedure in steps 14-16 to convert this distance to km. 18. Using a similar technique, determine the greatest distance that Mars and the earth can be from one another. Remember, the two planets are not always on the same side of the Sun. 19. Calculate the eccentricity of the orbit of Mars. Eccentricity is a number that indicates by how much an orbit deviates from a circle. Hence, a perfectly circular orbit would have an eccentricity of 0 and a long, narrow orbit might have an eccentricity of 0.9. The eccentricities of ellipses run from 0 up to, but not including 1.0. 20. To compute the eccentricity divide the distance from the Sun to the midpoint divided by the length of the semimajor axis. You do not need to convert from centimeters to AU or km before dividing. 21. Compare this to the accepted value of eccentricity of 0.093 by calculating the percent error (your instructor will show you this computation) Orbit of Mars 3/7
Orbit of Mars 4/7
Data Sheet Name Partners Turn in this data sheet and the answers to the questions. Make sure your name is on your group's drawing and that it is turned in with one of the labs. (step 13) SCALE SEMIMAJOR AXIS = cm. (step 14) SEMIMAJOR AXIS aexp = A.U. (step 15) %error = a acc a exp a acc 100% = % (aacc = 1.52 A.U) (step 16) SEMIMAJOR AXIS = km (1A.U. = 1.5 x 10 8 km) (step 17) Distance of closest approach = km (step 18) Distance of greatest separation = km (step 20) Your value for eccentricity eexp= (step 21) %error = e acc e exp e acc 100% = % (eacc = 0.093) References Gingerich, Owen, Laboratory Exercises in Astronomy - The Orbit of Mars, in Sky and Telescope, Sky Publishing Corporation, October 1983, pp. 300-302 This work is licensed under a Creative Commons non-commercial, share-alike license Orbit of Mars 5/7
Questions: Questions 1-3 refer to figure 5 (below), which represents the elliptical orbit of a planet orbiting a hypothetical star. Each tick mark on the line represents a distance of one Astronomical Unit (A.U.). FIGURE 5 1. What is the length of the semimajor axis in A. U. in the figure above? A.U. 2. What is the eccentricity of the orbit depicted in the figure 5? 3. The heliocentric longitude of the planet in the figure 5 is about degrees. 4. A certain planet (B) has an eccentricity of 0.005, another planet (C) has an eccentricity of 0.034. Which planet B or C has the more circular orbit? Use the diagram that you constructed in parts 1-10 of the lab to answer the following questions: 5. Oppositions that occur during which month provide the best opportunities to view Mars? 6. Mark an "E" on your diagram at the position that the Earth would occupy about Jan 1st every year. 7. Make an "O" at the position Mars would occupy if it were in opposition on Jan Ist. 8. Make a "C" at the position that Mars would occupy if it were in conjunction on Jan 1st. Orbit of Mars 6/7
FIGURE 6a FIGURE 6b 9. Make a "Q" at each position that Mars would occupy if it was at quadrature on Jan 1st. (see figure 6b). 10. For an observer on Mars, Earth is an inferior planet. Place an "I" at the two places that Earth could be located to be at greatest elongation if viewed from position 1 on your diagram (see figure 6a). EXTRA CREDIT (5 points) The greatest elongation of Mercury as seen from the earth is about 18 near perihelion and about 28 near aphelion. Draw and label the orbit of Mercury, to the appropriate scale, on your diagram. Assume that both the largest and smallest greatest elongations are observed on August 06 (of different years), and that the larger of the two elongations is westward while the smaller of the two is eastward. Orbit of Mars 7/7
Version Updated Author NOTES 1.1 June, 2004 John LaBrasca Revised and updated graphics. Created one-sheet instructions for construction of Mars positions. 1.0 August, 2003 John LaBrasca Revised graphics, updated procedure 0.0?? Dick Shamrell Orbit of Mars 8/7