Laboratory Exercise - Astronomy. The Orbit of Mars Name Date: Materials: protractor, compass, centimeter ruler, paper (graph paper is helpful) and calculator. Object: To reconstruct the approximate orbit of Mars by using some of Tycho's data. Tycho Brahe collected a number of observations of the positions of Mars as seen from Earth during the latter part of the 16 century. In order to plot the orbit of Mars it is necessary not only to know where Mars was in relation to the earth but also to know where the earth was in relation to the Sun. Mars's period of revolution about the Sun is 687 days. So any two observations of the position of Mars separated by exactly 687 days would view Mars at the same position in its orbit. Where those two lines of sight (from Earth to Mars) crossed would be Mars's position in space. The table below lists pairs of Mars positions observed by Tycho and arranged by Kepler to be 687 days apart. These were recorded in Keplers book of 1609, Astronomia Nova (\"The New Astronomy\"). Date Heliocentric Longitude of Geocentric Longitude of Earth 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 Procedure: You may work in groups. However, EACH student will turn in the results of the following procedures. 1. Place the graph paper in front of you with the long side horizontal. Place a small dot, representing the Sun, near the center of your graph paper. Label the dot "Sun". Draw a straight line from the center (the Sun) to the right-hand edge. Label this line the \"autumnal equinox.\" This line represents the 0 direction in space. All angles should be measured counter-clockwise from this direction. 2. Draw a circle of radius 5.0 cm centered on the Sun. This circle represents the orbit of Earth. In reality, the Earth's orbital eccentricity is 0.016, or 1.6% deviation from perfect circularity. Note: This scale means that 5.0 cm. = 1 A.U. Where 1 A.U. is the average distance between the Earth and Sun (93 million miles) Label this circle "Earths Orbit". Laboratory Exercise - Astronomy. 3. Using the protractor centered on the Sun with 0 toward the autumnal equinox and using the heliocentric longitude of the Earth (as given in the table above) plot the positions of the Earth with dots on the Earth's orbit. You should label the date next to each of the 10 dots. 4. The observations of Mars are paired. Go to the first entry. Move the protractor so that the Earth is at the protractor's center, but the 0 direction is still parallel to the autumnal equinox line. Find the geocentric longitude of Mars observed for that date and mark it. Draw a line in this direction starting from Earth and proceeding nearly to the edge of the page (as in figure 1). Figure 1 5. Repeat for the second entry of the first observation pair, then for the rest of the observation pairs. 6. For each observation pair Mars is located at the intersection of the two lines. Puta conspicuous dot on each of the five intersections (as in figure 2). ars L Ea Figure 2 7. Kepler chose the first two sets of data to represent aphelion and perihelion respectively for Mars. Draw a line from the aphelion to the perihelion for Mars. This line should go through (or pass close to) the Sun. If not, something has gone wrong. This line is called the major axis of the orbit. Measure the major axis in centimeters to the nearest millimeter (tenth of a cm) cm. 8. Find the middle of the major axis by dividing the length of the major axis by 2. This length is defined as the semimajor axis. cm. Mark the center of the major axis and label it \"midpoint\". Laboratory Exercise - Astronomy. 9. Using the compass draw a circle representing Mars' orbit by placing the point of the compass on the midpoint and the pencil part either on the perihelion or aphelion points and making a circle. If you found the midpoint correctly your orbit should pass through both perihelion and aphelion points. Your diagram should look somewhat like the one in figure 3. The other three points of the orbit should pass quite close to the circle that you drew. No wonder Kepler and others initially thought that planetary orbits were circular! Figure 3 10. Calculate the length of the semimajor axis of the orbit of Mars in A.U. and in miles. Remember that on our drawing that 5 cm =1 A.U. = 93 million miles. Show your work. Semimajor axis = AU. Semimajor axis = miles 11. Look up the accepted value for Mars's semimajor axis length in AU in the appendix of your textbook. Compare your calculated value against the accepted one by calculating the percent deviation, using the following formula. Show your work clearly. (Note that to multiply by 100% means you multiply by 100 and attach the percent sign to the answer) percent deviation = [ (measured accepted)/accepted ] x 100% 12. What is the very closest that Mars can get to Earth in miles? Hint: Measure the closest distant that the two orbits get on your diagram in centimeters, then convert to AU, then to miles. Show your work. Closest distance = miles Laboratory Exercise - Astronomy. 13. What is the very farthest apart that Mars and Earth can be in miles, not necessarily at opposition but anytime? Hint: Earth and Mars are not always on the same side of the Sun! Show your work. Farthest distance = miles 14. Finally we will calculate the eccentricity of the orbit of Mars. The eccentricity is a number that tells us how oval an ellipse is, for example a perfectly circular orbit would have an eccentricity of zero and a flattened-out oval would have an eccentricity of 0.9. Eccentricities of all ellipses lie between 0 and up to, but not including 1. To find the eccentricity follow this simple formula: The eccentricity equals the distance from the Sun to the midpoint divided by the length of the semimajor axis. Both numbers should have the same units (for example, centimeters) so that after the division, the eccentricity has no units. Show your work. Your value of eccentricity = 15. In the textbook or on the internet, look up the accepted value for the eccentricity of Mars's orbit. Accepted value of eccentricity = Percent deviation = Synthesis questions: 16. 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 most circular orbit? 17. In your own word describe eccentricity: Laboratory Exercise - Astronomy. Use the diagram that you constructed in parts 1-9 18. \"Opposition\" is when a planet-Earth-Sun line-up occurs, with the Earth between the planet and the Sun. Oppositions that occur during which month provide the best oppositions to view Mars, because Mars is closer to the Earth than at other oppositions. See your chart for the answer to this. 19. Make an "E" on your graph paper at the position that the Earth would occupy about Jan 1st every year. 20. Make an "Q" at the position Mars would occupy if it were in opposition on Jan 1st. 21. Make a "C" at the position that Mars would occupy if it were in conjunction on Jan 1st. \"Conjunction\" is when a planet-Sun-Earth line-up occurs, with the Sun between the planet and the Earth. 22. If you lived on Mars, Earth would be an inferior planet. Place an "I" at the two places that Earth could be located to be at greatest elongation if viewed from Mars when Mars was located at position 1 (aphelion). Greatest elongation occurs when an inferior planet makes the largest possible angle with respect to the Sun from the observer's viewpoint