The Orbit of Mars C

The Orbit of Mars Big Idea Tycho Brahe made a number of observations of the positions of Mars during the latter part of the 16 th century. Despite not having a telescope, Brahe was able to obtain the most accurate measurements of the positions of Mars of his time. His assistant, a young mathematician named Johannes Kepler, devised a method of triangulation to determine the orbit of Mars around the Sun. In this experiment, you will:  Recreate Kepler’s measurements using Tycho Brahe’s data;  Analyze the properties of Mars’ orbit, and;  Investigate a modern claim about Mars’ appearance in the night sky. Setup You will need:  Ruler  Protractor  Compass  Pencil (tip: if you can use a colored pencil in addition to a regular pencil, that would be great)  Calculator (optional) Part I: Brahe’s data Kepler knew from Brahe’s observations that the sidereal period of Mars is 687 days , so every 687 days, Mars would return to the same position among the fixed stars. He also knew that the Earth completes two orbits around the Sun in 730 days . That means by the time Mars completes one full orbit around the Sun, Earth will not quite have completed two full orbits. Below is a table of Brahe’s data, grouped into five pairs of dates, each 687 days apart: Pair Date Pair (687 days apart) Heliocentric longitude of Earth Geocentric longitude of Mars 1 February 17, 1585 January 5, 1587 159  23' 115  21' 135  12' 182  08' 2 September 19, 1591 August 6, 1593 5  47' 323  26' 284  18' 346  56' 3 December 7, 1593 October 25, 1595 85  53' 41  42' 3  04' 49  42' 4 March 28, 1587 February 12, 1589 196  50' 153  42' 168  12' 218  48' 5 March 10, 1585 January 26, 1587 179  41' 136  06' 131  48' 184  42' Every Martian year (687 days) Mars returns to the same point in its orbit around the Sun, thus if we view Mars at these intervals we can, by triangulation, determine that point. You should follow the procedure below to get the first point, then repeat four more times to get the orbit. 1 Name: ________________________ ________________________ Date: ________________________

Part 2: Plot the orbit Attached is a diagram of the Sun with the orbit of Earth drawn around it (the orbits of Mercury and Venus are drawn in as well, to help show their relative distances.) The dashed horizontal line indicates where the Sun would appear to an observer from Earth (on the opposite side of the Sun) on the March equinox (March 21). This position represents 0 degrees of heliocentric longitude. 1. With the protractor and Sun as the center, plot the heliocentric longitude of the Earth as a point on the Earth's orbit as given in the table (159 degrees). 2. Now with the protractor and using the Earth as the center plot the geocentric position of Mars (135 degrees). You can use the horizontal lines to help make sure your protractor is lined up at 0 degrees longitude. Your drawing should be similar to Figure 1: Figure 1 3. Now repeat for the Jan. 5 th 1587 date. First mark the position of Earth from its heliocentric longitude, and from that point draw a line to the geocentric longitude of Mars. The point of intersection is the position that Mars had on these two dates. Draw a dot there to represent Mars. Label this as position P1 . Your drawing should be similar to Figure 2: Figure 2 4. Repeat the above steps for the remaining four pairs of dates in the data table. Label the positions of Mars as P2 , P3 , P4 , and P5 . Kepler chose the first two sets of data to represent aphelion and perihelion , respectively for Mars. 5. Draw a line from the first position for Mars P1 to the second position for Mars P2 . This line should pass close to the Sun (if your line passes nowhere near the Sun, your measurements for the 2