Project1-Instructions-FALL2023

2 Part II. Kepler’s 3rd La w Kepler’s third law of planetary motion states that the cube of the semimajor axis 𝑎 of the orbit of a planet equals the square of the orbital period 𝑝 , 𝑝 2 = 𝑎 3 (equation 1) The semimajor axis 𝑎 is half the longest diameter of the elliptical orbit, measured in astronomical units (the average distance between the Sun and the Earth). The orbital period 𝑝 is the time the planet takes to complete exactly one orbit, measured in Earth years. In this section, you will use the PhET simulation to verify Kepler’s third law . You will measure the semimajor axis and period of three different orbits and confirm equation 1 is satisfied. Complete the following steps. STEP 1. Reset the simulation so that it is set to reproduce the Sun-Earth system (see Part I, STEP 3). STEP 2. Shorten a bit the length of the velocity arrow to decrease the initial speed. The velocity arrow must be vertical on the screen before the simulation starts, correct it if it looks inclined. Keep the original star mass, so that Kepler’s 3 rd law has the form shown on of equation 1. Run the simulation and pause it when the planet completes exactly one orbit . STEP 3. Use the “Measuring Tape” tool t o determine the length of the longer axis of the orbit, in kilometers. If you placed the velocity vector vertical at the start of the simulation, the longer axis of the elliptical orbit must be oriented horizontally. If that is not the case, you will need to figure out the orientation of the longer axis of the orbit. However, the longer axis always will pass through the position of the star. The figure below shows how to measure the length of the major axis. Note that the measuring tape extends from side to side across the oval enclosed by the orbit. Divide the width of the longer axis by 2 to get the length of the semimajor axis (half the width) “ 𝑎 ” in units of kilome ters. Record this value on the worksheet, Table 1, column 2. STEP 4. Record the orbital period “ 𝑝 ” of the planet, in units of Earth days, in the worksheet, Table 1, column 4. This information is displayed at the bottom right of the simulation, just above the “Clear” button. STEP 5. Take a screenshot of the simulation making certain that it shows the orbit, the Measuring Tape tool reading the length of the longer axis, and the orbital period. Paste the image on the designated location of the worksheet . STEP 6. Repeat steps 2-5 to get measurements for a total of three different orbits. STEP 7. Now we need to convert the widths to astronomical units (AU) and the periods to years. To convert the widths to AU, divide the width in kilometers by 149 600 000, and record that value in the worksheet Table 1, column 3. To convert the period to years, divide the period in days, by 365.25 . Record the results in the worksheet Table 1 , column 5. STEP 8. Next, calculate a 3 and p 2 , and record the results in the worksheet , Table 1 , columns 6 and 7.