Copy of Kepler's 3rd Law Investigation

Kepler’s Third Law Investigation AET PHYSICS Introduction: Purpose : In this lab you will use publicly available data on the moons and planets to analyze three bodies in our solar system (two planets and the sun). From the period of the orbits and their orbital radius, you will determine the mass of Jupiter, another planet of your choosing, and the Sun using Kepler’s Third Law. Background: Kepler’s Third Law can be used to determine the mass of an object that is being orbited by another, usually smaller, object. In this case you will use the periods of several moons to determine the masses of Jupiter and one other planet. Kepler’s Third Law can be written as r 3 = MG 4 π 2 T 2 where G is the gravitational constant ( 6.67 ⋅ 10 − 11 N m 2 / k g 2 ), M is the mass in kg of the object the moon is orbiting, T is the period of the moon in seconds, and r is the orbital radius of the moon in meters. If we gather data on the radii and periods of the moons of a given planet, their interdependence should contain enough information to determine the mass of the planet! To do this we will create a scatter plot with the radius cubed ( r 3 ) on the vertical axis, and the period squared (T 2 ) on the horizontal axis. The slope of this graph should a line, which contains the value of the mass: y = mx + b r 3 =( MG 4 π 2 ) T 2 slope = MG 4 π 2 M = slope ⋅ 4 π 2 / G Methods: Procedure: Please note: Once completed, step 11 will instruct you to repeat steps 1-10 for a new planet. 1. Go to https://nineplanets.org/solar-system-data/ . If you scroll down you should see a large data table consisting of information on the various orbiting bodies in the solar system. 2. Highlight and copy the moons of Jupiter, and all their data. 3. Make a copy of “ Kepler Investigation Data Sheet ” and add your last name to the title. (For example: Kepler’s Investigation Data Sheet - Wissler” . Make sure you are on the first tab - Sheet 1: “ Jupiter” 4. Paste the data for Jupiter’s moons. ○ Remove all columns containing information beyond Name, Distance, and Period, align the data to the appropriate columns.

○ Remove any rows for which data on Distance and Period is not known. You should have data for 16 moons remaining. 5. Calculate column D values for orbital radius, r, in meters, based on the distance values given in kilometers. The given unit is x1000 km or 1x10 6 m. Note: The distance given is ‘Mean distance (semimajor axis) between centers’ so no compensation for the diameter of Jupiter is necessary. 6. Calculate column E values for orbital period, T, in seconds, based on the period values given in days. Use the absolute value function in Sheets, ABS( ), to include any moons that may be in retrograde. 7. Calculate column F and G values for T 2 and s 3 . 8. Create a scatter plot using columns F and G. Fit a linear trendline to the plot and display the equation of the trendline. (Use the Customize tab in Chart Editor and select “Series”.) 9. Kepler’s 3rd Law predicts the slope of this line should have a value MG 4 π 2 . Use this to determine the mass of Jupiter ( M = slope ⋅ 4 π 2 / G ). 10. Calculate the % error for the mass of Jupiter using 1.8981x10 27 kg as an accepted value. 11. Repeat steps 2-10 for another planet and its moons. Choosing one from: ○ Saturn, Mass = 5.683 × 10 26 kg ○ Uranus, Mass = 8.681 × 10 25 kg ○ Neptune, Mass = 1.024 × 10 26 kg Complete your work in Sheet 2 of the Kepler Investigation Data Sheet document. Results: Data: Paste a copy of your Jupiter graph, with the trendline and equation shown, below:

Record your experimental mass of Planet 2 here: M Planet = 5.6879699127E26 Record your percent error for the mass here: %Error = 0.0874522734471% Summary Questions 1. Two students are having a discussion about the relationship between radius and period for an orbiting body.

Student A: “The radius is directly proportional to the period because when we made a graph of r 3 vs. T 2 , the graph was linear” . Student B: “I disagree, this means that the radius is directly proportional to T 2 because we had to square the period to get a linear graph”. Are either of the students correct? If so, explain why. If not, explain what a correct interpretation would be. Neither is correct, the correct interpretation would be that strictly r^3 and T^2 are directly proportional to each other because r^3 and T^2 have a linear relationship, but the radius itself is not directly proportional to the period alone. 2. A student gathers data that is supposed to represent objects in orbit around the sun. The data are listed below: Distance (AU) Period (yrs) A 2.7658 4.59984 B 39.4821 248.0208 C 52.889 498.5 D 67.67 557.2 One of these bodies actually does not orbit the sun. Which is it, and why? Please be careful with the units!!