PH105 Kepler’s Third Law of Planetary Motion

According to Kepler’s 3^rd law of planetary motion, the square of the period of revolution of a planet about the sun is proportional to the cube of the mean distance between the planet and the sun. This is a consequence of Newton’s law of universal gravitation and Newton’s second law of motion:

(1)

or, (2)

In this in-class exercise you will use data given in the textbook for the planets to verify Eq. (2) and to determine the mass of the sun.

1. Enter the orbital periods of the planets and the mean distances between the planets and the sun into separate columns in an Excel spreadsheet. Plot T versus r and note that the resulting plot is not linear.

2. Now calculate T² and r³ in two adjacent columns (let Excel do the calculations) and plot T² versus r³. Is the plot linear? Do a regression analysis and determine the slope of the plot. According to Kepler’s 3^rd law,

, or (3)

Calculate M_s using the above formula and compare with the value given in the textbook.

3. A better way to analyze the planetary data is to do a log-log plot of T versus r. In this case,

(4)

Calculate log(T) and log(r) in two additional columns in Excel and plot log(T) versus log(r). Do a regression analysis and determine both the slope and the intercept. According to the above equation, the slope of the plot should be 1.5 and the intercept should be

(5)

What slope does your analysis give?

Use Eq. (5) to calculate M_s from the intercept and compare with the textbook value of the mass of the sun.

Each group should turn in this worksheet containing the results and calculations and a printout of the Excel data and graphs.