Kepler's Third Law

 

Background

 

At the end of the 16th century, the great astronomer Tycho Brahe spent decades making the most complete and accurate measurements of the positions and movements of the planets in the history of the world.  His data were as accurate as possible before the invention of the telescope (which occurred after his death).  Contained within Tycho's data lay the truth of the motions of the planets.  It was Tycho's assistant, the German mathematician Johannes Kepler, who extracted this truth.  Kepler developed a mathematical model for the motion of the planets which fit Tycho's data.

 

Kepler discoveries are now known as Kepler's Laws.  The first and second laws were published in 1609.  The first law describes the shape of planetary orbits:

 


The orbit of a planet about the Sun is an ellipse with the Sun at one focus.

 

 


The second law describes how a planet's orbital speed varies as it travels around the sun.  Kepler realized that planets move more quickly when they are closer to the Sun.  His second law makes this precise:

 

A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

 

Kepler's first two laws describe the motion of a single planet.  His third law compares the motions of different planets.  The third law says that the sidereal period of a planet (the time it takes the planet to travel once around the Sun) is related to the semimajor axis of the orbit (the average distance of the planet from the Sun).  Your job is to rediscover Kepler's third law.

 

Data and Questions

 

You will deduce Kepler's third law from the data below (don't cheat and look it up!).  We want to know whether the relationship between the sidereal period and the semimajor axis is linear, exponential, logarithmic, a power function, or none of these.  We will measure the sidereal period in years and the semimajor axis in millions of miles.

 

Planet

Sidereal Period (years)

Semimajor axis (106 mi)

Mercury

0.24

36.25

Venus

0.61

66.92

Earth

1.00

92.96

Mars

1.88

141.29

Jupiter

11.86

483.37

Saturn

29.46

886.80

Uranus

84.01

1783.83

Neptune

164.79

2794.26

Pluto

248.54

3674.55

 

1.      Graph the data points.  Eyeballing the data, is there a relationship between the two variables?  What kind of relationship might it be?

2.      Find the line which best fits the data (using linear regression).  How good a fit is this line?

3.      Graph the log plots of the data, and use it to find the exponential and logarithmic functions which best fits the data.  How good are the fits?

4.      Graph the log-log plot of the data, and use it to find the power function which best fits the data.  How good is the fit?

5.      Graph the best of your models together with the data.  What is Kepler's third law?  Would another choice of units make the law simpler?

 

(The background information and data for this project were adapted from Universe, Kauffman and Freedman, 5th edition.)