Chapter 4. Kepler's Laws

4.1 Introduction

Our Solar System

The goals of this module: After completing this exercise, you should be able to:

1. Describe the properties of an ellipse.
2. Explain how a planet's orbital speed varies throughout its orbit.
3. Describe the relationship between a planet's period of revolution and its average distance from the Sun.

In this module you will explore:

1. The orbital motions of the planets around the Sun.
2. What Kepler's three laws predict for planetary motion.

Why you are doing it: As citizens of a heliocentric system, it is important for us to understand and appreciate the laws that explain the world we live in. In addition, understanding the orbital motion of the planets is key to understanding not only our own Solar System, but also the orbital properties of spacecraft, moons, binary stars, binary galaxies, etc!!

4.2 Background

Nicolaus Copernicus

Tycho Brahe in his observatory

From the time of the ancient Greeks until the early 16^th century, the prevailing belief among both scientists and the general public was that everything in the known universe revolved around the Earth. Then, in 1543, Nicolaus Copernicus revived a theory first proposed by the ancient Greek astronomer Aristarchus, which stated that the Earth revolves around Sun. This revolutionary idea was merely a hypothesis until the late 1500s when an observer named Tycho Brahe began to make careful and fairly accurate measurements of the orbital positions of the planets. Johannes Kepler studied Brahe's data, and used it to develop a model of planetary motions that still holds true today. This was the first scientific model of the Solar System as we now understand it to be!

Johannes Kepler

From Brahe's data, Kepler created three laws describing the orbital motions of the planets around the Sun. His first law describes the shapes of the planets' orbits. His second law explains how a planet's orbital speed varies as it travels around the Sun. His third law describes a relationship between the time it takes a planet to make one complete revolution around the Sun and the planet's average distance from the Sun.

In this activity, you will delve deeper into the significance of each of these three laws and how they apply to our Solar System.

4.3 Kepler's First Law

Kepler's first law states:

An ellipse looks something like a squashed circle. Because it is not perfectly round, it has two axes, one longer (called the major axis) and one shorter (called the minor axis). Half of the major axis is called the semimajor axis, often represented by a lower case "a"; the semimajor axis is the average distance of the planet from the Sun. Instead of a center (such as a circle has), an ellipse has two foci. Press the play button to look at the animation below to learn more about elliptical orbits firsthand.

4.4 Ellipses

Ellipses can be very long and thin or very nearly (or even exactly!) circular - in fact, a circle is just a special case of an ellipse. The amount of elongation an ellipse has is described by its eccentricity. Eccentricity is simply the difference in length between the major and minor axes divided by their sum:

The eccentricity of a circle is zero. In our Solar System, most planetary orbits have eccentricities very close to 0. As an ellipse becomes more elongated, the eccentricity approaches a value of 1. Comets usually have very highly eccentric orbits. As you can see below, even though these ellipses have major axes of the same length, the eccentricity affects the shape of the ellipse.

4.5 Kepler's Second Law

Kepler's second law states:

This next animation shows a planet orbiting the Sun. The eccentricity of this orbit is exaggerated to show Kepler's second law easily. Click on the play arrow to begin.

4.6 Kepler's Third Law

Kepler's third law states:

This law is often written as \(P^{2}\propto a^{3} \) , where P is the period, a is the semimajor axis of the orbit and \(\propto\) means "is proportional to." The equation becomes an equality when we use units of years for P and AU (AU stands for an "astronomical unit", which is the average distance between the Earth and the Sun) for a; then we can write \(P^{2}= a^{3} \)

Planet	Sidereal period P (years)	Semimajor axis a (AU)	p 2	a 3
	0.24	0.39	0.06	0.06
	0.61	0.72	0.37	0.37
	1.00	1.00	1.00	1.00
	1.88	1.52	3.53	3.51
	11.86	5.20	140.7	140.6
	29.46	9.55	867.9	871.0
	84.10	19.19	7,072	7,067
	164.86	30.07	27,180	27,190

Table : Table Caption

4.7 Quick Check Quiz

Indepth Activity: Kepler's Laws