Kepler and the Laws of Planetary Motion – Lab 3
Read Chapter 3
Introduction
Johannes Kepler was a brilliant young high school teacher when the Copernican revolution was taking place. He was a gifted mathematician and physicist. During this time he analyzed the data of another astronomer, Tycho Brahe, who had amassed extremely premise data on the orbit of Mars. By utilizing this data he was able to formulate three general laws on planetary motion that influenced the history of science.
Kepler’s First Law of Planetary Motion – Planets Move in Elliptical Orbits
Kepler discovered that elliptical orbits allowed him to more accurately reproduce the motion of the planets in the solar system. An ellipse can be thought of as a squashed circle. A circle can be drawn from its center by extended a piece of string and making a complete loop around the center. Similarly to the circle with it’s center, an ellipse has two points called foci. A focus is one one of two points within an ellipse that help to define its shape. An ellipse can be drawn by taking a piece of string, attaching it to both foci within the ellipse, and then drawing a line around the foci. We will explore elliptical motion in this lab.
In addition there are also some concepts that we should be familiar with regarding ellipses. Each ellipse has two axes. The longer axis is know as the major axis (half of the major axis is know as the semimajor axis). The shorter axis is known as the minor axis (half of the minor axis is know as the semiminor axis). These quantities are shown in Figure 1.
Figure 1. Diagram of the components of an ellipse.B is the semiminor axis and a is the semimajor axis.
Another important quantity with an ellipse is the eccentricity of the ellipse, or the distance from the center of the ellipse to its foci divided by its semimajor axis. An eccentricity of 0 gives a circular orbit, wile an eccentricity of 1 gives essentially a straight line of oscillation. Through these developments Kepler enabled physicists to accurately predict the orbit of planets around the Sun.
Kepler’s Second Law of Planetary Motion – Planets sweep out equal areas in equal times.
Kepler had shown that he could account for the motion of the planets around the Sun, but now he had to account for the conservation of angular momentum and the velocity of the planets. Kepler theorized that if you fill in the area that a planet sweeps out of the ellipse as it travels around the sun, it will be equal to a second area swept out by the same planet, as long as the amount of time between the two points is the same. This is shown in Figure 2.
Figure 2. Illustration of Kepler’s Second Law. The blue areas are plotted over the same time period, and are all equal to one another.
This was an important observation because it allowed scientist to describe the velocity of a planet as it orbits the sun. With the second law Kepler showed that planets speed up when they are closer to the sun, and slow down when they are further away from the sun. This concept also supports the conservation of angular momentum. This principle states that as an object rotates around another body, its speed changes depending on the distance from that body. For instance a skater with her arms out will spin slower than a skater with his arms close to his body. In orbital mechanics this is demonstrated in that a planet at perihelion (closest point to the Sun) is moving faster than a planet at aphelion (furthest point from the Sun).
Kepler’s Third Law of Planetary Motion – The Period Radius Rule
The third law discusses the relationship between a planet’s average distance from the Sun (it’s radius) and its orbital period (the time it takes for one complete revolution of the Sun). Kepler’s third law is defined as:
P2 = R3 (1)
When the period is expressed in years and the average radius is expressed in AU, this relationship is true. This is a very powerful tool for physicists because it is universal, and it allowed physicists to calculate the average radius of any planet in the solar system from the Sun. With these three laws developed, Kepler had elucidated the Copernican model of the solar system.
Materials
Online Orbit Simulator
Procedure
In this lab we are going to explore Kepler’s three laws through modeling the orbital motion (Well loosely based on Kepler’s laws, the simulation itself looks more like a circular orbit). First, we want to model the Earth/Sun system. Start the simulation and have the the Earth and the Sun trace out their respective orbits. Answer the following questions:
What direction is the velocity vector pointed in as the Earth orbits the Sun? What direction is the acceleration oriented toward?
If you double the mass of the Earth what happens? Why might this occur?
What if you double the mass of the Sun? Why might this take place?
What is one interesting effect that you were able to produce by changing the variables of the system? Why do you think this occurred?
Repeat this activity for the Earth/Moon system.
Once you have completed the simulation answer the additional questions contained in the next section. When you submit your report, it should be in paragraph form. Answer the questions below in the introduction. Discuss the results from the experiment in the Results/Conclusions section.
Questions:
Describe Kepler’s first law and its impact on the field of astronomy.
Discuss Kepler’s Second law. Describe the law and its importance.
Discuss Kepler’s Third Law. Define the law and describe why it is important to astrophysics.
You are an astronomer studying a distant exoplanet, Kepler 1625b around the yellow star kepler 1625. You determine that the average radius (the distance from the star) is 4 AU. What is the period of this exoplanet?