Kepler’s Laws describe the motion of one object in orbit around a much more massive object. This is the case for the planets that orbit the Sun whose mass is 1000 times greater than the most massive planet Jupiter. It is also true for the moons that go around the planets in our Solar System. It is not true for Pluto whose moon Charon is only 1/6 its mass. But it is true for all satellites that have been launched from Earth. Kepler’s Laws can be derived from Newton’s laws of motion, which apply to all non-relativistic motion. The are the following:
- The orbits of the planets are ellipses with the Sun at one focus.
- The speed of the planet as it goes about its orbit increases or decreases so that the area between it and the Sun over a given time stays the same.
- The amount of time it takes for a planet to complete its orbit, called its period, squared is equal to the distance of the planet from the Sun, actually the semi-major axis of the ellipse, cubed.
$P^2 = a^3$
Laboratory Tools
In this lab we will make use of the Planetary Orbit Simulator, developed by the University of Nebraska-Lincoln. This simulator demonstrates Kepler’s three laws of planetary motion. The simulator has a box for Orbit Settings which allows you to select a planet or choose a distance (semi-major axis) and eccentricity (how squished is the ellipse) of your choice. The Animation Controls box starts and stops the animation and controls the speed, while the Visualization Options box will allow you to add the orbits of the planets for comparison. There are four tabs under the diagram that can be selected. Three correspond to the different laws of Kepler and the fourth is Newtonian Features which we will not use.
Assignment
First use the Kepler’s 1st Law tab. Check all of the boxes so that all features are shown. Now vary the eccentricity of the orbit with the slider in the Orbit Settings box. Describe how the semi-major and semi-minor axes change with eccentricity and how the focus and center change too,
Next use the Kepler’s 2nd Law tab. Click on the sweep continuously box and then start sweeping. Change the parameters to Venus and then Earth in the Orbit Settings box. What do you notice about the areas swept out by the planets motion? Then try Mercury and Pluto, how do the colored regions change? Play with the eccentricity slider in the Orbit Settings box. How do the swept out regions change with eccentricity?
Now use the Kepler’s 3rd Law tab. The plot shows period versus semi-major axis. According to Kepler’s 3rd Law all orbits will fall on a single curve in this plot. If you set the orbit to Jupiter’s in the Orbit Setting box you will see that the period is 11.9 years. What semi-major axis would give a period of exactly 10 years? What semi-major axis gives a period of exactly 100 years?
Questions
- If you discovered a new comet, what would you need to know to determine its period from Kepler’s Laws?
- What does Kepler’s second law tells us about the speed of a comet when it is far from the Sun?
- What property of an orbit determines the distance between the Sun and the center of the orbit?
