Kepler's Second Law
- A JAVA Interactive Tutorial

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For any two bodies interacting via a force acting along the line connecting them, the line joining the two bodies sweeps equal areas in equal times. The gravitational force is one such force.

This is the modern formulation of Kepler's second law. For a historical perspective of the work of Johannes Kepler and Tycho Brahe (whose careful observation of the motion of Mars were used by Kepler to deduce the law above as well as two other laws) click here.

Suppose these two bodies are the Sun and a planet (see the "applet" in the separate window). First, let us test your intuition. From Kepler's first law, we know that: the orbit of a planet around the Sun is an ellipse . In the panel of the applet window, choose an eccentric orbit, that is one in which one of the semiaxis is much bigger than the other, by adjusting the Eccentricty slider, and then pressing Submit.

Question If Kepler's second law holds, do you think that the speed of the planet will be constant along its orbit around the Sun?

Now run the "applet" (press Continue) and watch for the motion of the planet (Don't touch the other buttons yet). Does it have constant speed? If not, where along the orbit is it slower/faster? Pressing Show Velocity will display a green arrow which indicates the direction, and magnitude of the velocity.

Do you need help in running the applet?
Let us now test Kepler's second law. Press the Start Sweeping button; after a few seconds, press the Stop Sweeping button. In the panel you can read the amount of time taken to complete that fraction of the orbit as well as the area swept by the line joining the Sun and the planet. To verify Kepler's second law, press the Start Sweeping button again. The area traced by the line joining the Sun with the planet will become visible. After the amount of time given by the first box has elapsed, the tracing will stop automatically. You can compare the two areas which have been traced in equal times.

For a strongly elliptic orbit, it is hard to visually verify that equal areas are swept in equal times; but the panel gives you the actual areas swept. As you can see, they are the same.

Now adjust the parameters of the orbit and press Submit to observe a different orbit.
Last modified . Written by David McNamara and Gianfranco Vidali . Department of Physics, Syracuse University. This project was partially supported by a NSF- Curriculum Course Development Grant.