During the last quarter of the 16 centry, the Danish bobleman Tycho Brahe provide careful observation of the planets over an extended period of time (20 years). He hoped to use his data to verify his own model of the solar system, in which the sun orbited th earth and all other planets orbited the sun. At Brahes death at 1601, his assistant Johannes Kepler inherited the data that Brahes had accumulated. Kepler spend some 20 years analyzing these data, looking for mathematical regularities.  He came to the conclusion that the ideal of circular orbits should be discarded and replaced with elliptical orbits. Kepler summerized his laborious study of planetary motion with the following three laws:

1. The planets move in elliptical orbits with the sun at one focus.
2. A line from the sun to any planets sweeps out equal areas in equal time.
3. The square of a planet's period is proportional to the cube of the planet's mean distance from the sun.

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Your browser can't run 1.0 Java applets, so here's a picture ofthe window the program brings up:

The red circle at the center of the screen represent the sun.
The moving yellow circle is the planet.
The initial condition is represented by the blue arrow.
The planet start from the starting point of the blue arrow,
and its initial velocoty is proportional to the length of the arrow.
You can drag the blue arrow to change its initial condition in three different modes:
1. fixed kinetic energy
2. fixed angular momentum
3. arbitrary

Find out the relation between shape and size of the trajectory with the above parameters.

Right click the mouse button to suspend the animation. Click it again to resume.

What if you click it with the left mouse button? Find out by yourself.

Use reset button to clear the screen. (Try to compare different trajectorys before press it!)

You can select four different mode to play with it.

For the energy mode or while you drag the mouse button:

The lower green curve is the potential energy : U(r)=-GMm/r
another one is the effective potential energy U_eff(r)=-GMm/r + L²/(2mr²)
Two horizontal red lines show the total energy of the particle.
(watch the small moving dots! )

The period of the particle motion is also shown in real time unit.

Any suggestions? Please click hwang@phy03.phy.ntnu.edu.tw

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