Back in the day, some guys who went by Newton and Kepler (because those were there names) decided to be really intellegent and figure out the laws governing planetary motion. Between the two of them, they were able to devise rules that we still use today to predict the motions of heavenly bodies. Now, here is an opportunity for you, the reader to behold the power of Newton's and Kepler's laws.

What you see before you is an applet. No, I'm not refering to a small apple, although if you happen to be holding a small apple, you may see that before you. What I'm refering to is a Java applet - a small application that has been embedded in a web page (this web page!) so that it can easily be run on any computer with a decent browser and access to the internet. If you do not see an applet, you need to download a Java patch that will allow you to view Java2 applets on your outdated browser.

In order to use this applet you need to read no further. In order to learn something from it, you might want to.

Now, you're thinking, "Well, that's snazzy, but I could have drawn a blue circle on my computer monitor if I wanted to." Yes, you could have, provided you own a blue marker, but you probably couldn't do other things that this applet is capable of. For instance, try changing the value of "Y init" to 100. This changes the starting position of the planet, moving it vertically by 100 unimportant length units. Now, press "Start/Stop" to stop the animation, "Clear" to clear the screen and "Reset" so that the applet knows to take in the new value of "Y init." If you start the animation again, you'll see that the planet is now orbiting in an elongated ellipse. This illustrates one of Kepler's three laws. A planets orbits a star in an elliptical orbit with the star at one of the focci of the ellipse. In the case of most of the planets in our solar system, the ellipses are very close to circles (which are just a special type of ellipse where the two focci are on top of each other), and it took centuries before Kepler came along and told astronomers that planets orbit in ellipses, not circles. It is much more obvious when looking at comets, rather than planets, that satellites orbit Sun in ellipses.

You can change other parameters, like the initial x and y velocities. If you make the initial x velocity 100 by changing "Vx init," you will see an even more elongated ellipse that goes well off the screen. Notice how the dots are farther appart when the planet passes close to the star. This is because the planet is moving (visibly) faster. This is true of real planets orbiting real stars. If the orbit is very eliptical, the planet will move very fast as it sling shots around the star, and very slowly when it is far away. This inverse speed to distance ratio was apparent to Kepler, and another one of his laws states it more precisely. A planet orbiting a star sweeps out equal area in equal time. To see what this means, let's use our applet.

Click on the "Paint Area" check box. Now, change the value of "Vx init" back to 0 for best results. You'll have to click "Reset" again to make the changes take hold. Now, if you start the animation, you'll see a sector of the orbit is painted in a glorious color. Keep clicking the "Start/Stop" button. Kepler's law tells us that every one of those patches of color is the same area. The time elapsed between each blue dot is the same, and Kepler's law states that equal area is swept out in equal time. You can change the number of time units that are swept out by changing the "Painted Time Steps" text box.

The last really fun thing to do with this applet is to change the "Force Exponent" text field. Newton's law of gravitation states that the force between two massive bodies is calculated by F = G M m/(r^2). In this equation, G is Newton's gravitational constant, M is the mass of one object, m is the mass of the other, and r is the distance between them. Well, with this applet, we can create a universe where the law of gravity is anything we want. Change the "Force Exponent" field to 2.05. This in effect changes Newton's law to F = G M m/(r^2.05). Turn off the "Paint Area" option, and change "Vx init" to 50 and "Y init" to 50 also. Click "Reset" and "Clear." Now, start the animation, and be wowed as the planet traverses a path you never thought possible. The planet should no longer be orbiting in an ellipse. Now, its orbit looks like it came from a spirograph as the planet precesses around the star. This is all very nice, but what are the odds we are goind to encounter a universe with a 1/(r^2.05) force law? Well, probably not too good. However, it is not out of the realm of possibilites to encounter a star emersed in a dust cloud. In such a situation, the effective mass of the star would be constantly channging, from the point of view of the planet, and it might feel a force law that is similar to 1/(r^2.05). But, for the most part it is just fun to look at the pictures, smile, and imagine a beautiful universe where Isaac Newton was wrong.