Kepler's Second Law

Kepler's Second Law characterizes the the velocity of a planet along its elliptical path.

The planet's speed varies -- but it does so in a completely regular way, a way that can be expressed in a simple mathematical formula.

Kepler's Second Law says says that a line running from the sun to the planet sweeps out equal areas of the ellipse in equal times. This means that the planet speeds up as it approaches the sun and slows down as it departs from it. Let's use the diagram above to think through how this is so.

Suppose it takes a certain time T for the planet to travel from Position 1 to Position 2. The line from the sun to the position of the planet during this journey covers the area bordered by Sun-P1/P1-P2/P2-Sun. This is the same area as found in the region bordered by Sun-P3/P3-P4/P4-Sun. So the time it takes for the planet to travel from Position 3 to Position 4 will be the same time, T. The path along the ellipse from P1 to P2 (while the planet is closest to the sun) is longer than the path along the ellipse from P3 to P4 (when the planet is farthest from the sun). In other words, the planet is travelling fastest while it is close to the sun and slowest when it is furthest away from the sun. For the areas to be the same, P3-P4 has to be shorter than P1-P2, because S-P3 and S-P4 are longer; conversely, P1-P2 has to be longer (if the same area is to be swept out), in order to compensate for the shortness of S-P1 and S-P2. But if a body covers a shorter distance in a given stretch of time than some other body covers in the same stretch of time, it's going slower. The planet will be traveling fastest at the point on the ellipse midway between P1 and P2, after which it begins to decelerate. And it will be traveling slowest at the point on the orbit midway between P3 and P4, after which it begins to pick up speed.

Kepler's law describes the motion not only of planets around the sun but of moons around planets. If your browser is equipped with Shockwave, check out Raman's orbit simulator to look at a moon traveling in an ellipse around a planet traveling in an ellipse around the sun! (And then think of how much paper and ink Kepler must have gone through in the eleven years it took him to figure out that the way Mars seems to us to travel across the background of the fixed stars how Mars would be produced for an observer on a planet traveling in an elliptical orbit arount the sun by a planet traveling on a different elliptical orbit around the sun (more distant from the former, and out of phase with it) -- provided we imagine the right average distances from the sun for the respective orbits, and pick the right point on each orbit corresponding for an initial observation of the more distant planet from the one closer to the sun. Whew!)

The designer of this clever orbit simulator is ahead of Kepler, because he's working from Newton's laws, which are a later part of our story. You can nevertheless have some informative fun here by seeing what happens when you tinker with the various variables. For simplicity's sake, set the velocity of the sun itself at zero. You might also want to click on the grid.

Now experiment with different values for the mass of the sun and for the gravitational constant. If you set these high enough, you'll get some interesting results. Sometimes the moon gets so accelerated as it passes the sun that it escapes the planet's gravitational field and travels on out into space. (Maximize the mass of the star and set the value for the gravitational constant as high as you can on this program, and you'll get this.) Sometimes it comes so close to the sun that the sun captures it away from its planet, and turns it into another planet, whipping around on a different ellipse. (If you set the mass of the star at 90000 and pretend that the gravitational constant is about 1.64 times what it is in our world, you'll witness this happening after about 3 planetary revolutions.) Sometimes it ends up plunging into the sun. Fiddling with the variables gives you a more intuitive feel for what the unchanging formula in which the variables are imbedded means.

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This page last updated 11 October 2000.