Planetary Motion & Kepler's Laws (Page 2 of 2)

Take a trip back to those geometry classes. The mathematical definition of an ellipse involves having two focii, one at each side of the center of the ellipse in perfect symmetry. The object that is being orbited around must be at one of these focii, even if there is absolutely nothing at the other.

The amount of “ellipseness” in a given orbit as opposed to “circleness”, referred to as a planet's eccentricity, varies tremendously between the planets. Some are nearly circular, such as Venus, while others are as dramatically elliptical as (the non-planet) Pluto's. However, most planetary orbits appear to look very similar to circles, and given the relative crudeness of their observations, it's startling that Kepler was even able to tell the difference between a circular orbit and an elliptical one with his data.

This one might be slightly difficult to visualize. Make a mental line between the orbiting and the orbited, and then take an arbitrary amount of time and have the line move along. During this time along any interval of the orbit, the same amount of area should be swept out. The consequence of this law is that when the planet is closer to the sun, the planet will move faster, but if the planet is further away, it will move slower so it does not move as far along the orbit so that these two intervals will indeed sweep out the same amount of area.

Geometry whizzes will note that this is an obvious consequence of the First Law; indeed, all of Kepler's Laws are really just restatements of each other after a little geometry. However, with this phrasing it is more obvious what Kepler's Laws indicate as far as the kinetic and potential energy of the planets are—and thus how their velocity changes. This is important, because it explained the slight differences in speed that Kepler noted in the planets during different parts of their orbit.

Ouch!

Read that statement over a few times, and it'll start to make sense. Mathematically, it can be simplified to: P² is proportional to a³. As with the others, this law is just a restatement of the previous laws . The difference here is that it's more mathematical, which may or may not make more sense. However, it's very elegant, and has served countless astronomers well over the years in figuring out the distance to a variety of objects orbiting our sun.

That proportionality constant implied in the law is the same for any planet that orbits the same body—so, from Venus to Uranus, everything that orbits the sun, they all share that same constant. Using this, one may find the period or semi-major axis of an unknown object by using a known one in the same system, by comparing P²/a³ for each object.

Not everything works out perfectly, however: the universe has a lot of variables in it, and planetary motion is no exception. One of the biggest things effecting the accuracy of Kepler's predictions is the gravitational pull that the planets exert on each other, which can cause subtle variations in orbit known as gravitational perturbations. In addition, the atmospheric drag of a planet as it hurtles through not-quite-empty space, collisions and other effects can cause small, but relevant changes to a planet's orbit. These may all be taken into account, however, with today's mathematical modeling of the universe. Thanks to Kepler and other astronomers, we finally have a complete understanding of how celestial objects move about each other.