Kepler lived from 1571 to 1630. In this time, he used Tycho's data to formulate three descriptive laws of motion. He didn't explain why things worked, just how. Kepler based his studies on Copernicus' theories. But Kepler couldn't make Tycho's data match up with Copernicus' system of circular orbits. Finally, to make the math fit Tycho's observations, Kepler realized that he would have to change the Copernican model. Kepler was able to keep the sun as the center of the solar system, but to change the Copernican model enough to make the math work, he discovered that the orbits of the planets aren't circles at all. Planets orbit the sun in ellipses. With this knowledge, Kepler discovered three laws of planetary motion. Law One The orbits of planets are ellipses, with the sun at one focus. What's a Focus? Try this activity. Put down two thumbtacks. Encircle the two thumbtacks and a pencil with a loop of string. Pull the string tight with the pencil, and draw a line around the thumbtacks. There's your ellipse. Each thumbtack is one focus of your ellipse. With the orbits of planets, the sun is at the center of one of those foci. To sum up: the orbits of planets are ellipses, with the sun at one focus of the ellipse. Law Two, the Law of Areas An imaginary line from a planet to the sun will sweep over equal areas of the ellipse in equal intervals of time. Huh? That's kind of a confusing statement. What it really means, is that the planet moves faster in its orbit when it is closer to the sun. Kepler's law just explains things mathematically. Kepler discovered an inverse relationship between how far a planet is from the sun and how fast a planet is traveling. As the planet's distance from the sun decreases, its speed increases and vice versa. If a planet traveled in a perfectly circular orbit, the planet would always be the same distance from the sun and would always travel at the same speed. In an ellipse, though, the planet's distance from the sun varies, and, Kepler figured, so does its speed. Just remember Kepler's second law in terms of this: a planet moves faster in its orbit when it's closer to the sun. When the planet is farther from the sun, it moves more slowly. If you're wondering why all this happens, then remember that Kepler didn't know either. He just knew that it did happen because of his studies of the orbits of Mars and the other planets. We'll get to how it works when we learn about Newton. Law Three A planet's period squared is proportional to a cubed. Formula: P2 is proportional to a3 What's a Period? A period is the amount of time, in Earth years, that a planet takes to orbit the sun once. The period is measured in Earth years, not days, so the Earth's period is one year and not 365 days. Then what's a? A is the length of the semi-major axis of a planet's orbit. The semi-major axis is half the length of the longest diameter of the orbit, which is an ellipse. How's that work? Well, if we draw the longest possible line from one end of an ellipse to the other end, we have the major axis. If we cut that in half, we have the semi-major axis. The only catch here is that you have to measure the semi-major axis in a unit called AU's. AU stands for "astronomical unit." One AU is equal to the length of the Earth's semi-major axis. We also use the semi-major axis to measure the Earth's distance from the sun. The long and the short of Kepler's third law is that a planet's distance, in AU's, raised to the third power, is equal to the time it takes that planet to orbit the sun, squared. How is that useful? Well, slap our pappy happy, we're gonna tell you. If you know the distance a planet is from the sun, you can figure out its period of orbit. Or, if you know its period of orbit, you can figure out its distance from the sun. If you know one, you can figure out the other. Example Let's try a problem. Mars takes 1.9 years to complete one orbit of the sun. So the period of Mars's orbit is 1.9. How far away from the sun is Mars? Using Kepler's third law, the distance cubed (a3) is equal to the period squared (1.92) squared. That gives us a3 = 3.61. Now we take the cubed root of both sides of the equation so we can find "a" To do that, you'll probably need a calculator. The cubed root is just the opposite of raising something to the third power, so we just have to tell the calculator that. On most calculators, that's accomplished by hitting the inverse button, followed by the y to the x button to get the exponent, and then 3, for the third power. So, punch the following sequence into your calculator: 3.61; inverse; yx; 3. The calculator should tell you "1.53 something," which we'll just round to 1.5. If we look in an astronomy book, we'll see the distance from Mars to the sun is 1.5 AUs. Just like Kepler said, the distance cubed is equal to the period squared. Kepler's third law has come in especially handy, because it's easier to use Kepler's law and do a little math than to actually measure the distance between the sun and all the planets. I mean, think of what kind of measuring tape you'd need.