Kepler's Third Law

The final law, Kepler's third law, is one of the most useful relations in astronomy. It states that the period of time it takes a planet to orbit the sun, squared (that's period*period), is proportional to its distance from the sun, cubed (distance*distance*distance). Or, as astronomers would say: P^2=a^3, where P is period and a is semi-major axis (i.e. distance).  The graph above shows the period and orbital distance of some planets in our solar system. The line going through all the points corresponds to the spot where P^2=a^3. The fact that all the planets fall on this line means that Kepler's third law is correct, and that we can predict the orbital time if we know the orbital distance, or vice versa. This relationship can be applied to most objects orbiting a larger object in space. Astronomers use it to estimate the period of exoplanets orbiting stars, and stars orbiting galaxy centers. 

And there you have it! Kepler's three laws of planetary motion!

Image Credit: Kevin Brown, Reflections on Relativity

Kepler's Second Law

Kepler's second law states:  The line joining the planet to the Sun sweeps out equal areas in equal intervals of time. 

 

This law is often referenced as the "law of equal areas" . So what does it mean? In the diagram above we have a planet going around the sun (or any star) following an elliptical path (as the 1st law states). When the planet is at point A, we draw an imaginary line towards the star. The planet continues to orbit the star, and lets assume one month passes. The planet is now at point B, and we draw another imaginary line towards the star. The area shaded in blue is the imaginary triangle in space that is created by the two lines we drew. We can calculate the area of this triangle because we know the length of the two lines we just drew. Now we repeat this scenario for when the planet is at points X and Y, and again it took the planet one month to go from point X to point Y. Notice that it traveled a much shorter distance on its orbit, and that the imaginary triangle we made is a lot thinner. But, again we know the length of the lines we drew, and if you calculate the area of this green triangle, you should get exactly the same amount as for the blue triangle! So in one month, the planet sweeps out a path of equal area!

Why is this the case? When the planet is closer to the star, it feels a stronger gravitational force from the star. The star sort of whips the planet around the corner closest to it, and has a weaker effect when the planet is farther away. All planets that orbit their host star in an ellipse will follow this rule.