It’s in the headlines we read every week: Scientists have found a new planet. It’s a “Hot Jupiter”, and its orbit last only 4 days, skimming the surface at a scant 4.6 million miles… You read it again, and wonder, how do astronomers know how far a planet is away from its host star? It’s because of Kepler’s laws of planetary motion!

## Historical Background

Johannes Kepler began working with the Danish astronomer Tycho Brahe in the late 1500’s. Having access to Brahe’s meticulously detailed records of planetary positions, Kepler was able to deduce that the orbit of Mars (and thus all planets) was elliptical in 1605[2].

Before his work, astronomers and mathematicians had several competing theories that were constantly being debated. The most popular, and successful one at the time was the use of Epicycles in a geocentric model – meaning that Earth was the center of the universe. Epicycles explained why planets had an apparent retrograde motion in the sky by using a ‘Deferent’, or larger, circular orbit. The epicycles themselves were small circular orbits centered along the line of the deferent. In the image to the right, the deferent is the larger black circle centered around ‘T’, and the epicycle is the smaller black circle centered around ‘P’. The observer is at ‘O’, and the observed orbit is the red circle. Point ‘S’ moves around the epicycle and deferent, tracing a path which is the large red circle. From time to time, Point ‘S’ will have to reverse direction. Confused yet? You’re not the only one!The Copernican Theory correctly placed the Sun in the center, and also correctly ordered the planets in distance from the Sun. His theory was able to explain the retrograde motion of other planets in the sky because of the way that the Earth moved through its orbit as well. He also stated that all of the motions are uniform, eternal, and circular. Though he was unable to completely eliminate the need for epicycles.If all of the motions were uniform, that would mean that the Earth would take 182.5 days to complete one half of an orbit, and all 4 seasons would be the same length. In fact, as I discuss below, Earths orbit is not evenly divided. our seasons are not the same length… Kepler was able to explain this using his first two laws of planetary motion.

## Kepler’s first law:

“Planets move in ellipses with the Sun at one focus”.

ɛ = (ᴨ(186.5-178.5) / 4(186.5+178.5))

ɛ = 25.1327 / 1460

ɛ = 0.0172

The actual value of Earths eccentricity is 0.0167, so this method gets us very close. It also proves that the orbit cannot be circular. If we were to replace the 186.5 and 178.5 values with 182.5, (half of 365), then the eccentricity would be 0. This mathematical proof was the deathblow to all Geocentric models, and it was the crucial piece of the puzzle that evolved the Copernican Theory into the current model we have today. Earths orbit is an ellipse, and so is every other planet in our solar system.Using an ellipse instead of circular orbits alone almost completely eliminates the need for complex geometry such as epicycles and deferents. However, it does not fully predict the motion of the planets in our skies. The reason for this is that the Copernican Theory expects that the planets velocity is constant in its orbit, and by keeping the velocity constant in an elliptical orbit, planets tend to either lead in front of a calculated position, or lag behind. It’s clear that the planets velocity changes during an orbit.

Animation showing Kepler’s second law of planetary motion |

## Kepler’s Second Law:

“A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.”

This law is more or less describing the fact that planets move at a higher velocity the closer they are to Suns focus of the elliptical orbit. But why does a planet move faster when it’s closer to the Sun? Gravity, of course! But it wasn’t until Newton published his work on universal gravitation in his book “The Principia” in 1687 that gravity found to be responsible for the changes in velocity. Actually, in my research, I could find no real reasoning for how the velocity in an orbit was thought to change over time. Instead, I’ll jump ahead to Newton. What we do know is that Kepler was able to predict the speed at any point within an orbit, as well as what that rate of change was by working out how to find the area of an ellipse by calculating the area of a triangle within the ellipse. The triangle used the Sun as one vertex, and to points along Mars’s orbit that were one arc minute apart [5].

Without going into any math, here is how we understand it to work today: assuming that a planet starts off at its furthest point from the Sun, and is moving along its orbit, bringing it closer to the Sun side of the ellipse, the Sun’s gravity pulls the planet a little stronger with each passing moment. This causes the planet to accelerate along the orbit. After reaching perihelion (closest approach), the planet has achieved maximum velocity, and starts moving away. The Sun, still pulling on the planet, starts slowing it down again. Viola! We now have an explanation for variable velocity in a planets orbit. Knowing how the velocity changes along the orbit, we now have a way to precisely predict the position of a planet in the sky.

## Kepler’s third law:

“The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of the orbit.”

The third law that Kepler derived is arguably the second most important one. If you happen to know exactly how long a planet takes to orbit the Sun (which ancient civilizations have known for thousands of years – look at the Mayan calendar!), then you immediately know the size of its orbit, and its distance from the Sun. The equation for this is P^2 = a^3, Where “P” is the orbital period in Earth years, and “a” is in Earth/Sun distances (Astronomical Units). Using Earth as an example:

1^2 = a^3

1 = a^3

CubeRoot(1) = CubeRoot(a^3)

1 = a

3.53 = a^3

CubeRoot(3.53) = CubeRoot(a^3)

1.52 = a

I will be revisiting the Third Law in a future science experiment. To Be Continued…

http://spaceplace.nasa.gov/review/dr-marc-solar-system/planet-distances.html, 2014[5] Math.org, Proofs 2,

http://plus.maths.org/content/origins-proof-ii-keplers-proofs, 2014