Kepler’s Third Law of Gravity

Kepler’s Laws of Gravitation – Law 3

So how was all that math? You now understand the math of an ellipse and the math needed to explain the motion of a planet along an elliptical orbit.

How about a mathematical formula to describe the actual motion of a planet around a star? That’s all that is left…right?

Well, almost.

The mathematical proof behind Kepler’s Third Law of Gravitation in beyond that of yourself (an 11th or 12th grade student), but the findings are simple enough to work with.

Kepler found mathematically that there was a proportionality between the distance a planet is from a star and the period of the orbit.

This is to say, that the square of the period (in years) is proportional to the distance (the semi-major axis) of the planet from the star (in Astronomical Units).

You now have the skills necessary to think in terms of Keplerian Motion – reference Carl Sagan’s video below to refresh your memory visually of the three laws governing planetary motion:

This attached link is a Excel file of observations made of an exo-solar planet. What do you notice about the path the planet is taking around the parent star? Use your understanding of Kepler’s Laws to describe the statistics!

http://www.u.arizona.edu/~duboisr/Keplerian%20Motion%20Observations.xls

Kepler’s Second Law of Gravity

Kepler’s Laws of Gravitation – Law 2

So now you understand that planetary orbits are ellipses.

Kepler's First Law

So what more can be added to this. We now know the shape in which planets move around stars, but what does their motion look like?

To answer this, we need an understanding of the Conservation of Angular Momentum. This conservation law which is responsible for governing planetary motions is responsible for actions we see on Earth. For example, when an ice skater spins, they tuck their arms in to spin faster. Why is that?

As with before, let’s look at the math

http://csep10.phys.utk.edu/astr161/lect/solarsys/angmom.html

So to have a planet change a parameter such as distance from a star, one of two other parameters, either mass or velocity must change. Since the mass of a planet does not seem like a likely parameter to change constantly, we can hypothesize that the velocity of a planet will change as its distance from a star changes.

What Kepler found out was that a planet sweeps out equal areas in equal times as it orbits around a star. This is explained perfectly by using the Conservation of Angular Momentum.

Kepler’s First Law of Gravity

Kepler’s Laws of Gravitation – Law 1

Johannes Kepler’s helped shape our understanding of gravity. While the basic principles of gravity seem intuitive to us now in the 21st century, the reason for gravity and its workings were not understood in the 16th and 17th century.

How do planets move? Do planets travel in circles?

Kepler was able to determine how planets should move along a path. While we have a strong grasp of circles and appreciate the ease at which circular mathematics provides us, we simply can not count on all science to be easy.

What Kepler found was that planets do not travel around stars in circular orbits – rather they move around their parent stars in ellipses.

While we will investigate this further, it may be prudent to refresh yourself with the mathematics of ellipses. NASA has provided a tutorial on elliptical orbits which you may find useful – http://istp.gsfc.nasa.gov/stargaze/Skepl1st.htm.

Math of an Ellipse

Once you have mastered ellipses, you are ready to move on to Kepler’s next two laws of planetary motion.