Teaching students (home instruction, tutoring, etc.) about orbital motion, I explained that the earth goes in orbit around the sun because the gravitational attraction of the sun causes the velocity to change direction. The gravitational force is equal to the centrifugal force, and so the earth orbits the sun, rather than fall in. This is the standard stuff in the textbooks.

 

A student asked me a question.

 

“The earth's orbit is actually an ellipse, not a circle.”

 

I was stumped. A good teacher should be stumped occasionally. If the teacher always knows all the answers, the teacher is not a good teacher.

 

Researching it out, on physics forums and on the Internet, I found the answer:http://www.braeunig.us/space/orbmech.htm

 

The eccentricity e of an orbit is given by

where Rp is the periapsis (the closest point), and Vp is the velocity. For the orbit of the earth around the sun, periapsis is called perihelion.

 

For a certain Vp the eccentricity would be zero (a circle). The math is straightforward, and the source readable. Imagine the earth orbiting the sun in a circle. We give a kick to the earth, making it move faster. The orbit would become an ellipse. Other planets pull on the earth, speeding it up.

 

Sverre Aarseth from University of Cambridge wrote to me:

 

“The formula does not explain the reason why Earth's eccentricity changes with time. In science, we try to get an understanding of what is happening and this usually means some kind of calculation or estimate. Actually, my hunch proved right. I did the calculation, including the effect due to Jupiter, and got a maximum value around 0.05.”

 

These calculations are difficult numerical calculations. It is not possible to explain these calculations to students. If we just talk about the math of the circular orbit, and fail to mention the reality of the elliptical orbit and its ramifications, we are failing to convey to our students important ideas that they should know.

 

We can mention that if the orbit is a circle, and we make it move faster, it will become an ellipse. The extra speed the earth has in its orbit around the sun is due to the gravitational attraction of the large planets. Furthermore, the orbit changes in time. This is called Milankovitch Cycles, and may have a profound effect on the dying of species and the rise of new species, and on the ice ages.