Moon’s orbit around the sun #

Background #

The orbit of the earth around the sun and the orbit of the moon around the earth are both simple shapes. They are essentially circular. But if you think about it, the moon also orbits the sun while it is orbiting the earth, and the shape of this compound orbit is much more interesting. A few years ago I stumbled into an online forum debating the nature of this shape. We will use math to settle the debate.

Objective #

A few years ago, a reader posted the following question on Slashdot:

Question: Is the moon's orbit around the sun convex?

In a convex shape, the path between any two points stays within the shape, so things like circles and regular polygons are convex. Intuition suggests that the orbit in question should look something like the path below:

In the picture, the earth is blue, the sun is yellow, and the moon is red. The shape within the path traveled by the moon, also colored in red, is decidedly not convex. But is our intuition correct? What does the shape of this orbit look like? That’s exactly the object of this lab.

Procedure #

Our approach will use parametric curves and especially their vector forms. To keep things simple, let us make the following assumptions. First assume that the moon, sun, and earth all orbit each other in the same plane, so that our problem is just two-dimensional. Second, assume the sun is fixed at the origin, as it should be. Finally, assume that the orbits are circular. We will use a Mathematica notebook to plot the orbits.

A. First, plot the orbit of the earth around the sun. Assume that units are defined so that the distance between these two celestial bodies is 1 and it takes exactly one unit of time to complete a revolution. Begin by writing down the parametric coordinates of the orbit which we will call \(\vec{s}(t)\) and plot it in the notebook.

B. Now introduce the moon. For the time being, pick some nice numbers to work out the principles involved. Assume that:

• the distance of the moon from the earth is a quarter of the distance of the distance of the earth from the sun, and
• that the moon circumnavigates the earth exactly ten times each year.

Write down \(\vec{r}(t)\), the parametric coordinates of the moon orbit around the earth. Next, use the result of the previous question to write down the parametric coordinates of the moon’s orbit around the sun. Plot your results. What does the orbit look like?

C. Now use real data on the distances between these bodies and their rates of revolution to make an accurate plot of what happens. To make the numbers simple, assume that the average distance from the earth to the sun is 1 and work with distances based on this measurement. Also, work in terms of earth years as your measure of time. Explain what you see, and how is it different from what happened before?

D. Using the techniques introduced above, plot the orbit of Mars around the earth, again assuming that their orbits all occur on the same plane. This time, the earth should be at the origin in your diagram. Is the orbit convex?

Summarize your findings #