# The Orbit of the Moon around the Sun is Convex!

What does the orbit of the Moon around the Sun look like? Most people, even almost all mathematicians I've asked this question, tend to believe that it will have loops and look something like the picture below. In fact it looks like this picture! It is not a circle, but is close to a 12-gon with rounded corners. It is locally convex in the sense that it has no loops and the curvature never changes sign. (To be precise, it is more like a 13-gon, since we have to consider the sidereal month of 27.32 days instead of the synodic month of 29.54 days, and 365.25/27.32 = 13.37.)

There are several ways to see this. Since the eccentricities are small, we can assume that the orbits of the Earth around the Sun and the Moon around the Earth are both circles. The radius of the Earth's orbit is about 400 times the radius of the Moon's orbit. The Moon makes about 13 revolutions in the course of a year. The speed of the Earth around the Sun is about 30 times the speed of the Moon around the Earth. That means that the speed of the Moon around the Sun will vary between about 103% and 97% of the speed of the Earth around the Sun. In particular, the speed of the Moon around the Sun will never be negative, so the Moon will never loop backwards.

I like to visualize this as follows. Imagine you're driving on a circular race track. You overtake a car on the right, and immediately slow down and go into the left lane. When the other car passes you, you speed up and overtake on the right again. You will then be making circles around the other car, but when seen from above, both of you are driving forward all the time and your path will be convex.

Another approach is to compute the gravitational forces involved. It can be shown that the Sun's pull on the Moon is about twice the Earth's pull on the Moon. It follows that the Moon's orbit is primarily determined by the gravitation pull from the Sun, so the orbit of the Earth will always curve towards the Sun.

Here is a sketch of the details. (G MassSun MassMoon/Distance(Sun-Moon)^2)/(G MassEarth MassMoon/Distance(Earth-Moon)^2) = (MassSun/MassEarth) (Distance(Earth-Moon)/Distance(Sun-Moon))^2. But Distance(Sun-Moon) is close to Distance(Sun-Earth), and Distance(Earth-Moon)/Distance(Sun-Earth) is 1/390 while MassSun/MassEarth = 3.34X10^5, so we get 2.20.

You may also want to remember that it is the barycenter of the Earth-Moon system that moves in an ellipse around the Sun, and that the orbits of both the Earth and the Moon are perturbations of this ellipse.

• Moon orbit by Erich Neuwirth is very nice, but you must first get the Logotron plug-in.
• “The Sun, the Moon, and Convexity” by Noah Samuel Brannen in The College Mathematics Journal v. 32, 268-272 (2001) is excellent, but also look at “Why the Moon's Orbit is Convex” by Laurent Hodges in The College Mathematics Journal v. 33, 169-170 (2002).
• “Mad about Physics” by Jargodzki and Potter also discusses this problem.
• The National Maritime Museum mentions this on their fact file about orbits, but has a bad picture on their fact file about The Moon's movement.

If we assume that the ellipses are circles, the radius of the Moon's orbit is 1, the radius of the Earth's orbit is d, and that the Moon makes p revolutions while the Earth makes one, then we get curves of the form (d cos(t) + cos(p t), d sin(t) + sin(p t)). Above I used d=400 and p=13, but if we had changed the parameters, we could have gotten pictures like these.  It turns out that we loops when d < p, a wiggly path when p < d < p^2 and a convex path if p^2 < d. For details, see the article by Brannen.

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Helmer Aslaksen
Department of Mathematics
National University of Singapore
helmer.aslaksen@gmail.com

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