How to draw an elliptical orbit

Maybe you’ve heard of the Milankovitch cycles, one of which involves changes to the eccentricity of the Earth’s orbit, as it is perturbed by other objects in the Solar System. Suppose you want to depict this with a diagram, using a circle, and an ellipse of exaggerated eccentricity. You could just draw any old random ellipse, and go with it:

ellipticalorbits1

But if you want, you can do better.

Ellipse size

The area and circumference of the Earth’s orbit are not constant. But one parameter that doesn’t significantly change over time is the length of the major axis. The major axis is the longest line segment that you can draw inside an ellipse. The semi-major axis (one half of the major axis) is often used instead, since it is a kind of average radius.

Here are some ellipses with the same major axis length:

ellipticalorbits2

The (semi-)major axis determines the orbital period. So, the length of an Earth year (measured in, say, seconds) also doesn’t significantly change over time.

Ellipse position

The Sun, of course, lies at one of the foci of the Earth’s orbit. Close enough, anyway. The barycenter of the Sun-Earth system is close enough to the Sun’s center that it won’t make a difference in your diagram.

So, where are the foci of your ellipse? I get the impression that most people guess the foci are much closer to the center of the ellipse than they really are. For any fairly eccentric ellipse, the foci are pretty close to the extreme ends of the ellipse.

From a focus, the closest point on the ellipse is one of the endpoints of the major axis. To put it another way, draw the largest circle you can draw inside the ellipse, touching one of the endpoints of the major axis. A focus is at the center of that circle.

ellipticalorbits3

Summary

Putting this together, here’s a depiction of a circular orbit, and some properly-positioned elliptical orbits that have the same major axis length and orbital period:

ellipticalorbits4

If you’d rather not bother with all that, here’s an excuse you can use: perspective. All the preceding diagrams were from a top-down perspective. But if you view an orbit from an oblique angle, the focus will appear to be in the “wrong” place.

ellipticalorbits5

This is what tends to happen with real orbits as seen from Earth, for example the orbits of the stars near the Milky Way’s central black hole.

[Most of the diagrams in this post were made with GeoGebra.]

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