Charles Petzold

I recently saw an animation on Facebook that purported to show the “hidden beauty of the Moon’s orbit as it creates a flower-like pattern while revolving around Earth and Sun,” further referring to this motion as “celestial choreography” and an “intricate dance” (Brain Maze post from February 20, 2025).

Here is my re-creation of that animation as viewed from somewhere north of the North Pole:

The Sun is in the center. The cyan circle shows the Earth’s orbit around the Sun, and the little cyan dot is the Earth making a complete yearly revolution every 30 seconds. The white dot circling the Earth is the Moon, and its path leaves behind a white trail that does indeed exhibit a “flower-like” looping pattern.

At first it might seem a little off because the Earth is crossing the trail of the Moon’s orbit, but I think you’ll be able to convince yourself that the Moon maintains a constant distance from the Earth.

Yes, it is pretty cool.

After the Earth makes a complete revolution, the previous part of the looping pattern progressively disappears. The new trail doesn’t meet up with the previous trail because the sidereal period of the Moon (the revolution of the Moon relative to the fixed stars) is about 27.3 days, and that doesn’t divide evenly into the length of a year (about 365.26 days). Watch out: If you leave this page and return, the animation will be messed up somewhat but eventually correct itself.

Although the timing of the Earth and Moon revolutions is to scale (I think), the distances are not to scale. The manifestation of that “flower-like pattern” depends on the ratio between the orbital radius of the Earth around the Sun to the orbital radius of the Moon around the Earth. In this animation (and the one I copied it from), that ratio is about 6.7. But what if it’s different?

This animated graphic is the same as the first one except that it includes a slider that adjusts the orbital radius of the Moon and hence the ratio between the two orbital radii:

The initial slider value of 6.7 is what I calculated from the original graphic, but you can change that ratio from 2 through 25. Best results are obtained when you move the slider slowly; moving it too quickly causes discontinuities in the Moon’s orbit, Move it down to 2 and the looping is more extreme as the Moon gets perilously close to the Sun. Increase the ratio up to 12 or so, and the amazing “flower-like pattern” simply disappears, and at 25 (with the slider to the full right), it’s more like a gentle wave.

What is the actual ratio of those two orbital radii?

The distance from the Earth to the Sun is about 93,000,000 miles. The distance from the Earth to the Moon is about 239,000 miles, so the actual ratio is about 389.

Can that be visualized?

Not easily. There’s a very good reason why images of the Solar System are not rendered to scale. The following table shows the sizes of the Earth, Moon, and Sun, and distances between them, and the number of pixels required for each based on the Moon being rendered as 1 pixel:

To show the entire orbit of the Earth around the Sun, you’d need double that last number, and I doubt that many of us have monitors that size.

The two canvas elements that I’ve used so far in this blog entry are 720 pixels square. If I double that to 1440 pixels square, I can fix the orbital radius of the Earth at 700 pixels (leaving a little margin). The other pixels dimensions (rounded to the nearest integer) become:

So let’s forget about rendering the Earth and Moon and just show the Sun and the orbits:

And now, if you look real close, you can see the Moon’s white orbit shifting back and forth across the Earth’s cyan orbit.

I apologize for killing the flowers.