Can someone explain the mathematics of this formula for Venus' orbit

[Niaboc67]

I've always been interested by the pattern within venus' orbit around the sun. I found this image and the formula of its retrograde motion in the upper right hand corner which confused me. What does it mean? can someone break this down in layman's terms?

http://www.tokenrock.com/stock/harmonics_venus.jpg

Thank you

 

[D H]

That's numerology, pure and simple. Venus and Earth are close to being in, but not exactly in a 13:8 mean motion resonance. That graphic is pretending the relation is exact.

Orbital resonances are important, there's no doubt. However, whether a resonance exists becomes a bit numerological when the integers are large. When it comes to resonances, anything involving a number larger than five: That's large. Given any real number, one can always find a pair of integers whose ratio is close to that real number. The rationals are dense in R. That near 13:8 mean motion resonance might represent a real physics-based resonance, or it just be a numerological circumstance. From what I can tell, most solar system dynamicists view it as a numerological circumstance.

 

[Niaboc67]

Would it be fair to say this is pseudo-science?

 

[mfb]

I think so. Resonances like 2:1 are interesting and they often have some meaning, but an approximate 13:8 resonance... it is hard to find a ratio where no similar fraction gives a reasonable approximation.

 

[Ken G]

Yes, the graphic seems to claim that the ratio of the Earth's period to Venus' is the Golden ratio, but it isn't-- it's off by 0.46%. That's remarkably close, I'll give them that, but it still could be coincidence. As said above, it is also close to 13/8, which is at least a ratio of integers so sounds more like a resonance, though I don't know which would be more convincing-- that it is a kind of resonance, or that it is the Golden ratio. What is a larger error is that the graphic seems to indicate that the orbits return to their original configuration in 8 Earth orbits (which is what you'd need for a 13/8 ratio to make a figure with 5-fold symmetry as shown), but if one uses the Golden ratio as depicted, then the Golden ratio times 8 = 12.944, not an integer so the figure would not actually be closed. It's pretty close, but resonances usually play out over lots and lots of orbits, so it wouldn't matter much that it is nearly periodic over 8 Earth orbits-- it would matter that it is not exactly periodic over any number of Earth orbits, so it's not a resonance.

 

[tony873004]

The image is not realistic. That's not the path Venus would trace around the Sun as viewed from Earth. That's the path Venus would trace around the Earth in a top-down view of a Geocentric universe.

You can try it for yourself here: http://orbitsimulator.com/gravitySimulatorCloud/venusPentagram.html

This simulation runs in your browser, so you don't need to download anything to try it.

Watch as Venus traces a pentagram around Earth. It will take 8 simulated years. When it come back to repeat the pattern, it doesn't quite line up. This demonstrates what others here have said. The 13:8 ratio is almost, but not quite perfect. Therefore it is meaningless.

Although only the Sun, Earth, Moon and Venus are visible, the gravitational effects of all the planets are included in this simulation.