Mostly Closest Planet

It has been over 3.5 years since I last updated this dead blog. But I was inspired by a recent video by CGP Grey on Youtube. Here I answer his question with simulation and using data. His video can be found here: https://www.youtube.com/watch?v=SumDHcnCRuU 

What is the closest planet to the Earth? It's a question with answers that vary from Venus or Mars. While true that these are the planets that get closest to the Earth with Venus getting as close as 38M km at opposition. Mars gets as close as 58M km at opposition. However what planet is closest to the Earth on average? The answer is surprisingly Mercury. This is due to the fact that Mercury is never too far from the Earth. When at conjunction both Mars and Venus can be at their semi-major axis plus the semi-major axis of the Earth apart. Whereas Mercury with its short semi-major axis and fast revolutionary period is never too far from the Earth.

Here is a plot of the inner planets. For simplicity sake assume all planetary orbits are circular with zero eccentricity and nil tilt in the orbital plane. Both of which are reasonable assumptions. Mercury does have a large precession due to the curvature of space in the Sun's vicinity however I'll ignore that here and it won't change the results dramatically. To plot these orbits I plotted them on the complex plane. The vertical axis is actually the complex component the function I used to plot the orbit. For each orbit I used the equation r*e^(2*pi*i/T), where r is the orbital semi major axis and T is the orbital period.  

Here I took the differences between the orbit of Earth and the other inner planets. Notice that the curves are not sinusoidal, this is because the distance changes rapidly when the planets are approaching opposition and when at conjunction the orbital distance changes slowly. The dashed line is the average distance between the planets. We can see here that Mercury has the closest average distance from the Earth. 

Unsurprisingly here is the distance between Mercury and the other inner planets. Earth does get closer than Venus to Mercury however there is nothing interesting here as the distance between the other planets and Mercury is as expected. 

 Likewise for Venus and the other planets Mercury is the closest

Now lets have a look at the outer planets. Here are the orbits for scale of the gas giants. 

Now we can plot the difference between the Jupiter and the other planets. Unsurprisingly Saturn, Uranus and Neptune follow each other in average distance from Jupiter. However the inner planets are hard to see at this scale. 

Zooming in and adjusting the time to a few years we can see that once again Mercury is on average the closest planet to Jupiter. Now lets see what happens if we look at Neptune. 

The time is now in centuries this is accounting for Neptune's 165 year orbital period. Uranus is the closest planet that ever makes an approach to Neptune when the two planets are at opposition. The higher frequency curves of the other planets is attributed to their faster orbital speeds and catching up and surpassing Neptune once in their orbit. Jupiter's curve has a period of roughly 10 years which is its orbital period. Zooming in to look at the inner planets we have the following.

Once again the average distance is depicted but the dashed line. It is hard to see here which planet is the closest to Neptune on average. However Mercury just edges out the others. Venus and Earth are not very far apart. The average distance between Mercury and Neptune is essentially its semi major axis. Which is almost the average distance between Venus and Neptune as well as Earth and Neptune. This means for planets which are sufficiently far away and as a result will have a slower orbital period the average distance is approximately the semi major axis of the orbit of the larger planet. To determine if Mercury is in fact the closest planet to Neptune on average a more robust simulation accounting for precession, orbital eccentricity and inclination should be done. I will not do such a calculation as such an endeavour is quite complex.