Average distance between planets

What is the closest planet to Earth?

The planet whose orbit is closest to the orbit of Earth is clearly Venus. But what planet is closest? That changes over time. If Venus is between the Earth and the sun, Venus is the closest planet to Earth. But if Mercury is between the Earth and the sun, and Venus is on the opposite side of the sun, then Mercury is the closest planet to Earth.

On average, Mercury is the closest planet to the sun, closer than Venus! In fact, Mercury is the closest planet to every planet, on average. A new article in Physics Today gives a detailed explanation.

The article gives two explanations, one based on probability, and one based on simulated orbits. The former assumes planets are located at random points along their orbits. The latter models the actual movement of planets over the last 10,000 years. The results are agree to within 1%.

It’s interesting that the two approaches agree. Obviously planet positions are not random. But over time the relative positions of the planets are distributed similarly to if they were random. They’re ergodic.

My first response would be to model this as if the positions were indeed random. But my second thought is that maybe the actual motion of the planets might have resonances that keep the distances from being ergodic. Apparently not, or at least the deviation from being ergodic is small.

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