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 Earth, 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.

> On average, Mercury is the closest planet to the sun, closer than Venus!

I suppose you meant “closest planet to the Earth”? The statement wouldn’t seem particularly worthy of an exclamation otherwise.

This result was very surprising to me at first, but your note about planet positions being seemingly random made it easier to accept – treating orbits as circular, cocentric and coplanar as in the article, you quickly see that triangle inequality implies that the sun would be the closest object to any planet.

You’d think, being a reader of this blog, that by now I’d be used to the smack-thud of my hand against my forehead. But no, it always comes as a surprise. Of *course* Mercury is the closest planet, on average!

Just like the Sun is the closest star. You know what I mean.

Still, this post seemed to be a non-sequitur given the recent arc, until I looked a bit closer at the referenced article and read “E(x) is an elliptic integral of the second kind”.

It fits in just fine.

A less computationally intensive justification for this fact comes from the law of cosines:

c^2 = a^2 + b^2 – 2ab cos C

with the Earth at A, the other planet at B, and the Sun is at C (so a is the distance of the other planet to the Sun, b is the distance of the Earth to the sun, and c is the distance between the planets).

If the cosine term averages to zero (which it should if the orbits are not systematically related to each other in some way), then the smaller a is, the smaller c^2 is on average. (Okay, technically this shows Mercury has the lowest average *squared* distance to the Earth.)

Chatting with a bunch of friends a while ago, someone asked “What is the nearest planet to the Earth?” Quick as a flash, someone answered “the Earth.” You can tell it was a brunch of mathematicians, because nobody thought that was a silly answer.

A more serious comment. Non-inertial frames off reference are underused. You can’t do much physics in them, but for mathematical problems like this they can be useful. Here assume circular orbits and use a frame in which the Earth and Sun are stationary. The other planets move at constant speed in circular orbits (so are ergodic even if there is resonance.) Using Astronomical Units, a planet in an orbit of radius r clearly has median distance √(1+r^2) from the Earth, clearly smallest at Mercury. The mean distance is a horrible integral, but we can expect it not to be not too far from the median. Numerical integration confirms this: very close for small and large r, and pretty close everywhere. Definitely increasing with increasing r .

Original author here. Thanks for your blog. We had considered non random distribution of the planets. An easy way to understand its randomness (still assuming circular coplanar concentric) is to think in an inertial reference frame. If you sit on the Earth, the orbits of the other planets only seem to slow down. Their velocity is still constant and thus equally likely to be occupied over time. The only exception is if both planets have the same angular velocity which is impossible with real planetary orbits.

It breaks down when we consider elliptical orbits, though. Planets slow as they reach their apoapse, so a poaitional average would need to be weighted by the biased likelihood of each position. It’s a fun problem, but unnecessary for our solar system.

Thanks! Really enjoyed your article. It’s a surprising result, and you can look at it from a lot of different angles.