Venus Is Not the Earth's Closest Planetary Neighbor

Venus is the almost unanimous response, by both amateur and professional astronomers, to the question, Which planet is closest to Earth? But this overlooks a subtle point. Planets can be quite far apart when they are in opposition (on opposite sides of the Sun). When the distance between two planets is averaged over both their orbits, it turns out that Mercury has the smallest average distance to every other planet in the solar system. It is the closest to the Sun, and therefore distances when it is in opposition to another planet are considerably reduced.
For simplicity, assume circular, concentric and perfectly coplanar orbits for all the planets. With the Earth's orbital radius equal to 1 au (astronomical unit) by definition, the orbit of Mercury is 0.387 au, while that of Venus is 0.723 au. However, a calculation of the average Earth–Mercury distance works out to 1.038 au, while the Earth–Venus average equals 1.136. Thus Mercury is, on average, the closest planet to Earth! The calculations are worked out in the Details section.

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DETAILS

The average distance between two planets with orbital radii and can be obtained by averaging over the circular orbit of one of the planets while the other is fixed at a point in its orbit. This is shown in the diagram:
The average is given by
where (·) is a complete elliptic integral of the second kind. (This is a correction of the result given in [1].) The average distances work out to:
(Earth–Mercury)=1.03781 au, (Earth–Venus)=1.13567 au, (Earth–Mars)=1.69303.
Reference
[1] T. Stockman, G. Monroe and S. Cordner, "Venus Is Not Earth’s Closest Neighbor." Physics Today, PT.6.3.20190312a. (Mar 22, 2019) physicstoday.scitation.org/do/10.1063/PT.6.3.20190312a/full.
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