Positions of Mercury

Back in the '70s I assumed that whenever an inferior planet reaches maximum elongation (east or west) the radius vector of the planet is perpendicular to that of Earth. Logically, it seemed, the two radius vectors form a right angle whenever the angular distance of Mercury (or Venus) from the sun, as seen from Earth, is at its greatest. For example, if Mercury is at greatest western elongation when Earth is at heliocentric longitude 270 degrees, Mercury "must" be at heliocentric longitude 180 degrees. To appear farthest from the sun, I imagined, the radius vector of Mercury couldn't be less than 90 degrees away from Earth's, since the planet wouldn't be fully elongated to one side of the sun.

As the diagram above makes clear, this reasoning was fallacious. Maximum elongation doesn't occur when the radius vectors form a right angle. It occurs when they form an acute angle. The separation of the two radius vectors, at farthest elongation, is far less than the "logical" 90 degrees. It's only about 60 degrees. A line from Earth to Mercury (when at maximum elongation) and beyond is farther from the sun than an Earth-Mercury line when the radius vectors are perpendicular. That proves the angle between Mercury and the sun, as seen from Earth, is not at its greatest when the radius vectors are 90 degrees apart.

Why does reality, in this case, seem to defy logic? It's due to varying distances. When the radius vector of Mercury is perpendicular to Earth's, the planet is farther away from us than it is when at greatest elongation. When the planet is farther its angular distance from the sun appears less. Conversely, when it is nearer its angular separation is greater. If Mercury was much farther away its separation from the sun would be much smaller; if the distance were vast the two bodies would appear to merge. So it's not surprising the angular sun-Mercury distance appears greatest when the planet is closer to Earth, instead of when its radius vector is 90 degrees from ours.

Addendum I

When Mercury is at full elongation its radius vector is perpendicular to a line from Earth to Mercury. But that's irrelevant. A line from Earth to Mercury is not a radius vector (a line from planet to the sun).

Addendum II

This fall marks the 20th anniversary of the blog! It's great to see it's still extant and receiving more pageviews than ever. Thanks to all who contributed over the years!