Here comes the sun, or at least its shadow, at Dot Physics: When is the Sun directly overhead?

There’s a nice little video that accompanies the post which also demonstrates some of the foibles of doing experiments

What I want to do is change the question a little bit. Rhett points out that one of the common answers is “Everyday at noon,” which can never be correct if you are outside of the tropics. But let’s change this to “When is the sun on the N-S line that goes directly overhead?” The answer still isn’t “everyday at noon.”

Why? Because the earth’s orbit isn’t a circle, and we orbit the sun fastest when we are near perihelion (in January) and slowest near aphelion (in July), with the difference being about a kilometer per second. If we define our time in terms of the sun being on that overhead line — i.e. we use a sundial — then the length of the day will vary, and this is why we generally don’t use solar time (when we’re using solar time) without modifying it. What we do is apply the equation of time, which gives rise to the figure-eight-ish analemma (found on globes as well as sundials). This takes into account both the inclination of the sun and the eccentricity, to give a correction to solar time and correct the reading. While this makes our day 24 hours again, it also means that the sun will be on that overhead line as much as 15 minutes or so before or after actual noon, as kept by our clocks.

My memory was just jogged, bringing this to the fore. A series of cool analemma pictures taken in Greece, showing ruins in the foreground, with the additional feature of some being multiple exposures on a single frame of film.

The analemma — which you can often see on a globe — describes the variation in the position of the sun over the course of the year. Besides the pretty obvious vertical motion you get from the axial tilt of the earth, there is also the effect from the noncircular nature of the earth’s orbit. This causes the earth’s speed to change over the course of the year, so we do not sweep out exactly the same angle from day to day, if you were to assume that noon is when the sun reaches some position in the sky (like straight overhead, or on the vertical line that includes that point). In other words, the time you would read on a sundial will not agree with the time kept by a very good clock. The variation over the course of the year is on order of ± 15 minutes, or almost 4º of arc in each direction. The math that describes this is known as the equation of time.

More detail here or here or many other places on the web.