Analemmas And The Equation Of Time: Why The Path Of The Sun Traces Out An 8 On Earth

Generally speaking if you go outside and take a photo of the Sun every day at the same time (and with the proper equipment) and overlay the photos, you will find that the Sun makes a large figure of eight pattern in the sky.

This is known as an "analemma", and of course is created by the Earth and Sun's relative motion. But it also has a lot to do with how we measure and keep time.  So, what makes this pattern, and would it be the same on other planets?

The simple answer taught in textbooks is that the Earth's 23.5° tilt causes the Sun to appear higher or lower in our sky throughout the year (affecting the vertical axis) while the Sun's elliptical orbit affects the horizontal position.

"The Sun will appear at its highest point in the sky, and highest point in the analemma, during summer. In the winter, the Sun is at its lowest point," the Stanford Solar Center explains. "The in-between times generate the rest of the analemma pattern. Analemmas viewed from different Earth latitudes have slightly different shapes, as do analemmas created at different times of the day."

But while the orbit and tilt of the planet play their part, this is not the only factor involved. In fact, what analemmas are measuring really is how far local time (e.g. UCT) differs from mean solar time. 

"Introductory textbooks and other pedagogical sources sometimes explain the analemma’s horizontal displacement as resulting solely or primarily from the eccentricity of the Earth’s orbit," a paper on the topic explains.

"This is clearly incorrect: Kepler’s second law implies that a planet’s orbital angular velocity is faster than the mean when the planet is near the perihelion and slower when it is near the aphelion. If this were the dominant factor in determining the equation of time, then the azimuth of the Sun, observed at noon, would be greater than 180◦ during half of the year and smaller than 180◦ during the other half, giving a figure-zero rather than a figure-eight for the analemma. Orbital eccentricity does play an important role in determining the precise form of the equation of time, but the analemma would still be a figure-eight if the Earth’s orbit were perfectly circular."

The reason for the figure of eight analemma is the difference between the actual position of the Sun in the sky and the mean solar time measured by your own clock.

"Measuring the apparent variation in the position of the Sun has been intensively studied by astronomers, and is well understood. It is the cause of variations in the length of a solar day and the difference between solar time and mean time," another paper on the topic explains.

"The variations are encapsulated in an expression called the Equation of Time (the term ‘equation’ is used here in a historical sense, meaning a correction or adjustment)."

For example, 12pm is noon every day of the year going by our clocks. But solar noon, when the Sun is actually highest in the sky, varies throughout the year. Take a photo of the Sun at 12pm every day, and you are mapping the difference between the time measured on your watch vs the actual position of the Sun at that point. 

On Earth, depending on where you are, the analemma can be at different angles, though it always traces out this figure-of-eight pattern. But on other planets, with their own orbits and mean solar times, they can take on different shapes.

Neptune and Uranus get their own figures of eight, while on Venus and Jupiter (should you be able to keep your mind on space photography while being burned and/or crushed to death), they trace out an ellipse shape. Mars gets a teardrop shape, the same as Saturn, while on tidally-locked Mercury (but not in the traditional sense, with its 3:2 spin-orbit resonance), the analemma would be a single point.