Analemma & Shadow Clock
A vertical stick — a gnomon — casts a shadow that sweeps across the ground as the sun moves. Over the course of a year, the noon shadow tip traces a figure-8 pattern called the analemma, caused by Earth’s axial tilt (23.44°) and orbital eccentricity. Set your latitude, time, and date to explore.
altitude = arcsin(sin φ sin δ + cos φ cos δ cos h) | EoT = −7.66 sin(B) + 9.87 sin(2B + 3.59°)
The analemma
If you photograph the sun at the same clock time every day for a year, it traces a figure-8 shape in the sky called the analemma. This pattern arises from two effects: Earth’s axial tilt (obliquity, 23.44°) and the eccentricity of Earth’s orbit. The tilt causes the sun to move north and south over the seasons, while the orbital eccentricity causes the sun to speed up and slow down along the ecliptic.
The equation of time
The equation of time is the difference between apparent solar time (read from a sundial) and mean solar time (read from a clock). It varies by up to ±16 minutes over the year. The two main components are: (1) the effect of orbital eccentricity, which shifts the sun ahead or behind by up to 7.7 minutes, and (2) the effect of obliquity, which adds a sinusoidal variation of up to 9.9 minutes. Their sum produces the characteristic double-humped curve.
Solar position
The sun’s position is described by its altitude (angle above the horizon) and azimuth (compass bearing). These depend on the observer’s latitude, the solar declination (which varies with the day of year), and the hour angle (which varies with the time of day). At solar noon, the sun reaches its highest point and the shadow is shortest, pointing due north (in the northern hemisphere) or due south (in the southern).
The gnomon and sundials
A gnomon is the vertical stick or fin that casts the shadow on a sundial. The simplest sundial is a vertical stick on a flat surface — a horizontal sundial. The shadow’s direction indicates the solar time, while its length indicates the sun’s altitude. However, because of the equation of time, a sundial rarely shows exact clock time — it can be off by up to 16 minutes.
Solstices and equinoxes
The summer solstice (around June 21) is the longest day, when the sun reaches its highest noon altitude and shadows are shortest. The winter solstice (around December 21) is the shortest day, with the longest noon shadows. At the equinoxes (around March 20 and September 22), day and night are approximately equal, and the sun rises exactly east and sets exactly west.