Analemma Calculator

The analemma is the distinctive figure-8 curve traced by the Sun's apparent position in the sky when it is photographed from the same location at the same clock time on many days throughout a year. Rather than sitting at a fixed point in the sky, the Sun drifts systematically north and south (due to Earth's axial tilt) and also shifts east and west (due to the variable speed of Earth's orbit). Together, these two independent effects create the characteristic asymmetric figure-8 shape.

The vertical axis of the analemma corresponds to the Sun's declination — how far north or south of the celestial equator the Sun sits. This ranges from +23.5 degrees at the June solstice (Sun farthest north) to -23.5 degrees at the December solstice. The horizontal axis corresponds to the Equation of Time — the difference in minutes between mean solar time and apparent solar time, which can be as large as +16.4 minutes in early November and as negative as -14.3 minutes in mid-February.

The analemma's two lobes are unequal in size. The northern lobe (summer half for the Northern Hemisphere) is smaller and rounder, while the southern lobe is larger and more elongated. This asymmetry arises because Earth moves faster near perihelion (early January, when it is closest to the Sun) than near aphelion (early July, when farthest away). The faster orbital speed in winter partially cancels the declination change, compressing that half of the figure.

Sundial makers, architects designing solar-passive buildings, photographers capturing multi-exposure composites of the sky, and astronomers all use analemma data. This calculator computes the Equation of Time, solar declination, altitude, and azimuth for every five days of the year so you can trace the full analemma shape for any latitude.

The Equation of Time (EoT) measures how many minutes the apparent Sun (as read by a sundial) is ahead of or behind the mean Sun (as kept by clocks). Positive EoT means the sundial runs fast relative to the clock. The formula below is the NOAA/Fourier series approximation accurate to within 0.5 minutes:

Solar declination is the angle between the Sun's rays and the plane of Earth's equator. It changes throughout the year as Earth orbits the Sun with its axis tilted 23.44 degrees relative to the orbital plane. At the vernal and autumnal equinoxes, declination is approximately 0 degrees. At the summer solstice it peaks near +23.44 degrees, and at the winter solstice it reaches approximately -23.44 degrees.

For this calculator, the declination formula used is the Spencer (1971) Fourier approximation, which expresses declination as a function of the fractional year angle γ. This approximation has an accuracy of better than 0.035 degrees throughout the year, sufficient for sundial design and solar panel orientation calculations.

At your entered latitude, the solar altitude at the observation hour changes dramatically across the year because the higher the declination, the higher the Sun climbs above the horizon (for Northern Hemisphere observers). The maximum altitude difference between summer and winter solstice equals the full range of declination — approximately 47 degrees.

For each point on the analemma, the calculator derives the Sun's altitude (elevation above the horizon) and azimuth (compass bearing, measured clockwise from north) at your chosen observation hour. At solar noon the azimuth for a Northern Hemisphere observer is due south (180 degrees), and the altitude equals 90 degrees minus the observer's latitude plus the current declination.

Observing the analemma at any hour other than solar noon shifts the figure horizontally. At 9 AM the analemma appears west of its noon position; at 3 PM it appears east. The shape remains the same figure-8, simply offset in azimuth and reduced in altitude depending on the hour angle.

Composite analemma photographs require dozens of exposures taken at the same clock time (usually an hour before noon) on the same film or sensor. Each exposure captures the Sun at a slightly different position in the sky. After roughly 18 to 25 exposures spread throughout the year, the figure-8 becomes recognizable. Famous analemma composites have been published from locations in the USA, Greece, and Turkey, often with a landmark in the foreground.

For architecture and urban planning, analemma data helps predict when a building will shade a neighboring structure, whether a proposed window orientation will receive direct sunlight in winter, or how to orient solar thermal collectors for optimal year-round performance. Sundial correction tables are also derived directly from the Equation of Time values that form the horizontal axis of the analemma.