Analemma Simulator

What is an analemma

If you photograph the Sun from the same place and at the same clock time throughout a year, its position in the sky does not repeat: it rises and falls with the seasons and, in addition, runs a few minutes ahead of or behind the clock. Superimposed, the shots trace a closed figure-eight curve called an analemma.

The vertical extent is due to the tilt of Earth's axis: the Sun's declination swings between +23.4 and −23.4 degrees, so the figure eight is about 47 degrees tall. The sideways shift is the equation of time, the difference between clock time, based on a mean Sun that always moves at the same rate, and the position of the true Sun. That difference reaches about 16 minutes, equivalent to 4 degrees. It has two combined causes: the same axial tilt and the eccentricity of the orbit, which makes Earth move faster near perihelion, in early January. That is why the two loops of the figure eight are not equal.

Clock time and daylight saving time

All shots must be taken at the same time in a fixed time zone. If the clock moves forward an hour for daylight saving time halfway through the year, the Sun appears 15 degrees further along its daily path and the analemma splits into two sections. That is why the simulation works with a constant time zone, and the camera clock must stay on standard time all year.

The chosen time defines the orientation. Near solar noon the figure eight stands upright, and in mid-morning or mid-afternoon it appears tilted and closer to the horizon, which makes it possible to include the landscape in the shot. The mean solar time value shows which moment of the solar day the chosen clock time corresponds to at that longitude.

Solstices, equinoxes and the figure-eight crossing

The top and bottom ends of the figure eight roughly correspond to the solstices, but not exactly: with a fixed clock time, the equation of time shifts the highest or lowest point by a few days. The dates in the simulation are the astronomical instants, computed from the Sun's ecliptic longitude.

Nor are the equinoxes at the crossing of the figure eight, as is often believed: with zero declination, they sit halfway up, on the sides of the larger loop. The crossing occurs in mid-April and late August, when the Sun passes through the same declination with the same value of the equation of time.

Temporal and spatial distribution

With temporal distribution the shots are taken at a fixed interval of days. Since the Sun's declination changes much faster near the equinoxes, up to 0.4 degrees per day, than near the solstices, the discs end up spread apart on the sides of the figure eight and bunched together at its ends.

Spatial distribution spaces the shots according to the distance traveled along the curve, so that the discs are evenly separated. The cost is an irregular calendar, with shorter intervals near the equinoxes and longer ones near the solstices. In both cases the dates are rounded to whole days, so the spacing is never perfectly uniform.

A shot lost to clouds is not always serious: near the solstices a one-day difference barely moves the disc, while near the equinoxes it shifts it by almost a diameter.

The camera view and the focal length

The simulation uses a gnomonic projection, which is the one produced by a rectilinear lens. The camera is assumed to be level, so the horizon comes out straight. The Sun's disc is drawn to real scale, half a degree across, and the altitude includes atmospheric refraction.

The maximum focal length is the longest one that frames the complete analemma, with a margin, for the chosen sensor and orientation. When possible, it also looks for the tilt that keeps the horizon inside the frame. If the analemma is very high, near the zenith, no rectilinear lens can cover it together with the horizon, and the focal length is suggested for the analemma alone.

Practical tips

Never point the camera at the Sun without a proper solar filter, always placed in front of the lens, and do not look through the optical viewfinder. The camera must stay in exactly the same position and orientation all year: ideally on a fixed mount, or otherwise with reference marks that allow it to be placed precisely. It is best to work in manual mode, with focus, exposure and white balance fixed.

Most analemmas are composed with an additional shot of the landscape, taken without the filter and from the same camera position, over which the discs are superimposed. The shooting schedule can be downloaded in CSV format to load it into a calendar or print it.