What is the analemma?

What is the analemma?

The analemma is the figure-eight path the Sun appears to trace over a year when it is observed from a single fixed spot at the same clock time each day. Photograph the Sun at, say, 9 a.m. on forty dates spread through the year, superimpose the frames, and the forty Suns do not land on a single point — they fall along a slim, slightly lopsided figure-eight. The U.S. Naval Observatory describes how the curve is constructed: "If the Equation of Time is plotted as the x-axis and the Sun's noon height is plotted as the y-axis, the result is a figure-8 curve for the year called the analemma."¹

The two coordinates of every point on the curve have plain physical meaning, and each is the subject of its own concept on this site. The horizontal position is the equation of time — how far ahead of or behind the steady clock the real Sun runs on that date. The vertical position is the Sun's declination, its angular distance north or south of the celestial equator, which is the seasonal part of the Sun's position. Sampling both at the same instant each day, and stacking a year of samples, folds the two annual cycles into one closed loop.¹³

The name is older than the photograph. Analemma comes from the Ancient Greek ἀνάλημμα, "support" or "pedestal," and through Ptolemy's treatise of that title it came to mean a geometric method for projecting the celestial sphere onto a plane. The modern, figure-eight sense — the Sun's own annual trace — is the one in use today, and it is the entity Wikidata catalogues as Q484737.⁴⁵

Why does the Sun trace a figure-eight?

The figure-eight is the sum of two separate annual rhythms, one vertical and one horizontal, sampled at the same clock time every day. Neither alone would produce a figure-eight; it is the way they combine that closes the loop and pinches its waist.

The vertical rhythm is the Sun's declination. Over a year it rises and falls once, smoothly, between −23.44° at the December solstice and +23.44° at the June solstice, passing through zero at the two equinoxes; the 23.44° figure is the tilt of Earth's axis itself.³ On its own, this up-and-down motion would draw a vertical line in the sky, retraced once each way per year. It is what sets the height of the analemma and puts the solstices at the very top and bottom. NASA's Astronomy Picture of the Day states the placement directly: the solstice Suns sit "at the top and bottom of the analemma curve."²

The horizontal rhythm is the equation of time, the gap between the time the real Sun keeps and the time a uniform clock keeps. Unlike declination, the equation of time changes sign four times a year: the Sun runs as much as about 16 minutes ahead of the clock in early November, about 14 minutes behind in mid-February, with two smaller excursions in between.¹ Because the Sun swings east and then west of the mean clock position more than once a year while it climbs and falls only once, the horizontal coordinate doubles back on itself partway through each half of the year. That doubling-back is what folds the trace into two loops instead of a single oval, and it is why the curve crosses itself once near the middle.

The equation of time is itself the sum of two physical causes, and the same two causes shape the analemma. One is the 23.44° tilt of Earth's axis, which makes the Sun's apparent motion along its yearly path project unevenly onto the celestial equator; this contributes a twice-a-year term that the U.S. Naval Observatory notes "can reach ±10 minutes."¹ The other is the slight ellipticity of Earth's orbit, which speeds the planet up near its closest approach to the Sun in early January and slows it near its farthest point in early July; this contributes a once-a-year term reaching about ±7.5 minutes.¹ The page on perihelion and aphelion treats the orbital-speed side of that story.

Why are the two loops unequal?

The analemma's two loops are visibly different sizes — a small one toward the top, a larger one toward the bottom — and the asymmetry traces back to a single fact of timing: Earth's closest and farthest points from the Sun do not line up with the solstices. Wikipedia's reference article puts the cause plainly: the difference in the size of the lobes "arises mainly from the fact that the perihelion and aphelion occur far from the equinoxes," falling "a couple of weeks after the solstices, which in turn causes a slight tilt of the figure eight and its minor lateral asymmetry."⁴

The mechanism is the phase relationship between the two terms of the equation of time. The tilt-driven term completes two cycles a year and the orbit-driven term completes one; their peaks would coincide neatly only if perihelion fell exactly on a solstice. It does not — perihelion currently falls in early January, about two weeks after the December solstice — so in the northern-winter half of the year the two terms reinforce each other and the equation of time swings wide (the early-November maximum of about +16 minutes), while in the northern-summer half they partly cancel and the swings stay gentle.¹ Wider horizontal swings make a wider loop. The result is that the lower loop, drawn out during the northern winter, is the larger one, and the upper loop, drawn during the northern summer, is the smaller. Seen from the Northern Hemisphere the smaller loop sits at the top, above the larger.⁴

How big is the analemma, and which way does it lean?

The analemma's height is fixed and the same everywhere on Earth. Its long axis runs from the June-solstice point to the December-solstice point, a span of twice the Sun's maximum declination — twice 23.44°, or "about 47°," as the reference article gives it.⁴³ That figure does not depend on latitude: an analemma photographed from the tropics and one photographed from a polar station are the same 47° tall. Its width, the full sweep of the equation of time, is far smaller — about 30 minutes of time from the November extreme to the February extreme, which at four minutes of time per degree of sky is under 8° of arc. The analemma is a tall, narrow figure, several times higher than it is wide.

What does change with location is the figure's orientation. The analemma is the same shape wherever it is seen, but it is tipped at a different angle depending on the observer's latitude and on the time of day chosen for the daily photograph. Seen from high latitudes near local noon it stands nearly upright; seen from the equator it lies on its side; at intermediate latitudes, or at hours away from noon, it leans at an angle in between.⁴ Whether the small loop appears at the top, the bottom, or the side is a matter of where and when the camera is pointed — but the loops, the 47° length, and the asymmetry travel with the Sun, not the observer.

How do you photograph an analemma?

Photographing an analemma takes a fixed camera, a single frame of film or one stacked digital composite, and a year of patience. The method is exact and unforgiving: the camera must not move between exposures, and every exposure must be made at the same clock time, so that the only thing changing from frame to frame is the Sun's own position. NASA's example was assembled from frames "taken at exactly the same time of day" on dates spread from one September to the next.² Modern practitioners lock a camera to a wall or tripod mount, return to it on clear days through the year, and add one short, heavily filtered exposure of the Sun each time, often with a final landscape frame to anchor the figure-eight above a horizon.

It is harder than it sounds, mostly because of the weather. Clear skies at the chosen minute on enough well-spaced dates are rare, and a camera left fixed for a year is exposed to everything that year brings. The first person to capture the whole figure on a single frame was the astrophotographer Dennis di Cicco, who over 1978 and 1979 made 48 separate exposures of the Sun on one piece of film through a fixed window, a project written up in the June 1979 issue of Sky & Telescope.⁶ The technique it pioneered — same camera, same clock time, one accumulating frame — is still the template, and the few dozen people who have since repeated it remain a small club.

What is the analemma good for?

For most of its history the analemma was a practical correction tool, not a photograph. Because its horizontal axis is the equation of time, the figure-eight encodes, for every date, exactly how far a sundial's reading differs from clock time. Sundial makers have exploited this for centuries: Wikipedia's reference article notes that analemmas "have been used in conjunction with sundials since the 18th century to convert between apparent and mean solar time."⁴ A sundial inscribed with a figure-eight — or a gnomon whose shadow tip falls on a printed analemma — lets the reader convert the dial's apparent solar time into ordinary clock time without a separate correction table.

The same figure-eight turns up on globes, in the blank ocean of the Pacific, for the same reason: it is the compact graphical statement of where the Sun is overhead at clock noon through the year, the declination scale doubling as a guide to the latitude of the subsolar point. And as a teaching object the analemma is unmatched: it makes two abstractions — the equation of time and the Sun's seasonal march in declination — visible at once, in a single shape that the Sun draws for itself.

Frequently asked questions