The Analemma and the Equation of Time: Why Sun Time Drifts from Clock Time

Set up a camera on a tripod, point it at the sky, and photograph the sun at exactly the same clock time — say, 9:00 a.m. — once a week for a full year. You might expect the sun to appear in the same spot in every frame, or perhaps to slide straight up and down as the seasons change. Instead, when you stack the images, the sun traces a lopsided figure-8 across the sky: tall, narrow at the top, wide at the bottom, tilted to one side. This shape is called the analemma, and it’s one of the most information-dense pictures in astronomy.

The vertical stretch of the figure-8 is easy to explain: the sun rides higher in summer and lower in winter because of Earth’s axial tilt. The east-west width is the interesting part. It exists because the sun does not keep steady time. Over the course of a year, true solar time — the time a sundial reads — runs as much as about 16 minutes ahead of clock time and about 14 minutes behind it. That drift is called the equation of time (“equation” in the old sense of “correction”), and it quietly shapes things you can observe from your own driveway, including why the earliest sunset of the year lands weeks before the winter solstice.

What a Clock Assumes That the Sun Doesn’t Deliver

Your clock keeps mean solar time: it assumes every day is exactly 24 hours long, which is the average length of a solar day over the whole year. A solar day — noon to noon, the time it takes the sun to return to its highest point in the sky — is close to 24 hours, but almost never exactly. Through the year, the true solar day stretches and shrinks by up to about 30 seconds either side of 24 hours.

Thirty seconds sounds trivial, but the discrepancies accumulate day after day in the same direction for weeks at a time. By mid-February the sundial has fallen roughly 14 minutes behind the clock; by early November it has pulled about 16 minutes ahead. The equation of time is simply this running total — the accumulated difference between where the sun actually is and where a perfectly uniform clock-sun would be.

Two independent effects cause the drift, and the analemma’s odd shape comes from how they combine.

Cause One: Earth’s Elliptical Orbit

Earth’s orbit is not a circle. It’s a slight ellipse, and we’re closest to the sun (perihelion) in early January and farthest (aphelion) in early July. Kepler’s second law says a planet moves faster when it’s closer to its star — and Earth does, sweeping along its orbit noticeably faster in January than in July.

Here’s why that changes the length of the day. A solar day is longer than one rotation of the Earth, because while we spin, we also advance along our orbit; the Earth has to rotate a little extra — about one degree, roughly four minutes’ worth — for the sun to return to the same position in the sky. When Earth is moving faster along its orbit in January, it covers more orbital ground each day, so the extra rotation needed is larger and the solar day runs longer than 24 hours. Around perihelion, true solar days exceed 24 hours by several seconds each, day after day, and the sun steadily loses ground against the clock. Around aphelion in July the effect reverses, more weakly, since the orbit’s speed variation is gentler there.

This orbital effect cycles once per year, with an amplitude of roughly 7 to 8 minutes in the accumulated equation of time.

Cause Two: The Tilt of Earth’s Axis

The second effect is subtler and has nothing to do with orbital speed. Earth’s axis is tilted about 23.4° relative to its orbit — the same tilt that gives us solstices and equinoxes. Because of that tilt, the sun’s apparent path through the stars (the ecliptic) is inclined to the celestial equator, which is the reference circle that clock time effectively measures against.

Think of it this way: what matters for timekeeping is the sun’s eastward progress measured along the celestial equator, because that’s the direction Earth’s rotation carries the sky. Near the equinoxes, the sun is crossing the equator at an angle — a chunk of its daily motion is “spent” moving north or south rather than east, so its eastward progress is slower than average and solar days run slightly short. Near the solstices, the sun travels parallel to the equator, and there’s a second, less obvious boost: at the solstices the sun sits far from the equator, where the projection geometry compresses its motion onto fewer degrees of the equator per degree traveled — the equatorial “lines of longitude” converge — so its eastward progress runs faster than average and solar days run long.

Because this pattern repeats between each equinox-solstice pair, the tilt effect cycles twice per year, with an amplitude of roughly 10 minutes in accumulated time.

Adding the Two Waves Together

The equation of time is the sum of one annual wave (ellipticity, ~7–8 minutes) and one semiannual wave (tilt, ~10 minutes), out of phase with each other. Two waves with different periods don’t add up to anything symmetric, which is why the totals swing unevenly:

• Mid-February: the two effects reinforce on the slow side; the sun runs about 14 minutes behind the clock.
• Mid-May: a modest fast excursion, sun about 3–4 minutes ahead.
• Late July: a modest slow excursion, about 6 minutes behind.
• Early November: the strongest reinforcement on the fast side; the sun runs about 16 minutes ahead of the clock.

The equation of time crosses zero — sundial and clock briefly agreeing — four times a year, around mid-April, mid-June, September 1, and December 25.

This mismatch is also why the analemma is lopsided. The figure-8’s two lobes reflect the semiannual tilt wave; the annual orbital wave inflates one lobe and shrinks the other. The bottom loop (winter, when the sun is low) is big and wide because the February and November extremes are the year’s largest; the top loop (summer) is small and narrow because May and July excursions are mild. If Earth’s orbit were perfectly circular, the analemma would be a symmetric figure-8; if Earth also had no axial tilt, it would collapse to a single point.

Why This Explains December’s Early Sunsets

The equation of time isn’t a curiosity — it’s the reason sunrise and sunset times behave in ways that surprise people every year. Solar noon, the moment the sun is highest, doesn’t sit fixed at 12:00 (or 1:00 during daylight saving time). It wanders through the year by that ±14–16 minute swing, on top of the fixed offset that depends on where you live within your time zone — the full story is in solar noon vs. clock noon.

In late autumn and early winter, solar noon is drifting later by clock time, and fast — moving nearly 30 seconds later per day in December, since solar days near perihelion run longest. Sunrise and sunset are roughly centered on solar noon, so both are being dragged later too. Meanwhile, the day length is shrinking toward the December solstice and then growing after it. Superimpose the two motions: before the solstice, the shrinking-daylight effect and the drifting-later effect push sunset in opposite directions, and the drift wins first — sunset stops getting earlier around December 7–8 at latitudes near 40°N, days before the solstice. Sunrise keeps getting later well into early January, because for sunrise the two effects reinforce until the lengthening days finally overpower the drift. The result: earliest sunset in early December, latest sunrise in early January, shortest day in between on the solstice. We walk through the full mechanics in why the latest sunset doesn’t fall on the longest day — the equation of time is the engine behind it.

Sundials, Corrections, and Reading Old Instruments

For most of human history, the sundial was the clock, and true solar time was simply what time meant. The trouble started when mechanical clocks became accurate enough to disagree with the sun consistently. A well-made clock keeps uniform time; the sun doesn’t. By the 18th and 19th centuries, almanacs printed equation-of-time tables so people could set their clocks from a sundial reading: take the dial time, apply the day’s correction, get mean time.

Serious sundials still handle this today. Many engraved garden sundials include a small table or a figure-8 graph of the correction by date. Some elaborate dials build the correction into the gnomon’s shape or use an analemma-shaped hour line so the dial reads clock time directly. If you ever check a sundial against your phone and find it 10 minutes off, it’s probably not broken — check the date against an equation-of-time table (and remember to add an hour during daylight saving time, plus your longitude offset within the time zone).

Capturing Your Own Analemma

Photographing an analemma is a genuine project — the first widely published one took a full year of effort in the late 1970s — but the method is simple and the failure modes are all about discipline:

1. Fix your camera position permanently. Same tripod position, same aim, same lens, all year. Most people mount a camera bracket to a wall or windowsill facing the southern sky (in the Northern Hemisphere).
2. Shoot at the same clock time. Pick one time and stick to it all year — and do not shift for daylight saving time. Use standard time year-round, or the figure-8 will break into two offset pieces.
3. Shoot every 7–10 days. Around 40 well-spaced exposures trace the full figure. Use a strong solar filter or extreme underexposure for the sun images, plus one normal exposure for the foreground.
4. Composite the frames. Stack the solar images over the foreground shot in any photo editor.

A mid-morning or mid-afternoon time gives the classic tilted figure-8; shooting at exactly solar-noon-ish time gives a vertical one. If a year of photography sounds like too much, plot it instead: look up your location’s solar noon time (our site lists it daily) once a week and graph it against the date. The wobble you chart is the equation of time, measured from your own address — the same drift Ptolemy knew about nearly two thousand years ago, written in the sky over your house.