The Equation of Time and the Analemma

How a sundial drifts from the clock

If you set up a perfect sundial in your garden and a perfect clock beside it, you might expect them to agree at noon every day. They almost never do. On some dates the Sun reaches its highest point in the sky several minutes before your clock strikes twelve, and on others several minutes after. The gap, measured day by day, is called the equation of time. Here the word equation keeps its old meaning of a correction you add to make two things equal, not a formula with an equals sign.

Across a year the gap is never large, but it is surprisingly stubborn. It reaches about 16 minutes ahead of the clock in early November and about 14 minutes behind in mid February, and it passes through zero four times. None of this is the Sun misbehaving. It comes from two simple facts about Earth's motion that the clock quietly ignores, and the live curve on the left shows each one as its own gentle wave.

Two kinds of time

There are really two different Suns in this story. The first is the actual Sun in the sky. The moment it crosses due south and stands at its highest is called apparent solar noon, and a sundial faithfully tracks this apparent solar time. The trouble is that apparent solar days are not all the same length. From one real noon to the next can be a little more or a little less than 24 hours, and those small differences pile up.

So for everyday timekeeping we invented a second, imaginary Sun that moves at a perfectly steady pace all year. The time it would keep is called mean solar time, and that is what your clock shows. The equation of time is simply the first minus the second: apparent time minus mean time, the sundial minus the clock. When it is positive the sundial is ahead, meaning the real Sun reaches the meridian before clock noon. When it is negative the sundial lags behind. The readout above turns this into a plain sentence, including the clock time at which the Sun really crosses the meridian, its highest point, each day.

Cause one: Earth's lopsided orbit

Earth's orbit is not a perfect circle but a slightly flattened ellipse. That means our distance from the Sun changes a little through the year, and by Kepler's second law our speed changes with it. Earth sweeps along fastest at perihelion, its closest approach around the 3rd of January, and slowest at aphelion, its farthest point around the 4th of July. Because we are the ones moving, the Sun appears to slide across the background sky faster in January and slower in July.

The steady mean Sun cannot keep up with these changes, so the true Sun runs ahead of it for half the year and behind it for the other half. This is the eccentricity effect, drawn in green on the curve. It is a single smooth wave that rises and falls once a year, about 7.7 minutes at its largest, and it passes through zero right at perihelion and aphelion, where Earth's speed is momentarily at its average.

Cause two: the tilt of the axis

The second cause is the 23.4 degree tilt of Earth's axis, the same tilt that gives us the seasons. Because of it the Sun does not travel along the celestial equator, the line our clocks effectively measure against, but along the tilted circle of the ecliptic. Even if the Sun moved along the ecliptic at a perfectly even speed, its progress measured eastward along the equator would not be even.

Near the equinoxes the Sun's path slants steeply across the equator, so a day's worth of motion adds up to less eastward progress and the Sun seems to fall behind. Near the solstices the Sun's path runs almost parallel to the equator, and the Sun sits far from it, where the sky's lines of longitude crowd closer together. A single step along the path then counts for more eastward progress, so the Sun runs ahead. This obliquity effect, drawn in gold, is a faster wave that rises and falls twice a year, about 9.9 minutes at its largest, and it passes through zero at all four turning points of the seasons: both equinoxes and both solstices.

Adding the two waves

The white curve is just the green wave plus the gold wave, added day by day. Two ordinary waves of different rhythm combine into the lopsided shape that has puzzled sundial readers for centuries. It has four turning points: the deep minimum near the 11th of February, around 14 minutes behind; a small hump near the 14th of May, only about 4 minutes ahead; a shallow dip near the 26th of July, about six and a half minutes behind; and the tall maximum near the 3rd of November, about sixteen and a half minutes ahead. In between it crosses zero, when sundial and clock briefly agree, around the 15th of April, the 13th of June, the 1st of September, and the 25th of December.

Tap the turning-point buttons to jump straight to each of these moments, or press Play to let a year roll past. Notice how the tall November peak and deep February trough happen when the two waves push the same way at once, while the gentle summer stretch happens when they pull against each other.

The analemma: a year of noon Suns

Now look at the right panel. Imagine photographing the Sun from the same spot at exactly the same clock time, say noon, on many days through the year, then laying all the photos on top of one another. The Sun does not land in the same place each time. It drifts up and down as its declination climbs toward summer and sinks toward winter, and it drifts left and right by exactly the equation of time. Trace all those positions and you get a tall, slender figure-eight called the analemma.

The height of the figure-eight comes from the Sun's declination swinging between plus and minus 23.4 degrees, so the top of the loop is the high June Sun and the bottom is the low December Sun. The width comes from the equation of time. The two loops are not the same size: the lower loop is fat and the upper loop is pinched. That is because near the December solstice the orbit and tilt effects reinforce each other and the equation of time swings widely, while near the June solstice they nearly cancel and the swing is small. The two loops do not meet in the center either. Their crossover rides high, above the midline at about nine degrees of declination, so the fat December loop hangs well below it while the pinched June loop sits just above. That upward offset traces back to Earth reaching perihelion in early January, a few weeks after the December solstice rather than exactly on it. The same figure-eight is why globes often carry a curious figure-eight mark out in the empty ocean.

Why the earliest sunset beats the shortest day

The equation of time is not just a curiosity for sundial builders. It explains a puzzle many people notice in December: in the northern hemisphere the earliest sunset arrives a week or two before the shortest day, and the latest sunrise a week or two after it. The solstice still has the least daylight overall, but because solar noon keeps sliding later on the clock through that stretch, the whole solar day shifts and sunset bottoms out early. The same effect runs in reverse around the June solstice.

It is also why precise timekeeping abandoned the Sun long ago. Clocks, time zones, navigation, and the satellites in the global positioning system all run on smooth mean time, with the Sun's wandering folded into corrections rather than lived with directly. The sundial in the garden keeps the older, truer, but less convenient time, and the equation of time is the bridge between the two.

How this visual works

The curve and the analemma are built from a compact day-of-year approximation: an annual sine wave for the orbit-shape effect plus a twice-yearly sine wave for the tilt effect, with the Sun's declination from the standard cosine approximation. By construction the green and gold waves add up to the white total, and the model reproduces the real turning points and zero crossings to within a few days, which is more than enough to show how the cycle works. For minute-accurate values on a specific date, an almanac or a precise tool such as the NOAA Solar Calculator uses the full astronomical expressions. The distances and sizes in the diagrams are not to scale.