Analemma

What you are actually looking at

Pin a camera to a wall, point it at a fixed patch of sky, and trip the shutter at exactly the same clock time — say 12:00 local mean time — on thirty or forty mornings spread evenly across a year. Stack the frames. The Sun does not land in the same spot each time. Instead the dots trace a long, lopsided figure-8 leaning across the sky. That curve is the analemma.

It surprises people because intuition says "same time, same place" should put the Sun in the same position. It does not, for two independent reasons that happen to overlap on Earth. The Sun drifts up and down over the year because Earth's spin axis is tilted, and it drifts left and right because Earth does not move around its orbit at a constant speed. The up-down swing repeats once per year; the left-right swing repeats roughly twice per year. Layer a once-a-year wiggle on top of a twice-a-year wiggle and you get a curve that crosses itself — a figure-8 — rather than a simple loop.

The vertical axis: axial tilt and declination

Earth's rotation axis is tilted 23.4° (the obliquity) relative to the plane of its orbit. As Earth circles the Sun, this tilt points the Northern Hemisphere toward the Sun in June and away in December. From the ground, the Sun's declination — its angular height relative to the celestial equator — rides up and down like a sine wave:

• June solstice (~June 21): declination +23.4°, the Sun reaches its highest noon altitude.
• Equinoxes (~Mar 20, Sep 23): declination 0°, the Sun sits on the celestial equator.
• December solstice (~Dec 21): declination -23.4°, the Sun's lowest noon altitude.

This declination swing is the tall axis of the figure-8. Its full sweep is 46.8° — more than 90 Sun-widths, since the Sun spans about half a degree. If axial tilt were the only effect, the analemma would be a perfectly vertical line segment: the Sun rising and falling along a single column with no sideways motion at all.

The horizontal axis: the equation of time

The width of the figure-8 comes from the equation of time: the difference between where the Sun actually is (apparent solar time, what a sundial shows) and where a perfectly regular clock says it should be (mean solar time). When the real Sun is "ahead" of the clock, the noon dot shifts one way; when it lags, it shifts the other. The total swing is about ±16 minutes, which on the sky translates to roughly ±4° of east-west displacement.

The equation of time is itself the sum of two separate contributions:

• The eccentricity term (once a year). Earth's orbit is an ellipse with eccentricity e = 0.0167. At perihelion (early January) Earth moves fastest, so the Sun appears to race ahead; at aphelion (early July) it dawdles. This is a single sine cycle peaking near ±7.7 minutes.
• The obliquity term (twice a year). Even on a circular orbit, projecting the Sun's steady motion along the tilted ecliptic onto the celestial equator speeds it up near the solstices and slows it near the equinoxes. This is a double-frequency sine peaking near ±9.9 minutes.

Add a once-per-year curve to a twice-per-year curve of comparable size, and the result is asymmetric: it does not cross zero at evenly spaced moments. That asymmetry is exactly what pinches the analemma into a figure-8 with one large loop and one small loop, rather than a symmetric "8" or a plain oval.

Decomposing the figure-8

The clearest way to understand the shape is to treat it as a parametric curve in two coordinates over the year:

Component	Axis	Source	Period	Amplitude
Declination	Vertical (N–S)	Axial tilt (obliquity)	1 / year	±23.4°
Eccentricity term	Horizontal (E–W)	Elliptical orbit, e = 0.0167	1 / year	±7.7 min
Obliquity term	Horizontal (E–W)	Ecliptic-to-equator projection	2 / year	±9.9 min
Equation of time	Horizontal (E–W)	Sum of the two terms above	mixed	-14.2 to +16.4 min

The vertical axis is dominated by one big, clean sine. The horizontal axis is a smaller sum of two sines. Where the equation of time crosses zero four times a year (around April 15, June 13, September 1, and December 25), the curve briefly aligns vertically — and the two crossings nearest April and September are where the loops of the 8 pass through each other.

How latitude changes the picture

The figure-8 itself is a property of Earth's orbit and tilt, so its intrinsic shape is the same for every observer. What changes with latitude is its orientation and altitude on your local sky:

Latitude	How the noon analemma appears
Equator (0°)	Nearly vertical and very high, crossing close to the zenith; both loops well above the horizon.
Mid-northern (~40°N)	Tilted "8" in the southern sky; large loop low (winter), small loop high (summer).
High-northern (~65°N)	Strongly leaned over toward the south; lower loop may dip near or below the horizon in winter.
Southern Hemisphere	Appears flipped — the large loop points toward the December (summer) solstice, in the northern sky.

Choosing a different clock time also rotates and shears the whole figure: an analemma shot at 8 a.m. leans differently from one shot at noon or 4 p.m., because the Sun's daily arc tilts the east-west and altitude axes relative to the horizon.

Analemmas on other worlds

Because the shape is built entirely from a planet's obliquity and orbital eccentricity, every planet has its own signature analemma. The ratio between the eccentricity term and the obliquity term decides whether you get a figure-8, a teardrop, or an ellipse.

Body	Axial tilt	Eccentricity	Analemma shape
Earth	23.4°	0.0167	Asymmetric figure-8 (loops nearly balanced)
Mars	25.2°	0.0934	Teardrop — eccentricity dominates, no crossover
Jupiter	3.1°	0.0489	Slim ellipse — tilt too small for a tall 8
Saturn	26.7°	0.0565	Skewed teardrop / fat 8

Mars is the most famous case: its eccentricity is more than five times Earth's, so the once-a-year term overwhelms the twice-a-year term and the loops never cross — the Martian analemma is a single droplet. The Opportunity and later rovers effectively confirmed this from the surface.

Why the analemma matters

• Timekeeping. The analemma is the geometric record of why sundials disagree with clocks — the equation of time was printed on quality sundials for centuries as a correction table.
• Navigation and surveying. Reducing a Sun sight to longitude requires the equation of time; the analemma is its visual summary.
• Solar engineering. Heliostats, solar trackers, and analemmatic sundials must account for the same up-down and side-to-side drift to keep pointing accurately.
• Planetary science. A measured analemma directly reads out a world's tilt and eccentricity — two of the most fundamental orbital parameters.
• Photographic milestone. Capturing one demands a full year of disciplined, fixed-frame imaging — a rite of passage for astrophotographers.

Common misconceptions

• "The Sun draws an 8 in a single day." No — the analemma is a composite of dozens of dates across a whole year, all at the same clock time.
• "It's caused only by the tilt." Tilt gives the height; without the equation of time it would be a straight vertical line, not an 8.
• "The two loops are the same size." They are unequal because the eccentricity (once-a-year) and obliquity (twice-a-year) terms add asymmetrically.
• "The widest point is at a solstice." The east-west extremes occur near early November and mid-February, set by the equation of time, not the solstices.
• "Earth is closest to the Sun in summer." Perihelion is in early January — Northern-Hemisphere winter; seasons come from tilt, not distance.
• "Every planet has a figure-8." Only worlds where the two terms are comparable; Mars gets a teardrop because eccentricity dominates.