The Sun's Analemma: Read the Figure-Eight with a Worked Diagram

What is the Sun's analemma?

The Sun does not trace the whole eight during one day. Each observation supplies one point; observations distributed across a year reveal the pattern. This guide uses an original coordinate diagram to separate the two changing quantities before discussing what a real sky view would require.

An original diagram with clearly named axes

Read the vertical coordinate as declination: the Sun's angular position north or south of the celestial equator. Read the horizontal coordinate as the equation of time E, defined here as apparent solar time minus mean solar time. A positive E means that apparent solar time is ahead.

The larger lower loop is the part of this plotted curve with more negative declinations. “Lower” describes the page, not an instruction about which way a camera must face. A southern-hemisphere observer, a different time of day or a rotated camera can produce a differently oriented image.

Why does a figure-eight appear?

Earth's axial tilt makes solar declination change through the year. Tilt also matters to timekeeping: even motion along the tilted ecliptic would not project into uniform motion around the celestial equator. Meanwhile, Earth's motion around its elliptical orbit is not uniform. Together these effects produce the changing apparent-versus-mean solar time.

The two variations do not have identical phases. Combining their seasonal progress in two coordinates produces the familiar unequal loops. A useful idealized comparison is a circular orbit with no axial tilt: it removes the seasonal declination variation and the two effects discussed here. That comparison helps identify the ingredients without pretending it is a model of the actual Earth.

Do not attribute the entire shape to eccentricity alone, or confuse the analemma with Earth's orbital path viewed from space. The plotted eight concerns the apparent Sun sampled against a clock. NASA's linked discussion compares analemmas on different planets to show why their shapes need not be identical.

How the numbers in this guide were calculated

The diagram and table use the approximation in NOAA's solar-position calculation sheet. For a declared 365-day year, evaluate its fractional-year angle at hour 12: γ = 2π(n − 1)/365, where n is the day number beginning with January 1 as one. Trigonometric inputs are in radians.

E = 229.18[0.000075 + 0.001868 cos γ − 0.032077 sin γ − 0.014615 cos 2γ − 0.040849 sin 2γ], in minutes.

δ = 0.006918 − 0.399912 cos γ + 0.070257 sin γ − 0.006758 cos 2γ + 0.000907 sin 2γ − 0.002697 cos 3γ + 0.00148 sin 3γ, initially in radians. Multiply δ by 180/π to obtain degrees.

These are approximate model outputs, not measured positions or a precision ephemeris for a particular year. The plot uses days 1–365 and repeats the first model point at day 366 to close the curve. That closing point does not introduce a leap day. The model's denominator and calendar indexing must be reconsidered together when adapting it to a leap year.

Twelve monthly values you can compare with the curve

The decimal places make calculations inspectable; they do not claim observational accuracy to a hundredth of a degree or minute. The day numbers are non-leap-year positions of each month's fifteenth. Between monthly markers the diagram uses daily samples, so it is not simply twelve straight segments.

For February's marker, E ≈ −14.25 minutes and δ ≈ −12.95°. Apparent solar time is behind mean solar time, and the Sun is south of the celestial equator. November's marker instead has E ≈ +15.14 minutes while δ remains negative. A negative declination therefore does not determine the sign of E.

The row near zero in June is particularly useful. E is approximately −0.03 minutes, but declination is about +23.29°. Agreement between two solar clocks does not imply that the Sun is on the celestial equator.

Why four clock minutes correspond to one degree of hour angle

Hour angle describes angular displacement from the observer's meridian, conventionally positive westward. A full 360° corresponds to 24 hours, so one time minute corresponds to 0.25° of hour angle. At local mean solar noon, the equation-of-time contribution gives H = E/4 in degrees.

For an invented E = +8 minutes, H = +2°: the apparent Sun is west of the meridian at mean solar noon. For E = −8 minutes, H = −2° and it is east. These are angular positions relative to the meridian, not changes of two degrees in azimuth along the horizon.

The final column in the monthly table applies this conversion. It is not the hour angle at 12:00 on every civil clock. Civil noon also depends on longitude relative to the time-zone reference. The solar noon guide develops that separate offset and its worked calculations.

A coordinate plot is not a local sky photograph

To convert declination and hour angle to geometric altitude h at latitude φ, use sin h = sin φ sin δ + cos φ cos δ cos H. This already shows that the same declination can produce different altitudes for different observers or times. Azimuth needs its own calculation and stated directional convention.

For a simple invented case, take φ = 30°, δ = 0° and H = 0°. The altitude is 60°. Keep the same latitude and declination but use H = 2°; the altitude is about 59.94°. The familiar meridian expression 90° − |φ − δ| belongs to H = 0; it should not be silently applied to every same-clock-time sample.

A camera adds another layer through lens projection, framing and orientation. This article's horizontal scale is in time minutes, so measuring its apparent width with a ruler cannot yield a local angular sky width. Start with the numerical coordinates when making a further transformation.

What “the same time” must mean

For a consistent series, hold the location and the clock's UTC offset fixed, or convert every observation to a common time basis. A civil-clock change can otherwise shift part of the sequence even when each notebook entry says the same hour.

Suppose a fictional observing schedule uses 12:00 at UTC+1 during one part of the year, then 12:00 at UTC+2. Those entries refer to different UTC instants, one hour apart. A one-hour time change corresponds to 15° of hour angle, far larger than the equation-of-time offsets in the table. This is an illustrative offset change, not a claim about any place's current clock rules.

To keep the same UTC instant in that example, 12:00 at UTC+1 corresponds to 13:00 at UTC+2. Record the offset explicitly rather than relying on a phone screenshot or an ambiguous abbreviation. Also keep a stable observing position; moving to another longitude changes the relation between clock time and the Sun.

Three events that should not be confused

A zero of declination places the Sun on the celestial equator. A zero of E means apparent and mean solar time agree under the stated definition. The crossing of an analemma's two branches means that two parts of the yearly sequence share the same plotted coordinate pair, approximately or within the model's resolution.

These are different conditions. An equinox is not defined by E = 0, and the crossing point is not automatically an equinox. Nor does a zero of E guarantee that a city's civil clock shows 12:00 at solar noon: the longitude and time-zone offset remain.

Twelve monthly samples cannot establish the exact dates of roots, extrema or branch intersections. Even interpolation between daily samples should be described as a model estimate. For an actual year's event timing, use an appropriate ephemeris and a specified time scale instead of promoting this teaching diagram into a calendar authority.

A six-step shadow-recording worksheet

1. Choose a fixed vertical stick and a stable, level recording surface where a shadow can be measured.
2. Record the location, the stick's height and the orientation of the surface.
3. Choose a repeatable clock time and write its UTC offset beside it.
4. At each sample, record the date and shadow-tip position without moving the stick.
5. Mark missing or cloudy days as missing; do not invent observations to smooth the curve.
6. Compare the pattern with an appropriate local shadow projection, keeping it separate from this E–δ plot.

Shadow recording lets you study the changing solar direction without looking at the Sun. The shadow tip is not a direct copy of the diagram above: its location depends on solar altitude, azimuth and the geometry of the stick and surface. At some locations and seasons the chosen time may also give an unsuitable or absent shadow.

Keep observations and model values in separate columns. A deviation is a reason to inspect timing, alignment, weather and modelling assumptions. It is not a reason to adjust the notebook until it reproduces a picture.

Use the analemma to connect several solar questions

The annual curve brings together two quantities that are often taught separately: changing declination and unequal apparent solar days. It helps explain why fixed-clock observations shift both vertically and horizontally through a year, while preserving the distinction between angular coordinates and clock minutes.

Continue with sunrise direction and azimuth for horizon geometry, or equinox and equal day length for a different seasonal question. Use the local sunrise and sunset tool for place-specific times. An annual teaching curve cannot supply those local answers by itself.

Frequently asked questions