Armillary Sphere (2D): Sun-Path Chart, Meridian Section and Day Length

Presets

Live readouts

Graphs

Sun altitude through the day

Day length through the year

Solar declination through the year

Numerical checks

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How to use

Choose a place and a date, then press the day or year auto-play. Try: (1) Kyiv, summer solstice: the gold path is long and high, the day lasts about 16.5 h; (2) Equator, equinox: the Sun passes through the zenith at noon and the path is a straight line through the centre of the chart; (3) Arctic circle, midsummer: the Sun just touches the northern horizon at midnight; (4) North Pole: the Sun circles at constant altitude equal to its declination; (5) change the tilt and watch the tropic circles and the seasonal swing of the sunrise point. Space pauses, R resets.

The science

What an armillary sphere models. It is a skeleton of the celestial sphere: a fixed horizon ring and meridian ring for the observer, and a tilted set of rings that carry the sky—the celestial equator, the ecliptic (the Sun's yearly path, inclined by the obliquity ε = 23.44°) and the polar axis. For an observer at latitude φ the pole stands φ above the northern horizon, so the whole sky turns about an axis inclined at φ.

The Sun's coordinates. The Sun's ecliptic longitude advances uniformly, λ = 2π(d - 80)/365.2422 (d = day of year, equinox about day 80). From it:

sin δ = sin ε sin λ, tan α = cos ε tan λ

δ is the declination (the Sun's latitude on the celestial sphere) and α the right ascension. The local solar time h gives the hour angle H = 15°(h - 12), positive towards the west.

Altitude and azimuth. The two classic spherical-triangle results are

sin a = sin φ sin δ + cos φ cos δ cos H cos a sin A = -cos δ sin H, cos a cos A = sin δ cos φ - cos δ sin φ cos H (A from north through east)

The 3D page does not use these directly: it builds a rotation matrix from the sidereal angle θ = H + α and the colatitude 90° - φ and turns the whole brass sphere with it, so the sun bead sits where the rotated ecliptic ring puts it. A check compares the bead's world position against the closed form.

Sunrise, day length, noon. The Sun is on the horizon when a = 0, so cos H₀ = -tan φ tan δ and the day lasts 24 H₀/π hours. If the right-hand side is below -1 the Sun never sets (midnight sun); above +1 it never rises (polar night). Noon altitude is 90° - |φ - δ|. A vertical pole of height 1 casts a shadow cot(a).

The tilt slider. Setting ε = 0 removes the seasons: the Sun then follows the equator all year, every day is 12 h at every latitude and the tropic circles collapse. Larger ε widens the tropics and the polar circles.

Checks (run in the page). (1) altitude and azimuth from the closed form against the rotation-matrix route on a grid of 11 x 5 x 13 inputs; (2) noon altitude sampled over the day against 90° - (φ - δ); (3) day length at the equinox against 12 h; (4) day length on the solstice at 50° N, from a bisected horizon crossing, against the sunrise equation; (5) solstice declination against ε; (6) in 3D the bead position against the closed form.

Simplifications. Circular orbit at uniform speed (no eccentricity), so the equation of time is ignored and "solar time" is mean solar time at the observer; no atmospheric refraction (the real Sun rises about 4 minutes earlier at mid latitudes); the Sun is a point, so sunrise is the centre of the disc on the horizon; the observer sits at the centre of the sphere (no parallax). The globe in the centre is decorative.