Pantheon Oculus Sun Beam

What is modelled

The sun is a parallel beam. The oculus is a circular hole at the top of a hemispherical dome whose radius equals the height of the drum wall (21.65 m), so the whole room is 43.3 m wide and 43.3 m high. The section is cut in the vertical plane that contains the sun direction, so the beam is drawn as a strip of width equal to the oculus diameter, travelling away from the sun.

Equations

• Declination δ = 23.44° sin(2π(284 + day) / 365), hour angle H = 15° (t - 12 h).
• Altitude sin α = sin φ sin δ + cos φ cos δ cos H at latitude φ = 41.9° N.
• Beam centre falls by tan α per metre of horizontal travel: it reaches the wall at height 43.3 - 21.65 tan α, and the floor when α > arctan 2 = 63.4°.
• Beam power P = DNI · π d²/4 · sin α, the lit patch on a horizontal floor has the same area A as the hole (a parallel beam is only shifted), so the floor irradiance is DNI sin α.

Checks run in the page

• Oculus area from a 128-sided polygon against pi r².
• Noon altitude found by scanning the day against 90 - latitude + declination.
• Solar altitude from an independent rotation of the equatorial sun vector.
• Lit floor area counted on a grid against the aperture area.

Honest simplifications

No atmospheric refraction, no scattered skylight, no diffuse light reflected from the coffers, and the dome is an ideal hemisphere. The drawing is only a slice: the real lit patch on the dome and wall is an ellipse wrapped on a curved surface. The sun azimuth is shown but the slice always faces the sun.

• Choose June solstice noon: the beam is steep enough to reach the floor near the centre.
• Choose December solstice noon: the beam lands on the wall high above the floor.
• Widen the oculus to 11 m and watch the beam power and the patch size grow with the square of the diameter.