Forty-two degrees, and who chose it
Nobody chose it. A raindrop is a sphere, light bends by Snell's law, and water's refractive index is 1.333 — a number measured in a laboratory, with no rainbow in sight. Everything below follows from those three things: the angle, the dark band above it, the second bow with its colours reversed, two more bows behind your head that nobody ever sees, and the size of raindrop at which the extra fringes stop being visible. Nothing here is fitted to an observation. The observations are at the bottom, keeping score.
One drop, one ray
Aim the ray at the drop, from dead centre (b = 0) to a graze
(b = 1). It refracts in, bounces once off the inside,
refracts out, and leaves at some angle from the point in the sky directly opposite the sun.
Sweep the slider and watch that angle: it goes up, and then it turns around and comes
back. It never gets past 42.08°.
The turn-around is the bow
A drop sends light everywhere between 0° and that limit, and nothing at all
past it. Rays crowd up against the edge — a whole range of b leaves at almost
exactly the turning angle, because that is what a turning point is — so the edge is bright and
beyond it is dark. That is the bow: not a ring of drops doing something special, but the place
where every drop in the sky runs out of angle. Each colour runs out at its own angle, because
each colour has its own index, and that is the only reason a rainbow is coloured.
The sky, from the same rays
Now fire the rays properly: 8 000 impact parameters
per wavelength, 24 wavelengths, weighted by how much of the beam each one carries
(b·db — a drop is a disc, not a line) and by the Fresnel fraction that
actually gets through each surface rather than reflecting off it. Bin what comes out by angle,
divide by sin θ because a ring at angle θ is that much sky, and convert the
spectrum in each bin to a colour through the CIE curves. No rainbow was put in. This strip is
what falls out.
Every bow this drop can make
The measured column is what people with instruments report, and it was not used to compute anything in the other columns. The third and fourth bows are real and computable and sit around the sun, not opposite it — the third was first photographed in 2011, a hundred and fifty years after it was predicted, because you are looking into the glare to find something a hundred times fainter than the primary.
Where the geometry gives up
Geometric optics says the brightness at the turning angle is
infinite, which is a page announcing its own failure. Airy's fix in 1838 was to stop
counting rays and add up the wavefront, which is cubic near a turning point, and the integral
of a cubic phase is the Airy function. It says three things a ray cannot: the bright edge is
not at 42° but a little inside it, there are extra fringes below the bow, and both
effects scale as a−2/3 with the drop radius. Slide the drops from mist
to downpour. When the fringe spacing falls below 0.53° — the width of the sun itself,
which smears every one of these curves by that much — the fringes stop existing as far as your
eye is concerned. That happens at 240 µm, and it is why supernumerary
bows are a thing you see in fine mist and not in a thunderstorm.
What the page checked before showing you
Every line above is recomputed in the browser each time this page loads. The
same claims are checked again, by a different route and at forty digits, in
tests/test_bows.py — the closed forms there are derived rather than sampled,
the arithmetic is decimal rather than doubles, and the numbers the outside world
supplies (the index of water, the Abbe number, the zeros of the Airy function, the angles
people have measured in the sky) are typed in there again from print.