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The common optical halo has a radius of approximately $22^{\circ }$ around the Sun or Moon. Refraction through a $60^{\circ }$ ice-crystal prism supplies a minimum-deviation edge. Nearby incident directions are concentrated near that edge, forming a caustic.^[ref]

An optical halo is a ring, arc or other light pattern formed by refraction or reflection in atmospheric ice crystals. It is distinct from a dark-matter halo.

A common optical halo is produced by refraction through hexagonal atmospheric ice crystals. For a $60^{\circ }$ prism path and ice refractive index about $1.31$, the minimum deviation relation givesRandom suitable crystal orientations produce a ring around the Sun or Moon, with concentration near that minimum angle. Optical dispersion puts red near the inner edge. Other crystal paths or orientations produce other halos and arcs; the explanation should not treat every ring as the water-drop mechanism of a rainbow.^[ref]

A waterspout is a rotating atmospheric column over water. Converging air and an updraft can stretch and intensify existing rotation; cooling and condensation make part of the funnel visible, while strong surface winds raise spray. The visible funnel is not simply a solid column of sea water sucked intact to the cloud. Fair-weather waterspouts and thunderstorm-associated tornadic waterspouts have different developmental settings; a single thunderstorm-only explanation does not cover both.^[ref]

Hoar frost forms by deposition of atmospheric water vapour as ice on a sufficiently cold surface. The surface must be below freezing and below the relevant frost point. Radiative cooling on a clear quiet night can make a surface colder than air at ordinary thermometer height, so a positive reported air temperature does not rule out surface frost. Freezing previously formed liquid dew is frozen dew, a different formation route.^[ref]

Hail grows in a thunderstorm when ice embryos collect supercooled liquid droplets that freeze on them. A strong updraft increases residence time and can support growing stones; they fall out when their trajectories and weight take them out of adequate support. Clear and opaque ice layers reflect changing growth and freezing conditions. They are not proof that every layer required a complete repeated ascent and descent: horizontal movement and variations across an updraft matter too. Nor is the very cold cloud top necessarily the best growth region, because appreciable growth requires liquid water.^[ref]

Snell's law is $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, with angles measured from the interface normal. The Huygens-Fresnel principle explains it by constructing the transmitted wavefront from secondary wavelets: during the same time interval, distances travelled in the two media are proportional to $c/n_1$ and $c/n_2$. The geometry of the advancing plane wavefront then gives $\sin {\theta }_1/\sin {\theta }_2=(c/n_1)/(c/n_2)=n_2/n_1$.

For incidence from index $n$ into vacuum, the transmitted angle is $\alpha =\arcsin (n\sin \theta )$. The ray bends away from the normal, so its positive deviation isFor $0<\theta <{\theta }_c=\arcsin (1/n)$,Thus the graph rises monotonically from zero to a finite endpoint with a vertical tangent:At the endpoint the transmitted ray grazes the interface. Beyond the critical angle for total internal reflection, no propagating transmitted ray exists and total internal reflection occurs.

Inside a prism, the two ray angles with the face normals sum to its apex angle. For the equilateral cross-section, they are $\theta$ and $\pi /3-\theta$. The entry deviation equals the corresponding exit deviation under ray reversal. Adding the two changes of direction therefore givesThis is symmetric about $\theta =\pi /6$. Symmetry alone gives a stationary point; to establish that it is a minimum, differentiate once more:within the transmitted-ray interval. The sum is therefore strictly convex, andThese expressions require $1<n<2$ for a non-grazing path through both faces; $n=2$ is a grazing limit, and $n>2$ admits no such two-face transmitted path. More explicitly, both internal angles must be at most ${\theta }_c$, which is possible only if $2{\theta }_c\ge \pi /3$.

Atmospheric hexagonal ice crystals can present pairs of faces with a $60^{\circ }$ angle, acting as small prisms. Their orientations turn the minimum-deviation cone around the Moon into an optical halo. No transmitted ray of this family is deflected less than the minimum deviation of a prism. Moreover, near the minimum, $\Delta -{\Delta }_{\min }$ is quadratic in the departure from the symmetric ray, so a range of incident directions is compressed into a small outgoing angular range. This caustic makes the inner rim sharp and bright, with the ray density decreasing farther outside. The common 22-degree halo has an angular radius near $22^{\circ }$ about either the Sun or Moon, hence a diameter near $44^{\circ }$.^[ref]