Parhelia

Halos Around a Sun 20° High

Tracing

Sundog Distance
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Circumzenithal Arc
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Halo Edge
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Rays Traced
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Your photo stays on this device. It is drawn under the arcs, never uploaded.

Set the Sky

The Sun, the ice, and how it rocks

Crystals

Shares of the light, not of the count: the page divides by each population's own rays.

How far plates and columns rock from level: one standard deviation.

View

Arcs and Their Faces

Every arc named on the sky, by its faces

    Press an arc to see the crystal and the two faces behind it.

    What Did I See

    Tick what you saw; the crystals follow

    In the Sky

    Nothing ticked yet. Start with the brightest thing in the photo.

    How It Is Traced

    Seven wavelengths, one index each, checked against published values

    A halo is sunlight bent by ice crystals shaped as hexagonal prisms: six side faces and two end faces. Light that enters one face and leaves another is bent by an amount fixed by the ice's refractive index and the angle between the two faces, and it piles up at the angle of minimum deviation. A side face and the side face two along make a 60° wedge, and the 22° halo, the sundogs and the tangent arcs are that wedge seen in three ways of holding the crystal. An end face and a side face make a 90° wedge: the 46° halo and the circumzenithal arc.

    Each ray here picks a crystal and its orientation, a point on the Sun's 0.53° disc and one of seven wavelengths. It enters a face chosen by that face's area as the beam sees it. At every face the Fresnel equations, averaged over the two polarisations, split it: the reflected share and the transmitted share both go on, the transmitted share leaving the crystal and the reflected share bouncing inside, up to the set number of internal reflections. Every share that leaves lands on the sky with its weight and its wavelength's colour, in one of twelve buffers kept apart by population and by face pair, so any arc can be shown alone.

    minimum deviation D = 2·asin(n·sin(A/2)) − A

    skew ray (Bravais) n′ = √(n² − sin²e) / cos e

    sundog azimuth Δφ = 2·asin(n′/2) − 60°, at the Sun's altitude

    circumzenithal arc sin e = √(n² − cos²h), only while h < 32.20°

    circumhorizontal arc cos e = √(n² − sin²h), only while h > 57.80°

    Fresnel, unpolarised R = (Rs + Rp)/2, T = (Ts + Tp)/2, R + T = 1

    Receipts: the page's figures beside the published ones
    CheckPublishedThis PageSource
    Ice, Warren and Brandt (2008)
    WavelengthIndex22° Edge

    The test suite traces the same rays against the same published values: the 22° edge from 200,000 random crystals lands 0.0006° from 21.84°, the sundog edge at a Sun 40° high 0.0013° from the closed form, and every circumzenithal ray within a millionth of a degree of its formula. The worst disagreement is with the sundog table at a Sun 50° high, 0.43° against a table printed to the whole degree.

    What It Does Not Do

    • Single scattering only: each ray meets one crystal. In a thick cloud light passes through several, and the arcs soften.
    • The crystals are ideal prisms with flat faces: plates 0.3 as long as they are wide, columns 2.0, Parry columns 1.5, random crystals 1.0. Odd-radius halos need pyramids, and Lowitz arcs need plates spinning about a horizontal axis; neither is here.
    • Colour comes from seven wavelengths, 400 to 700 nm, not a full spectrum, and polarisation is not carried from one face to the next.
    • The sky behind the arcs is drawn, not computed: no Rayleigh scattering, no cloud, no ground.
    • The published 46° figure found is "about 46°", so that receipt is good to half a degree, not a hundredth.

    Sources