Abstract

A radiance formula shows that the halo arcs produced by ice crystals with two degrees of rotational freedom, the “tangent arcs,” are loci on which a mathematical function, the Jacobian, is zero. This enables complete analytic theories of many halo arcs to be given. In other cases, points of contact with the circular halos, and intersections with the solar vertical and parhelic circle, can be found analytically, giving a qualitative description of the forms of the arcs.

© 1976 Optical Society of America

Full Article  |  PDF Article

Corrections

Richard White, "Erratum: An analytic theory of certain halo arcs," J. Opt. Soc. Am. 66, 1446_1-1446 (1976)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-66-12-1446_1

References

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  1. W. J. Humphreys, Physics of the Air (Dover, New York, 1964).
  2. M. Minnaert, Light and Colour in the Open Air (Bell, London, 1952).
  3. C. S. Hastings, “A General Theory of Halos,” Mon. Weath. Rev. 48, 322–330 (1920).
    [Crossref]
  4. R. G. Greenler and A. J. Mallman, “Circumscribed Halos,” Science 176, 128–131 (1972).
    [Crossref] [PubMed]
  5. G. A. Gibson, Advanced Calculus (MacMillan, London, 1931), Eq. (3), p. 138.
  6. A. Bravais, “Memoire sur les halos,” J. l’Ecole R. Polytech. 18, (31), 1–270 (1847).
  7. H. S. Uhler, “On the deviation produced by prisms,” Am. J. Sci. Ser. 4 35, 389–423 (1913).
    [Crossref]
  8. E. C. W. Goldie, “A graphical guide to halos,” Weather 26, 391–393 (1971).
    [Crossref]
  9. E. C. W. Goldie and J. M. Heighes, “The Berkshire Halo Display of 11 May 1965,” Weather 23, 61–69 (1968).
    [Crossref]
  10. Measured from the source vertical.

1972 (1)

R. G. Greenler and A. J. Mallman, “Circumscribed Halos,” Science 176, 128–131 (1972).
[Crossref] [PubMed]

1971 (1)

E. C. W. Goldie, “A graphical guide to halos,” Weather 26, 391–393 (1971).
[Crossref]

1968 (1)

E. C. W. Goldie and J. M. Heighes, “The Berkshire Halo Display of 11 May 1965,” Weather 23, 61–69 (1968).
[Crossref]

1920 (1)

C. S. Hastings, “A General Theory of Halos,” Mon. Weath. Rev. 48, 322–330 (1920).
[Crossref]

1913 (1)

H. S. Uhler, “On the deviation produced by prisms,” Am. J. Sci. Ser. 4 35, 389–423 (1913).
[Crossref]

1847 (1)

A. Bravais, “Memoire sur les halos,” J. l’Ecole R. Polytech. 18, (31), 1–270 (1847).

Bravais, A.

A. Bravais, “Memoire sur les halos,” J. l’Ecole R. Polytech. 18, (31), 1–270 (1847).

Gibson, G. A.

G. A. Gibson, Advanced Calculus (MacMillan, London, 1931), Eq. (3), p. 138.

Goldie, E. C. W.

E. C. W. Goldie, “A graphical guide to halos,” Weather 26, 391–393 (1971).
[Crossref]

E. C. W. Goldie and J. M. Heighes, “The Berkshire Halo Display of 11 May 1965,” Weather 23, 61–69 (1968).
[Crossref]

Greenler, R. G.

R. G. Greenler and A. J. Mallman, “Circumscribed Halos,” Science 176, 128–131 (1972).
[Crossref] [PubMed]

Hastings, C. S.

C. S. Hastings, “A General Theory of Halos,” Mon. Weath. Rev. 48, 322–330 (1920).
[Crossref]

Heighes, J. M.

E. C. W. Goldie and J. M. Heighes, “The Berkshire Halo Display of 11 May 1965,” Weather 23, 61–69 (1968).
[Crossref]

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964).

Mallman, A. J.

R. G. Greenler and A. J. Mallman, “Circumscribed Halos,” Science 176, 128–131 (1972).
[Crossref] [PubMed]

Minnaert, M.

M. Minnaert, Light and Colour in the Open Air (Bell, London, 1952).

Uhler, H. S.

H. S. Uhler, “On the deviation produced by prisms,” Am. J. Sci. Ser. 4 35, 389–423 (1913).
[Crossref]

Am. J. Sci. Ser. 4 (1)

H. S. Uhler, “On the deviation produced by prisms,” Am. J. Sci. Ser. 4 35, 389–423 (1913).
[Crossref]

J. l’Ecole R. Polytech. (1)

A. Bravais, “Memoire sur les halos,” J. l’Ecole R. Polytech. 18, (31), 1–270 (1847).

Mon. Weath. Rev. (1)

C. S. Hastings, “A General Theory of Halos,” Mon. Weath. Rev. 48, 322–330 (1920).
[Crossref]

Science (1)

R. G. Greenler and A. J. Mallman, “Circumscribed Halos,” Science 176, 128–131 (1972).
[Crossref] [PubMed]

Weather (2)

E. C. W. Goldie, “A graphical guide to halos,” Weather 26, 391–393 (1971).
[Crossref]

E. C. W. Goldie and J. M. Heighes, “The Berkshire Halo Display of 11 May 1965,” Weather 23, 61–69 (1968).
[Crossref]

Other (4)

Measured from the source vertical.

G. A. Gibson, Advanced Calculus (MacMillan, London, 1931), Eq. (3), p. 138.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964).

M. Minnaert, Light and Colour in the Open Air (Bell, London, 1952).

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Figures (5)

FIG. 1
FIG. 1

Hastings’ A′ (left) and B′ (right) crystals, with the R1 axis vertical, and the R2 axis horizontal. Lettering of faces referred from Table I.

Fig. 2
Fig. 2

Ray incident along IP on a triangular prism. Refracting edge at right. Note x, y are coordinates here, not as in the text.

FIG. 3
FIG. 3

Quantities used in derivation of Eq. (4). ζ, λ are altitudes. O is the crystal at the center of the celestial sphere. S is the source, and the refracting edge and R2 axis extended meet the celestial sphere at P, R2, respectively. AR2 is part of the horizon.

FIG. 4
FIG. 4

Production, on the solar vertical, of arcs corresponding to all solutions of the equation of the arc, by multiple internal reflection. (Supralateral arc left.)

FIG. 5
FIG. 5

Crystal and ray path for the Goldie arc, on the solar vertical.

Tables (1)

Tables Icon

TABLE I Halo arcs of the second kind. Reciprocal arcs are those in which the roles of the entry and exit faces, and hence of l′, m′, and l, m defined in the text, are interchanged. Lettering of faces refers to Fig. 1.

Equations (40)

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J , α η ψ ρ - α ρ ψ η , = 0.
( F , G ) ( η , ρ ) = ( α , ψ ) ( η , ρ ) ( F , G ) ( α , ψ ) ,
F ( ψ , α , η , ρ ) = 0 ,             G ( ψ , α , η , ρ ) = 0 ,
( F , G ) ( η , ρ ) = 0 ,
x = x ,
y 2 + y 2 + 2 y y cos 2 δ = ( μ 2 - x 2 ) sin 2 2 δ ,
sin γ = cos ω cos ζ cos η + sin η sin ω sin ρ cos ζ - sin ζ sin ω cos ρ ,
x = x 1 ( η , ρ ) ,             x = x 2 ( η , ρ , ψ , α ) ,
x 1 - x 2 = 0.
x = x ( η , ψ , α ) .
f 1 = cos η cos ζ ,
f 2 = cos ( η + α ) cos ψ .
O S = ( cos γ sin β , cos γ cos β , sin γ ) ,             O R 2 = ( l , m , n ) ,
f 2 = l cos γ sin β + m cos γ cos β + n sin γ , = l y + m ( 1 - x 2 - y 2 ) 1 / 2 + n x .
f 2 = l y + m ( 1 - x 2 - y 2 ) 1 / 2 + n x ,
H ( x , η , ψ , α ) = 0.
( F , G ) ( η , ρ ) = H η ( x 1 - x 2 ) ρ - H x ( x 1 , x 2 ) ( η , ρ ) .
H η + ( x η ) ψ , α H x = 0 ,
cos γ sin β = cos η cos ζ ,             cos γ cos β = cos ( η + α ) cos ψ .
cos γ cos β = cos η cos ζ ,             cos γ sin β = cos ( η + α ) cos ψ .
p + cos 2 η cos 2 ζ = cos 2 ( η + α ) cos 2 ψ ,
[ 1 + cos ( T + 2 α ) ] cos 2 ψ = ( 1 + 2 p + cos T ) cos 2 ζ .
sin ( T + 2 α ) cos 2 ψ = sin T cos 2 ζ ,
cot T = cos 2 ζ sec 2 ψ csc 2 α - cot 2 α .
sin 2 α = p ( p - cos 2 ψ + cos 2 ζ ) cos 2 ψ cos 2 ζ             or             cos 2 ψ = p ( p + cos 2 ζ ) p + sin 2 α cos 2 ζ .
sin 2 α p ( p - cos 2 ψ + cos 2 ζ ) cos 2 ψ cos 2 ζ .
sin γ = cos η cos ζ ,             sin γ = cos ( η + α ) cos ψ ,
sin ζ = cos γ cos ( β + ρ ) ,             sin ψ = cos γ cos ( 2 δ + ρ - β ) , sin η cos ζ = cos γ sin ( β + ρ ) , sin ( η + α ) cos ψ = cos γ sin ( 2 δ + ρ - β ) ,
[ sin ( η + α ) cos ψ - sin η cos ζ ] [ 1 + ( β β ) γ ] .
β = β = arcsin [ ( μ 2 - sin 2 γ ) 1 / 2 sec γ sin δ ]
sin 2 γ = μ 2 - { ( μ 2 - 1 ) sin 2 δ + sin 2 1 2 θ } 2 csc 2 2 δ csc 2 1 2 θ .
cos ϕ = { ( f 2 - f 1 cos θ ) f 1 tan ζ ± ( 1 - f 1 2 sec 2 ζ ) 1 / 2 × [ ( 1 - f 1 2 ) sin 2 θ - ( f 2 - f 1 cos θ ) 2 ] 1 / 2 } csc θ ( 1 - f 1 2 ) - 1 ,
β = β = arcsin ( μ sin δ ) ,             γ = 0 ,             θ = 2 ( β - δ ) .
cos ϕ = [ - 3 4 μ ( 2 - μ 2 ) 1 / 2 sin ζ ± ( 1 - 3 8 μ 2 - sin 2 ζ ) 1 / 2 ] / 2 ( 1 - 3 8 μ 2 ) cos ζ ,
cos ϕ = [ 3 4 μ ( 2 - μ 2 ) 1 / 2 sin ζ ± ( 1 4 + 3 8 μ 2 - sin 2 ζ ) 1 / 2 ] / 2 ( 1 4 + 3 8 μ 2 ) cos ζ .
d A = | ( r ψ η ζ , r α η cos ψ d η ) × ( r d ψ ρ d ρ , r α ρ cos ψ d ρ ) | = r 2 cos ψ J d η d ρ ,
P N c T sec ψ i J i ,
d x = x i η d η + x i ρ d ρ ;             i = 1 , 2.
( x η ) ψ , α = ( ρ η ) ψ , α x i ρ + x i η ;             i = 1 , 2 ,
( x 1 , x 2 ) ( η , ρ ) = - ( x 1 - x 2 ) ρ ( x η ) ψ , α .