Abstract

Divergent-light halos are produced when light from nearby light sources is scattered by ice crystals in the atmosphere. We present a theory of divergent-light halos leading to an improved algorithm for the simulation of such halos. Contrary to the algorithm that we presented earlier for simulating such halos, the new algorithm includes a mathematically rigorous weighting of the events. The computer implementation is very compact, and the whole procedure is elegant and conceptually easy to understand. We also present a new simulation atlas showing halos produced by crystals of different shapes and orientations for a set of elevations of the light source.

© 2005 Optical Society of America

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References

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  1. L. Gislén, J. O. Mattsson, “Observations and simulations of some divergent-light halos,” Appl. Opt. 42, 4269–4279 (2003).
    [CrossRef] [PubMed]
  2. L. Gislén, “Procedure for simulating divergent-light halos,” Appl. Opt. 42, 6559–6563 (2003).
    [CrossRef] [PubMed]
  3. J. Moilanen, Ursa Astronomical Association, Raatimiehenkatu 3 A 2, 00140 Helsinki, Finland (personal communication, 2004).
  4. J. O. Mattsson, L. Bärring, E. Almqvist, “Experimenting with Minnaert’s Cigar,” Appl. Opt. 39, 3604–3611 (2000).
    [CrossRef]
  5. A beta version of the algorithm unit is available at http://www.thep.lu.se/∼larsg/ .

2003 (2)

2000 (1)

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Figures (9)

Fig. 1
Fig. 1

Minnaert’s cigars for scattering angles of (a) 22°, (b) 45°, and (c) 150°. The eye of the observer is located at one apex of the cigar, the light source at the other apex.

Fig. 2
Fig. 2

Cigar geometry. The light source is located at the left apex; the observer’s eye is located at the right apex. The ice crystal is located on the surface of the cigar and scatters the incident ray with scattering angle ω.

Fig. 3
Fig. 3

Crystal located at a point on the surface of the cigar by specifying angles θ and ϕ.

Fig. 4
Fig. 4

Restricting the cigar to crystals located within a sphere with radius r around the observer. If r < R, the distance between the observer and the light source, we have θ max = arcsin ( r R sin ω ) .

Fig. 5
Fig. 5

Atlas for oriented plate crystals with a c/a ratio of 0.3 and an axis tip of 1° from the horizontal.

Fig. 6
Fig. 6

Atlas for singly oriented column crystals with a c/a ratio of 2.0 and an axis tip of 0.5° from the vertical.

Fig. 7
Fig. 7

Atlas for Parry-oriented crystals with a c/a ratio of 2.0, an axis tip of 1.5° from the horizontal, and one side face tilted 1.5° from the horizontal.

Fig. 8
Fig. 8

Path of reflected light within a crystal. Dots and crosses, reflections in the upper and lower endfaces, respectively.

Fig. 9
Fig. 9

Halo produced by oriented plate crystals for a light source elevation of 30°, where the rays in the rear hemisphere of the light source (behind the lamp) have been blocked out.

Equations (31)

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d 2 b ̂ p ( b ̂ | â , U ) = σ ( â , U ) / σ 0 ~ 1 .
p ( b ̂ | â , U ) = p ( U 1 b ̂ | U 1 a , U 1 U ) = p ( U 1 b ̂ | U 1 a , 1 ) ,
P 1 = A σ 0 P 4 π a 2 b 2 p ( b ̂ | â , U ) .
P O = A σ 0 P n 4 π d 3 a a 2 b 2 D U Q ( U ) p ( b ̂ | â , U ) .
p ( b ̂ | â , U ) = σ ( â , U ) 4 π σ 0 1 / 4 π ,
P iso = A σ 0 P n 16 π 2 d 3 a a 2 b 2 D U Q ( U ) = A σ 0 P n 16 π 2 d 3 a a 2 b 2 = A σ 0 P n π 16 R .
Z = 4 R π 2 d 3 a a 2 b 2 D U Q ( U ) p ( b ̂ | â , U ) .
a = R sin ( ω θ ) sin ω ,
a = R sin ( ω θ ) sin ω ( sin θ cos ϕ , sin θ sin ϕ , cos θ )
0 θ ω π , 0 ϕ 2 π .
b = R a = R sin θ sin ω [ sin ( ω θ ) cos ϕ , sin ( ω θ ) sin ϕ , cos ( ω θ ) ] .
d 3 a = R 3 sin 2 θ sin 2 ( ω θ ) sin 4 ω d ω d θ d ϕ .
d 3 a a 2 b 2 = 1 R d ω d θ d ϕ .
d 2 â 0 4 π d 2 b ̂ 0 p ( b ̂ 0 | â 0 , 1 ) 1 .
Z = 4 R π 2 d 3 a a 2 d 3 b b 2 δ ( a + b R ) D U Q ( U ) p ( b ̂ | â , U ) .
D U = d 2 â 0 d 2 b ̂ 0 δ ( â 0 · b ̂ 0 â · b ̂ ) .
Z = 4 R π 2 d 3 a a 2 d 3 b b 2 δ ( a + b R ) × d 2 â 0 d 2 b ̂ 0 δ ( â 0 · b ̂ 0 â · b ̂ ) Q ( U ) p ( b ̂ 0 | â 0 , 1 ) .
Z = d 2 â 0 4 π d 2 b ̂ 0 p ( b ̂ 0 | â 0 , 1 ) W ( â 0 , b ̂ 0 ) ,
W ( â 0 , b ̂ 0 ) = 16 R π d 3 a a 2 d 3 b b 2 δ ( a + b R ) δ ( â 0 · b ̂ 0 â · b ̂ ) Q ( U ) .
U = â â 0 T + b ̂ b ̂ 0 T cos ω ( â b ̂ 0 T + b ̂ â 0 T ) + ( â × b ̂ ) ( â 0 × b ̂ 0 ) T sin 2 ω ,
W ( â 0 , b ̂ 0 ) = 16 π 0 π d ω 0 ω d θ 0 2 π d ϕ δ ( cos ω â 0 · b ̂ 0 ) Q ( U ) .
W = 32 ω sin ω 0 ω d θ ω 0 2 π d ϕ 2 π Q ( U ) ,
d θ ω d ϕ 2 π
Z d 2 â 0 d 2 b ̂ 0 p ( b ̂ 0 | â 0 , 1 ) × ω sin ω × 0 ω d θ ω 0 2 π d ϕ 2 π × Q ( U ) .
d Z d 2 â 0 d 2 b ̂ 0 p ( b ̂ 0 | â 0 , 1 ) × ω sin ω × d ψ ω d ϕ 2 π × Q ( U ) ,
d θ ω d ϕ 2 π .
W = 32 ω sin ω 0 arcsin ( ( r / R ) sin ω ) d θ ω 0 2 π d ϕ 2 π Q ( U ) 32 ω sin ω 0 ( r / R ) sin ω d θ ω 0 2 π d ϕ 2 π Q ( U ) h = ( r / R ) sin ω = × 32 r R 0 1 d h 0 2 π d ϕ 2 π Q ( U ) .
θ max = arcsin ( r R sin ω ) .
P tot = P n σ 0 4 π d 3 a a 2 = P n σ 0 r .
P tot / P = r / d .
d Z d 2 b ̂ = R π 3 0 d b ( R b b ̂ ) 2 = π ψ π 3 sin ψ .

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