Abstract

The formation of many ice crystal halos can be visualized in an appropriate coordinate system on the sphere. A given crystal orientation is first represented by a point on the sphere. When the same sphere is regarded as the celestial sphere, it is easy to find the point of light on the sphere that results from the given crystal orientation. The analysis gives crude information on intensities of halos, not just along the caustic curve but for the entire sky.

© 1979 Optical Society of America

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References

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  1. M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), has an enjoyable description of these and other halos.
  2. J. M. Pernter and F. M. Exner, Meteorologische Optik (Wilhelm Baumuller, Vienna and Leipzig, 1910).
  3. R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).
  4. R. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128–131, 1972.
    [Crossref] [PubMed]
  5. R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, “Form and origin of the Parry Arcs,” Science 195, 360–367, 1977.
    [Crossref] [PubMed]
  6. W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1929).
  7. Precisely, Y is a critical value of F if for some X, F(X) = Y and the Jacobian matrix of F at X has less than maximal rank. Thus, at a critical value there has been an unusual amount of bunching, at least infinitesimally.This terminology is not restricted to halos resulting from two-dimensional sets of crystal orientations. For instance, a sun dog has a single critical value while a circumzenithal arc has none. The caustic of the 22° halo is the inner circular boundary of the halo.
  8. R. A. R. Tricker, “Observations on certain features to be seen in a photograph of haloes taken by Dr. Emil Schulthess in Antarctica,” Q. J. R. Meteorol. Soc. 98, 542–562, 1972.
  9. If we replace R̂ by −P̂ then the resulting circle C is unchanged but the μ coordinate on C changes sign. The sense of μ is well-defined if P̂ is restricted to lie on one half of the equator.
  10. For a given crystal and a given path of light through the crystal (see Fig. 13, for instance), there are two choices for P̂, one choice being the negative of the other. Throughout this section we assume that P̂ lies on the right half of the equator; that is, we assume that the y component of P̂ is positive. This convention insures that the sign of α is well-defined, once a crystal and light path are chosen.
  11. To see analytically that for the upper and lower tangent arcs the caustic coincides with the minimum deviation locus, recall that F(θ,μ) = (θ,D(θ,μ)). So the Jacobian matrix of F, expressed in (θ,μ) coordinates, has determinant|JF|=|∂θ∂θ∂θ∂μ∂D∂θ∂D∂μ|=|10∂D∂θ∂D∂μ|=∂D∂μ.Thus for the upper and lower tangent arcs, |JF| = 0 if and only if ∂D/∂μ = 0. [This argument breaks down at (cos Σ, 0, −sin Σ), which is a singularity of the (θ,μ) coordinate system.]
  12. If θ and μ are coordinates for a point on the curve φ = 0°, then a computation shows cos μ = (sin Σ)/|sinθ|. For nonzero values of φ notice that μ(θ,φ) = μ(θ,0) + φ.
  13. The (ξ,μ) coordinate system is defined analytically in the Appendix.

1977 (1)

R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, “Form and origin of the Parry Arcs,” Science 195, 360–367, 1977.
[Crossref] [PubMed]

1972 (2)

R. A. R. Tricker, “Observations on certain features to be seen in a photograph of haloes taken by Dr. Emil Schulthess in Antarctica,” Q. J. R. Meteorol. Soc. 98, 542–562, 1972.

R. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128–131, 1972.
[Crossref] [PubMed]

Exner, F. M.

J. M. Pernter and F. M. Exner, Meteorologische Optik (Wilhelm Baumuller, Vienna and Leipzig, 1910).

Greenler, R.

R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, “Form and origin of the Parry Arcs,” Science 195, 360–367, 1977.
[Crossref] [PubMed]

R. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128–131, 1972.
[Crossref] [PubMed]

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1929).

Mallmann, A. J.

R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, “Form and origin of the Parry Arcs,” Science 195, 360–367, 1977.
[Crossref] [PubMed]

R. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128–131, 1972.
[Crossref] [PubMed]

Minnaert, M.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), has an enjoyable description of these and other halos.

Mueller, J.

R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, “Form and origin of the Parry Arcs,” Science 195, 360–367, 1977.
[Crossref] [PubMed]

Pernter, J. M.

J. M. Pernter and F. M. Exner, Meteorologische Optik (Wilhelm Baumuller, Vienna and Leipzig, 1910).

Romito, R.

R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, “Form and origin of the Parry Arcs,” Science 195, 360–367, 1977.
[Crossref] [PubMed]

Tricker, R. A. R.

R. A. R. Tricker, “Observations on certain features to be seen in a photograph of haloes taken by Dr. Emil Schulthess in Antarctica,” Q. J. R. Meteorol. Soc. 98, 542–562, 1972.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

Q. J. R. Meteorol. Soc. (1)

R. A. R. Tricker, “Observations on certain features to be seen in a photograph of haloes taken by Dr. Emil Schulthess in Antarctica,” Q. J. R. Meteorol. Soc. 98, 542–562, 1972.

Science (2)

R. Greenler and A. J. Mallmann, “Circumscribed halos,” Science 176, 128–131, 1972.
[Crossref] [PubMed]

R. Greenler, A. J. Mallmann, J. Mueller, and R. Romito, “Form and origin of the Parry Arcs,” Science 195, 360–367, 1977.
[Crossref] [PubMed]

Other (10)

W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1929).

Precisely, Y is a critical value of F if for some X, F(X) = Y and the Jacobian matrix of F at X has less than maximal rank. Thus, at a critical value there has been an unusual amount of bunching, at least infinitesimally.This terminology is not restricted to halos resulting from two-dimensional sets of crystal orientations. For instance, a sun dog has a single critical value while a circumzenithal arc has none. The caustic of the 22° halo is the inner circular boundary of the halo.

If we replace R̂ by −P̂ then the resulting circle C is unchanged but the μ coordinate on C changes sign. The sense of μ is well-defined if P̂ is restricted to lie on one half of the equator.

For a given crystal and a given path of light through the crystal (see Fig. 13, for instance), there are two choices for P̂, one choice being the negative of the other. Throughout this section we assume that P̂ lies on the right half of the equator; that is, we assume that the y component of P̂ is positive. This convention insures that the sign of α is well-defined, once a crystal and light path are chosen.

To see analytically that for the upper and lower tangent arcs the caustic coincides with the minimum deviation locus, recall that F(θ,μ) = (θ,D(θ,μ)). So the Jacobian matrix of F, expressed in (θ,μ) coordinates, has determinant|JF|=|∂θ∂θ∂θ∂μ∂D∂θ∂D∂μ|=|10∂D∂θ∂D∂μ|=∂D∂μ.Thus for the upper and lower tangent arcs, |JF| = 0 if and only if ∂D/∂μ = 0. [This argument breaks down at (cos Σ, 0, −sin Σ), which is a singularity of the (θ,μ) coordinate system.]

If θ and μ are coordinates for a point on the curve φ = 0°, then a computation shows cos μ = (sin Σ)/|sinθ|. For nonzero values of φ notice that μ(θ,φ) = μ(θ,0) + φ.

The (ξ,μ) coordinate system is defined analytically in the Appendix.

M. Minnaert, The Nature of Light and Color in the Open Air (Dover, New York, 1954), has an enjoyable description of these and other halos.

J. M. Pernter and F. M. Exner, Meteorologische Optik (Wilhelm Baumuller, Vienna and Leipzig, 1910).

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

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Figures (25)

FIG. 1
FIG. 1

Ice crystals and the corresponding point of light on the celestial sphere. For crystals all of a given orientation, the outgoing rays R ̂ will be parallel. The observer will see light at point F ̂ = R ̂ on the celestial sphere.

FIG. 2
FIG. 2

Refraction in the normal plane. The vector P ̂ is pointing out of the paper.

FIG. 3
FIG. 3

Deviation D as a function of angle of incidence μ. (α = 60°).

FIG. 4
FIG. 4

Function D as a folding and bunching operation. The upper segment is folded and bunched at μM and then placed on the lower segment. (α = 60°).

FIG. 5
FIG. 5

Projected deviation D as a function of θ and the projected angle of incidence μ. (α = 60°).

FIG. 6
FIG. 6

Ice crystal and light ray involved in forming a sun dog. The vector N1 is the inward normal to the rear face of the crystal.

FIG. 7
FIG. 7

Point A representing the crystal orientation with axial vector P ̂ and outward normal BA to the “first” crystal face. (See text.)

FIG. 8
FIG. 8

F ̂ = F ( A ), the point of light on the celestial sphere due to crystal orientation A in Fig. 7.

FIG. 9
FIG. 9

Formation of a sun dog. The parallel of latitude on the left sphere represents crystal orientations with the axial vector vertical. The sphere on the right is the celestial sphere. The halo function folds and bunches the shaded arc on the left and places it on the shaded arc on the right. The shaded arc on the right is the resulting sun dog.

FIG. 10
FIG. 10

Point A representing the crystal orientation with axial vector P ̂ and outward normal BA to the “first” crystal face. (See text.)

Fig. 11
Fig. 11

θ = const coordinate curves on the sphere. They are circles in vertical planes passing through Ŝ.

FIG. 12
FIG. 12

Halo function acting on the circle θ = θ0. The sphere on the left represents horizontal crystal orientations. The sphere on the right is the celestial sphere. The halo function folds and bunches the shaded arc on the left and places it on the shaded arc on the right.

FIG. 13
FIG. 13

Ice crystal and light ray involved in forming an upper tangent arc.

FIG. 14
FIG. 14

(a) Horizontal crystal orientations. The region bounded by the curves μ = μL(θ) and μ = −90° represent crystal orientations responsible for the upper tangent arc. (∑ = 23°) (b) Celestial (hemi-) sphere. The upper tangent arc is the region between the curves μ = Dmin(θ) and μ = Dmax(θ). The 22° halo and the horizon are shown for reference. (∑ = 23°.) Compare Figs. 14(a) and 14(b) with Fig. 12.

FIG. 15
FIG. 15

Upper and lower tangent arcs when the sun is on the horizon.

FIG. 16
FIG. 16

Upper and lower tangent arcs (actually a circumscribed halo) when the sun elevation is 32.5°.

FIG. 17
FIG. 17

Ice crystal (side view) and light ray involved in forming a Parry arc.

FIG. 18
FIG. 18

Parry arc and the responsible crystal orientations. (Σ = 20°).

FIG. 19
FIG. 19

Ice crystal and light ray involved in forming a circumzenithal arc.

FIG. 20
FIG. 20

Circumzenithal arc and the responsible crystal orientations. (Σ = 10°).

FIG. 21
FIG. 21

Contribution to the common halo due to crystal orientations with axial vector in plane π. The common halo is a superposition of such tangent arcs for 0 ≤ β ≤ 180°.

FIG. 22
FIG. 22

Part of the coordinate grid and tangent arcs to the 22° halo when crystals are tilted 5° to the horizontal. (Σ = 23°).

FIG. 23
FIG. 23

Upper tangent arcs when ν = ±5°. (Σ = 23°).

PLATE 109
PLATE 109

(Walter Tape, p. 1122). Upper tangent arc when the sun elevation is about 23°. There is also a 22° halo, a sun dog on the left, a circumzenithal arc in the dark area at the top of the picture, and a faint Parry arc.

PLATE 110
PLATE 110

(Walter Tape, p. 1122). Upper tangent arc when the sun elevation is about 32.5°.

Tables (1)

Tables Icon

TABLE I Values of μL(θ), μM(θ), Dmax(θ), and Dmin(θ) when α = 60°

Equations (3)

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sin μ L = sign α cos α ( n 2 1 ) 1 / 2 sin α , sin μ M = n sin ( α / 2 ) , D max = α + μ L 90 ° sign α , D min = α + 2 μ M , sign α = α / | α | .
( ξ , μ ) = ( cos cos 2 ξ + cos μ cos sin 2 ξ + sin μ sin ξ sin , ( 1 cos μ ) cos cos ξ sin ξ sin μ cos ξ sin , cos μ sin sin μ cos sin ξ )
|JF|=|θθθμDθDμ|=|10DθDμ|=Dμ.