Abstract

The relation between the symmetry in halo displays and crystal symmetry is investigated for halo displays that are generated by ensembles of crystals. It is found that, regardless of the symmetry of the constituent crystals, such displays are always left-right (L–R) symmetric if the crystals are formed from the surrounding vapor. L–R symmetry of a halo display implies here that the cross sections for formation of a halo arc on the left-hand side of the solar vertical and its right-hand side mirror image are equal. This property leaves room for two types of halo display only: a full symmetric one (mmm-symmetric), and a partial symmetric one (mm2-symmetric) in which halo constituents lack their counterparts on the other side of the parhelic circle. A partial symmetric display can occur only for point halos. Its occurrence implies that a number of symmetry elements are not present in the shape of the halo-making crystals. These elements are a center of inversion, any rotatory-inversion axis that is parallel to the crystal spin axis P, a mirror plane perpendicular to the P axis, and a twofold rotation axis perpendicular to the P axis. A simple conceptual method is presented to reconstruct possible shapes of the halo-generating crystals from the halos in the display. The method is illustrated in two examples. Halos that may occur on the Saturnian satellite Titan are discussed. The possibilities for the Huygens probe to detect these halos during its descent through the Titan clouds in 2005 are detailed.

© 2003 Optical Society of America

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References

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  1. W. Tape, Atmospheric Halos, Vol. 64 of the Antarctic Research Series (American Geophysical Union, Washington, D.C., 1994).
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    [Crossref]
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    [Crossref]
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    [Crossref]
  6. J.-P. Lebreton, European Space Agency/ESTEC, Noordwijk, The Netherlands (personal communication, 2001).
  7. R. C. Weast, ed., CRC Handbook of Chemistry and Physics, 62nd ed. (CRC Press, Boca Raton, Fla., 1981).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  13. G. P. Können, A. A. Schoenmaker, J. Tinbergen, “A polarimatric search for ice crystals in the upper atmosphere of Venus,” Icarus 102, 62–75 (1993).
    [Crossref]

2000 (1)

1999 (1)

1998 (2)

1993 (1)

G. P. Können, A. A. Schoenmaker, J. Tinbergen, “A polarimatric search for ice crystals in the upper atmosphere of Venus,” Icarus 102, 62–75 (1993).
[Crossref]

1984 (1)

1978 (1)

G. J. H. van Nes and A. Vos, “Single-crystal structures and electron density distributions of ethane, ethylene and acethylene,” Acta Crystallogr. Sect. B 34, 1947–1956 (1978).
[Crossref]

1925 (1)

H. Mark, E. Pohland, “Über die Gitterstruktur des Äthans und des Diborans,” Z. Kristallogr. 62, 103–112 (1925).

Bézard, B.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Combes, M.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Coustenis, A.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

deBergh, C.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Doose, L. H.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Gliem, F.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Keller, H. U.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Können, G. P.

Lebreton, J.-P.

J.-P. Lebreton, European Space Agency/ESTEC, Noordwijk, The Netherlands (personal communication, 2001).

Lellouch, E.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Luukonen, I.

Mark, H.

H. Mark, E. Pohland, “Über die Gitterstruktur des Äthans und des Diborans,” Z. Kristallogr. 62, 103–112 (1925).

McLaurin, G. E.

Moilanen, J.

Pekkola, M.

Pohland, E.

H. Mark, E. Pohland, “Über die Gitterstruktur des Äthans und des Diborans,” Z. Kristallogr. 62, 103–112 (1925).

Riikonen, M.

Rosenqvist, J.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Ruoskanen, J.

Saint-Pé, O.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Salinpää, M.

Schmitt, B.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Schoenmaker, A. A.

G. P. Können, A. A. Schoenmaker, J. Tinbergen, “A polarimatric search for ice crystals in the upper atmosphere of Venus,” Icarus 102, 62–75 (1993).
[Crossref]

Smith, P. H.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Solderblom, L. A.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Sullivan, D.

Tape, W.

W. Tape, G. P. Können, “A general setting for halo theory,” Appl. Opt. 38, 1552–1625 (1999).
[Crossref]

W. Tape, Atmospheric Halos, Vol. 64 of the Antarctic Research Series (American Geophysical Union, Washington, D.C., 1994).

Thomas, N.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Tinbergen, J.

G. P. Können, A. A. Schoenmaker, J. Tinbergen, “A polarimatric search for ice crystals in the upper atmosphere of Venus,” Icarus 102, 62–75 (1993).
[Crossref]

Tomasko, M. G.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

van Nes and A. Vos, G. J. H.

G. J. H. van Nes and A. Vos, “Single-crystal structures and electron density distributions of ethane, ethylene and acethylene,” Acta Crystallogr. Sect. B 34, 1947–1956 (1978).
[Crossref]

Vira, L.

West, R. A.

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

Whalley, E.

Acta Crystallogr. Sect. B (1)

G. J. H. van Nes and A. Vos, “Single-crystal structures and electron density distributions of ethane, ethylene and acethylene,” Acta Crystallogr. Sect. B 34, 1947–1956 (1978).
[Crossref]

Appl. Opt. (4)

Icarus (1)

G. P. Können, A. A. Schoenmaker, J. Tinbergen, “A polarimatric search for ice crystals in the upper atmosphere of Venus,” Icarus 102, 62–75 (1993).
[Crossref]

J. Opt. Soc. Am. A (1)

Z. Kristallogr. (1)

H. Mark, E. Pohland, “Über die Gitterstruktur des Äthans und des Diborans,” Z. Kristallogr. 62, 103–112 (1925).

Other (5)

W. Tape, Atmospheric Halos, Vol. 64 of the Antarctic Research Series (American Geophysical Union, Washington, D.C., 1994).

J.-P. Lebreton, European Space Agency/ESTEC, Noordwijk, The Netherlands (personal communication, 2001).

R. C. Weast, ed., CRC Handbook of Chemistry and Physics, 62nd ed. (CRC Press, Boca Raton, Fla., 1981).

J.-P. Lebreton, ed., “Huygens, science payload and mission,” ESA SP-1177 (European Space Agency Publication Division, ESTEC, Noordwijk, The Netherlands, 1997).

M. G. Tomasko, L. H. Doose, P. H. Smith, R. A. West, L. A. Solderblom, M. Combes, B. Bézard, A. Coustenis, C. deBergh, E. Lellouch, J. Rosenqvist, O. Saint-Pé, B. Schmitt, H. U. Keller, N. Thomas, F. Gliem, “The descent imager/spectral radiometer (DISR) aboard Huygens,” in Huygens, Science Payload and Mission, J.-P. Lebreton, ed., ESA SP-1177 (European Space Agency, Publication Division, ESTEC, Noordwijk, The Netherlands, 1997), pp. 109–138.

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Figures (10)

Fig. 1
Fig. 1

The five halo symmetry groups. Each panel shows the pole diagram associated to a halo symmetry group, the resulting display of point halos, and a description of the crystal symmetry conditions that are required for generating these halo symmetries. P is the crystal spin axis. According to the Crystal Orientation Corollary, only halo symmetry groups that result in L–R symmetric composites (mmm, mm2) can arise from crystals that are formed from the surrounding vapor. They are depicted in the upper row, which are panels a and b. The three halo symmetry groups that lack L–R symmetry (2/m, 222, 2-symmetric; panels c–e) are not expected to occur in natural circumstances. See Table 1 for more details. The depicted halos are for wedge angle 60°, refraction index 1.31, solar elevation 40°. The composites consist of the point halo with P u = (1/42,1/23,-1/42) ≡ B(30°,45°), together with the halos from the induced poles according to the mmm, mm2, 2/m, 222, and 2 symmetry. These halos are the 22° equivalents of the 46° Parry infralateral and supralateral arcs. Solid circles in the pole diagrams refer to halo poles that are on the front hemisphere of the halo sphere; open circles, to halo poles on the rear hemisphere. In a mm2-symmetric halo display that consist of halo composites from several wedges, the missing halos may be the upper arcs for a given halo-making wedge (like the situation in panel b), whereas for other halo-making wedges, the missing arcs may be the lower arcs instead. Similar arguments hold for 2/m, 222, and 2-symmetric halo composites.

Fig. 2
Fig. 2

Halo poles for point halos in the Tape display. The radius of the associated circular halo is indicated at each pole. On the y = 0 plane are the poles of the circumzenith arc (CZ) and the upper suncave Parry arc (USCP). The (0, ±1,0) poles are those of the left and the right parahelia (LPH and RPH). The remaining poles are those of the left and the right Parry supralateral arcs (LPS and RPS) and the left and the right Parry infralateral arcs (LPI and RPI). Around the halo sphere are four solutions to the minimum configuration of the halo-making crystals, as inferred from the occurrence of the halos (spin axis P vertical; crystal main axis normal to the paper). Solid circles in the crystal diagrams refer to face normals pointing to the front hemisphere of the diagrams or that are parallel to the paper; open circles, to face normals pointing to the rear hemisphere. The face numbering is according to Tape’s1 system. The Tape display is on the cover of his book1; a simulation of the display is on the back cover.

Fig. 3
Fig. 3

Computer simulation of the Sturm display. The halos result from pyramidal ice crystals. The innermost circular halo is the 9° halo; the most prominent circular halo is the 22° halo. The mm2-symmetry of the display is apparent from the fact that some halo arcs (e.g., 9°, 24°) lack their counterparts on the other side of the parhelic circle. The simulation is shown in Fig. 10–19 of Ref. 1. A photograph of the display is shown in Ref. 1, Fig. 10–17. The pole diagram of the point halos in the display is shown in Fig. 4.

Fig. 4
Fig. 4

Halo poles for point halos in the Sturm display (Fig. 3). The radius of the associated circular halo is indicated at each pole. The small diagram is a single-crystal solution to the minimum configuration of the halo-making crystals, as inferred from the occurrence of the halos (spin axis P and crystal main axis normal to the paper). Solid circles in the crystal diagram refer to face normals pointing to the front hemisphere of the diagrams or that are parallel to the paper; open circles, to face normals pointing to the rear hemisphere. The face numbering is according to Tape’s1 system. Faces in brackets cannot be inferred from the display. The Sturm halo is shown in Fig. 10–17 of Tape’s book.1

Fig. 5
Fig. 5

Atlas for mmm-symmetric point-halo composites for solar elevation 40°. The wedge angle α is 60°; refraction index is n = 1.31. Because the shapes of the halos are only weakly dependent on α and n, the atlas gives a fair impression how point halos associated to a certain halo pole are approximately shaped. The approximate shapes of Titan refraction halos that may be detected during the descent of the Huygens probe in Jan 2005 can be found by looking at the spots where the Titan halo poles (Fig. 6) appear. The coordinate θ indicates the Bravais colatitudes on the halo sphere.

Fig. 6
Fig. 6

Poles of Titan halos resulting from truncated octahedrons of methane (M) and from hexagonal crystals of ethane (E) with basal faces. The truncations apply to all vertices of the octahedrons. It is assumed that the methane and ethane crystal main axes are vertically oriented. Halos that are empty for solar elevation 40° are in brackets. However, for these halo angles, circular halos (not depictable in the pole diagrams) are always possible if the crystal orient randomly. The M 48° halos are due to two cubic faces, the M 29° halos to two octahedral faces, and the M 20° halos to one cubic and one octahedral face. Halos that are within the reach of the Huygens imagers are marked with *. These are the upper 20° methane Parroid arc and the lower 29° methane Parroid arc. The 32° ethane parhelia and the 29° methane halos from poles with z = 0, marked with **, are out of reach of the imagers, but their subhorizon counterparts (viz. the 32° ethane subparhelia and two halos that are shaped as the reflections of the two 29° halos in question at a horizontal mirror plane) are within reach. Figure 5 visualizes how an mmm-symmetric point halo display associated to a certain halo pole is approximately shaped. Figures 7 10 give a Monte Carlo simulation of a Titan halo display.

Fig. 7
Fig. 7

Halo display that may occur in the Titan atmosphere during the decent of the Huygens probe. In this Monte Carlo ray-tracing simulation, there are four populations of methane crystals: square pyramids with cubic face up and with cubic face down, randomly oriented equidimensional cube-octahedrons with all vertices truncated, and equidimensional cube-octahedrons with the fourfold rotation symmetry axis vertical and all vertices truncated. Additionally, there are two ethane crystal populations: plate oriented and randomly oriented hexagonal crystals with basal ends. The standard deviation of the tilts of axes of the preferentially oriented crystals is 1°. The figure is uplooking with a field of view of 180°: The zenith is in the center, and the circle that surrounds the simulation is the horizon. The symbol S marks the position of the Sun, which is at 40° elevation. The prominent circular halo is the 20° methane circular halo; the three other possible circular halos are barely visible. Another choice for the populations or for the crystal parameters would result in different relative intensities of the various halos in the display. Fig. 10 is a legend to the refraction halos. The three above-horizon regions that are in the fields of view of the Huygens imagers consist of a 6° wide vertical band through the Sun extending from 15° till 65° above the horizon; a similar vertical band straight opposite to the Sun; and the entire region between the horizon and a height of 6°.

Fig. 8
Fig. 8

Halo display that may occur in the Titan atmosphere. As in Fig. 7, but now downlooking: The nadir is in the center, and the circle that surrounds the simulation is the horizon. The symbol SS marks the position of subsun, which is the reflected image of the Sun at horizontal crystal faces. Hence the subsun is directly below the Sun and is as far below the horizon as the Sun is above. With exception of a circular region of 6.5° radius centered at the nadir, the entire subhorizon sky can be photographed by the Huygens probe.

Fig. 9
Fig. 9

Halo display that may occur in the Titan atmosphere. As in Fig. 7, but now facing to the point at the horizon straight below the Sun. The horizontal line is the horizon. The symbol S marks the position of the sun; the symbol SS marks the position of the subsun. The bright subhorizon halos emerge from light paths refraction-reflection-refraction, where the reflecting face is a horizontally oriented crystal face. The two subhorizon spots are the 32° ethane subparhelia, which appears as the reflected images of the 32° ethane parhelia from a horizontal mirror plane. The two other subhorizon arcs appear as the reflected images of the two 29° methane halos from poles with z = 0. Contrary to their above- horizon counterparts, they may arise from crystals as simple as square pyramids (with cubic face down).

Fig. 10
Fig. 10

Key to the refraction halos in Figs. 7, 8, 9. The radius of the circular halos that are associated with the various refraction halos are indicated. Halo angles printed in the solar vertical refer to Parroid arcs (poles on the y = 0 plane in the pole diagram, Fig. 6). The lower 20° methane Parroid arc consists of two disconnected halo arcs for this solar elevation. Halo angles printed near the parhelic circle (which is the solar almucanter) refer to halos from z = 0 poles. These include the 32° ethane parhelion. The remaining refraction halos are from the poles that are not on a coordinate plane of the halo sphere. The symbol S marks the position of the Sun.

Tables (1)

Tables Icon

Table 1 Symmetry of Crystals and Symmetry in Halo Displaysa

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