Halo arcs from airborne, pyramidal icecrystals falling with their c axes in vertical orientation

Marko Pekkola, Marko Riikonen, Jarmo Moilanen, and Jukka Ruoskanen

Many halo arcs are caused by pyramidal crystals that have $1 0 21 1% crystal faces. We treat halo arcsarising from pyramidal crystals that fall in the air with their c axes vertically oriented. To our knowl-edge only 6 of the 12 possible halo phenomena that belong to this category have been dealt with in theliterature. Surprisingly the yet undiscussed halos are predicted to be of comparable intensity with thosealready treated. They are produced by reflections from pyramidal crystal basal faces. A theoreticalsummary and computer simulations are presented of the mentioned 12 halo phenomena and of theindividual arcs into which they break in the sky. We give an overview to the current level of documen-tation of these phenomena by listing the first published photographs of each phenomenon known to theauthors. © 1998 Optical Society of America

OCIS codes: 010.1290, 010.2940.

1. Introduction

Halo rings of unusual radii, i.e., with radii other thanthe common 22° and 46°, have shown up regularly inthe optical literature since the late 19th century.Odd-radii rings were attributed to pyramidal crys-tals, but for a long time the exact form of the pyramid~e.g., the top pyramid angle! was not known for cer-tain. Steinmetz and Weickmann did simultaneousairborne ice-crystal collecting and halo observingover Germany during and after the Second WorldWar.1,2 Their results determined the pyramid topangle as 56°. On Easter Sunday, 14 April 1974, anusually well-developed odd-radius halo display oc-curred over England and the Netherlands.3 Apply-ing results of x-ray analysis, a Bragg unit cell, andthe Easter Sunday display photographs, Tricker con-firmed the angle of pyramid top faces as 56.142°.4This angle fits with $1 0 21 1% crystal faces inthe Miller indices system. From pyramidal crystalswith $1 0 21 1% faces in random orientation, sixsolar concentric odd-radii halos should result, withradii expected at 9.0°, 18.3°, 19.9°, 22.9°, 23.8°, and34.9° from the Sun. By convention, the same phe-nomena are often ascribed to full degrees in the lit-

The authors are with the Ursa Astronomical Association, Raa-timiehenkatu 3A2, 00140 Helsinki, Finland.

Received 20 June 1997; revised manuscript received 21 October1997.

0003-6935y98y091435-06$15.00y0© 1998 Optical Society of America

erature, i.e., 9°, 18°, 20°, 23°, 24°, and 35° halos.Each of the six halos have been photographed duringthis century. The first ones were photographicallydocumented by Barkow as early as 1916.5 The lastone, the faint and genuinely rare 35° halo was pho-tographed by Neiman in 1986 and published in 1989.6

Studies and observation series from the past twodecades in Alaska, Antarctica, Finland, and the Phil-ippines indicate that pyramid crystal displays, atleast in these locations, are by far not as rare as hadearlier been assumed.7–9 In general the data pointto the conclusion that a well-motivated and skilledobserver may photograph several displays, includingthe 9°, 18°, or 23° halos, during an average year inany of the geographic locations listed above. Thesame might be true for many midlatitude locations aswell. In the same regions faint pyramidal halos arecontinually missed by inexperienced, less motivatedobservers: this is the human factor that we believeexplains much of the exaggerated reputation of raritythat has been ascribed throughout the twentieth cen-tury to all odd-radii halos. A few of these halos aresurprisingly common, some are genuinely rare.

Many of the older reports, as well as the photo-graphs of the Easter Sunday display, contained evi-dence of arcs associated with rings of abnormal radii.By involving pyramidal crystals with $1 0 21 1%faces, falling in the sky with their c axes maintainedvertically, Tricker4 was able to identify the pyramidalarcs that had shown up in the 14 April 1974 display.He also foresaw the possibility of arcs from pyramidalcrystals with their c axis maintained in a horizontal

20 March 1998 y Vol. 37, No. 9 y APPLIED OPTICS 1435

position. The arcs at the minimum distance of 22.9°were left unconsidered by Tricker. In 1990 and 1994Tape7,10 gave the first computer simulations of pyra-midal arcs and ample field evidence from Antarcticaand Alaska that correlated simultaneous ice-crystalsamples with odd-radii rings and arcs visible in thesky. Among his results, he extended the explana-tion to the phenomenon left unconsidered by Tricker.

The possibility of additional basal face reflectionshas long been known with regard to halos from reg-ular, hexagonal plates.7 The parhelia, parhelic cir-cle, circumzenithal arc, and 120° parhelia are due tobasal reflections, which are duplicated below the ho-rizon, in a league of similar phenomena. Thus wehave the subsun escorted with subparhelia, subpar-helic circle, 120° subparhelion, and the subcir-cumzenithal arc ~in practice the circumnadir arc!.

To our knowledge similar reflections from the basalfaces of pyramidal crystals, ~$1 0 21 1% faces!falling with their c axes in the vertical orientation,have not yet been discussed. The new phenomenafrom this mechanism result mainly in arcs below thehorizon, but partly as arcs above it. Interestingly,many of these halos are of comparable intensity withthose that have been observed and photographed innature, and for which no additional reflections areinvolved.

2. Grouping of the Halo Arcs

Arcs from pyramidal crystals with their c axes in thevertical direction may be divided into two maingroups: parhelia11 ~of odd radii! and antiparhelia ~ofodd radii!. Arcs arising without basal reflection andthose with two basal reflections belong to the parhe-lia. These arcs are grouped around the Sun. Tothis group belong all arcs treated in the literature.Arcs arising from one basal face reflection belong tothe antiparhelia. These arcs organize themselvesaround the subsun. The antiparhelia are perfectmirror images of the parhelia. They bear analogousshapes and are always found as far below the horizonas the corresponding parhelia are above.

However, the arcs belonging to parhelia are notalways located above the horizon. At low solar ele-vations ~,30°!, some of these arcs are found below thehorizon. At the same time, for the same solar ele-vations a few antiparhelia may be located above thehorizon. As the Sun rises higher than 30°, the low-est of the parhelia finally emerges above the horizonand the last of the antiparhelia sinks below the ho-rizon.

The minimum distance from the Sun or subsun ofeach arc is rounded off to full degrees. The arcs thatlie above the altitude of the Sun are described asupper and those below are lower. Equally for theantiparhelia, the altitude of the subsun divides thearcs into upper and lower arcs. The 18° arcs arealways found in the solar ~or subsun! horizontalplane, and thus the division into upper and lower arcsis not necessary in this case. The 9°, 18°, and 24°arcs have inner and outer components; the outer com-

1436 APPLIED OPTICS y Vol. 37, No. 9 y 20 March 1998

ponents, as far as we know, have not been discussedbefore.

The components that are at most elevations nearerto the Sun ~or the subsun! are referred to as inner ~orA! components. Similarly the components that atmost elevations are found farther away from the Sun~or the subsun! are referred to as outer ~or B! compo-nents. As far as we are aware, no observations orphotographs of any of the outer components haveshown up in the literature.

For some of the arcs discussed here, the ray pathsexperience changes as functions of solar elevationangles. This involves all arcs that arise from raysentering into the lower pyramidal prism and thatrequire basal reflections. When the solar elevationis ,9.9°, the ray from the Sun entering the lowerpyramid prism face refracts upward with respect tothe horizontal, and the basal face reflection can occuronly from the upper basal face. When the solar el-evation is .9.9°, the ray refracts downward with re-spect to the horizontal plane, and the basal facereflection can occur only from the lower basal face.At a solar elevation of 9.9°, the ray path entering thelower pyramidal prism continues parallel to the hor-izontal and thus the ray cannot experience any basalreflections. At that Sun elevation the arcs that oc-cur as the ray first hits the lower pyramid face cannotbe seen. All halo arcs and their ray paths, includingthe solar-elevation-dependent changes, are given inTable 1. The numbering of pyramidal crystal facesis given in Fig. 1.

In the situation in which the light ray producingthe arcs experiences more than one internal reflec-tion from the basal faces, we can simplify the ray pathby omitting from consideration the pairs of successivereflections that involve opposite basal faces.7 Theray path 23-2-1-6 of the 9° lower parhelion B, forexample, can be simplified to 23-6 and the ray path3-2-1-16 of the 9° upper parhelion B to 3-16. We cancompare the ray paths of the antiparhelion arcs withparhelion arcs by considering that the light originat-ing from the former comes from the subsun. Thenthe internal reflection from the bottom basal face canbe excluded. The ray paths 23-1-15 and 23-2-15 ofthe 35° upper antiparhelion attain the form 13-15and the ray path 3-2-16 of the 9° upper antiparhelionA reduces to 3-16.

3. Observability of the Arcs

Because the intensities are comparable, an obviousquestion arises: how is it possible that there stillseem to be no observations or photographs availableof the antiparhelia or of the outer parhelia? Theconditions for observing the antiparhelia are limitedby the practical issue of their location. Most of theyet undiscovered forms are, at most solar elevations,completely below the horizon. Hence observationsfrom airplanes flying above ice-crystal clouds are re-quired. The alternative possibility involves rela-tively high observation points ~hills, mountains,towers, or bridges! in cold climates where adequate

Table 1. Different Pyramidal Crystal Halo Forms and Arcs into which They Divide

9° Parhelia 9° AntiparheliaUpper A 13-6* ~1! Lower A 13-2-6 ~17!Upper B 3-2-1-16 ~2! Lower B 3-2-26 ~18!Lower A 3-26* ~3! Upper A 3-2-16 ~19!Lower B 23-6 ,9.9° ~4! Upper A 23-1-6 ,9.9° ~20!

23-2-1-6 .9.9° 23-2-6 .9.9°24° Parhelia 24° Antiparhelia

Upper A 13-5* ~5! Lower A 13-2-5 ~21!Upper B 3-2-1-15 ~6! Lower B 3-2-25 ~22!Lower A 3-25* ~7! Upper A 3-2-15 ~23!Lower B 23-5 ,9.9° ~8! Upper B 23-1-5 ,9.9° ~24!

23-2-1-5 .9.9° 23-2-5 .9.9°20° Parhelia 20° Antiparhelia

Upper 13-16* ~9! Lower 13-2-26 ~25!Lower 23-26* ~10! Upper 23-1-16 ,9.9° ~26!

23-2-16 .9.9°35° Parhelia 35° Antiparelia

Upper 13-15* ~11! Lower 13-2-25 ~27!Lower 23-25* ~12! Upper 23-1-15 ,9.9° ~28!

23-2-15 .9.9°18° Parhelia 18° Antiparhelia

A 13-25* ~13! A 13-2-15 ~29!B 23-15 ,9.9° ~14! B 23-1-25 ,9.9° ~30!

23-2-1-15 .9.9° 23-2-25 .9.9°23° Parhelia 23° Antiparhelia

Upper 1-23* ~15! Lower 1-2-13 ~31!Lower 13-2 ~16! Upper 13-2-1 ~32!

aThe arcs that are already documented with photographs are marked with an asterisk. The responsible ray path for each arc is given.If the ray path changes with solar elevation, it is marked in the third column. Each halo arc is numbered for identification in Figs. 2 and3.

amounts of ice crystals may be found below the hori-zon line, visible against the landscape.

Moreover, the history of atmospheric optics observ-ing in practice gives good reason to doubt an aware-ness as to what to look for as a secondary, if not themost important, factor. Repeatedly, previously un-recorded phenomena have been redocumented withinrelatively short time intervals as soon as the phenom-enon has been brought to the attention of the observ-ing community. Such was the case, e.g., with bothpollen coronas12 and elliptical halos.13–15 Exactknowledge and anticipation as what to look for, as

Fig. 1. Numbering of the pyramidal crystal faces ~adapted fromRef. 7!.

well as lack of those qualities, make a large differencein the observation results. Crystal-growth-relatedfactors7 may also affect the few appropriate field sit-uations encountered so far. These factors include~1! a lack of a sufficient quantity of thin crystals, ~2!a too strong tilting angle of the crystals, and ~3! poorcrystal development.

Parhelia observed so far in odd-radii displays have,according to simulations, arisen from thick pyramidalplate crystals, which are close to equidimensionalshape. Thick pyramidal crystals are ineffective informing arcs that require basal reflections because ofthe small proportional size of the basal faces. Thesecrystals can form reflected arcs mainly at high solarelevations, when light rays meet the basal face at asteep angle. Thin pyramidal crystals, which have rel-atively larger basal faces, are more favorable for mak-ing inner reflection halos. There is, however, notmuch evidence available of the existence of the thinpyramidal crystals in nature. Pyramidal crystals aremainly thought to be formed from the rapid freezing ofwater droplets, which makes these crystals relativelyequidimensional.16,17 Ohtake has published a photo-graph of a thin pyramidal crystal with large basal facesthat was sampled under ice–fog conditions.16 Per-haps under some conditions thin pyramidal crystalscould occur in sufficient quantities to form halo arcsrequiring basal face reflections. The effects of thickand thin pyramidal crystals for the halo arcs for solarelevations of 2° and 20° are presented in Fig. 2.

20 March 1998 y Vol. 37, No. 9 y APPLIED OPTICS 1437

Fig. 2. Effect of the crystal aspect ratio and crystal tilting for the halo arcs for Sun elevations of 2° and 20°. Thin crystals @~a!, ~c!, ~e!,~g!# are effective in forming halo arcs that require basal face reflections; thick crystals @~b!, ~d!, ~f !, ~h!# form only arcs that require no basalface reflections. Tilting of the crystals weakens the basal reflection arcs @~c!, ~g!# more markedly than the arcs requiring no basal facereflections @~d!, ~h!#. Tilts of 10° were used on simulations ~c!, ~d!, ~g!, ~h! and 0° for ~a!, ~b!, ~e!, and ~f ! ~refer to Table 1 for the identificationof the numbered halo arcs!. Parhelia are marked with P, subparhelia with SP, Sun with S, and subsun with SS. For simulations ~a!,~c!, ~e!, and ~g!, the number of rays considered is 70,000 and two populations of pyramid crystals with aspect ratios of 0.05y0.05y0.05 and0.03y0.03y0.03 were used. For simulations ~b!, ~d!, ~f !, and ~h!, the number of rays is 50,000 and the aspect ratios are 0.25y0.25y0.25 and0.20y0.20y0.20. The aspect ratio of the pyramid crystal is given in the form pyramid prismymiddle prismypyramid prism. Thesimulations were made with a program developed by Frank Pattloch and Eberhard Trankle.

1438 APPLIED OPTICS y Vol. 37, No. 9 y 20 March 1998

Fig. 3. Distances of the arcs from the Sun and subsun found from the solar vertical as a function of solar elevation. The numbers nextto each curve indicate the halo arcs that the curve represents. Refer to Table 1 for the halo arc identification. For readability, two curvesare dashed.

Even a small tilting of the horizontal crystal angleweakens those arcs that require basal face reflec-tions, whereas the arcs not requiring basal face re-flections remain relatively more intense ~Fig. 2!.According to Fraser18 and Sassen,19 the crystal tiltingis dependent on the crystal size. If pyramidal crys-tals are too small they cannot assume the very smalltilt angles needed for antiparhelia. Large and thincrystals may orient poorly as well. This was notedduring the halo expedition carried out by three of theauthors in Eastern Siberia during the winter of1996–1997. On every occasion in which large thinplates were sampled, the crystals formed a uniformlytwinkling field around the sky and produced poorhalos: often a mere pillar was visible. Apparently,the same tendency has been noted by Tape in fieldconditions of the Antarctic interior.20

In addition to crystal size, the magnitude of thecrystal tilting angle is also dependent on the aspectratio of the pyramid crystal: the closer to equidi-mensional shape the crystal is, the bigger the tiltangle. As a rule, simulations of pyramidal crystalhalo displays require relatively strong tilt angles andalmost equidimensionally shaped crystals. The yetundiscovered parhelia B arcs that do not requirebasal reflections at solar elevations ,9.9° should alsobe observable with thicker crystals. Identifyingthese arcs at solar elevations ,9.9° might, however,be difficult, as at these solar elevations they occurmerged or almost merged with the corresponding Aarcs. In Fig. 3 the Sun–subsun distances of the arcs

situated in the solar vertical are given as functions ofsolar elevation.

Poor crystal development affects the visibility ofthe halos that require inner basal reflections.7 In-ner structures, such as air bubbles and cavities21 inthe pyramid crystal, might weaken the basal reflec-tion arcs, whereas the arcs requiring no basal facereflections are still observable.

The visibility of all arcs from pyramidal crystalswith their c axes in the vertical direction is naturallyalso affected by the overall structure of the pyramidcrystal. Halo display simulations, laboratory exper-iments, and pyramidal crystal samples have shownthat the structure of these crystals is quite variable.The crystal in question may be a combination of amiddle prism and both pyramid prisms. Anotherchoice combines just one pyramid prism and the mid-dle prism. Then there is the option of no middleprism. These alternatives are enhanced by thevarying sizes of each structural part and by theamount of truncation in each pyramid. Such struc-tural aspects influence the visibility and the bright-ness of different pyramid-crystal-originated arcs inthe sky. For example, when the crystals have nomiddle prism faces, all arcs found at minimum dis-tances of 9° and 24° are entirely absent.

4. Observations

Where exactly then lies the distinction between doc-umented and as yet undocumented parhelia? Asmentioned above, none of the antiparhelia, nor any of

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the outer parhelia, are recorded as far as we areaware. Hence all parhelia listed below representthe inner ~A! ones. The first 9° ~upper! parhelionwas documented by Scorer in the 1974 Easter Sundaydisplay3 and published by Tricker in 1979.4 To ourknowledge, the earliest published picture of a lower9° parhelion was taken by Riikonen in Joensuu, Fin-land, in 1993, and published by Pekkola in 1993.22

The first 18° parhelia were apparently photographedby Thompson in the United States and published in1980 by Greenler.23 The first 20° ~upper! parhelionwas captured by Scorer in the Easter Sunday display3

and published by Tricker.4 Potentially the first 20°lower parhelion was recently photographed by Tape,and it was presented in the Light and Color in theOpen Air topical meeting in Santa Fe, New Mexico, inFebruary 1997. The first 23° ~upper! parhelion wasphotographed by Frank Nieuwenhuys in Hague, TheNetherlands, in 1974 and published by Hattinga Ver-schure in 1975.24 The lower 23° parhelion is stillmissing from the photographic record. The firstpublished 24° ~lower! parhelia were captured by Ri-ikonen in Joensuu, Finland, in 1993 and published byPekkola in 1993.22 The first published 24° upperparhelia were photographed by Tape in 1989 andpublished by Tape in 1994.7 The first published 35°~lower! parhelion was taken by Klaus Sturm in 1987in the Antarctica and was published by Tape in1994.7 Photographs containing the 35° upper par-helia have been taken by Tape in Alaska, but to ourknowledge these have not yet been published.20

5. Conclusions

Pyramidal crystals with their c axes oriented verti-cally give rise to 12 halo phenomena, of which 6 havebeen documented so far. These 12 halos divide intoa wealth of 32 individual arcs in the sky, of whichonly 10 have been photographed. The new arcsawait their first documentation, and although thereare many factors that seem to weaken the possibilityof their existence in nature, we anticipate that in thefuture some of these halo forms will be observed inthe sky.

We express our gratitude to Walter Tape andGunther P. Konnen and the anonymous reviewers fortheir useful and interesting advice and corrections.We thank Finnish ~Finnish Halo Observing Net-workyUrsa!, German ~FG, Sektion Halobeobachtung!and Dutch ~Werkgroep Weeramateurs! halo observ-ing networks, and their observers for the data thatwere available for this research. We are grateful tothe following leaders and correspondents of foreignnetworks: Wolfgang Hinz ~Germany!, ClaudiaHetze ~Germany!, Gerald Berthold ~Germany!, SirkoMolau ~Germany!, Hattinga Verschure ~The Nether-

1440 APPLIED OPTICS y Vol. 37, No. 9 y 20 March 1998

lands!, Stefan Jak ~The Netherlands!, and FrankNieuwenhuys ~The Netherlands!.

References and Notes1. H. Steinmetz and H. Weickmann, “Zusammenhange zwischen

einer seltenen Haloerscheinung und der gestalt der Eiskri-stalle,” Heidelb. Beitr. Mineral. 1, 31–36 ~1947!.

2. H. Weickmann, “The ice phase in the atmosphere,” Royal Air-craft Establishment Library translation 273 ~Ministry of Sup-ply, London, 1948!.

3. E. C. W. Goldie, G. T. Meaden, and R. White, “The concentrichalo display of 14 April 1974,” Weather 31, 304–311 ~1976!.

4. R. A. R. Tricker, “Arcs associated with halos of unusual radii,”J. Opt. Soc. Am. 69, 1093–1100 ~1979!.

5. E. Barkow, “Eine seltene Haloerscheinung,” Meteorol. Z. 33,476 with plate ~1916!.

6. P. J. Neiman, “The Boulder, Colorado, concentric halo displayof 21 July 1986,” Bull. Am. Meteorol. Soc. 70, 258–264 ~1989!.

7. W. Tape, Atmospheric Halos, Vol. 64 of Antarctic ResearchSeries ~American Geophysical Union, Washington, D.C., 1994!.

8. V. Makela, ed., Ursa Minor, Bulletin of the Ursa AstronomicalAssociation ~Helsinki, Finland, 1986–1997!.

9. M. Riikonen, “Hamaransateita ja haloja Filippiineilla,” Tahdetja Avaruus 24, 45–47 ~1994!.

10. W. Tape, “Pyramidal ice crystals and odd radius halos,” inLight and Color in the Open Air, Vol. 12 of 1990 OSA TechnicalDigest Series ~Optical Society of America, Washington D.C.,1990!, pp. 64–66.

11. The extension of the word parhelia to ascribe arcs associatedwith odd radius halos was introduced by Tape in 1994 ~Ref. 7!.For convenience we apply the same term in this paper.

12. P. Parviainen, C. F. Bohren, and V. Makela, “Vertical ellipticalcoronas caused by pollen,” Appl. Opt. 33, 4548–4551 ~1994!.

13. J. Hakumaki and M. Pekkola, “Rare vertically elliptical halos,”Weather 44, 466–473 ~1989!.

14. M. Pekkola, “Finnish halo observing network: search for rarehalo phenomena,” Appl. Opt. 30, 3542–3544 ~1991!.

15. M. Riikonen and J. Ruoskanen, “Observations of verticallyelliptical halos,” Appl. Opt. 33, 4537–4538 ~1994!.

16. T. Ohtake, “Unusual crystal in ice fog,” J. Atmos. Sci. 27,509–511 ~1970!.

17. C. Magono, S. Fujita, and T. Taniguchi, “Unusual type of singleice crystals originating from frozen cloud droplets,” J. Atmos.Sci. 36, 2495–2501 ~1979!.

18. A. B. Fraser, “What size of ice crystals causes the halos?,” J.Opt. Soc. Am. 69, 1112–1118 ~1979!.

19. K. Sassen, “Remote sensing of planar ice crystal fall attitudes,”J. Meteorol. Soc. Jpn. 58, 422–429 ~1980!.

20. W. Tape, Department of Mathematical Sciences, University ofAlaska Fairbanks, Fairbanks, Alaska 99775 ~personal commu-nication, 1997!.

21. K. Sassen, N. C. Knight, Y. Takano, and A. J. Heymsfield,“Effects of ice crystal structure on halo formation: cirruscloud experimental and ray-tracing modeling studies,” Appl.Opt. 33, 4590–4601 ~1994!.

22. M. Pekkola, “Kolme komeaa halonaytelmaa,” Tahdet ja Ava-ruus 6, 40–41 ~1993!.

23. R. G. Greenler, Rainbows, Halos and Glories ~Cambridge U.Press, Cambridge, 1980!.

24. P.-P. Hattinga Verschure, “Bijzondere halo’s waarnemingen in1973 and 1974,” Zenit 2, 364–365 ~1975!.