260 MONTHLY WEATHER REVIEW SEPTEMBER 1911
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FIGURE 7.
THE GEOMETRICAL THEORY OF HALOS-VI
By EDGAR W. ,WOOLARD
[Weather Bureau, Washington, D. O., September 19411
PART 3. THE OPTICAL METEORS PRODUCED BY ICE
CRYSTALS IN THE ATMOSPHERE
Among the innumerable crystalline forms produced by
the condensation of water vapor in the atmosphere at
temperatures below freezing, as illustrated, e. g., in frost-
work and by snowflakes, there are two or three quite simple ones from which all the others may be built up,
viz, hexagonal columns with or -without pyramidal caps
(complete or truncated) and hexagonal plates; the columns are sometimes capped with plates, and the pyramids may
occur unattached to columns.
These elementary forms (figure 19) are frequent1 ob-
polar regions; they often are present in the atmosphere at the surface of the earth when a halo display is witnessed,
and there is every reason to believe that it is some one or
more of them, or simple combinations thereof, which ordinarily produce halos, and not the complicated crystal
groups and patterns shown in general by snowflakes-in
fact, the majority of authent'icated halos do not require
anything more complicated than a simple hexagonal right,
prism (column or plate). The present investigation will therefore be restricted to
hexagonal right prisms (in the form of either columns or
disks) , hexagonal right pyramids (complete or truncnt ed),
and simple combinations of these two forms.
served in snow and frost at low temperatures, especia s ly in
1 The previous papers have appeared in the MONTIFLP WEATHER REYIEW as follows: I 64:331-326 1836' I1 65:4-8' I11 66:55-57' IV 65:1W192' V 65:301-302 1837. The fig-
& in the &sm't pAper a r l nu&bered coksechvely with those in the &her papers.
From the six lateral faces and two bases of a hesagonal
right prism, taken two at a time, may be formed 28 pos- sible combinations. Of these combinations, one consists
merely of the two bases, which form a refracting angle of
0' and do not produce any resultant deviation; 15 are com- binations between lateral faces, of which 3 are between
opposite faces and again form 0' angles, and G are between
adjacent faces and form angles of 120' through which no
transmission is possible; 6 are between alternate faces, and
.all form truncated 60' refracting angles; 12 are combha- tions of a lateral face with a base, forming in nll cases a
refracting angle of 90'.
To determine all the halos w-hich a hexagonal right prism
with plane bases is capable of producing, it is necessary to calculate, for each of all orientations of the prism in space,
the images obtained by refraction through the six GOo angles between alternate lateral faces and through the twelve 90' angles between lateral faces and bases, together
with the images formed by reflection (external, and in-
ternal with or without accompanying refraction) from the
six lateral faces and two bases. The refracting edges of
the 60' angles are parallel to the principal axis of the crystal, and those of the 90' angles are perpendicular to
the principal crystal axis.
In pyramidal crystals, the triangular faces may, accord-
ing to the laws of crystallography, have any one of several different inclinations to the principal axis of the crystal,
these different values being connected by simple numerical
relations. Unfortunately, the possible inclinations can be
SEPTEMBER 1941 MONTHLY WEATHER REVIEW
Alternate faces of same pyramid (1, 3). .
~ ~
A pyramidal face (1) and-
~ ~
~. . .. .. ........-
Opposite faces of same pyramid (1,1) ....
~
Opposite face of hexagonal prism (4) ... .-....... . .. ... .... .-.
-4lternate face of hexagonal prism (3).. .. ....................
A pyramidal face and base of hexagonal prism .____...._...._.__.
(3) ............................................................
~
.. . . ~. . . ... .. .......
A face of one pyramid (1') and alternate face of opposite pyramld
261
76'24' 42'48'
40 42 90
24 51 90 63 01 2808
65 09 90
2808 53 58
determined only by actual measurement of the particular
one of them that some actual crystal happens to have; an ice crystal is a difficult thing to measure-few such meas-
urements have been mttde, and they necessaril are more or less inexact. Hence the values of the possi g le, and of
the actually occurring, inclinations are somewhat un-
certain. The most probable va.lue-one determined in
part from the evidence of the halos themselves-seems to
be the one deduced by Humphreys, viz, 24'51', and it
FIGURE 19.-Elenieiitary Forms sf Ice Crystals.
will be adopted here; the culculntions for pyramicliil crystals nay easily be repented by anyone who desires,
w-ith any other given value of this angle.
Then, in a crystal with a p,vramidal element, the re-
fracting angles through which transmission is possible (i. e., the diheclral angles less than 99'31') are as follows (see figure 20):
Planes
Inclination
YIGUKE %.-Refracting angles in a pyramidal crystal: The figure is obtained by passing planes through the center of a sphere. parallel Jo the faces of a hexagonal right prism and the faces of8 hexagonal right pyramid, cutting the spherein great circles; theangles at. which these circles intersect one another include all the dihedral angles formed by any two faces of any of the crystal forms in figure 19, and the points of intersection are thelocationsofthe refracting edges. Pairs 0; opposite faces in the prism, or in B bi- pyramid, are parallel and cut the sphere in the same great circle. Given VIV;= V$V,= . . . =VsV1=60~ and ha=24"51', any desired arc or angle in the figure may tie computed.
The present series of papers is confined to only the geo-
metrical formation of theoretically possible halos; but it
is necessary to introduce a few physical considerations as
a general guide. On any occasion, there will always be
some crystals in each of all geometrically possible orienta-
tions in space. At times the distribution of the crystals will be entirely random-no larger proportion in one orien-
tation than in any other; a t other times, one or more
pltrticular orientations will more or less predominate.
Each separate crystal will always procluce a set of images, but ~liether or not the aggregate effect of all the crystals
in the same orientation will be distinguishable depends jointly on the number and the proportion of crystals in
this particular orientation and on the relative intensity of the light transmitted and reflected by that orientation 2 W. J. Humphreys Physics of the .4ir 3 ed. p 523 528 1940' MONTELY WEATHER
(cf. pnper I, p. 324).
REVIEW 5O:535-536, lCl22, and 51:255-256, iY23. i'f Bessoi, IViONT&,Y WEATEER REVIEW 51:?54-?55, 1923.
262 MONTHLY WEATHER REVIEW SEPTEMBER 1941
In the formal geometrical theory, we shall take into account only those orientations of each crystal form that
may reasonably be expected under natural conditions to
lead to sufficient concentration of light for the production
of readily observable effects: that is, only orientations that (1) correspond to the minimum minimorum, or (2 )
predominate as a result of a restrictive influence that deprives the crystal of one of its degrees of freedom and
a t the same time correspond to minimum deviation, or
(3) predominate because of a restrictive influence that
deprives the crystal of two of its degrees of freedom, in
which case all deviations must be considered. In general,
only reflections that are total need be taken into account.
The different orientations that are to be taken into account in any case, for deriving the collective effect of
all the crystals, may be conveniently specified by the positions in which the principal crystallographic axis may
lie in space, and the extent to which rotation of the crystal
may take place around the axis.
THE OPTICAL METEORS PRODUCED BY CRYSTALS ORIENTMD
AT RANDOM
The case in which the crystals have three degreesof
freedom and are oriented completely at random-as many crystals lying with their axes in one position as in any
other, and rotating freely around their axes-is easily
disposed of. The only important relative concentration
of light into a limited region of the sky is produced by refraction a t and very near the minimum minimorum.
Any plane through the line from observer to luminayv
will intersect some of the crystals; a certain proportion of these crystals will happen to be so oriented that the sec- tion by the plane is a principal plane of some one of the
refracting angles, or very nearly so. All rays in this plane
that are incident on such crystals will be refracted in or
near a principal plane; the sections themselves will be
randomly oriented in the intersecting plane, so that all
possible values of the angle of incidence, and hence of the deviation, will occur. Of the crystals that produce any given deviation D, all those on a line through the observer
a t an angle D with the line from observer to luminary will
send the refracted ray to the observer; the observer will therefore see an image on the sky a t an angular distance
from thc luminary equal to the deviation, and in a direc- tion from the luminary on the great circle where the in-
tersecting plane cuts the celestial sphere. The images
corresponding to thc different deviations in any such plane will collectively form an arc extending along this
great circle from the minimum minimorum to the maxi- mum deviation, but fading rapidly in brightness with
increasing deviation. The same effect will be produced
in all planes through the line from observer to luminary, all of which may be obtained by revolving a plane around
this line; hence a circular ring of light will appear, centered at the luminary, with a sharp inner edge (contrasting with
a comparatively dark sky within) of radius equal to the
minimum minimorum, and a diffuse outer border merging into a general sky glare beyond.
The concentration of light ncar minimum deviation in the principal plane is so strong that these circular halos
may be distinguishable even when particular orientations
predominate among the crystals sufficiently to give other
arcs also.
Each refracting angle can produce such a circular halo; and it is to phenomena of this type that the generic name
halo properly applies (Gr., iixws). The radii of all these halos that can be produced by the crystal forms we have
enumerated are as follows:
Refrac-
21 50 IIesagonal prism.
1 p~ ~ ~~~
.
~~
The 22’ halo is by far the commonest of all halo phe- nomena; nearly all the others in this table have been
observed with certainty, though most of them are very
rare.3
3 Scc W. J. Humphrrys Physics nf the A i r , 3 od., pp. 534-536, 1940, and the further referencrs thcre given. Cf.’Besson, MONTHLY TI’E~T~ER REVIEW, 42: 443,1914,and 51: 254,
1923.
RECALIBRATION OF INSTRUMENTAL EQUIPMENT AT SOLAR RADIO STATIONS
BY IRVING F. HAND AND HELEN F. CULLINANE
[U. 6 . Weather Bureau Solar Radiation Supervisory Station, Blue Hill Observatory of Harvard University, Milton, Mass., October 19411
The desirability of recalibrating the equipment used at
stations where records of solar radiation are now being
made has long been recognized. The original calibrations of the pyrheliometers were made by three separate agencies: (1) the Solar Radiation Investigations section
of the United States Weather Bureau (2 ) the Eppleg Laboratory, and (3) the National Bureau of Standards.
Calibrations by (1) were made by occulting the sun a t regular intervals 011 clear days, subtracting the values of
the sky radiation thus determined from the total radiation on a horizontal surface, and obtaining the ratio between
this result and the otherwise measured value of the normal
incidence radiation reduced to a horizontal surface by means of the sine law. Calibrations by (2 ) and (3) were
obtained by direct comparison against standards furnished by the Weather Bureau. It was obvious that great im-
provement would be obtained if all instruments wcre recalibrated against a single carefully standardized pair
of pyrheliometers; and the need for this increased after a
more thorough study of the Epplry pyrheliometer had shown that the cosine law failed to hold with low sun.‘
Moreover, some stations had not been inspected for over
10 years, and it was thought best to check not only the pyrheliometers but also thc recording equipmcnt and
other uccessorics.
Between March and July 1941, all stations listed in table 1 were therefore visited; the pyrheliometers were carefully leveled, where necessary, and clieckcd against
citlicr the 10- or the 50-junction standards, which pre- viously had been standardized directly against the stand-
ard Smithsonian silver-disk normal incidence pyrheliom-
eter. Wc may now bc confident that all these stations
nre on tlic same standard, and as close to the Smithsonian scnlc of pyrlicliometry as we are able to place it. Table 1
gives tlic avt’rage monthly c. 111. f . of d l pydirliomctcrs
checkd, aid iilso the percrntiigc1 c*haLngt. frnrii the rntwn
1 Byron €€. Woertz and Irvmg B IIand The Charnr*teristics o1 L ~P Epplcy Eyrhrhonr ter. MONTHLY WEATEER REVIEW, 69 146-148. 1941, May