The corona is the result of scattering of light by particles ranging in size from about 10 μm to 100 μm. In nature, these particles can be ice needles or cloud droplets. Scattering in the classic corona is a result of diffraction, which will be discussed in detail below. Corona appears as rings of colored light surrounding the luminous source, usually the moon or sun. These rings are not as large as the rings of the halo, and are usually more strikingly colored. The angular diameter of the typical corona is less than 5°. Like the rainbow, the corona can be observed on a small scale as well, with terrestrial sources of light and particles close to the observer.
If the density of particles is high, and if they are large or of a wide range of sizes, the corona degenerates into a fuzzy bright patch at the location of the source. This happens quite often in altostratus and altocumulus clouds. The corona is often seen in thin altostratus translucidus, or in cirrostratus, where it may be combined with the halo. The halo gives evidence of larger ice crystals, while the corona reveals particles of about 20 μm diameter. Corona is more common than the halo, but really good, colorful coronae are rather rare.
The classic corona consists of a bright aureole, bluish in the centre and browinish on the periphery, surrounded by one or more rings of lesser intensity that are bluish on the inside and red on the outside, passing through green and sometimes yellow on the way. Usually only one ring is visible, but up to three rings have been observed. Corona is usually observed around the moon, since the moon is easy to look at, but it occurs as often around the sun, where is is much brighter. Since looking at the sun is uncomfortable, it is necessary to screen off its brilliance when looking for the corona. This can be done by putting the sun behind a building or other screen, but better by looking in a silvered garden globe with your head blocking the sun. Welder's goggles can also be used. The problem here is the same as in viewing eclipses, and the same cautions are necessary.
Compared to the rainbow and the halo, the corona is a relatively neglected part of meteorological optics. However, it is one of the most frequently observed phenomena of this type, and can furnish intellectual challenge, useful information and visual pleasure.
The scalar Huygens-Fresnel-Kirchhoff diffraction theory will be accurate enough for our purposes here. In this theory, the light intensity at any point is the absolute value squared of a scalar light amplitude, which can be found by adding coherently the contributions from a previous wavefront, assuming the wavefront to be a new source of waves. A more accurate theory takes into account the vector nature of light amplitudes as electromagnetic fields. Since most diffraction phenomena are insensitive to polarization effects, this much more complex theory is not required for explanation of the corona.
The geometry of the diffraction problem is shown at the right. A source S emits monochromatic light of wavelength λ. dA is a portion of the wavefront of this light at some time, and n is a unit normal to the wavefront element in the direction of propagation. The light travels a distance r from S to dA, and r is a unit vector in this direction. We desire the contribution to the amplitude at point P a distance s from dA, and let s be the unit vector in the direction from P to dA. The contribution dU from dA is the product of several factors. First, it is proportional to dA. Next, it is proportional to the obliquity factor nr - ns. The phase and amplitude factor for the propagation from S to dA, and from dA to P, is eikr/r times eiks/s. Finally, there is a factor -Ci/2λ multiplying everything. C allows for the strength of the source, while (-i/λ) is a factor from the wave equation that describes the propagation of light. The contribution is inversely proportional to the wavelength, and has a phase shift of -π/4 radians. If you start from a plane wavefront at time t, this formula accurately predicts a plane wavefront of the same amplitude at time t + dt, which has progressed a distance c dt.
Our problems will involve propagation close to a straight-line direction from source S to observer P, and for a small aperture A. The obliquity factor in this case is about 2, and the distances r and s are roughly constant, the major variation coming in the exponents if 1/k is large compared to the size of the aperture. This gives the much simpler formula shown in the diagram. In fact, we may usually consider r a constant (plane wave incident on the aperture), and get a still simpler formula where only an integral of e eiks over the area of the aperture is required.
We shall make use of a relation called Babinet's Principle, illustrated in the figure at the left. At the top we have an aperture A' that we imagine to be an opening in an opaque screen, which produces a certain amplitude U' at the point P. This has been shown for the usual arrangement for viewing Fraunhofer diffraction, in which a lens of focal length f focuses parallel rays on a screen at a finite distance. In the middle, there is an obstacle A" complementary to A'. In this case, we have an amplitude U" at the same point P. Finally, at the bottom, there is no obstacle nor aperture, and the amplitude at P is U. Quite generally, we have that U' + U" = U, as can be seen from considering the sum of the contributions from the diffraction integral in the two cases.
With the Fraunhofer setup as shown, the lens makes U = 0, so that U' = -U". Thus, the amplitudes from the obstacle and its complementary screen are equal in magnitude but opposite in phase. Therefore, if we have obstacles in the light path, we can find the diffraction pattern by considering the complementary apertures.
We consider two kinds of obstacles in connection with the corona. First, we may have ice needles, whose complementary screen is a single slit of, say, width a and length c. Next, we have spherical droplets, which act like circular obstacles of diameter D. Their complementary screen is a circular aperture of diameter D. The evaluation of the diffraction integral for the two problems is straightforward, but we use the results only, leaving the derivations to the References.
The amplitude for a slit of infinite length is sinc u = (sin u / u), with u = (πa/λ)sin θ. If the slit is finite, the amplitude is the product of two such factors, one for the width and one for the length. The amplitude for a circular aperture of diameter D is 2J1(u)/u, where u = (πD/λ)sin θ. J1(u) is the Bessel function of order 1 that is finite at the origin. 2J1/u = J0 + J2, which may be easier for numerical evaluation. This is called the Airy pattern, after G. B. Airy, who first derived it in 1835. The intensity, of course, is the square of these expressions.
The first zero of the slit pattern occurs when u = π, or sin θ = λ/a. The first zero of the Airy pattern occurs when u = 1.22π, or sin θ = 1.22λ/D. In each case, we have a strong central maximum surrounded by considerably weaker subsidiary maxima separated by dark spaces, and the size of the pattern is inversely proportional to the size of the object. For an angular radius to the first minimum of the pattern of 4°, we can estimate the size of the object as about D = 1.22λ/sin 2° = 17.4 μm, assuming that λ = 0.5 μm. This is close to the conditions in the corona as commonly observed.
Now suppose we have many obstacles, each diffracting the incident light. If the obstacles are randomly distributed, then the superposition of the amplitudes from each obstacle is incoherent. This means that we can add the intensities of the contributions from each obstacle, since cross terms will vanish. The resultant intensity will have the same angular distribution as the intensity from one obstacle, but its intensity will be proportional to the number of scatterers in the direction of observation. On the other hand, if there is any regularity in the distribution of the obstacles, the amplitudes must be added instead, and the cross terms may not vanish. This can greatly change the observed diffraction pattern.
Lycopodium powder, a fine beige dust, consists of the spores of the Lycopsid Lycopodium clavatum, a modest club moss, shown in the photo at the right, that is the survivor of very ancient plants first appearing in the Devonian. It was used by pharmacists to coat pills by rolling them (the pills) in it, when pharmacists made their own pills. It was used in Kundt's Tubes to show the location of acoustical nodes. Today its principal uses seem to be in homeopathic medicine quackery and as a chemistry demonstration of the inflammability of fine powders. The chemophobes have already made it a hazardous substance, since it appears to cause a dermatitis, and should be kept out of the eyes, but it is, of course, innocuous. As a fire hazard, it is as dangerous as flour. The small, spherical spores are very closely the same size. When dusted on a glass plate, they give very nice Airy patterns that completely agree with theory, with a bright aureole and colored rings. Unfortunately, lycopodium is about the only naturally available powder that will do this. Other powders, such as flour, are too irregular to give anything but a featureless aureole. Very rough observations gave me a diameter of about 60 μm. Lycopodium powder is available from www.sciencelab.com at $10.52 for 25g, which is should be an ample amount.
The corona can be seen in a glass plate misted by the breath (and in other similar cases), but the diffraction pattern is quite different from the proper corona produced by lycopodium powder. Instead of the Airy pattern, a dark aureole with colored rings is obtained, the bright aureole being missing. In this case, the superposition of amplitudes is coherent, and the size of the pattern depends on the regular separation of the droplets of the mist, which are not by any means spherical or even circular. The size of the droplets has little effect on the size of the diffraction pattern. Tricker made diffracting screens of dots drawn by hand photographically and obtained similar patterns with dark centres. His patterns show a central dot surrounded by a ring whose radius is independent of the size of the dots. Apparently, there is very little intensity at small angles of deflection in this case. This also happens with the coronae seen in misty café windows at night, which will be observed to have no aureole.
Natural coronae are usually observed in white light, so they are a superposition of patterns of different sizes in the different colors, red patterns being smaller than blue ones, since the wavelength is greater. For circular obstacles of the same size, such as Lycopodium powder or cloud particles, this explains the usual coloring. The colors will be mixed in less regular coronae, and so they will be typically white. In monochromatic light, the Airy pattern will show distinctly, and more rings will be visible.
Altocumulus translucidus, or thin altocumulus, is the typical cloud of the corona. Sometimes the moon lights a patch of 5° radius or more, as an irregular white area, surrounded by cloud of a greyer general tone, that represents the cloud lighted from above by the moon. The light patch is produced by multiple scattering in the relatively dense cloud. When a thicker patch of cloud floats by, it may be opaque and quite dark. In areas that may seem free of cloud, the moon often shows a coronal aureole with a reddish or brownish outer border, of about 1° diameter. This is caused by the same drops as those in the thick cloud, but now single scattering predominates rather than multiple. Although invisible, there is still some cloud in this area. The best coronas are produced in this thin altostratus. Sometimes it helps to mask off the bright part of the moon to eliminate entopic coronas produced in the eye. They will disappear, but the real corona will remain. Estimate the size of the corona by comparing it with the size of the moon, which is about half a degree.
The corona varies greatly in size, depending on the diameter of the particles causing it. This is often seen near the edge of a cloud, where the cloud particles are evaporating, and are smaller than particles deeper in the cloud. The corona is not the work of a single particle, but the cooperative effort of all the particles that return light to you. The part of the corona in front of a deeper part of the cloud will be smaller in diameter, while that in front of the edge of the cloud will be larger. The corona will be distinctly noncircular, resembling an ellipse with the moon or sun in one focus. A cloud must be quite thin in order for a good corona to be seen.
Altocumulus, either produced as such by mid-level afternoon instability, or as a decay product of cumulus, is a very common cloud type in Denver, seen practically every evening in the summer. Whenever the moon is between first and last quarter, corona phenomena of some type are frequently visible. A good corona with one or more colored rings is, however, always a rare event. A brown-tinged aureole is the usual corona manifestation.
Clouds whose thin edges are iridescent are very often seen in Denver in the autumn and winter. These are the orographic altostratus lenticularis that form in the lee of the mountains, and remain fixed in location. These colors are simply parts of the corona, and depend on the distance from the sun and the size of the cloud particles. Within a few degrees of the sun, the light is white and brilliant, since these parts correspond to the aureole. The colors farther away, up to as much as 30°, are not spectral colors, but for every size drop and solar distance the color can be calculated and assigned CIE coordinates. By noting the color and the distance to the sun, the size of the droplets at that point can be calculated.
On 12 August 2003, the full moon at Denver rose into altostratus translucidus, through which the moon's limb was clearly distinguished. The aureole of the corona was about 1° in radius, and tinged with brown on the outside. The effective angular radius was 0.75° or 0.01309 radian. If the effective wavelength is taken as 571 nm (the usual value), then the diameter of the cloud particles was (1.22)(.571)/(0.01309) = 53 μm. No halo phenomena were seen.
M. Born and E. Wolf, Principles of Optics (London: Pergamon Press, 1959). Chapter VIII, especially pp. 369-400.
R. A. R. Tricker, Introduction to Meteorological Optics (London: Mills and Boon, 1970). Chapter V.
M. Minnaert, The Nature of Light and Colour in the Open Air (New York: Dover, 1954). pp. 208-230.
The photograph of Lycopodium clavatum is taken from a photograph by Paul Hackney of the Ulster Museum.
Composed by J. B. Calvert
Created 2 August 2003
Last revised 12 August 2003