The Halo


Contents

  1. Introduction
  2. Theory of the Halo
  3. References

Introduction

The halo includes the phenomena of meteorological optics arising from light scattering by refraction and reflection in atmospheric ice crystals. Of the other phenomena, the rainbow and the glory are due to light scattering by water droplets, and the corona is due to diffraction in small particles of any kind, whether water droplets, ice crystals or dust.

The word halo comes from the Greek 'alws, the word for a (round) threshing floor, that also came to be used for the disc of the sun and moon, or of a round shield. It should not be confused with 'als, which is "salt." Our word applies mostly to an effulgent ring, which is how the meteorological phenomenon got its name.

Halos are seen almost exclusively in thin cirrostratus cloud, which forms a veil, often scarcely perceptible, through which the sun or moon shines clearly. Of course, halos can be seen in other forms of clouds of the same nature, but this is usually only in patches. A halo exists only where there are ice crystals to scatter light to the eye. The various halo phenomena reflect the different shapes, sizes and orientations of ice crystals.

Ice crystals form from supercooled water, and the form of the crystal depends on the temperature and degree of supersaturation. The principal forms giving rise to halos are hexagonal plates and hexagonal prisms or needles. The prisms may be "capped" with a plate at one or both ends when during its growth it resides in regions of different characteristics. Bullet-shaped crystals with pyramidal terminations also exist, usually in groups joined at the points (to the small seed crystal on which they began). Dendritic and other crystal forms do not have the smooth faces necessary for halos. The hexagonal axis of plates and columns is called the c-axis. Growth normal to the c-axis produces plates, while growth in the direction of the c-axis produces needles and columns.

Some of the observed halos are shown in the figure at the right. Usually, only a selection of them occur at any one time. The parhelia and the 22° halo are the most common (by far). The circumscribed halo can be continuous from top to bottom, but usually is represented only be brighter areas at the top and bottom of the 22° halo. The Lowitz arcs and the Parry arcs are much rarer. Not shown are the infrequent 46° halo and the bright spots associated with it, which are similar to those for the 22° halo.

The 22° halo is caused by refraction in randomly oriented small (20 μm) plates and needles, as will be explained below. Such small crystals are strongly battered around by Brownian motion, which gives them their random orientation. This halo may be reddish toward the sun, and bluish on the other side. Larger plates and capped columns are subject to aerodynamic forces in their fall, and orient themselves for maximum drag--that is, crossways so the plates are horizontal. The parhelia are caused by refraction in these oriented crystals. The circumscribed halo, Parry arcs, and Lowitz arcs are caused by such crystals with a rocking or vibrating motion, the details of which are subjects for argument among halo enthusiasts.

The sun pillars and parhelic circle are caused by reflection, not refraction. The parhelic circle is caused by crystals with the c-axis vertical, usually capped columns, but it has been argued that needles can fall this way, and not turn crossways. Very rarely is a complete parhelic circle observed. There may be increased illumination at an anthelion, and two paranthelia at 120° from the sun. Things happen at the antisolar point as well, but this is usually below the horizon. Sun pillars are caused by plates floating horizontally, though the exact cause is also a subject of discussion. The reflection features are not colored, of course.

When the sun is lower than 32.2°, the colorful circumzenithal arc, or at least a segment of it, can be seen around the zenith, distant from the other halo phenomena. When the sun is at 32.2° elevation, light entering the top of a capped column strikes a vertical face at the critical angle, and a spot of light appears at the zenith, since the light is coming directly downwards. For a higher sun, the light is lost in the lower crystal termination. As the sun sinks lower, the light exits at an increasing angle, creating a colored arc around the zenith. When the sun is at 22.2°, the light exits an an angle of 67.9° from the normal, so that the deviation will be 45.7°, the same as for the 46° halo. In this case, the circumzenithal arc is tangent to the 46° halo. All this can be easily worked out by Snell's Law. The reversed path gives a circumhorizon arc for a high sun, but this is rarely observed, if at all.

I look for halos whenever I suspect that cirrostratus is present. In Denver, this seems to be chiefly in the winter, and in the late afternoon sky. Parhelia, or sun dogs, are often seen when the sun is low under these conditions. There is a picture of parhelia in Sky Observations. In that picture, only short fragments of the 22° halo are visible near the parhelia. This is the most common halo occurrence in Denver. My attempts to view other solar halos have so far been unsuccessful, since the necessary clouds seem to be rare in Denver during the day. Lunar halos are visible now and then, but not frequently, and sometimes only short arcs are seen. The lunar halos show only the 22° halo, and no "moon dogs."

Halos are more difficult to view than rainbows, because they occur toward the source of light, whose brilliance competes with them. The contrast is much better with the lunar halo against a dark sky; probably most solar halos are missed because they are too dim. The source of illumination is at the centre of the halo, and the halo is quite often seen as a complete circle. Halos can be reflected in water, and can be produced by light that has been reflected, just as rainbows can. For an explanation of the differences between the reflection of the halo and of a physical object, see The Rainbow.

Halos are polarized, but not nearly as strongly as the rainbow. The direction of polarization is tangential, not radial as with the rainbow.

Theory of the Halo

The theory of the halo is the calculation of the intensity of light scattered by small ice crystals of various shapes and orientations. This is a very difficult problem in general, but some headway can be made by considering the paths of light through the crystals by ray-tracing. A computer is then necessary to average over all the possible orientations of the crystals, and all types of crystals. There are people who have done this, and their results are quite pleasing, reproducing most of the features of the halo, especially the rare features. A link to Bob Fosbury's halo website appears in the References.

Let's consider the simpler ray-tracing problems, which will at least give a qualitative explanation of halo features. We will consider rays in a principal section (perpendicular to the refracting edge) for simplicity, but these are by no means the only rays important in halos. The diagram shows the ray in a principal plane with a refracting angle α, drawn as 60°, which is right for rays in planes perpendicular to the c-axis of plate and column crystals. The angle of incidence at the first face is φ1, and the angle of refraction is φ1'. From Snell's Law, sin φ1 n sin φ1'. The ray travels across the crystal and the angle of incidence is φ2'. From geometry, α = φ1' + φ2'. At the second face, the angle of refraction is φ2, given by sin φ2 = n sin φ2'. The deviation of the ray, δ = φ1 + φ2 - α. These formulas can be used in this order to find the deviation of the ray.

It is a distinct complication that a ray may meet a different face than the one indicated at certain angles of incidence. Also, a ray inside the crystal may meet a face at greater than the critical angle for total reflection, and be totally reflected instead of refracted out of the crystal. Further complication arises since the whole bundle of rays meeting the crystal from a certain direction may be split into parts taking different paths. How this happens depends on the size of the crystal as well.

The index of refraction of ice is n = 1.307 for 656.3 nm, and 1.317 for 404.7 nm. A Cauchy formula, n = 1.310 + 2643/λ2, where λ is in nm, gives reasonable values for the visible range. Ice is a uniaxial doubly-refracting crystal, but the birefringence is small and is not significant.

For the mean index of refraction n = 1.310 of ice, corresponding to orange light, the deviation is 43.46° when the angle of incidence is 13.46°. At smaller angles of incidence, the ray is totally reflected if it meets the same opposite face. At an angle of incidence of 90°, grazing incidence, the deviation is again 13.46°, though no light gets through because there is no aperture. The deviation is less at intermediate angles, smoothly passing through a minimum of 21.84° for an angle of incidence of 40.92°. The minimum deviation ray is quite special, since it passes through the crystal symmetrically, with φ1 = φ2, and φ1' = φ2' = α/2, or 30° for the hexagonal crystal. Also, the whole entering bundle of rays also gets out the other side.

If we differentiate the expression for δ with respect to φ1 and set it equal to zero, which is the condition for an extremum, we get dφ2/dφ1 = -1. This shows that there is unit angular magnification at minimum deviation. The index of refraction, the refracting angle, and the angle of minimum deviation are related by the formula n = sin[(δmin + α)/2]/sin(α/2).

We can discover by explicit calculation that if the angle of incidence varies from 30° to 53°, the deviation only varies in the range 22° to 23°. This suggests that light entering from a wide range of directions (imagine the crystal rotating, but the direction of the incident light constant) is mostly deviated by about 22° or 23°. Since this is near the angle of minimum deviation, we also know that other rays will not be deviated by smaller amounts. Therefore, if we are facing in the direction of the light source, randomly oriented hexagonal crystals will send us light from a cone of directions of about a 22° angle. This is quite analogous to the case of the rainbow, except there our back was to the light source and the rainbow came to us at an angle of 42° with the anti-source point. Our eyes will interpret the light from the 22° cone as coming from a luminous ring surrounding the source of light, which will be the 22° halo. This qualitative argument is upheld by arduous computer calculations.

The ray paths in observing the 22° halo are shown at the right. Note that the two crystals shown are at different distances from the observer O, but at about the same level in the cirrostratus cloud. The crystals contributing to the halo seen by observer O are on an ellipse in the cloud. Other observers will see a halo coming from different crystals. Even the two eyes of an observer will see slightly different halos, though in perception they may be fused stereoptically. The inside of the halo is darker than the outside, since light scattered by this process is not present inside (22° is a minimum deviation). In the rainbow, one drop included all angles of incidence, so that a caustic surface was created from rays surrounding the ray of minimum deviation. In the case of the halo, the different angles of incidence are in different crystals, so no coherent superposition is possible, and there are no caustic surfaces. This means that there is no analogue to the supernumerary bows, and no picking out of colors by the peaks in the Airy function. In this respect, at least, the theory of the halo is simpler than the theory of the rainbow.

The parhelia are created the same way. If the rays were in planes perpendicular to the axis, they would lie on the 22° halo. In fact, they do so when the sun is low, and its rays almost horizontal. When the sun is higher, the rays take an oblique path, and the angle of minimum deviation is increased. The parhelia still lie at the same level as the sun, of course. When the elevation of the sun is 50°, the parhelia are 10° 36' outside the halo. At 30°, the displacement is only 2° 59', and at 10°, only 20'.

The angle of minimum deviation for a 90° refracting angle is given by the formula above to be 45.73°, agreeing well with the angular diameter of the large 46° halo. This is the angle between the top or bottom of a plate or column and the lateral sides. It does not appear as often as the 60° angle in ice crystals, and the scattering is, therefore, smaller. Whether parhelia exist for the 46° halo is an open question, since the required orientation of the crystals would be a difficult one. Lynch shows the 46° parhelia in his diagrams, but I have not seen a photograph showing them. The 46° halo is usually only partial and much weaker than the 22° halo.

The effects of diffraction have been neglected in our analysis, because even the small crystals responsible for the 22° halo are still much larger than the wavelength of light (20 μm vs. 0.5 μm). Still, there must be some diffusion of the light from this cause. Still smaller crystals are responsible for the corona, which is a diffraction phenomenon, and more common than the halo. It is quite possible for the halo to be combined with the corona; all that is required is cloud particles of the necessary sizes.

Minnaert reports that halo phenomena have been observed in ice crystals lying on the ground. Just as in the case of the similar rainbow phenomenon, the dew bow, the intersection of the cone of the 22° or 46° halo with the plane of the ground is a hyperbola. The sun must be low (lower than 22°) to observe this halo. As explained in connection with the dew-bow, this is an effect of the visual sense trying to make the most sensible interpretation possible of what it receives.

References

R. A. R. Tricker, Introduction to Meteorological Optics (London: Mills and Boon, 1970). Chapter IV.

M. Minnaert, The Nature of Light and Colour in the Open Air (New York: Dover, 1954). pp. 190-208.

D. K. Lynch, Atmospheric Halos, Scientific American, April 1978.

Bob Fosbury's Halo Site is excellent. Look under Personal Items on the index page.


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Composed by J. B. Calvert
Created 2 August 2003
Last revised 6 August 2003