A theory for the visibility of halos and rainbows is presented. The light reaching the observer’s eye from the direction of the halo or rainbow is assumed to consist of two parts: (1) a beam of singly scattered sunlight (or moonlight) from a cloud of ice crystals or a rainswath, which, in turn, has suffered depletion by scattering or absorption in its passage to the observer, and (2) the general background brightness. The model is able to account for several long-known qualitative observations concerning halos, namely, that the brightest halos are produced by optically thin cirrostratus clouds (i.e., for which the cloud optical depth τc ≤ 1) and that when the sun is low in the sky the top of the halo is visible much more frequently than the bottom. (This is shown to result in good part from extinction by the turbid atmosphere.) With the rainbow the brightness of the beam increases monotonically with the optical depth τR of the sunlit part of the rainswath, but the increase is quite small for τR ≥ 1. On the other hand, the brightness of the background increases more rapidly with τR for τR > 1 so that the rainbow appears most easily visible for τR ≲ 1. This implies that the most easily visible rainbows are produced by light or moderate showers rather than heavy downpours. Finally, suggestions are made for applying the theory to other atmospheric optical phenomena, such as coronas and glories.
© 1980 Optical Society of America
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