Subsun & Subparhelion
Bart De Bruyn saw these from high over the Atlantic Ocean. To the left glints a bright subsun with a trace of lower sun pillar. On the right is an elusive prismatic subparhelion, the subhorizon equivalent of a sundog.
  ©Bart De Bruyn, shown with permission.
Atmospheric
Optics

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Multiple internal reflections rule when plate crystals form subsuns and subparhelia.

The often blindingly bright subsun is seen in clouds directly beneath the sun and an equal distance below the horizon.

It is formed by the large and almost horizontal hexagonal faces of plate crystals reflecting the sun back upwards. The reflection can be external from the uppermost face. But more often it is an internal one from the lower face. Some rays are even reflected several times between the two faces.

The simplest sundog forming ray is that passing between two plate crystal side faces inclined 60 to one another but most ray paths are more complicated. Light is internally reflected twice or more between the two large horizontal faces before emerging form the crystal.

The difference between an above horizon sundog and below horizon subparhelion is simply in the number of internal reflections that sunlight undergoes before it escapes. An even number of reflections gives a sundog. An odd number and the result is a subparhelion 22 or more from the subsun.