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How can horizontal column crystals make pillars? Pillars are created by reflections from approximately horizontal crystal faces but the side faces of columns are inclined at all angles.
The key is that the crystals have orderly orientations - they do not tumble. A 'singly oriented column' drifts with its long axis nearly horizontal and has two rotational freedoms. (1) Its long axis can point in all azimuthal, NSEW, directions as indicated by the large arrowed red circle. (2) The crystal adopts all rotational positions around the long axis as shown by the smaller red circle.
Let's look at the resulting tilt directions of a single side face. These are most easily illustrated by the black arrow, the "normal vector", which points in a direction perpendicular to the face.
Imagine the crystal fixed at one azimuthal angle, say pointing east, and rotating around its long axis. The normal vector's tip sweeps along a line of longitude and passes through the zenith. Other azimuthal orientations similarly cause the vector to sweep along lines of longitude and always through the zenith. The zenith is a preferred normal vector direction. That is, the face spends more time nearly horizontal!
We can explore the vector tip positions by computer simulation. Choose a random azimuth, combine it with another randomly generated rotational position about the crystal long axis and plot the resulting tip position as a black dot on the sky sphere. The figure shows 10,000 dots generated this way.
Each slice of latitude (altitude) contains the same number of dots but the area of the slices decreases dramatically towards the zenith. The dots concentrate at the zenith, they are not evenly distributed over the sky. In other words, side faces spend more time nearly horizontal than tilted at other angles. And near horizontal faces make sun pillars!
To summarise, the combination of two constrained
crystal rotations cause an arrow perpendicular to a side face to spend
more time pointing near to the zenith than elsewhere. This means that
side faces are therefore more often nearly horizontal than tilted at
other inclinations. Nearly horizontal faces produce sun pillars and
so, counter intuitively, column crystals generate pillars.
crystals probably produced this pillar imaged at Oslo, Norway by Erik
The pillar is unusually tall, some 20º, and is topped by an intense
upper tangent arc also generated by singly oriented columns.
©Erik Brenna, shown with permission.
|Each dot represents the tip of a line perpendicular to a side face of the crystal as it rotates about two axes. The dots concentrate at the zenith. Side faces spend more time nearly horizontal and so produce pillars.|