The National Aeronautics and Space Administration (better known as NASA), (see also here), maintains a database (archived here) of fascinating pictures of the universe. Every day a new picture is featured and added to the database. You can see today's picture at their "Astronomy Picture of the Day" (APOD) site, which can be found here. The picture (to the right), which was featured on January 12, 2009, shows some beautiful "light pillars" which were seen standing "above" the brighter lights in Latvia. The APOD editors explained, these columns of light probably involved falling ice crystals reflecting back these lights. They also explained that it was currently unknown why these pillars fanned out at the top.
The APOD editors invited readers to take a crack at explaining the mystery! Hundreds of comments were submitted; they can be read here. After reading what the others had written, I took a crack at it myself. My thoughts were posted here. Since I (and I presume the others as well) thought it was fun taking on a previously unsolved scientific problem, I would like to thank APOD and encourage them to present other similar challenges in the future. Many of those who responded to APOD's invitation made very good observations: First, a responder using the name "logmark" made the observation that the pillars exist on a vertical plane between the source and the observer. (A similar observation was made by Stephen McDonald). Next, "chadair" pointed out that the stars in the pictures could be used to provide angular scale; he then provided some estimates which, as he pointed out, ignored distortion caused by the camera's wide-angled lens; however, his estimates were still both necessary and sufficient for sorting out what was going on. And finally, a responder who called himself "PiTHON" presented many detailed observations which I would certainly have missed without his posts.
Before I try to explain the trumpet-shaped fans at the tops of the pillars, I should take a little time to explain the math and physics of the pillars themselves. When light passes through hexagonal ice crystals, it refracts at a twenty-two degree angle, causing some interesting effects such as halo rings which can sometimes be seen around the sun or moon (twenty-two degrees away for them). This angle can be calculated using Snell's law, the 1.31 index of refraction for ice, and the sixty degree angle between the light-transmitting faces of the hexagonal Ice crystals.
If the ice crystals all happen to be horizontal (because they are falling), only two spots will be seen (instead of the circular halo which is seen when the crystals are turned is different directions around the observer's line of sight. If the ice crystals are flat, these spots will be seen to the right and left of the source; if the crystals are rod shaped, the spots appear above and below the light source. In this picture, it appears that the ice crystals are all horizontally oriented rod shaped crystals; but the refracted light (not reflected, contrary to APOD's suggestion) appears to be spread out along a vertical line because, in this case, many different light paths are involved. When a very distant light source, such as the sun or moon, is involved, this refracted light is only observed at twenty-two degrees away from the source; but when the source is near by, and inside the cloud of crystals (which nearly filled the Latvia night air), there is a whole family of paths which involve a single twenty-two degree downward refraction. Since there is a single refraction, and the angle is always the same, geometry tells us that the crystals which are involved are all located along a circular arc (since all angles inscribed in the same arc are of the same size). This arc makes an arch on logmark's vertical plane with one leg planted at the light source and the other at the observer. The top (peak) of the arch is seen (by the observer) eleven degrees (not twenty-two) above the source. This is because the relevant light path starts up at an eleven degree angle and gets bent down twenty-two degrees so it approaches the observer dropping at eleven degrees.
This circular arch is comprised of forty-four degrees (twice the refraction angle) worth of a full circle (see the solid black arch in the illustration below). The tail-end of this arch hits the observer at the full twenty-two degree angle; so nothing can be seen above the twenty-two degree point (see the thin dashed black line in the illustration below). Since chadair has roughly placed the bottom of the fan at seventeen degrees (safely past the eleven-degree middle) the "top" of the fan appears to be coming from the half of the arch which is next to the camera lens! This effect is caused by a distribution of ice crystals which are surprisingly close to the camera. (Unfortunately, this would not be observable with stereoscopic photography, or by photographing from a different angle, because the "mirage" moves with the observer.)
This means the trumpet-shaped tops of these pillars are probably an effect of light refracted from the "glow" around the light source. Depending on the exact path (through moving tree branches, over lower structures, etc.) the column's "trumpet" may assume many different forms - including either strong or weak middle streaks (from the primary source) or missing "fans" (because the glow around the source is occulted). The "glow" itself may also chang shape with time owing to differences in dust density. As PiTHON has pointed out, these trumpets seem to come and go quite quickly. An inspection of the photograph of the light sources reveals they are partly occulted by tree branches which are probably moving with minor changes in wind speed.
Once again, I would like to thank NASA and APOD for the opportunity to let the rest of us work on interesting problems. I would also like to encourage readers to take advantage of the wonderful source of information they provide!