Twinned rainbow over Dresden, May 11th, 2012
During the late afternoon of May 11th, 2012, a mild thunderstorm moved from west to east over Dresden. I was located at the University campus (51° 02′ N, 13° 44′ E) south of the city and seemingly at the southern edge of the cloud. At 17:30 CEST I noticed the beginning of rainfall, and at 17:32 the cloud already gave way for the sun again in the west, while the peak intensity of the rainfall was just reached at my place. I could trace the primary rainbow in the glittering of nearby drops at this time, similar to the appearance of halos in close ice crystals which are frequently observed at cold placed during winter. Already at this stage, the primary looked unusual, and the idea of twinning occurred to me. So I rushed through the building to fetch my camera and find a suitable location to take photos. I arrived at my preferential observation window at 17:36. At this time, local rainfall had almost completely ceased, and a beautiful double rainbow display could be seen in the east (Fig. 1, 17:37:44, Pentax K-5 + Zenitar 16 mm).
The primary rainbow showed a rather broad single supernumerary around its top, which could be interpreted as a weak form of twinning. Within the next minute, some cloud (maybe a part of the cumulonimbus that was still above my place, though not producing rain anymore) did cast a localized shadow on a part of the veil of raindrops in the east, and a very distinct twinning of the primary bow near its top became visible (Fig. 2, 17.39.06, Pentax K-5 + Pentax DA 18-55 mm at 18 mm, with boosted contrast and unsharp masking)
The most convincing theory for twinned bows so far is the assumption of a mixture of smaller and larger raindrops within the shower, with the larger ones being more flattened or squeezed in the vertical due to the resistance of air during falling . According to this, the primary rainbow produced by the larger drops is shifted inwards (i.e. towards the antisolar point), especially near the top. However, as calculations show, the secondary rainbow remains almost undisturbed. This behavior of the secondary was exactly what I could record during my observation (Fig. 3, 17:39:12, Pentax K-5 + Pentax DA 18-55 mm at 40 mm, with boosted contrast)
The twinning was visible up to 17:41, when the top of the rainbow faded away due to the growing shadow of larger clouds in the west. The left part remained visible for some more minutes (Fig. 4, 17:43:44, Pentax K-5 + Pentax DA 18-55 mm, single frame extracted from video file). At 17:42, a complete and intense double rainbow could be observed also from a location 5 km north of my place, with a well-developed supernumerary inside the primary but without any signs of twinning .
For the image taken at 17:39:06, I performed a detailed analysis of the position of the twinned primary and the un-twinned secondary rainbow. The sun was located at this very moment at 26,9° in elevation and 265,9° in azimuth. To calibrate the photo with respect to focal length, elevation and azimuth of the image center as well as rotation around the optical axis, I took a star field image in the late evening of May 14th, 2012 from the very same place. From the position of two stars, the star field photo can easily be calibrated, which allows to calculate the positions of two landmarks at the horizon. Their positions allow in a second step the calibration of the original rainbow image . Lens distortion was not considered, i.e. perfect rectilinear projection was assumed, since all pictures with the zoom lens were taken with activated real-time distortion compensation (camera feature). The results are very convincing, indicating the high accuracy of this calibration method when carried out carefully. This can be illustrated nicely by an overlay of the actual image with the theoretical positions of rainbows made by spheres (Fig. 5)
Only geometric optics was used here, and from it only the Descartes angles for the monochromatic wavelengths of 600 nm (red), 530 nm (green), and 460 nm (blue). Furthermore, I transformed the image into a sun-centered equirectangular projection, in which the rainbows from spherical drops have to appear as straight horizontal lines, indicated by the marks on both the left and right side of the image (Fig. 6)
The inward shift of the lower branch of the primary is obvious, with no part of it extending into Alexander’s dark band, i.e. beyond the Descartes angle. This is in accordance with the theory for flattened drops, as well as the secondary not showing any measurable shift or distortion and furthermore sticking closely to the Descartes angle all along its visible extension. It should be noted that more exotic splittings of the primary have been observed  whose explanation requires a different theoretical approach.
Taking a closer look at the shadow edge near the top of the primary, it seems that coming from the left the “ordinary” rainbow, i.e. the upper branch, is dimmed in the shadow region, but can be traced up to +20° in clock angle. On the other hand, one marks a smooth transition from the supernumerary in the bright region on the left into the inner branch of the twinned bow in the shadow region on the right. It is obvious that this localized shadow which is cast by a smaller cloud segment will prevent a certain set of drops in this very direction from contributing to the primary rainbow, and that the remaining sunlit drops are in the right mixture to give rise to a clear twinned bow. Very likely the drops a few degrees to the left will not exhibit a drastically changed size distribution. However, the drops in the foreground are illuminated there and add to the overall primary brightness, thus covering unfortunately the twinned bow. One can speculate that there might be many twinned bows in nature that are hidden from us due to the contribution of “ordinary” drops in front or behind the interesting region along the rainbow cone.
Author: Alexander Haußmann, Dresden, Germany