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Fake sun observations are stacking up! There is much more than I thought in the beginning. First, Mike Hollingshead sent some of his old photos which look much like a fake sun. In the first two photos the fake sun is above the sun and in the third photo, where real sun is completely hidden, below. And today I found out, while visiting the repository Library in Kuopio to check the Dutch Weather amateur's journals Onweders, optische verschijnelselen, that there is a regular column devoted to observations of double suns and moons, before rainbows and halos. Plenty of sightings are listed from about two decades starting from the end of 19th century, with two drawing published from years 1897 and 1899, shown in the last two images. Yesterday also one fresh appearance was reported in Finland.
The effect is not really as forgotten as I thought, because it is mentioned in Minnaert's Light and color in the open air, which is not that old book (my Minnaet has been lost for many years). Mike suggests that the effect can't be at all rare. This maybe true. Possibly people just don't look for it because one has to look near the glare of the sun, or even when they look, they don't think it is anything special. And if one lives in a place where low sun can not be seen, this limits the oppotunities to observe the phenomenon.
Some additional issues are worth emphasizing about the fake sun. First, the sun need not be at all hidden for it to be seen. This is seen nicely in Johannes Hevel's draving from 5 February 1674, the fourth image above. Also my observation from 10 May 1987 (fifth image) is a situation where sun is shining unobstructed. The note for the observation says that the fake sun above the real sun was only SLIGHTLY fainter than the real sun. Second point that emerges from many images above is that the fake sun can appear some distance apart from the sun. And finally, not every pillar has a fake sun; there is my other observation (sixth image), from April 1987, where the situation would look fruitful for the fake sun, but only pillar was reported. For me fake sun is something that is or can be potentially mistaken for a real sun. Since I started to write about this, some people have been dubious about fake sun in general or not really understanding how it should look. I can only say that when one sees a true fake sun, there is no need to question anymore.
Two names have been used for this effect, fake sun and double sun. The French use the former (faux soleil) and the Dutch use the latter (dubbele zon). Both have their advantages and disadvantages depending on the situation. The Dutch have also observations of fake moons. I wonder if there are any sightings of fake planets or stars.