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Shadow bands appear right before and right after totality during an eclipse. According to the Wikipedia article, this phenomenon occurs because all the light coming through from the sun at that time is collimated and susceptible to atmospheric scattering effects.

Does a crescent moon have the same properties as a "crescent sun," and if so, would it at least in theory be possible to detect extremely faint shadow bands during a crescent moon?

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Based on first principles, I would say "in principle yes" - but then you have to think about the mechanism of shadow bands. These bands appear when the sunlight becomes almost completely (spatially) coherent - the angular size of the source becomes so small that atmospheric fluctuations can focus the light more in some places than others.

For the moon to be a "sliver of a crescent", the sun needs to be essentially directly behind it: this means that any shadow bands would be completely drowned out by the intensity of the sun light. As such, any fluctuations would be no different than the ones you can observe with full sunlight: in principle, there will be shadow bands for any size of source, but they will be convolved with the size of the object (which makes them effectively disappear except when the sun is reduced to a tiny sliver). So we can safely say they would be invisible.

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