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If the moon reflects exactly the sun spectrum, then the blue photons part of the spectrum should scatter in the atmosphere during the night (assuming a perfect dark night sky). Why don't we see a bluish light during the night, the equivalent of a blue sky during the day ?

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Answer to this question is very well given at Physics forum pointed out by @gerrit. In short, Light should be strong enough in order to notice its colour by human eyes. Even on full moon night, light from the moon is 10 times less luminous than the dark limit of civil twilight hence we don't perceive sky color or even moon colour like we do during the day.

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