If you look around the web for the answer to this question, you'll find a lot of conflicting information. A number of seemingly authoritative sources declare flat out that colors cannot be seen under moonlight, because the eye's cone cells are not active. The same sites also often mention that normal sized printed text cannot be read in moonlight, because the central fovea is packed with cones that won't be active. Both of these claims are false.
Since this is something that's really easy to test, there's no reason for this misconception to persist. All you have to do is wait for a night with a full moon, find a place outside that doesn't have any direct lamp light (e.g. a park), take something colorful with you (e.g. a children's book), and see if you can recognize the colors or not.
I did it last night, with Mr. Forgetful:
I found out the answer right away: I was able to distinguish the red hat and blue body not as different shades, but as "red" and "blue". I did have more trouble on some other pages. For example, on the page below I was pretty sure that the "grass" wasn't green, but I had no idea what color it was. When I went back inside I found out it was not actually grass, but brown dirt:
I also had no trouble at all reading the text, which is probably about 14 point font. I tried reading a text with a smaller font, and while I had a bit more difficulty, it was certainly still possible.
But please don't take my word on this! This is a science experiment that can be performed by anyone without vision impairment, so the next time there's a full moon, go out and try it yourself!
You might ask why any of this matters. It turns out that this actually came up one time in court! A witness
claimed to have seen a red car under moonlight, and experts called to
testify disputed the fact that such a feat was even possible. The case motivated some Australian researchers to do a controlled test.
It turns out that certain colors are easier than others (red is the
easiest), and the color of larger objects is easier to discern than
smaller objects.
But there's another reason why it's important. The confusion online demonstrates the extent to which vision at night is poorly understood. It's almost certainly the case that lighting levels in our cities could be dramatically reduced without meaningful effects on visual performance. Doing so would save money, energy, might help people sleep better, and would bring many stars back to our urban skies. But how low can we go and still be able to see well on city streets? The world spends something close to 100 billion dollars per year on outdoor area lighting, so spending a few million to get the right answer could result in a massive payback in reduced electricity bills.
Did the same experiment in Namibia 10 years ago, with only the full moon high up and no artificial light: color vision, especially red, was fine, and large-font text could be read.
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