The UV index is not the warm sensation of sunlight on bare skin

The warmth of sunlight is not a reliable indicator of how quickly you will sunburn in a given situation. On one hand, you can still get sunburned on a cooler cloudy day, while on the other hand, the sun can feel scorching hot on your skin in the morning and yet you won't sunburn (very quickly).
You probably knew this already, but it still feels unintuitive to me. I'm going to look at numbers and think about them until I convince myself, and perhaps you, that this is true.
Sunburn is triggered when your DNA is damaged from light absorption - in particular, UV-B (and to a limited extent, UV-A) is absorbed in a way that alters DNA's molecular structure 1 .
Image: DNA UV mutation by Mouagip (derivative of a NASA / David Herring original), released into the public domain.
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