Sunburned by height

“Maybe you got more sunburned because you were closer to the sun.”
—Seen on Facebook last week.

Random act of math: At what point would height determine which of two people gets more sunburned?
It’s not going to be literally closeness to the sun that determines it, because then you’d have to be tall enough to reach some significant percentage of the way to the Sun. Instead, you just have to be tall enough to be some significant percentage of the way through Earth’s atmosphere. Then it’ll block less of the sun’s rays and interfere less with the sunburning. The “significant fraction of the distance into the atmosphere” bit says right away that it won’t matter much under normal circumstances.

Assume the two people are at the same latitude, same time of day and year, and so on. Also assume they have precisely the same tolerance to UV. And despite this being one of the times when “how’s the weather up there” is actually a reasonable question, we’ll just say that’s the same, too. A perfectly clear sunny day, both at ground level and head level.

There’s a way of measuring sunburn danger, of course, and of course nobody uses it. The UV index measurement, unusually for this kind of thing, is linear instead of logarithmic. So a 9 on that scale actually is about three times as much sunburn risk as a 3 for the same amount of time, and a being out in the sun when it’s a seven actually is equivalent to the amount of UV as a six plus a one. Which is really shockingly convenient and it’d be nice if more scales did this.

A 0-1 on the UV index means there’s no UV radiation to speak of that day. Maybe wear sunglasses if you’re walking on a reflective surface like snow or a giant mirror, but you don’t really have to worry about sunburn. In the spirit of erring on the side of conservative assumptions, I’ll use 1 to mean the difference between slight risk of a small sunburn versus definitely no sunburn. Thanks to the really convenient linearity property of the UV index, the difference between a 0 and a 1 is about the same as the difference between a 3 and a 4 or a 16 and a 17. Sure, that sounds obvious, but don’t ever make that assumption if you’re working with the Richter or Kardashev scales. Not if you want your city to continue to exist.

The day in question, there was a UV index of about 8. That’s for a clear, sunny day with sunburning amounts of UV B radiation. So for the tall person to get more burned than the vertically challenged one, she’d have to be high enough that there’s an index of maybe around 9. Since the difference between slight and none should be the same as the difference between some and slightly more.

That means there’d be an eighth more radiation, an increase of 12.5% While there is more to the UV index than just measuring amount of radiation, most of that “more” (according to the compendium of human knowledge) is just involved in weighting the appropriate types of radiation. We care more about some than others because only some are dangerous. So we multiply them all by different amounts to reflect that. But as we go upward, they all increase by the same percentages.

According to the World Health Organization, UV levels increase by about ten percent for each additional kilometer of elevation. This actually sounds pretty questionable, because I can’t think why it would be a constant ratio at each kilometer instead of something like not much change until you reach the ozone layer, and then more UV quickly as you start going through that. But it’s the WHO, so they’re probably right.
So to increase the UV index from eight to nine by being tall, your height would have to be 42.6 metric PB (Paul Bunyans). And that’s the Disney Paul Bunyan. Moral of the story: No. Just no.

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