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If I wanted to make an air shield around the earth to contain exactly half the planet's atmosphere (by mass), how far above sea level would I have to build it?

It's slightly complicated to me because you have to take into consideration the change in pressure as well as the geometric growth of the volume of a sphere as you increase the radius.

Thank you.

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    $\begingroup$ How precisely do you need to know? ±500 metre, ±50 metre, ±5 metre? $\endgroup$ – gerrit May 8 '17 at 15:45
  • $\begingroup$ Closest 1000' would be great, but 5k would be fine. Basically any answer that states its assumptions is fine. $\endgroup$ – xaxxon May 8 '17 at 15:46
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Roughly 5.5 km, although the exact value depends on the weather. All you need to do is to solve the hydrostatic equation and find the height at which the pressure is 50% of the height at sea level (or perhaps average elevation of Earth surface instead of sea level). Any answer actually performing the calculation deserves upvotes more than I do ;-)

See, for example, Thickness of Earth's Atmosphere:

Using boundary layer theory as an example, we can define the thickness of the atmosphere to be the altitude that encloses 99 percent of the total mass of the atmosphere. Looking at the chart, we can see that this seems to be about 31 kilometers. The halfway point, where half the mass of the atmosphere is below and half above occurs at 5.5 kilometers. Another interesting fact is that when you are cruising in a modern jet transport at 11 kilometers, you are above 77.5 percent of the atmosphere. The total mass of the atmosphere turns out to be 5.3 zettagrams (5.3 Zg).

I'm not entirely sure how the calculations in the linked article are performed, but I recall calculating this as a student and ending up somewhere around 5 km, so it sounds right. Simple calculations often assume an isothermal atmosphere, which is of course not accurate, but good enough if you don't need to know the answer more precisely than within several hundred metre.

This page at Stanford reaches the same number.

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It turns out that the answer is a bit simpler than you think.

Atmospheric pressure (neglecting small weather variations) is due to the weight of the column of air above a given point. If we divide the pressure at a given altitude by the pressure at sea level, we get the amount of atmosphere left above us. To find the height at which half the atmosphere is below us, we just need to find the height at which the pressure is half of the sea level pressure, which turns out (from standard atmosphere charts) to be slightly less than 18,000 feet (about 5500 metres).

If you want to work out the expression for the change in pressure with height, that can be done from assuming that air is an ideal gas and using hydrostatic pressure.

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    $\begingroup$ For the latter you usually also want to assume an isothermal atmosphere, or it quickly becomes very complicated. $\endgroup$ – gerrit May 8 '17 at 16:12
  • $\begingroup$ Usually what you do is take the ISA temperature profile, which assumes that temperature decreases linearly with height in the troposphere. $\endgroup$ – Chris W May 8 '17 at 16:18

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