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I'm building a world consisting of floating landmasses located entirely within an indestructible rectangular prism that is 10,000km wide, 10,000km long, and 4,000km tall. If I want air pressures to be around 1 bar at the midpoint altitude (2,0000km), how can I determine what the air pressure will be at the 'ceiling' and 'floor' of the container?

The container is made of an arbitrarily strong material.

I imagine the landmasses themselves take up an arbitrarily small percent of the containers volume.

I was thinking that the world wouldn't experience gravity in the classical sense, as that would cause everything to be attracted to the centre of the container, but instead everything feels a constant downward force of 1G, almost as if the entire container was accelerating upward at 1G.

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    $\begingroup$ Scale height. (But as an intuitive shortcut, assuming that the air at midpoint is about as warm as Earth air, about 100 km above the midpoint the air pressure will be effectively zero. Scale height for Earth air with sea-level Earth air temperature at 1 g is about 8.5 km, so every 8.5 km the pressure will increase or decrease by a factor of 2.72; so that about 30 km below the midpoint air will be a supercritical fluid.) $\endgroup$
    – AlexP
    Apr 1, 2023 at 17:36
  • $\begingroup$ Artifexian has a spreadsheet for calculating atmospheres, which could be useful. Pretty sure it includes a scale height calculator too. $\endgroup$ Apr 2, 2023 at 18:45
  • $\begingroup$ how are the landmasses floating? the same magic could affect the air in which case the pressure can be anything. $\endgroup$
    – ths
    Apr 5, 2023 at 11:04
  • $\begingroup$ Can you comment on whether or not the mass of the box, air, land, etc., has any gravitational effect? Or is magic straightening that out making the force vectors all parallel and of same length? If the whole box is accelerating at 1g, that would still leave the masses to impart additional gravitational forces. Also, you said the land masses 'float' - in magic, or in, e.g. mercury? If the materials have their ususal gravitic effect, you need to tell us more about the makeup of the box - what is the floatation medium, for instance. $\endgroup$
    – bukwyrm
    Jun 21, 2023 at 6:57
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    $\begingroup$ You might have a greatly expanded "habitable area" in that volume if you do use classical gravity instead of a uniform 1G field. (Or if you used a uniform 0.2 G field) The lower gravity would let the pressure be more uniform, and perhaps give the inhabitants more ability to fly from one landmass to another. $\endgroup$
    – Brianorca
    Aug 2, 2023 at 17:56

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What you're looking for is the Barometric formula that gives the pressure P at a given height h in an ideal gas:

Barometric formula

Where:

Symbols and notations

The "temperature lapse rate" is the proportionality coefficient between variations in temperature with variations in pressure so if the gas inside your container is not earth air, you'll have to find it's temperature lapse rate and molar mass.

Applying this to your case, assuming that all values are at hb=2000km as they are on Earth's surface, we have:

Values of constants

So at h1=-2000km, we get the minimum pressure of about 500 million bars and at h2=2000km, we get the maximum pressure of about [error: division by 0]. Obviously, at these scales, we can no longer suppose that we are dealing with an ideal gas and the formula is to be thrown out the window. I don't have the math to back me up on this but I think that the bottom of your box would see atmospheric gas turn solid in order to carry that much mass.

However, at h1=-30km, we get a pressure of about 14.5 bars and at h2=30km, we get a pressure of about 0.0033 bars (300 times less that atmospheric pressure), which is already a pretty good spread.

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