# How would air pressure gradients be affected by controlled gravity?

My current story is set inside a hollowed out dwarf planet (i.e. Ceres) with an artificial gravity generating "mesh" within the shell. This Gravity Mesh can generate a gravitational field that can be modified, anywhere between 0.01g and 10g in strength.

On Earth, sea level air pressure is 101.33 Kpa, and the Armstrong limit (the pressure at which water boils at human body temperature [37 degrees celsius]) is 6.26 kPa, which usually occurs at around 19,000 meters. The elevation at which humans usually need supplemental oxygen is 4,500 meters, which has an air pressure of 57.73 kPa.

What I would like to know is, how would changing the strength of the gravitational field affect the air pressure.

• What would the air pressure be at "sea level" at 2g? 5g? 10g? 0.01g?

• Would the different gravities alter the elevation at which humans would need supplemental oxygen? If so, how?

• At what elevation would the Armstrong limit occur at these different gravities?

• What is the highest gravity before humans breathing the air would go into a state of hyperoxia (too much oxygen)?

Assume the atmosphere consists of a standard nitrogen/oxygen mix with Earth-like stats at 1g.

• Venus has pretty much the same mass (thus gravity) as Earth, but nearly 90 times the surface atmospheric pressure. Mars is a little lighter than Earth at about half the gravity, but has about 0.6% of Earth's surface atmospheric pressure (Wikipedia specifies 0.00628 atm). I think these three data points goes to show that there is far, far more to surface atmospheric pressure than merely gravity.
– user
Commented Jun 7, 2017 at 19:56
• You first need to determine how much air you have on your planet. If you want enough air to provide 1 atm at 1 g, then we can start somewhere, but right now you have one too many variables. Commented Jun 7, 2017 at 20:00
• @MozerShmozer Thanks for that input, I was assuming Earth-normal conditions at 1g, that is why I had Earths stats. I will clarify that. Commented Jun 7, 2017 at 20:06
• Is the outer shell completely sealed? Is your atmosphere held in place by gravity or by the outer shell? Is the artificial gravity acting inwards (towards the centre) or outwards (towards the shell)? Is the planet completely hollow, or is there a core inside the outer shell? Commented Jun 8, 2017 at 11:21
• @SteveES To answer your questions: 1. Yes, the outer shell is completely sealed. 2. The atmosphere is held within the shell, with the gravity pulling it towards the shell. 3. The planet is completely hollow, with a small artificial "sun" in the very center. Commented Jun 10, 2017 at 8:03

# The Physics

Since we're assuming an Earth-like system (1 atm at sea level with Earth's g), there's a convenient formula we can use for this:

As you can see, pressure is a function of:

1. Height, h. You have asked for a profile of pressures as h varies.
2. The gravitational acceleration, g. This is your free parameter.
3. The temperature, T. For simplicity, I assume a constant ambient T of 25 Celsius.

The other terms: k is the Boltzmann constant, and m is the mass of the air molecules. This is taken as the mean for gas mixtures. These are fixed parameters.

So now we can plot a few curves and take a look:

We can can calculate the heights at which the Armstrong Limit occurs for each value of g. They are:

• 1997.8km at 0.01x g
• 19.978km at 1.0x g
• 9.9889km at 2.0x g
• 3.9956km at 5.0x g
• 1.9978km at 10.0x g As you can see, the Armstrong Limit happens rather quickly at 10x g, just above 1200 miles high.

## Some Remarks

• The Armstrong limit is well-defined and occurs at P = 0.0618 atm, but hyperoxia has a much less defined threshhold.
• Likewise, the range of altitudes at which humans require supplemental oxygen varies widely due to variations in physiology. However, the altitudes will follow a similar trend as the Armstrong Limit.
• For similar reasons, it is very difficult to treat your final question, as the hyperoxia threshold for even a single person varies widely over many days.

# An Improved Plot

Thanks to SteveES for pointing this out. My original plot had each P(0) occurring at the same value; this is inaccurate for the label I used on the vertical axis, which should have been P(h)/P(0), not just P(h). Here's an updated plot with the true values. I had to cut off the upper part of the curves for higher g. Here is the updated plot:

• You appear to be assuming that sea-level pressure (P0) doesn't change with g. This seems like a poor assumption... Commented Jun 8, 2017 at 11:16
• Ah, I see that my Axis label is a bit misleading. Each curve is the ratio of P(h) to P(0). The P_0(g) scales linearly with g because P=F/A and that F is due only to gravity's pull on the vertical column of air above a unit area. I'll adjust this later today for clarity. Commented Jun 9, 2017 at 12:24
• At the very beginning you say this is an "Earth-like system". We make the assumption that gravity and air pressure would be the same as a result of the sci fi space magic, but wouldn't the total volume of air in the atmosphere be smaller, since the planet is smaller? Nothing in the barometric equation indicates to me it would make a difference (I even went and re-derived it from the ideal gas law just to make sure) but something makes me uncomfortable about that, and I don't know why. Thoughts? Commented Jun 9, 2017 at 15:44
• The pressure exerted by the atmosphere is governed by the same laws that determine any pressure; that is, the force exerted over a given area. This depends only on the weight of air directly above a unit area of the planet. To achieve "Earth-like conditions," this mini-planet's atmosphere might be thicker (relatively) than Earth's, but you're right-- these expressions assume the same amount of gas around it. This is "reasonable" because its denizens/builders are able to manipulate gravity. Your intuition is correct: an atmosphere with this volume is unnatural; it would be artificial. Commented Jun 9, 2017 at 20:34

Air pressure (atmospheric pressure) multiplies with gravity, so that is easy. Assuming a thorough mix of gas components in the atmosphere, at 0.5 g the Armstrong limit would be 1/2 as high and at 2g twice as high.

Now that is done, the excellent part of this is that by cyclically adjusting gravity and thereby air pressure, air could be made to flow in and out of any cavities in continuity with the atmosphere. Done correctly, humans and all other animals could breathe automatically and without any effort. It would be a worldwide iron lung. This would be useful for the very sick, the very tired and the very lazy all of whom would thrive in your subterranean paradise.

• Are you sure that atmospheric pressure increase with gravity? E.g: If our Earth would have 2 g atmospheric pressure would be 2 atm? Commented Jun 8, 2017 at 20:04
• @EnderLook, yes indeed it does; see my comment above. Though there are more factors that play into local atmospheric pressure, the bulk of it is governed by gravity. Commented Jun 9, 2017 at 14:11