What is the expression for air pressure below Earth's surface?

[Curl]

If you were to drill a hole in the ground, what is the expression for the air pressure below ground? Above ground it is exponential with distance (at least to an approximation), so what is it for below Earth's surface?

It has to increase but probably with a lesser absolute slope than above ground. Let's assume the usual "earth is a perfect sphere with uniform density" and that the air is isothermal.

 

[cjl]

Unless you start getting so far into the Earth that the gravitational force is changing, it will be an exponential, just like it is above sea level. There's nothing special about sea level that would cause its behavior to change.

 

[snorkack]

Unless you start getting so far into the Earth that the gravitational force is changing, it will be an exponential, just like it is above sea level. There's nothing special about sea level that would cause its behavior to change.

No, but you do not need to wait until gravitational force starts changing. The pressure change stops being exponential when air compressibility deviates from ideal gas - meaning, from couple hundred atmospheres.

 

[Curl]

The whole point of the problem is to take into account the changing gravitational field.

 

[jbriggs444]

But if that is the point of the problem then the model does not call for pressure to be exponential above ground either.

 

[cjl]

No, but you do not need to wait until gravitational force starts changing. The pressure change stops being exponential when air compressibility deviates from ideal gas - meaning, from couple hundred atmospheres.

That's a good point actually - Earth's atmosphere has a scale height of about 8km (if I remember right), so by 50km below the surface, the pressure would be about 400 atmospheres. Given the behavior of N₂ at high pressure, this would definitely be sufficient to notice deviations from the ideal gas behavior. On the other hand, 50km down, the gravitational change would be fairly negligible, especially since the gravitational acceleration inside Earth is fairly constant for the first 2000km or so.

One thing that you have to keep in mind when thinking about this problem: the atmosphere is extremely thin. Gravitational changes across the entire thickness of the atmosphere are pretty much negligible, so they can pretty safely be ignored, even if you were to extend the atmosphere significantly in either direction.

 

[DrZoidberg]

I wrote a little program to calculate the pressure above and below ground.
I assumed air is an ideal gas.
And I calculated gravity like this:

double newR = earthRadius+altitude;
if(altitude >= 0) {
gravity = earthGravity*Math.pow(earthRadius, 2)/Math.pow(newR, 2);
} else {
gravity = earthGravity/earthRadius*newR;
}

Here are some results.
alt: 0km, pressure: 101.312kPa
alt: 1km, pressure: 89.418kPa
alt: 2km, pressure: 78.924kPa
alt: 3km, pressure: 69.664kPa
alt: 4km, pressure: 61.492kPa
alt: 5km, pressure: 54.282kPa
alt: 6km, pressure: 47.919kPa
alt: 7km, pressure: 42.303kPa
alt: 8km, pressure: 37.347kPa
alt: 9km, pressure: 32.973kPa
alt: 10km, pressure: 29.112kPa
alt: 11km, pressure: 25.704kPa
alt: 12km, pressure: 22.696kPa
alt: 13km, pressure: 20.041kPa
alt: 14km, pressure: 17.697kPa
alt: 15km, pressure: 15.628kPa
alt: 16km, pressure: 13.802kPa
alt: 17km, pressure: 12.189kPa
alt: 18km, pressure: 10.765kPa
alt: 19km, pressure: 9.508kPa
alt: 20km, pressure: 8.398kPa
alt: 21km, pressure: 7.418kPa
alt: 22km, pressure: 6.552kPa
alt: 23km, pressure: 5.788kPa
alt: 24km, pressure: 5.113kPa
alt: 25km, pressure: 4.517kPa
alt: 26km, pressure: 3.991kPa
alt: 27km, pressure: 3.526kPa
alt: 28km, pressure: 3.115kPa
alt: 29km, pressure: 2.752kPa
alt: 30km, pressure: 2.432kPa
alt: 31km, pressure: 2.149kPa
alt: 32km, pressure: 1.899kPa
alt: 33km, pressure: 1.678kPa
alt: 34km, pressure: 1.483kPa
alt: 35km, pressure: 1.311kPa
alt: 36km, pressure: 1.158kPa
alt: 37km, pressure: 1.024kPa
alt: 38km, pressure: 0.905kPa
alt: 39km, pressure: 0.800kPa
alt: 40km, pressure: 0.707kPa
alt: 41km, pressure: 0.625kPa
alt: 42km, pressure: 0.552kPa
alt: 43km, pressure: 0.488kPa
alt: 44km, pressure: 0.432kPa
alt: 45km, pressure: 0.382kPa
alt: 46km, pressure: 0.338kPa
alt: 47km, pressure: 0.298kPa
alt: 48km, pressure: 0.264kPa
alt: 49km, pressure: 0.233kPa
alt: 50km, pressure: 0.206kPa
alt: 51km, pressure: 0.182kPa
alt: 52km, pressure: 0.161kPa
alt: 53km, pressure: 0.143kPa
alt: 54km, pressure: 0.126kPa
alt: 55km, pressure: 0.112kPa
alt: 56km, pressure: 0.099kPa
alt: 57km, pressure: 0.087kPa
alt: 58km, pressure: 0.077kPa
alt: 59km, pressure: 0.068kPa
alt: 60km, pressure: 0.060kPa
alt: 61km, pressure: 0.053kPa
alt: 62km, pressure: 0.047kPa
alt: 63km, pressure: 0.042kPa
alt: 64km, pressure: 0.037kPa
alt: 65km, pressure: 0.033kPa
alt: 66km, pressure: 0.029kPa
alt: 67km, pressure: 0.026kPa
alt: 68km, pressure: 0.023kPa
alt: 69km, pressure: 0.020kPa
alt: 70km, pressure: 0.018kPa
alt: 71km, pressure: 0.016kPa
alt: 72km, pressure: 0.014kPa
alt: 73km, pressure: 0.012kPa
alt: 74km, pressure: 0.011kPa
alt: 75km, pressure: 0.010kPa
alt: 76km, pressure: 0.009kPa
alt: 77km, pressure: 0.008kPa
alt: 78km, pressure: 0.007kPa
alt: 79km, pressure: 0.006kPa
alt: 80km, pressure: 0.005kPa

below sea level:
alt: -1km, pressure: 114.802kPa
alt: -2km, pressure: 130.069kPa
alt: -3km, pressure: 147.364kPa
alt: -4km, pressure: 166.955kPa
alt: -5km, pressure: 189.146kPa
alt: -6km, pressure: 214.283kPa
alt: -7km, pressure: 242.756kPa
alt: -8km, pressure: 275.007kPa
alt: -9km, pressure: 311.536kPa
alt: -10km, pressure: 352.911kPa
alt: -11km, pressure: 399.773kPa
alt: -12km, pressure: 452.849kPa
alt: -13km, pressure: 512.961kPa
alt: -14km, pressure: 581.041kPa
alt: -15km, pressure: 658.144kPa
alt: -16km, pressure: 745.463kPa
alt: -17km, pressure: 844.351kPa
alt: -18km, pressure: 956.339kPa
alt: -19km, pressure: 1083.158kPa
alt: -20km, pressure: 1226.770kPa
alt: -21km, pressure: 1389.396kPa
alt: -22km, pressure: 1573.550kPa
alt: -23km, pressure: 1782.077kPa
alt: -24km, pressure: 2018.198kPa
alt: -25km, pressure: 2285.560kPa
alt: -26km, pressure: 2588.291kPa
alt: -27km, pressure: 2931.061kPa
alt: -28km, pressure: 3319.161kPa
alt: -29km, pressure: 3758.574kPa
alt: -30km, pressure: 4256.076kPa
alt: -31km, pressure: 4819.336kPa
alt: -32km, pressure: 5457.032kPa
alt: -33km, pressure: 6178.988kPa
alt: -34km, pressure: 6996.319kPa
alt: -35km, pressure: 7921.608kPa
alt: -36km, pressure: 8969.095kPa
alt: -37km, pressure: 10154.893kPa
alt: -38km, pressure: 11497.240kPa
alt: -39km, pressure: 13016.772kPa
alt: -40km, pressure: 14736.844kPa
alt: -41km, pressure: 16683.885kPa
alt: -42km, pressure: 18887.799kPa
alt: -43km, pressure: 21382.428kPa
alt: -44km, pressure: 24206.064kPa
alt: -45km, pressure: 27402.035kPa
alt: -46km, pressure: 31019.367kPa
alt: -47km, pressure: 35113.534kPa
alt: -48km, pressure: 39747.301kPa
alt: -49km, pressure: 44991.682kPa
alt: -50km, pressure: 50927.025kPa
alt: -51km, pressure: 57644.233kPa
alt: -52km, pressure: 65246.154kPa
alt: -53km, pressure: 73849.142kPa
alt: -54km, pressure: 83584.832kPa
alt: -55km, pressure: 94602.145kPa
alt: -56km, pressure: 107069.552kPa
alt: -57km, pressure: 121177.634kPa
alt: -58km, pressure: 137141.988kPa
alt: -59km, pressure: 155206.499kPa
alt: -60km, pressure: 175647.045kPa
alt: -61km, pressure: 198775.696kPa
alt: -62km, pressure: 224945.446kPa
alt: -63km, pressure: 254555.576kPa
alt: -64km, pressure: 288057.715kPa
alt: -65km, pressure: 325962.692kPa
alt: -66km, pressure: 368848.284kPa
alt: -67km, pressure: 417367.979kPa
alt: -68km, pressure: 472260.880kPa
alt: -69km, pressure: 534362.909kPa
alt: -70km, pressure: 604619.467kPa
alt: -71km, pressure: 684099.754kPa
alt: -72km, pressure: 774012.955kPa
alt: -73km, pressure: 875726.542kPa
alt: -74km, pressure: 990786.962kPa
alt: -75km, pressure: 1120943.029kPa
alt: -76km, pressure: 1268172.365kPa
alt: -77km, pressure: 1434711.295kPa
alt: -78km, pressure: 1623088.635kPa
alt: -79km, pressure: 1836163.896kPa
alt: -80km, pressure: 2077170.461kPa
alt: -81km, pressure: 2349764.399kPa
alt: -82km, pressure: 2658079.637kPa
alt: -83km, pressure: 3006790.328kPa
alt: -84km, pressure: 3401181.343kPa
alt: -85km, pressure: 3847227.949kPa
alt: -86km, pressure: 4351685.864kPa
alt: -87km, pressure: 4922193.050kPa
alt: -88km, pressure: 5567384.754kPa
alt: -89km, pressure: 6297023.529kPa
alt: -90km, pressure: 7122146.188kPa
alt: -91km, pressure: 8055229.876kPa
alt: -92km, pressure: 9110379.766kPa
alt: -93km, pressure: 10303541.165kPa
alt: -94km, pressure: 11652739.223kPa
alt: -95km, pressure: 13178349.821kPa
alt: -96km, pressure: 14903405.681kPa
alt: -97km, pressure: 16853942.285kPa
alt: -98km, pressure: 19059388.765kPa
alt: -99km, pressure: 21553009.599kPa
alt: -100km, pressure: 24372403.713kPa
...
alt: -1000km, pressure: 9700986171002809000000000000000000000000000000000000.000kPa

Obviously some of those numbers are unrealistic since air is not really an ideal gas.

 

[jbriggs444]

A quick spot check indicates that your program is suspect.

The pressure ratio for consecutive kilometers in the reported figures deep under the surface is greater than the pressure ratio for consecutive kilometers at the surface. If modeled gravity were decreasing with depth, one would expect the ratio to decline with depth rather than increasing.

I cannot check the algorithm above surface as readily. The high altitude numbers are reported without enough significant figures to make a clear determination. It appears that the same error is present, however.

 

[mfb]

The numbers increase by ~13.3% near the surface, and ~13% at -100km altitude. That looks consistent, just a bit high (but I do not know which temperature you used). For positive altitudes, the ratio seems to go down a bit, that is fine.

 

[Darwin123]

That's a good point actually - Earth's atmosphere has a scale height of about 8km (if I remember right), so by 50km below the surface, the pressure would be about 400 atmospheres. Given the behavior of N₂ at high pressure, this would definitely be sufficient to notice deviations from the ideal gas behavior. On the other hand, 50km down, the gravitational change would be fairly negligible, especially since the gravitational acceleration inside Earth is fairly constant for the first 2000km or so.

One thing that you have to keep in mind when thinking about this problem: the atmosphere is extremely thin. Gravitational changes across the entire thickness of the atmosphere are pretty much negligible, so they can pretty safely be ignored, even if you were to extend the atmosphere significantly in either direction.

It would deviate from exponential behavior because air is compressible. Even if the air acted like an ideal gas, the density of the air will be changing due to the increased pressure. So the pressure would rise faster than an exponential due to change in density even if the air acted like an ideal gas.
However, the gravitational acceleration will be decreasing with depth. So that would push the rate of increase in the opposite direction.

 

[mfb]

Even if the air acted like an ideal gas, the density of the air will be changing due to the increased pressure.

Exactly that leads to an exponential shape. Pressure rise is proportional to the density multiplied with the gravitational acceleration - and in an ideal gas (and constant temperature), density is proportional to pressure.
Deviations from the exponential law come from:
- air is not an ideal gas (very important >50km below the surface)
- temperature is not constant (ignored here)
- gravitational acceleration is not constant (small effect)

 

[jbriggs444]

The numbers increase by ~13.3% near the surface, and ~13% at -100km altitude. That looks consistent, just a bit high (but I do not know which temperature you used). For positive altitudes, the ratio seems to go down a bit, that is fine.

Bah, my mistake. You are right.