## Air pressure below sea level

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.
 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.

 Quote by 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.
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.

## Air pressure below sea level

The whole point of the problem is to take into account the changing gravitational field.
 But if that is the point of the problem then the model does not call for pressure to be exponential above ground either.

 Quote by snorkack 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.
 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.
 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 modelled 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.
 Mentor 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.

 Quote by cjl 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.

Mentor
 Quote by Darwin123 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)

 Quote by 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.
Bah, my mistake. You are right.