Debate Magazine

Fun with Numbers: Gravity, Altitude, Pressure and Temperature.

Posted on the 12 March 2020 by Markwadsworth @Mark_Wadsworth

I hope we've all learned something new this week, it has been a slog!
If you don't understand why this relationship holds on the basis of commonsense/observation; or you can't imagine how a big cloud of "air molecules" all repelling each other but attracted by Earth's gravity behave, then you can revert to the "modified gas/pressure laws as they apply in a strong gravitational field over long vertical distances" (I don't know if there is a snappier term). This is a maths thing, so you don't really need to "understand" it. Clever physicists have worked it all out for us.
This formula give the relationship between height (i.e. above sea level); pressure; and temperature T (in Kelvin):

{\displaystyle P_{h}=P_{0}e^{-{\frac {mgh}{kT}}},}
Ph is the pressure at height h,
P0 is the pressure at reference point 0 (typically referring to sea level),
m is the mass per air molecule,
g is gravity,
h is height from reference point 0,
k is the Boltzmann constant,
T is the temperature in kelvins.

Pressure is easily estimated; it's highest at sea level (about 1,000 mbar) and half that (about 500 mbar) when you are about halfway up through the atmosphere, which (we are told) is on average 5.5 km.
There are three constants - "m", "g" and "k". Whatever they are, and whatever the units, you can multiply them together to get one constant. If you define height in kilometres and do the numbers, "mg/k" boils down to approx. 31.
We know, or are told, that the expected average temperature of earth's surface would be 255K in the absence of an absence of an atmosphere.
So it's reasonable to assume that the overall average temperature of the atmosphere is 255K, and that (average) halfway up, 5.5 km, it's 255K. Which turns out to be broadly true.
Plug all that into the formula and it all checks out:
1,000 x e^-((31 x 5.5)/255) = +/- 500
In Excel, you do =1000*EXP(-(31*5.5)/255), which comes out at 512 mbar, close enough to expected 500 mbar.
You can't plug height = zero into the formula, because e^0 = 1, so the answer would come out right whatever temperature you assume i.e. you can't use it to calculate temperature at sea level, but you can substitute, say 1 km altitude. T will come out at about 282K
This is 27K warmer for a 4.5km fall in height = 6K per kilometre, which is pretty close to typical "moist adiabatic lapse rate".
Extrapolate that down to sea level, and you get 288K, which, we are told, is the average temperature of the surface of the earth.
Extrapolate that up to the top of Mount Everest, and you know why it's so cold up there, i.e. colder than it would be if we had no atmosphere at all.
That's where the normal "greenhouse gas" explanation falls flat on its face - it's one possible explanation for why it's 33C warmer at sea level, but it doesn't explain why it's so cold up there. That's the thing that has been bugging me for a decade and nobody has ever even tried to tweak it.
So there we have it!
The extra 33C surface temperature is not down to back radiation from "greenhouse gases", it is just how the "modified gas/pressure laws as they apply in a strong gravitational field over long vertical distances" work.

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