Table of Contents:

download in PDF

1. Preface
2. Atmospheric Temperature in Relation to Altitude
3. Atmospheric Pressure Explained
4. Atmospheric Pressure in Relation to Altitude
5. Adjustments for Non-Standard Conditions
6. Examples with Real Numbers
7. Further Reading and Helpful Links


Atmospheric Pressure in Relation to Altitude

The ISA model doesn't actually give us a formula that relates pressure to altitude, at least one that's solved for us - instead it gives us one that relates pressure to atmospheric temperature. This is where our earlier temperature equation will come in handy, as we'll soon see. And secondly, it bears mentioning that the change in air pressure with respect to altitude is not linear but rather logarithmic, and what's more, the relationship differs depending on whether we're in an isothermal air layer or region (one where temperature remains constant), or in a non-isothermal region.

For these reasons, we're going to have to hold on to our neurons for the next page or so, as the formulas I'm going to unleash will consequently be a bit more involved than what we've seen heretofore.

This is the formula that relates air pressure to atmospheric temperature in the non-isothermal regions of the atmosphere (we'll deal with the isothermal formulas later):

Some of these variables should look familiar from our previous temperature formula, namely T, T1, and a. The others should already start to make a little bit of sense from what we've discussed previously, but I'll touch on each of them here.

Note that we have a variable (actually a constant) for the force of gravity, g. This makes sense. The force of gravity is what draws us towards the surface of the earth, it's what gives us weight. Without gravity our atmosphere would quickly fly off into space. The fact that we do have gravity is not only what keeps our atmosphere hanging around, but also what causes it to exert a pressure commensurate with its mass, as discussed earlier. So it makes sense this formula should include the gravitational constant.

The "ideal gas constant" R isn't something you need to know too much about, though if you want to do some reading about it there's plenty of interesting information you could learn. For now, it's enough to know that it deals in general with the mass of air, which we can also see would be important. To know how much pressure my feet are going to exert on the ground, we need to know how the force of gravity would act on my body's mass (as an aside, gravity and mass are two concepts we often conveniently lump together with the term weight). To measure the pressure of the atmosphere we need to know the same sorts of things. Above with g we have information about gravity, and R is our information about the mass of the atmosphere (for you brainos out there, yes, I know I'm oversimplifying).

That leaves p and p1 as the only other variables we've not yet discussed. The first is easy enough, p is the pressure of the air we are attempting to determine, or that we've measured. The second, p1, is the pressure at the lower bound of the given atmospheric layer/region that we're in. You're probably sitting in the Troposphere right now. The pressure at the lower bound of that layer is therefore simply the pressure at sea level. Above the Troposphere is the Tropopause, and the pressure at the bottom of the Tropopause is the same as the pressure at the top of the layer below it (the Troposphere), and so on. It's the same concept as our T1 discussed earlier in the temperature section.

Ok, so we have a dandy formula but it still doesn't give us altitude from air pressure. Also, it appears that it tells us what air pressure is, but we don't need to be told that because it's what we're going to be measuring with our cool pressure sensor. So let's rearrange Equation 2 like so:

Alright, that was quick and easy. Now, let's replace T with our earlier Equation 1 (I told you it would come in handy):

Nice. Now we're getting somewhere. We have a formula that has both altitude (h) and pressure (p) in it. All we have to do is rearrange it so we solve for h. I'll save you a lot of weeping and gnashing of teeth and just show you the final result:

And voila! For a given air pressure measurement p we can now solve for our altitude h. All the other variables we know from Table 1, except for the two constants R and g (they're just constants so we'll worry about their actual values later), and p1, which you can see in Table 2 below for each atmospheric layer.

As before pupper isn't used in any formula; I include it only to show you the range for the given region. Note also that although the pressures in Table 2 are listed in Pascals (Pa), the examples to follow will use kilopascals (kPa). To get kilopascals, simply divide Pascals by 1,000.

At this point we're pretty much through with our altitude formula. The Equation 3 we've just solved lets you plug in an air pressure reading and some constants, and out comes an altitude. We'll want to refine it a bit - such as an adjustment to compensate for conditions different than those on the so-called "standard day."

But before we go on to that, let's solve one more set of formulas. As mentioned earlier Equation 3 only applies in the layers of the atmosphere where temperature varies. This includes the Troposphere, Stratosphere, and Mesosphere. For the isothermal regions, where temperature remains constant, the formula is slightly different. Let's just run through it real quick-like.

Equation 4 is the equivalent of Equation 2 but for the isothermal regions of the atmosphere. Again we need to rearrange it, but one thing you'll notice that's different from Equation 2 is that height is already a part of the calculation, so we won't have to substitute in our Equation 1 (temperature/altitude equation). Another thing you'll want to notice is that although temperature T is included, it will actually just be a constant - this is by definition what the temperature does in an isothermal region, it stays the same. To get this T, you'll just look up the value in Table 1.

So really all we need to do is rearrange Equation 4 so we have altitude (h) on the left side, and we're through. Again, I'll spare you all the bloody details and just show you the result (yes, the ln is a natural log):

And there you have it. With Equation 5 for isothermal regions and Equation 3 for the rest, and with the standard assumptions for some variables listed in Table 1 and 2, you have everything you need to determine altitude from air pressure. There are further refinements that we'll get to next, but these two formulas are the core of what you need to know - and if you don't plan on going higher than the Troposphere, which extends to 11 kilometers (36,000 feet) above sea level, then you really only need to know Equation 3.

Next: Adjustments for Non-Standard Conditions