Atmospheric scale height

Why does air pressure halve every few kilometres instead of running out at a fixed height?

Ask most people where the atmosphere stops and they will name a height. It is a reasonable instinct: the air has weight, there is a finite amount of it, so pile it up and it should reach some level and stop, the way water in a glass does.

Yet nobody can tell you that height, and every table of the atmosphere is written in fractions instead — half the pressure by five and a half kilometres, a tenth by sixteen, a thousandth by about fifty. Why does the air keep offering fractions rather than an ending?

Begin with what pressure actually is at any level: the weight of all the air above that level, resting on each square metre. Climb, and you leave some air beneath you, so the pressure drops by exactly the weight of the air you passed through.

Now the crucial question. How much does the pressure drop per kilometre climbed? That depends on how much air is in that kilometre — that is, on the density. And what sets the density?

Here the argument closes on itself. A gas squeezed harder is denser. The density at any level is set by the pressure at that level. So the rate at which pressure falls with height is proportional to the pressure that is still there.

That is the whole answer, and it is a familiar shape. Whenever a quantity's rate of loss is proportional to how much remains, the quantity cannot reach zero at any finite point — it can only be reduced by a fixed fraction per fixed step, forever. It is the same mathematics as compound interest running backwards, or radioactive half-life. Water in a glass has a top because water barely compresses: its density does not care about the pressure above it, so it thins out at a constant rate and runs out. Air compresses, and that single difference removes the ending.

The step size in that fixed-fraction rule is the scale height. Work it out from the ideal gas law and you get temperature times the gas constant, divided by the mean molecular mass times gravity — around eight and a half kilometres for air at around fifteen degrees Celsius, a little over seven at the colder temperatures higher up. That is the distance over which pressure falls by a factor of e. The halving distance is a bit smaller, about five and a half kilometres, which is where the tables get their number.

Now let us test the intuition we discarded, because it deserves a proper burial. Suppose air really were incompressible at sea-level density. How high would the atmosphere then be? Divide sea-level pressure by density times gravity and you get about eight and a half kilometres — the scale height again. That coincidence is worth sitting with. The scale height is precisely how deep the atmosphere would be if you could pour it into a glass at uniform density. In the real, compressible atmosphere, that same distance is not the top; it is the distance in which things get a factor of e thinner, and then does it again, and again.

Two consequences fall straight out, and both are strange until you have the exponential in hand. First, half of the entire mass of the atmosphere lies below five and a half kilometres — below the summit of Kilimanjaro. Second, there is no point at which the air stops. Somewhere above about five hundred kilometres, collisions between molecules become so rare that individual particles simply follow ballistic arcs, and the fastest ones escape to space altogether. The atmosphere fades out; it has no lid.

The formula also tells you what would change it. Scale height rises with temperature and falls with molecular mass and gravity. That is why Mars, with a far thinner atmosphere, nonetheless has a larger scale height than Earth — around eleven kilometres — because its gravity is much weaker. Thickness of atmosphere and steepness of atmosphere are independent things.

Some honest corrections to the clean exponential.

The simple formula assumes a single temperature throughout, and the real atmosphere has nothing of the sort: it cools with height through the troposphere, warms again through the stratosphere as ozone absorbs ultraviolet, and so on. So the real pressure profile is better described as a stack of exponentials with different scale heights, each following the local temperature. The curve is still exponential in character everywhere, just not with one constant.

Higher still, above roughly a hundred kilometres, mixing gives way to diffusive separation and each gas begins to settle out according to its own molecular mass, so lighter gases develop their own, larger scale heights. The single mean molecular mass in the formula stops being a valid simplification there.

And one caution about what the exponential does not say. It says pressure never reaches zero mathematically. Physically, at a certain point the gas is so rarefied that speaking of pressure and density as smooth continuous quantities stops being meaningful, and you have to count particles instead. The atmosphere ends in a change of description rather than at a height.

• Why the temperature profile itself has structure, and what the lapse rate has to do with convection.
• Jeans escape — how the tail of the velocity distribution slowly strips light gases from a planet.
• Why pressure altimeters must be reset for the day's weather, given all this.

Every term the collection defines is gathered in the glossary.