The scale height is a useful way to describe how the atmospheric pressure decreases with altitude. From Wikipedia,
For planetary atmospheres, scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of $e$. The scale height remains constant for a particular temperature. It can be calculated by
$$H = \frac{RT}{Mg}$$
where $R$ is the gas constant, $T$ is the temperature in Kelvin, $M$ is the mass of 1 mole of atmospheric particles, and $g$ is the gravitational acceleration.
That formula assumes that $T$ and $g$ are constant. With a bit of calculus, we can produce a formula that handles the variation in $g$ with altitude, but that's not really necessary, since $g$ at the Kármán line isn't much lower than what it is at sea level. The variations due to $T$ are much more significant, and it's not easy to measure those from the ground, even with modern technology, but using weather balloons we did have a fairly good idea how the atmospheric temperature varies before the space era began.
Here are some typical scale height values for Earth, courtesy of Wikipedia:
Temp (K) |
Height (m) |
290 |
8500 |
273 |
8000 |
260 |
7610 |
210 |
6000 |
Eg, if the scale height is $H = 8500$ m, then $6H=51$ km, and $e^{-6}\approx0.00248$, so the air pressure at $51$ km is around $\frac14$% of the sea level pressure.
Curiously, that Wikipedia article gives no historical info regarding when we first learned about scale height. However, the article on the barometer does mention that
However, Pascal went even further to test the mechanical theory. If, as suspected by mechanical philosophers like Torricelli and Pascal, air had weight, the pressure would be less at higher altitudes. Therefore, Pascal wrote to his brother-in-law, Florin Perier, who lived near a mountain called the Puy de Dôme, asking him to perform a crucial experiment.
Perier was to take a barometer up the Puy de Dôme and make measurements along the way of the height of the column of mercury. He was then to compare it to measurements taken at the foot of the mountain to see if those measurements taken higher up were in fact smaller. In September 1648, Perier carefully and meticulously carried out the experiment, and found that Pascal's predictions had been correct. The column of mercury stood lower as the barometer was carried to a higher altitude.
In the subsequent years, variations on this experiment have been performed, measuring temperature as well as barometric pressure, and carefully recording the time of measurement.
By the dawn of the space era, sensitive barometric pressure measurements were the standard way for aviators to determine their altitude, so estimating the air pressure at orbital altitudes was a simple exercise for any aeronautical navigator. See Altimeter for details.
The change in temperature with altitude is generally known as the lapse rate. From Wikipedia,
The lapse rate is the rate at which an atmospheric variable, normally temperature in Earth's atmosphere, falls with altitude.
[...]
The environmental lapse rate (ELR), is the rate of decrease of temperature with altitude in the stationary atmosphere at a given time and location. As an average, the International Civil Aviation Organization (ICAO) defines an international standard atmosphere (ISA) with a temperature lapse rate of $6.50$ ºC/km from sea level to $11$ km. From $11$ km up to $20$ km the constant temperature is −56.5 °C, which is the lowest assumed temperature in the ISA. The standard atmosphere contains no moisture.
Unlike the idealized ISA, the temperature of the actual atmosphere does not always fall at a uniform rate with height. For example, there can be an inversion layer in which the temperature increases with altitude.
The behaviour of dry air is relatively simple to model. Moist air is much more complicated because water vapour is not well-modelled as an ideal gas. Water has a substantial latent heat of vaporisation, so it has a big effect on the air temperature when it vaporises and condenses.