To calculate the air density at an altitude of 38,969 meters using the Standard Atmosphere Model, we can use the barometric formula and the temperature profile of the atmosphere.
At this altitude, we are in the stratosphere, where the temperature is approximately constant. The standard temperature at 40,000 meters is around -50.5 °C, which is about 222.65 K.
Using the ideal gas law, the density of air can be calculated as follows:
\[ \rho = \frac{P}{R \cdot T} \]
Where:
- \(\rho\) = air density (kg/m³)
- \(P\) = pressure (Pa)
- \(R\) = specific gas constant for dry air (approximately 287.05 J/(kg·K))
- \(T\) = temperature in Kelvin (K)
From the Standard Atmosphere tables, the pressure at 38,969 meters is approximately 0.0000114 atm (or 0.00056 kPa).
First, convert pressure from kPa to Pa:
\[ P = 0.00056 \text{ kPa} \times 1000 = 0.56 \text{ Pa} \]
Having all variables:
- \(P = 0.56 , \text{Pa}\)
- \(R = 287.05 , \text{J/(kg K)}\)
- \(T = 222.65 , \text{K}\)
Substituting the values into the formula:
\[ \rho = \frac{0.56}{287.05 \times 222.65} \]
Calculating the denominator:
\[ 287.05 \times 222.65 \approx 63848.82325 \]
Now substitute back:
\[ \rho = \frac{0.56}{63848.82325} \approx 0.00000877 , \text{kg/m³} \]
Thus, the air density at 38,969 meters altitude is approximately:
\[ \rho \approx 0.000877 , \text{kg/m³} \text{ or } 0.877 , \text{g/m³} \]
Air density decreases significantly with altitude, and at 38,969 meters, it is very low.