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Aerospace
Pressure in an isothermal layer
Static pressure at height ΔH above the base of a layer of constant absolute temperature Tb, such as the standard stratosphere between 11 and 20 km.
- Ref
- AER-10
- Rev
- v1
- Field
- Aerospace
- Result
- p [Pa]
In an isothermal layer the hydrostatic equation integrates to an exponential: p = pb exp(−g₀ΔH/(RTb)). The pressure falls by a factor e over the scale height RTb/g₀, about 6.3 km at 216.65 K. Starting from the layer’s own base pressure matters, since each layer has a different base value.
Static pressure at height ΔH above the base of a layer of constant absolute temperature Tb, such as the standard stratosphere between 11 and 20 km. It decays exponentially with a scale height RTb/g0. Start from the pressure pb at the layer’s own base.
Calculator
- Scale height, RTb/g₀6,342 m
- Pressure ratio p/pb0.7295
- Height for pressure to halve, scale height × ln 24,396 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- After a rise of 2,000 m the pressure is 16.51 kPa, 72.95 % of the base value.
- The pressure halves every 4,395.7 m in this layer, so the pressure after 5 halvings (21.98 km) is 1/32 of the base.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| pb Base pressure | 1,000 to 100,000 Pa | 729.5 | 72,950 | 72,220 | +10% | +10 % |
| ΔH Height above the layer base | 0 to 20,000 m | 966.2 | 22,630 | 21,670 | +10% | −3.1 % |
| Tb Layer temperature | 180 to 300 K | 15,480 | 18,020 | 2,539 | +10% | +2.91 % |
| R Specific gas constant of air | 280 to 290 J/(kg·K) | 16,380 | 16,560 | 183.6 | +10% | +2.91 % |
| g0 Standard gravity | 9.7 to 9.9 m/s² | 16,460 | 16,570 | 106.2 | +10% | −3.1 % |
Sweep of Height above the layer base
| Height above the layer base (m) | Static pressure (Pa) |
|---|---|
| 0 | 22,630 |
| 2,500 | 15,260 |
| 5,000 | 10,290 |
| 7,500 | 6,936 |
| 10,000 | 4,676 |
| 12,500 | 3,153 |
| 15,000 | 2,126 |
| 17,500 | 1,433 |
| 20,000 | 966.2 |
Sweep one variable
Variables and how each one contributes
Static pressure
What the equation gives you. Everything else feeds it.
Base pressure
The pressure at the base of the layer, in Pa. Scales the result in direct proportion. Raising it by 10 % (to 24,900) moves the result by +10 %, from 16,510 to 18,160 Pa.
Standard gravity
The standard gravitational acceleration, in m/s². Raises the rate of decay and so lowers the pressure. Raising it by 10 % (to 10.79) moves the result by −3.1 %, from 16,510 to 16,000 Pa.
Height above the layer base
The height above the layer base, in m. Pressure falls exponentially with height. Raising it by 10 % (to 2,200) moves the result by −3.1 %, from 16,510 to 16,000 Pa.
Specific gas constant of air
The specific gas constant of air, in J/(kg·K). A larger R lengthens the scale height and slows the decay. Raising it by 10 % (to 315.8) moves the result by +2.91 %, from 16,510 to 16,990 Pa.
Layer temperature
The constant layer temperature, in K. A warmer layer has a taller scale height, so pressure falls more slowly. Raising it by 10 % (to 238.3) moves the result by +2.91 %, from 16,510 to 16,990 Pa.
Worked example with the values above
- Scale height: RTb/g₀ = 287.1 × 216.7 / 9.807 = 6,341.62 m.
- Exponent: −ΔH/scale height = −2,000 / 6,341.62 = −0.315377.
- p = 22,630 × e^−0.31538 = 16,510.38 Pa.
Assumptions and limits
- The 1976 U.S. Standard Atmosphere: dry air as an ideal gas, constant g₀ in geopotential height and hydrostatic balance.
- Heights inside one layer with constant temperature gradient, or constant temperature, as the entry states.
Common mistakes
- Using geometric height where the layer equations need geopotential height.
- Using the temperature gradient with the wrong sign, which is negative in the troposphere.
- Applying a layer equation outside the layer it was derived for.
Statistics and uncertainty
The standard atmosphere is a model, not the weather. Real temperature profiles differ from it by 10 K or more, which changes the pressure at height by 1 % to 3 % and the density by 3 % to 5 %. Use measured conditions for performance and flight-test work.
Where it is used
- Finding pressure, density and temperature at altitude for performance calculations.
- Calibrating altimeters and airspeed instruments.
Maths you need
- Hydrostatic equation: dp/dz = −ρg
- Ideal gas law: p = ρRT
- Lapse rate: the rate at which temperature changes with height