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Aerospace
Local pressure scale height
Height over which pressure falls by a factor of e in an isothermal hydrostatic ideal gas, defined as −p/(dp/dz).
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- AER-12
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- Hp [m]
If temperature and gravity were constant, hydrostatic balance would make pressure fall exponentially, p ∝ exp(−z/Hp), with scale height Hp = RT/g. Equivalently, Hp = −p/(dp/dz) defines the local scale height even when T varies. For air at 250 K it is about 7.3 km, which is why the atmosphere seems to thin out over a few kilometres.
Height over which pressure falls by a factor of e in an isothermal hydrostatic ideal gas, defined as −p/(dp/dz). T is the local absolute temperature, g local gravity and R the specific gas constant. It is a true exponential decay length over a finite interval only when T/g is constant.
Calculator
- Height for pressure to halve, Hp ln 25,072 m
- Height for pressure to fall by 90 %, Hp ln 1016,850 m
- Pressure fraction left at 10 km, e^(−10 000/Hp)0.255
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.
- The scale height is 7.3178 km, so pressure falls to 36.79 % of its value every 7.318 km of height.
- At 10 km the pressure would be 25.5 % of the ground value in an isothermal atmosphere at this temperature, close to the real value of 26 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| T Air temperature | 180 to 320 K | 5,269 | 9,367 | 4,098 | +10% | +10 % |
| R Specific gas constant | 280 to 290 J/(kg·K) | 7,138 | 7,393 | 254.9 | +10% | +10 % |
| g Gravitational acceleration | 9.6 to 9.9 m/s² | 7,249 | 7,475 | 226.5 | +10% | −9.09 % |
Sweep of Air temperature
| Air temperature (K) | Pressure scale height (m) |
|---|---|
| 180 | 5,269 |
| 197.5 | 5,781 |
| 215 | 6,293 |
| 232.5 | 6,806 |
| 250 | 7,318 |
| 267.5 | 7,830 |
| 285 | 8,342 |
| 302.5 | 8,855 |
| 320 | 9,367 |
Sweep one variable
Variables and how each one contributes
Pressure scale height
What the equation gives you. Everything else feeds it.
Specific gas constant
The specific gas constant, in J/(kg·K). A larger R lengthens the scale height in direct proportion. Raising it by 10 % (to 315.8) moves the result by +10 %, from 7,318 to 8,050 m.
Air temperature
The local absolute temperature, in K. A warmer atmosphere is taller, with scale height in direct proportion to T. Raising it by 10 % (to 275) moves the result by +10 %, from 7,318 to 8,050 m.
Gravitational acceleration
The local gravitational acceleration, in m/s². Stronger gravity compresses the atmosphere, shortening the scale height in inverse proportion. Raising it by 10 % (to 10.79) moves the result by −9.09 %, from 7,318 to 6,653 m.
Worked example with the values above
- Product: R T = 287.1 × 250 = 71,763.22 J/kg.
- Divide by gravity: 71,763.22 / 9.807.
- Hp = 7,317.812 m.
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