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Aerospace
Pressure in a constant temperature-gradient layer
Static pressure at height ΔH above the base of an atmospheric layer in which temperature changes linearly with geopotential height at the signed rate λ (K/m), found by integrating the hydrostatic equation.
- Ref
- AER-09
- Rev
- v1
- Field
- Aerospace
- Result
- p [Pa]
In a layer where temperature changes linearly with height, T = Tb + λΔH, the hydrostatic equation and the ideal gas law integrate to p = pb(1 + λΔH/Tb)^(−g₀/(Rλ)). For the standard troposphere, λ = −0.0065 K/m and the exponent is 5.2559. Pressure falls roughly by half every 5.5 km. The page uses a log form that stays accurate for small gradients.
Static pressure at height ΔH above the base of an atmospheric layer in which temperature changes linearly with geopotential height at the signed rate λ (K/m), found by integrating the hydrostatic equation. pb and Tb are the base pressure and temperature. Stay within one layer; λ must not be zero and the temperature must stay positive.
Calculator
- Temperature at this height, Tb + λΔH255.6 K
- Pressure ratio p/pb0.5331
- Pressure exponent, −g₀/(Rλ)5.256
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.
- At 5,000 m above the base the pressure is 54.02 kPa, 53.31 % of the base value, and the air has cooled to 255.65 K.
- Pressure falls faster at first in absolute terms but at a constant fraction per unit height only for an isothermal layer; in a gradient layer the fractional fall rate itself grows with height.
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 200,000 Pa | 533.1 | 106,600 | 106,100 | +10% | +10 % |
| ΔH Height above the layer base | 0 to 10,000 m | 26,440 | 101,300 | 74,890 | +10% | −6.5 % |
| Tb Base temperature | 180 to 350 K | 35,580 | 60,710 | 25,130 | +10% | +6.23 % |
| λ Temperature gradient (signed) | −0.01 to −0.002 K/m | 52,840 | 55,420 | 2,584 | +0.01 K/m | +5.49 % |
| R Specific gas constant of air | 280 to 290 J/(kg·K) | 53,170 | 54,370 | 1,196 | +10% | +5.88 % |
| g0 Standard gravity | 9.7 to 9.9 m/s² | 53,700 | 54,390 | 693.2 | +10% | −6.1 % |
Sweep of Height above the layer base
| Height above the layer base (m) | Static pressure (Pa) |
|---|---|
| 0 | 101,300 |
| 1,250 | 87,180 |
| 2,500 | 74,680 |
| 3,750 | 63,680 |
| 5,000 | 54,020 |
| 6,250 | 45,590 |
| 7,500 | 38,250 |
| 8,750 | 31,900 |
| 10,000 | 26,440 |
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 111,500) moves the result by +10 %, from 54,020 to 59,420 Pa.
Base temperature
The temperature at the base of the layer, in K. A warmer base lowers the relative temperature change and so slows the pressure drop with height. Raising it by 10 % (to 317) moves the result by +6.23 %, from 54,020 to 57,380 Pa.
Temperature gradient (signed)
The signed temperature gradient dT/dH, in K/m (negative in the troposphere). A larger magnitude cools the air faster and raises the exponent’s effect, so pressure falls faster. Raising it by 10 % (to −0.00715) moves the result by −0.394 %, from 54,020 to 53,810 Pa.
Height above the layer base
The geopotential height above the layer base, in m. A greater height lowers pressure steeply, close to exponentially. Raising it by 10 % (to 5,500) moves the result by −6.5 %, from 54,020 to 50,510 Pa.
Standard gravity
The standard gravitational acceleration, in m/s². Raises the exponent and so lowers the pressure. Raising it by 10 % (to 10.79) moves the result by −6.1 %, from 54,020 to 50,730 Pa.
Specific gas constant of air
The specific gas constant of air, in J/(kg·K). Lowers the exponent when it rises. Raising it by 10 % (to 315.8) moves the result by +5.88 %, from 54,020 to 57,200 Pa.
Worked example with the values above
- Exponent: −g₀/(Rλ) = −9.807 / (287.1 × (−0.0065)) = 5.25588.
- Base: 1 + λΔH/Tb = 1 + (−0.0065 × 5,000) / 288.1 = 0.8872115.
- p = 101,300 × 0.8872115^5.2559 = 54,019.89 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