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Density altitude in the standard troposphere
Geopotential height at which the standard troposphere has the measured air density ρ, with e = −g0/(Rλ) − 1 and λ the negative standard lapse rate.
- Ref
- AER-11
- Rev
- v1
- Field
- Aerospace
- Result
- H_ρ [m]
Density altitude is the height in the standard atmosphere at which the density equals the measured density. In the troposphere, ρ/ρ₀ = (T/T₀)^(−g₀/(Rλ) − 1) with T = T₀ + λH, so H = (T₀/λ)[(ρ/ρ₀)^(1/e) − 1], with e = −g₀/(Rλ) − 1 = 4.2559. A hot day lowers the density and raises the density altitude well above the field elevation, which reduces engine power and lift.
Geopotential height at which the standard troposphere has the measured air density ρ, with e = −g0/(Rλ) − 1 and λ the negative standard lapse rate. T0 and ρ0 are the sea-level standard values. Valid only inside the troposphere.
Calculator
- Exponent e = −g₀/(Rλ) − 14.256
- Density ratio ρ/ρ₀0.7347
- Standard temperature at that height, T₀ + λH268 K
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.
- A density of 0.9 kg/m³ is the standard density at 3,097.8 m, where the standard temperature is 268.01 K.
- The aircraft performs as if it were at 10,163 ft, so take-off distance and climb rate should be calculated for that height, not the airport elevation.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ρ Measured air density | 0.4 to 1.4 kg/m³ | −1,413 | 10,250 | 11,660 | +10% | −30.1 % |
| T0 Sea-level temperature | 250 to 320 K | 2,688 | 3,440 | 752.5 | +10% | +10 % |
| λ Temperature gradient (signed) | −0.01 to −0.002 K/m | 2,736 | 3,452 | 715.9 | +0.01 K/m | −22.8 % |
| ρ0 Sea-level standard density | 1.2 to 1.25 kg/m³ | 2,898 | 3,293 | 395.5 | +10% | +29.5 % |
| R Specific gas constant of air | 280 to 290 J/(kg·K) | 3,008 | 3,136 | 128.2 | +10% | +12.1 % |
| g0 Standard gravity | 9.7 to 9.9 m/s² | 3,063 | 3,138 | 75.36 | +10% | −10.6 % |
Sweep of Measured air density
| Measured air density (kg/m³) | Density altitude (m) |
|---|---|
| 0.4 | 10,250 |
| 0.525 | 8,003 |
| 0.65 | 6,133 |
| 0.775 | 4,521 |
| 0.9 | 3,098 |
| 1.025 | 1,818 |
| 1.15 | 653.2 |
| 1.275 | −418.7 |
| 1.4 | −1,413 |
Sweep one variable
Variables and how each one contributes
Density altitude
What the equation gives you. Everything else feeds it.
Sea-level temperature
The sea-level standard temperature, in K. Raises the density altitude in proportion for the same density ratio. Raising it by 10 % (to 317) moves the result by +10 %, from 3,098 to 3,408 m.
Temperature gradient (signed)
The signed standard temperature gradient, in K/m. Changes the exponent and the scale of the result. Raising it by 10 % (to −0.00715) moves the result by +1.94 %, from 3,098 to 3,158 m.
Measured air density
The measured air density, in kg/m³. Lower density gives a higher density altitude. Raising it by 10 % (to 0.99) moves the result by −30.1 %, from 3,098 to 2,164 m.
Sea-level standard density
The standard sea-level density, in kg/m³. A higher reference density lowers the density ratio and raises the altitude. Raising it by 10 % (to 1.348) moves the result by +29.5 %, from 3,098 to 4,011 m.
Standard gravity
The standard gravitational acceleration, in m/s². Changes the exponent slightly. Raising it by 10 % (to 10.79) moves the result by −10.6 %, from 3,098 to 2,768 m.
Specific gas constant of air
The specific gas constant of air, in J/(kg·K). Changes the exponent slightly. Raising it by 10 % (to 315.8) moves the result by +12.1 %, from 3,098 to 3,474 m.
Worked example with the values above
- Exponent: e = −g₀/(Rλ) − 1 = 4.25588; density ratio ρ/ρ₀ = 0.734694.
- Power: (ρ/ρ₀)^(1/e) = 0.9301204; minus 1 gives −0.0698796.
- H = (T₀/λ) × −0.0698796 = 3,097.818 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