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Aerospace
Geopotential altitude from geometric altitude
Geopotential height H, the height that makes gravity look constant, from geometric height z above the model reference surface.
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- AER-08
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- v1
- Field
- Aerospace
- Result
- H [m]
Gravity falls with height, so the atmosphere is tabulated in geopotential height H, the height that would give the same potential energy under constant sea-level gravity. For geometric height z, H = r z/(r + z), with r the effective Earth radius. The difference is small near the ground, 19 m at 11 km, and grows to over 1 km at 80 km. Use H to pick the layer and to apply the layer equations.
Geopotential height H, the height that makes gravity look constant, from geometric height z above the model reference surface. re is the effective Earth radius (6,356,766 m in the 1976 standard atmosphere). Use H, not z, to select the atmospheric layer and to apply the layer equations.
Calculator
- Difference z − H19 m
- Gravity at this height relative to g₀, (r/(r + z))²0.9965
- Fractional difference, (z − H)/z0.001727
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 geometric height of 11,000 m corresponds to a geopotential height of 10,981 m, 19 m lower.
- Gravity at this height is 99.655 % of its sea-level value, which is why geopotential height drops behind geometric height.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| z Geometric height | 0 to 86,000 m | 0 | 84,850 | 84,850 | +10% | +9.98 % |
| re Effective Earth radius | 6,300,000 to 6,400,000 m | 10,980 | 10,980 | 0.2991 | +10% | +0.0157 % |
Sweep of Geometric height
| Geometric height (m) | Geopotential height (m) |
|---|---|
| 0 | 0 |
| 10,750 | 10,730 |
| 21,500 | 21,430 |
| 32,250 | 32,090 |
| 43,000 | 42,710 |
| 53,750 | 53,300 |
| 64,500 | 63,850 |
| 75,250 | 74,370 |
| 86,000 | 84,850 |
Sweep one variable
Variables and how each one contributes
Geopotential height
What the equation gives you. Everything else feeds it.
Effective Earth radius
The effective Earth radius of the model, in m (6,356,766 m in the 1976 atmosphere). A larger radius makes gravity fall more slowly, so H gets closer to z. Raising it by 10 % (to 6,992,000) moves the result by +0.0157 %, from 10,980 to 10,980 m.
Geometric height
The geometric height above the reference surface, in m. Raises H, but less than one-for-one, as the difference grows with height. Raising it by 10 % (to 12,100) moves the result by +9.98 %, from 10,980 to 12,080 m.
Worked example with the values above
- Sum of radius and height: r + z = 6,357,000 + 11,000 = 6,368,000 m.
- Product: r z = 6,357,000 × 11,000 = 6.9924426×1010.
- H = 6.9924426×1010 / 6,368,000 = 10,981 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 in the layer equations instead of geopotential height.
- Using the standard 6,356,766 m radius with a different reference ellipsoid.
- Forgetting that the difference becomes large above 50 km.
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