The hypsometric equation, also known as the thickness equation, relates an atmospheric pressure ratio to the equivalent thickness of an atmospheric layer under the assumptions of constant temperature and gravity. It is derived from the hydrostatic equation and the ideal gas law.
The hypsometric equation is expressed as:
- = thickness of the layer [m]
- = geometric height [m]
- = specific gas constant for dry air
- = mean temperature in Kelvin [K]
- = gravitational acceleration [m/s2]
- = pressure [Pa]
In meteorology, and are isobaric surfaces. In altimetry with the International Standard Atmosphere the hypsometric equation is used to compute pressure at a given height in isothermal layers in the upper and lower stratosphere.
The hydrostatic equation:
This is combined with the ideal gas law:
to eliminate :
This is integrated from to :
R and g are constant with z, so they can be brought outside the integral. If temperature varies linearly with z (as it is assumed to do in the International Standard Atmosphere), it can also be brought outside the integral when replaced with , the average temperature between and .
or, eliminating the ln: