Hypsometric equation

Equation

The hypsometric equation is expressed as:

h = z 2 − z 1 = R ⋅ T ¯ g ⋅ ln ⁡ ( p 1 p 2 ) ,

where:

h = thickness of the layer [m], z = geometric height [m], R = specific gas constant for dry air, T ¯ = mean temperature in kelvins [K], g = gravitational acceleration [m/s²], p = pressure [Pa].

In meteorology, p 1 and p 2 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.

Derivation

The hydrostatic equation:

p = ρ ⋅ g ⋅ z ,

where ρ is the density [kg/m³], is used to generate the equation for hydrostatic equilibrium, written in differential form:

d p = − ρ ⋅ g ⋅ d z .

This is combined with the ideal gas law:

p = ρ ⋅ R ⋅ T

to eliminate ρ :

d p p = − g R ⋅ T d z .

This is integrated from z 1 to z 2 :

∫ p ( z 1 ) p ( z 2 ) d p p = ∫ z 1 z 2 − g R ⋅ T d z .

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 T ¯ , the average temperature between z 1 and z 2 .

∫ p ( z 1 ) p ( z 2 ) d p p = − g R ⋅ T ¯ ∫ z 1 z 2 d z .

Integration gives

ln ⁡ ( p ( z 2 ) p ( z 1 ) ) = − g R ⋅ T ¯ ( z 2 − z 1 ) ,

simplifying to

ln ⁡ ( p 1 p 2 ) = g R ⋅ T ¯ ( z 2 − z 1 ) .

Rearranging:

z 2 − z 1 = R ⋅ T ¯ g ln ⁡ ( p 1 p 2 ) ,

or, eliminating the ln:

p 1 p 2 = e g R ⋅ T ¯ ⋅ ( z 2 − z 1 ) .