# Hypsometric Equation

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.

From the Hydrostatic Balance , one form of the hydrostatic equation is:

**Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): g dz = d\Phi = -\frac{RT}{p} dp = -RT d \ln p **

Integration of this equation in the vertical yields a form of the hypsometric equation:

**Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \Phi(z_2) - \Phi(z_1) = g_0(Z_2 - Z_1) = R \int_{p2}^{p1} T d \ln p **

Here **Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): Z = \Phi(z)/g_0 **
is the geopotential height.

In terms of Z the hypsometric equation becomes:

**Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): T = Z_2 - Z_1 = \frac{R}{g_0} \int_{p2}^{p1} T d \ln p **

where T is the thickness of the atmospheric layer between the two pressure levels.