scale height

Scale height is a measure of the decrease of something that falls off exponentially by height, specifically, the height over which it falls by a factor of e (~ 2.718). Examples can include:

• Pressure of some planetary atmospheres at heights from the surface (a pressure scale height or atmospheric scale height; for example, Earth's pressure scale height is roughly 8.5 km, the distance over the pressure falls by a factor of e).
• Temperature of a stellar atmosphere (temperature scale height).
• Stellar density in a galaxy at distances from its galactic center (a density scale height).

Despite the word "height", it might be applied to dimensions not thought of as vertical, though the term scale length is sometimes used in such cases. When it is thought of as vertical (e.g., re density or pressure of an atmosphere above a body's surface, or density of stars in a galaxy above/below its plane), it may be referred to as a vertical scale height. Equation-forms associated with scale height:

A = A0e-z/H

   A       z
ln —— = - ——
   A0      H

dA
—— = -HA
dz

(The first two are forms of the solution to the differential equation.)

• z - the height, i.e., a coordinate on a (vertical) axis.
• H - the scale height.
• A - the quantity, a function of z for which H applies (e.g., pressure).
• A₀ - the value of A when z = 0.