Scale Height is a measure of the decrease of something that falls off exponentially, specifically, the distance over which it falls by a factor of e (~ 2.718). Examples can include:
For example, a planet's atmosphere might fall by a factor of e every 10km of height. Despite the word "height", it might be applied to dimensions not thought of as vertical. 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.)