Chapman function

A Chapman function describes the integration of atmospheric absorption along a slant path on a spherical Earth, relative to the vertical case.

It applies to any quantity with a concentration decreasing exponentially with increasing altitude.

To a first approximation, valid at small zenith angles, the Chapman function for optical absorption is equal to where z is the zenith angle and sec denotes the secant function.

[1] In an isothermal model of the atmosphere, the density

denotes the density at sea level (

The total amount of matter traversed by a vertical ray starting at altitude

towards infinity is given by the integrated density ("column depth") For inclined rays having a zenith angle

, the integration is not straight-forward due to the non-linear relationship between altitude and path length when considering the curvature of Earth.

is defined as the ratio between slant depth

, it can be written as A number of different integral representations have been developed in the literature.

Chapman's original representation reads[1] Huestis[2] developed the representation which does not suffer from numerical singularities present in Chapman's representation.

(horizontal incidence), the Chapman function reduces to[3] Here,

refers to the modified Bessel function of the second kind of the first order.

, the Chapman function converges to the secant function: In practical applications related to the terrestrial atmosphere, where

is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.