A Chapman function, denoted , describes the integration of an atmospheric parameter along a slant path on a spherical Earth, relative to the vertical or zenithal case. It applies to any physical quantity with a concentration decreasing exponentially with increasing altitude. At small angles, the Chapman function is approximately equal to the secant function of the zenith angle, .
The Chapman function is named after Sydney Chapman, who introduced the function in 1931. It has been applied for absorption (esp. optical absorption) and the ionosphere.
In an isothermal model of the atmosphere, the density varies exponentially with altitude according to the Barometric formula:
where denotes the density at sea level () and the so-called scale height. 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. Here, the integral reads
where we defined ( denotes the Earth radius).
The Chapman function is defined as the ratio between slant depth and vertical column depth . Defining , it can be written as
A number of different integral representations have been developed in the literature. Chapman's original representation reads
Huestis developed the representation
which does not suffer from numerical singularities present in Chapman's representation.
For (horizontal incidence), the Chapman function reduces to
Here, refers to the modified Bessel function of the second kind of the first order. For large values of , this can further be approximated by
For and , 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 60ð to 70ð, depending on the accuracy required.
For and , the approximation
is accurate to 2 % at and to 0.1 % at . The accuracy improves with increasing .