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


 * $$\sec(z),\ $$

where z is the zenith angle and sec denotes the secant function.

The Chapman function is named after Sydney Chapman, who introduced the function in 1931.

Definition
In an isothermal model of the atmosphere, the density $\varrho(h)$ varies exponentially with altitude $h$  according to the Barometric formula:
 * $$\varrho(h) = \varrho_0 \exp\left(- \frac h H \right)$$,

where $\varrho_0$ denotes the density at sea level ($h=0$ ) and $H$  the so-called scale height. The total amount of matter traversed by a vertical ray starting at altitude $h$ towards infinity is given by the integrated density ("column depth")
 * $$X_0(h) = \int_h^\infty \varrho(l)\, \mathrm d l = \varrho_0 H \exp\left(-\frac hH \right) $$.

For inclined rays having a zenith angle $z$, 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
 * $$ X_z(h) = \varrho_0 \exp\left(-\frac hH \right) \int_0^\infty \exp\left(- \frac 1H \left(\sqrt{s^2 + l^2 + 2ls \cos z} -s \right)\right) \, \mathrm d l$$,

where we defined $s = h + R_{\mathrm E}$ ($R_{\mathrm E}$  denotes the Earth radius).

The Chapman function $\operatorname{ch}(x, z)$ is defined as the ratio between slant depth $X_z$  and vertical column depth $X_0$. Defining $x = s / H$, it can be written as
 * $$ \operatorname{ch}(x, z) = \frac{X_z}{X_0} = \mathrm e^x \int_0^\infty \exp\left(-\sqrt{x^2 + u^2 + 2xu\cos z}\right) \, \mathrm du $$.

Representations
A number of different integral representations have been developed in the literature. Chapman's original representation reads
 * $$\operatorname{ch}(x, z) = x \sin z \int_0^z \frac{\exp\left(x (1 - \sin z / \sin \lambda)\right)}{\sin^2 \lambda} \, \mathrm d \lambda $$.

Huestis developed the representation
 * $$\operatorname{ch}(x, z) = 1 + x\sin z\int_0^z \frac{\exp\left(x (1 - \sin z / \sin \lambda)\right)}{1 + \cos\lambda} \,\mathrm d \lambda$$,

which does not suffer from numerical singularities present in Chapman's representation.

Special cases
For $z = \pi/2$ (horizontal incidence), the Chapman function reduces to
 * $$\operatorname{ch}\left(x, \frac \pi 2 \right) = x \mathrm{e}^x K_1(x) $$.

Here, $K_1(x)$ refers to the modified Bessel function of the second kind of the first order. For large values of $x$, this can further be approximated by
 * $$\operatorname{ch}\left(x \gg 1, \frac \pi 2 \right) \approx \sqrt{\frac{\pi}{2}x}$$.

For $x \rightarrow \infty$ and $0 \leq z < \pi/2$, the Chapman function converges to the secant function:
 * $$\lim_{x \rightarrow \infty} \operatorname{ch}(x, z) = \sec z$$.

In practical applications related to the terrestrial atmosphere, where $x \sim 1000 $, $\operatorname{ch}(x, z) \approx \sec z$ is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.