Chapman function

Representations

A number of different integral representations have been developed in the literature. Chapman's original representation reads

${\displaystyle \mathrm{ch}(x,z)=x\sin z{\int }_0^z\frac{\exp \left(x(1-\sin z/\sin \lambda )\right)}{{\sin }^2\lambda }\mathrm{d}\lambda }$.

Huestis developed the representation

${\displaystyle \mathrm{ch}(x,z)=1+x\sin z{\int }_0^z\frac{\exp \left(x(1-\sin z/\sin \lambda )\right)}{1+\cos \lambda }\mathrm{d}\lambda }$,

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

${\displaystyle \mathrm{ch}\left(x,\frac{\pi }{2}\right)=x{\mathrm{e}}^x{K}_1(x)}$.

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

${\displaystyle \mathrm{ch}\left(x\gg 1,\frac{\pi }{2}\right)\approx \sqrt{\frac{\pi }{2}x}}$.

For $x\to \mathrm{\infty }$ and $0\le z<\pi /2$, the Chapman function converges to the secant function:

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\mathrm{ch}(x,z)=\sec z}$.

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

Approximations

For $x\ge 50$ and $0\le z\le \pi /2$, the approximation

${\displaystyle \mathrm{ch}(x,z)=\sqrt{\frac{x\pi }{2}}\exp \left(\frac{x}{2}{\cos }^{2}z\right)\left(1-\mathrm{erf}\left(\sqrt{\frac{x}{2}}\cos z\right)\right)}$

is accurate to 2 % at $x=50$ and to 0.1 % at $x=800$. The accuracy improves with increasing $x$.