# 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 ϱ(h){\textstyle \varrho (h)} varies exponentially with altitude h{\textstyle h} according to the Barometric formula:
    $\varrho (h)=\varrho _{0}\exp \left(-{\frac {h}{H}}\right)$,
where ϱ$_0${\textstyle \varrho _{0}} denotes the density at sea level (h=0{\textstyle h=0}) and H{\textstyle H} the so-called scale height. The total amount of matter traversed by a vertical ray starting at altitude h{\textstyle h} towards infinity is given by the integrated density ("column depth")
    $X_{0}(h)=\int _{h}^{\infty }\varrho (l)\,\mathrm {d} l=\varrho _{0}H\exp \left(-{\frac {h}{H}}\right)$.
For inclined rays having a zenith angle z{\textstyle 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
    $X_{z}(h)=\varrho _{0}\exp \left(-{\frac {h}{H}}\right)\int _{0}^{\infty }\exp \left(-{\frac {1}{H}}\left({\sqrt {s^{2}+l^{2}+2ls\cos z}}-s\right)\right)\,\mathrm {d} l$,
where we defined s=h+R$_E${\textstyle s=h+R_{\mathrm {E} }} (R$_E${\textstyle R_{\mathrm {E} }} denotes the Earth radius).
The Chapman function ch⁡(x, z){\textstyle \operatorname {ch} (x, z)} is defined as the ratio between slant depth X$_z${\textstyle X_{z}} and vertical column depth X$_0${\textstyle X_{0}}. Defining x=s/H{\textstyle x=s/H}, it can be written as
    $\operatorname {ch} (x,z)={\frac {X_{z}}{X_{0}}}=\mathrm {e} ^{x}\int _{0}^{\infty }\exp \left(-{\sqrt {x^{2}+u^{2}+2xu\cos z}}\right)\,\mathrm {d} u$.

## 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=π/2{\textstyle 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){\textstyle K_{1}(x)} refers to the modified Bessel function of the second kind of the first order. For large values of x{\textstyle 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→∞{\textstyle x\rightarrow \infty } and 0≤z<π/2{\textstyle 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∼1000{\textstyle x\sim 1000}, ch⁡(x, z)≈sec⁡z{\textstyle \operatorname {ch} (x, z)\approx \sec z} is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.