Chapman function

In an isothermal model of the atmosphere, the density ${\textstyle \varrho (h)}$ varies exponentially with altitude ${\textstyle h}$ according to the Barometric formula:

${\displaystyle \varrho (h)=\varrho _{0}\exp \left(-{\frac {h}{H}}\right)}$,

where ${\textstyle \varrho _{0}}$ denotes the density at sea level (${\textstyle h=0}$) and ${\textstyle H}$ the so-called scale height. The total amount of matter traversed by a vertical ray starting at altitude ${\textstyle h}$ towards infinity is given by the integrated density ("column depth")

${\displaystyle X_{0}(h)=\int _{h}^{\infty }\varrho (l)\,\mathrm {d} l=\varrho _{0}H\exp \left(-{\frac {h}{H}}\right)}$.

For inclined rays having a zenith angle ${\textstyle z}$, 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

${\displaystyle X_{z}(h)=\varrho _{0}\exp \left(-{\frac {h}{H}}\right)\int _{0}^{\infty }\exp \left(-{\frac {1}{H}}\left({\sqrt {s^{2}+l^{2}+2ls\cos z}}-s\right)\right)\,\mathrm {d} l}$,

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

The Chapman function ${\textstyle \operatorname {ch} (x,z)}$ is defined as the ratio between slant depth ${\textstyle X_{z}}$ and vertical column depth ${\textstyle X_{0}}$. Defining ${\textstyle x=s/H}$, it can be written as

${\displaystyle \operatorname {ch} (x,z)={\frac {X_{z}}{X_{0}}}=\mathrm {e} ^{x}\int _{0}^{\infty }\exp \left(-{\sqrt {x^{2}+u^{2}+2xu\cos z}}\right)\,\mathrm {d} u}$.

A number of different integral representations have been developed in the literature. Chapman's original representation reads^[1]

${\displaystyle \operatorname {ch} (x,z)=x\sin z\int _{0}^{z}{\frac {\exp \left(x(1-\sin z/\sin \lambda )\right)}{\sin ^{2}\lambda }}\,\mathrm {d} \lambda }$.

Huestis^[3] developed the representation

${\displaystyle \operatorname {ch} (x,z)=1+x\sin z\int _{0}^{z}{\frac {\exp \left(x(1-\sin z/\sin \lambda )\right)}{1+\cos \lambda }}\,\mathrm {d} \lambda }$,

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

For ${\textstyle z=\pi /2}$ (horizontal incidence), the Chapman function reduces to^[4]

${\displaystyle \operatorname {ch} \left(x,{\frac {\pi }{2}}\right)=x\mathrm {e} ^{x}K_{1}(x)}$.

Here, ${\textstyle K_{1}(x)}$ refers to the modified Bessel function of the second kind of the first order. For large values of ${\textstyle x}$, this can further be approximated by

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

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

${\displaystyle \lim _{x\rightarrow \infty }\operatorname {ch} (x,z)=\sec z}$.

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

For ${\textstyle x\geq 50}$ and ${\textstyle 0\leq z\leq \pi /2}$, the approximation

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

is accurate to 2 % at ${\textstyle x=50}$ and to 0.1 % at ${\textstyle x=800}$.^[5] The accuracy improves with increasing ${\textstyle x}$.

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