# Chapman function

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[[file:Chapman function.svg|thumb|300px|right|[Graph](/source/graph_of_a_function) of ch(x, z)]]

A '''Chapman function''', denoted {{math|ch}}, 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](/source/physical_quantity) with a concentration [decreasing exponentially](/source/exponential_decay) with increasing [altitude](/source/altitude). At small [angle](/source/angle)s, the Chapman function is approximately equal to the [secant function](/source/secant_function) of the [zenith angle](/source/zenith_angle), <math>\sec(z)</math>.

The Chapman function is named after [Sydney Chapman](/source/Sydney_Chapman_(astronomer)), who introduced the function in [1931](/source/1931).<ref name="chapman">{{cite journal |last1=Chapman |first1=S. |title=The absorption and dissociative or ionizing effect of monochromatic radiation in an atmosphere on a rotating earth part II. Grazing incidence |journal=Proceedings of the Physical Society |date=1 September 1931 |volume=43 |issue=5 |pages=483–501 |doi=10.1088/0959-5309/43/5/302|bibcode=1931PPS....43..483C }}</ref>
It has been applied for [absorption](/source/absorption_(electromagnetic_radiation)) (esp. [optical absorption](/source/absorption_(optics))) and the [ionosphere](/source/ionosphere).<ref>Simple Comparative Ionospheres Using the Chapman Layer Model https://heliophysics.ucar.edu/sites/default/files/heliophysics/resources/presentations/2014_Lab_4.pdf</ref>

== Definition ==
In an isothermal model of the atmosphere, the density <math display="inline">\varrho(h)</math> varies exponentially with altitude <math display="inline">h</math> according to the [Barometric formula](/source/Barometric_formula):
:<math>\varrho(h) = \varrho_0 \exp\left(- \frac h H \right)</math>,
where <math display="inline">\varrho_0</math> denotes the density at sea level (<math display="inline">h=0</math>) and <math display="inline">H</math> the so-called [scale height](/source/scale_height).
The total amount of matter traversed by a vertical ray starting at altitude <math display="inline">h</math> towards infinity is given by the integrated density ("column depth")
:<math>X_0(h) = \int_h^\infty \varrho(l)\, \mathrm d l = \varrho_0 H \exp\left(-\frac hH \right) </math>.

For inclined rays having a zenith angle <math display="inline">z</math>, 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
:<math> 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</math>,
where we defined <math display="inline">s = h + R_{\mathrm E}</math> (<math display="inline">R_{\mathrm E}</math> denotes the [Earth radius](/source/Earth_radius)).

The Chapman function <math display="inline">\operatorname{ch}(x, z)</math> is defined as the ratio between ''slant depth'' <math display="inline">X_z</math> and vertical column depth <math display="inline">X_0</math>. Defining <math display="inline">x = s / H</math>, it can be written as
:<math> \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 </math>.

== Representations ==
A number of different integral representations have been developed in the literature. Chapman's original representation reads<ref name="chapman" />
:<math>\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 </math>.

Huestis<ref name="huestis">{{cite journal |last1=Huestis |first1=David L. |title=Accurate evaluation of the Chapman function for atmospheric attenuation |journal=[Journal of Quantitative Spectroscopy and Radiative Transfer](/source/Journal_of_Quantitative_Spectroscopy_and_Radiative_Transfer) |year=2001 |volume=69 |issue=6 |pages=709–721 |doi=10.1016/S0022-4073(00)00107-2|bibcode=2001JQSRT..69..709H }}</ref> developed the representation
:<math>\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</math>,
which does not suffer from numerical singularities present in Chapman's representation.

== Special cases ==
For <math display="inline">z = \pi/2</math> (horizontal incidence), the Chapman function reduces to<ref>{{cite journal |last1=Vasylyev |first1=Dmytro |title=Accurate analytic approximation for the Chapman grazing incidence function |journal=Earth, Planets and Space |date=December 2021 |volume=73 |issue=1 |pages=112 |doi=10.1186/s40623-021-01435-y|bibcode=2021EP&S...73..112V |s2cid=234796240 |doi-access=free }}</ref>
:<math>\operatorname{ch}\left(x, \frac \pi 2 \right) = x \mathrm{e}^x K_1(x) </math>.
Here, <math display="inline">K_1(x)</math> refers to the modified [Bessel function](/source/Bessel_function) of the second kind of the first order. For large values of <math display="inline">x</math>, this can further be approximated by
:<math>\operatorname{ch}\left(x \gg 1, \frac \pi 2 \right) \approx \sqrt{\frac{\pi}{2}x}</math>.
For <math display="inline">x \rightarrow \infty</math> and <math display="inline">0 \leq z < \pi/2</math>, the Chapman function converges to the secant function:
:<math>\lim_{x \rightarrow \infty} \operatorname{ch}(x, z) = \sec z</math>.
In practical applications related to the terrestrial atmosphere, where <math display="inline">x \sim 1000 </math>, <math display="inline">\operatorname{ch}(x, z) \approx \sec z</math> is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.

== Approximations ==
For <math display="inline">x \geq 50</math> and <math display="inline">0 \leq z \leq \pi/2</math>, the approximation
:<math>\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) </math>
is accurate to 2&nbsp;% at <math display="inline">x = 50</math> and to 0.1&nbsp;% at <math display="inline">x = 800</math>.<ref>{{cite journal |doi=10.1364/AO.3.000640 |last1=Fitzmaurice |first1=John A. |journal=Appl. Opt. |volume=3 |page=640 |language=en |year=1964 |title=Simplification of the Chapman Function for Atmospheric Attenuation}}</ref> The accuracy improves with increasing <math display="inline>x</math>.

== See also ==
* [Air mass](/source/Air_mass_(astronomy))
* [Atmospheric physics](/source/Atmospheric_physics)
* [Ionosphere](/source/Ionosphere)

== References ==
{{Reflist}}

== External links ==
* [http://scienceworld.wolfram.com/chemistry/ChapmanFunction.html Chapman function at Science World]
* {{cite journal |last1=Smith |first1=F. L. |last2=Smith |first2=Cody |title=Numerical evaluation of Chapman's grazing incidence integral ch(X,χ) |journal=J. Geophys. Res. |year=1972 |volume=77 |issue=19 |pages=3592–3597 |doi=10.1029/JA077i019p03592|bibcode=1972JGR....77.3592S }}

Category:Radio frequency propagation
Category:Special functions
Category:Vertical distributions

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Adapted from the Wikipedia article [Chapman function](https://en.wikipedia.org/wiki/Chapman_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Chapman_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.