Chapman function explained

thumb|300px|right|Graph of ch(x, z)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.^[1]

Definition

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

\varrho(h)=\varrho₀\exp\left(-

	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)=

	infty
\int
	h

\varrho(l)dl=\varrho₀H\exp\left(-

	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 thecurvature of Earth. Here, the integral reads

X_z(h)=\varrho₀\exp\left(-

	hH
	\right)

	infty
\int
	0

\exp\left(-

	1H
	\left(\sqrt{s

²+l²+2ls\cosz}-s\right)\right)dl

,where we defined

s = h + R_

(

R_

denotes the Earth radius).

The Chapman function $\operatorname(x, z)$ 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)=

	X_z
	X₀

=e^x

	infty
\int
	0

\exp\left(-\sqrt{x²+u²+2xu\cosz}\right)du

Representations

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

\operatorname{ch}(x,z)=x\sinz

	z
\int
	0

	\exp\left(x(1-\sinz/\sinλ)\right)
	\sin²λ

dλ

Huestis^[2] developed the representation

\operatorname{ch}(x,z)=1+x\sin

	z
z\int
	0

	\exp\left(x(1-\sinz/\sinλ)\right)
	1+\cosλ

dλ

,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^[3]

\operatorname{ch}\left(x,

	\pi
	2

\right)=xe^xK_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\gg1,

	\pi
	2

\right) ≈ \sqrt{

	\pi
	2

.For

x \rightarrow \infty

and

0 \leq z < \pi/2

, the Chapman function converges to the secant function:

\lim_x\operatorname{ch}(x,z)=\secz

.In practical applications related to the terrestrial atmosphere, where

x \sim 1000

\operatorname(x, z) \approx \sec z

is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.

External links

Chapman function at Science World
Smith . F. L. . Smith . Cody . Numerical evaluation of Chapman's grazing incidence integral ch(X,χ) . J. Geophys. Res. . 1972 . 77 . 19 . 3592–3597 . 10.1029/JA077i019p03592. 1972JGR....77.3592S .

Notes and References

Chapman . S. . The absorption and dissociative or ionizing effect of monochromatic radiation in an atmosphere on a rotating earth part II. Grazing incidence . Proceedings of the Physical Society . 1 September 1931 . 43 . 5 . 483–501 . 10.1088/0959-5309/43/5/302. 1931PPS....43..483C .
Huestis . David L. . Accurate evaluation of the Chapman function for atmospheric attenuation . Journal of Quantitative Spectroscopy and Radiative Transfer . 2001 . 69 . 6 . 709–721 . 10.1016/S0022-4073(00)00107-2. 2001JQSRT..69..709H .
Vasylyev . Dmytro . Accurate analytic approximation for the Chapman grazing incidence function . Earth, Planets and Space . December 2021 . 73 . 1 . 112 . 10.1186/s40623-021-01435-y. 2021EP&S...73..112V . 234796240 . free .