Chapman function

Graph of ch(x, z)

A Chapman function, denoted 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 with a concentration decreasing exponentially with increasing altitude. At small angles, the Chapman function is approximately equal to the secant function of the zenith angle, $\sec (z)$.

The Chapman function is named after Sydney Chapman, who introduced the function in 1931. It has been applied for absorption (esp. optical absorption) and the ionosphere.

Definition

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

$\varrho (h)={\varrho }_0\exp (-\frac{h}{H})$,

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")

$X_0(h)=\underset{h}{\overset{\mathrm{\infty }}{\int }}\varrho (l)\mathrm{d}l={\varrho }_{0}H\exp (-\frac{h}{H})$.

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 the curvature of Earth. Here, the integral reads

$X_z(h)={\varrho }_0\exp (-\frac{h}{H})\underset{0}{\overset{\mathrm{\infty }}{\int }}\exp (-\frac{1}{H}(\sqrt{s^2+l^2+2ls\cos z}-s))\mathrm{d}l$,

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

The Chapman function $\mathrm{ch}(x,z)$ 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

$\mathrm{ch}(x,z)=\frac{X_z}{X_0}={\mathrm{e}}^x\underset{0}{\overset{\mathrm{\infty }}{\int }}\exp (-\sqrt{x^2+u^2+2xu\cos z})\mathrm{d}u$.

Representations

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

$\mathrm{ch}(x,z)=x\sin z\underset{0}{\overset{z}{\int }}\frac{\exp (x(1-\sin z/\sin \lambda ))}{{\sin }^2\lambda }\mathrm{d}\lambda$.

Huestis developed the representation

$\mathrm{ch}(x,z)=1+x\sin z\underset{0}{\overset{z}{\int }}\frac{\exp (x(1-\sin z/\sin \lambda ))}{1+\cos \lambda }\mathrm{d}\lambda$,

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

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

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

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

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

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

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

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

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

See also

• Air mass
• Atmospheric physics
• Ionosphere

External links

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