thumb|300px|right|[[graph of a function|Graph of ch(x, z)]]

A Chapman function, denoted , 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, <math>\sec(z)</math>.

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 <math display="inline">\varrho(h)</math> varies exponentially with altitude <math display="inline">h</math> according to the 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.

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).

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 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

:<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 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>. The accuracy improves with increasing <math display="inline>x</math>.

See also

  • Air mass
  • Atmospheric physics
  • Ionosphere

References

  • Chapman function at Science World