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Chapman function

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Chapman function
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A Chapman function or Chapman layer, 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, .

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Graph of ch(x, z)

The Chapman function is named after Sydney Chapman, who introduced the function in 1931.[1] It has been applied for absorption (esp. optical absorption) and the ionosphere.[2]

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Definition

In an isothermal model of the atmosphere, the density varies exponentially with altitude according to the Barometric formula:

,

where denotes the density at sea level () and the so-called scale height. The total amount of matter traversed by a vertical ray starting at altitude towards infinity is given by the integrated density ("column depth")

.

For inclined rays having a zenith angle , 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

,

where we defined ( denotes the Earth radius).

The Chapman function is defined as the ratio between slant depth and vertical column depth . Defining , it can be written as

.
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Representations

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

.

Huestis[3] developed the representation

,

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

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Special cases

For (horizontal incidence), the Chapman function reduces to[4]

.

Here, refers to the modified Bessel function of the second kind of the first order. For large values of , this can further be approximated by

.

For and , the Chapman function converges to the secant function:

.

In practical applications related to the terrestrial atmosphere, where , is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.

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See also

References

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