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

Graph of ch(x, 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

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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

 .

Representations

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

 .

Huestis[2] developed the representation

 ,

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

Special cases

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For   (horizontal incidence), the Chapman function reduces to[3]

 .

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.

See also

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References

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  1. ^ a b Chapman, S. (1 September 1931). "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. 43 (5): 483–501. Bibcode:1931PPS....43..483C. doi:10.1088/0959-5309/43/5/302.
  2. ^ Huestis, David L. (2001). "Accurate evaluation of the Chapman function for atmospheric attenuation". Journal of Quantitative Spectroscopy and Radiative Transfer. 69 (6): 709–721. Bibcode:2001JQSRT..69..709H. doi:10.1016/S0022-4073(00)00107-2.
  3. ^ Vasylyev, Dmytro (December 2021). "Accurate analytic approximation for the Chapman grazing incidence function". Earth, Planets and Space. 73 (1): 112. Bibcode:2021EP&S...73..112V. doi:10.1186/s40623-021-01435-y. S2CID 234796240.
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