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Chapman function

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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 ) ,   {\displaystyle \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.

Definition

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

ϱ ( h ) = ϱ 0 exp ⁡ ( − h H ) {\displaystyle \varrho (h)=\varrho _{0}\exp \left(-{\frac {h}{H}}\right)} ,

where ϱ 0 {\textstyle \varrho _{0}} denotes the density at sea level ( h = 0 {\textstyle h=0} ) and H {\textstyle H} the so-called scale height. The total amount of matter traversed by a vertical ray starting at altitude h {\textstyle h} towards infinity is given by the integrated density ("column depth")

X 0 ( h ) = ∫ h ∞ ϱ ( l ) d l = ϱ 0 H exp ⁡ ( − h H ) {\displaystyle X_{0}(h)=\int _{h}^{\infty }\varrho (l)\,\mathrm {d} l=\varrho _{0}H\exp \left(-{\frac {h}{H}}\right)} .

For inclined rays having a zenith angle z {\textstyle 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 ) = ϱ 0 exp ⁡ ( − h H ) ∫ 0 ∞ exp ⁡ ( − 1 H ( s 2 + l 2 + 2 l s cos ⁡ z − s ) ) d l {\displaystyle X_{z}(h)=\varrho _{0}\exp \left(-{\frac {h}{H}}\right)\int _{0}^{\infty }\exp \left(-{\frac {1}{H}}\left({\sqrt {s^{2}+l^{2}+2ls\cos z}}-s\right)\right)\,\mathrm {d} l} ,

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

The Chapman function ch ⁡ ( x , z ) {\textstyle \operatorname {ch} (x,z)} is defined as the ratio between slant depth X z {\textstyle X_{z}} and vertical column depth X 0 {\textstyle X_{0}} . Defining x = s / H {\textstyle x=s/H} , it can be written as

ch ⁡ ( x , z ) = X z X 0 = e x ∫ 0 ∞ exp ⁡ ( − x 2 + u 2 + 2 x u cos ⁡ z ) d u {\displaystyle \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 {d} u} .

Representations

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

ch ⁡ ( x , z ) = x sin ⁡ z ∫ 0 z exp ⁡ ( x ( 1 − sin ⁡ z / sin ⁡ λ ) ) sin 2 ⁡ λ d λ {\displaystyle \operatorname {ch} (x,z)=x\sin z\int _{0}^{z}{\frac {\exp \left(x(1-\sin z/\sin \lambda )\right)}{\sin ^{2}\lambda }}\,\mathrm {d} \lambda } .

Huestis developed the representation

ch ⁡ ( x , z ) = 1 + x sin ⁡ z ∫ 0 z exp ⁡ ( x ( 1 − sin ⁡ z / sin ⁡ λ ) ) 1 + cos ⁡ λ d λ {\displaystyle \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 } ,

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

Special cases

For z = π / 2 {\textstyle z=\pi /2} (horizontal incidence), the Chapman function reduces to

ch ⁡ ( x , π 2 ) = x e x K 1 ( x ) {\displaystyle \operatorname {ch} \left(x,{\frac {\pi }{2}}\right)=x\mathrm {e} ^{x}K_{1}(x)} .

Here, K 1 ( x ) {\textstyle K_{1}(x)} refers to the modified Bessel function of the second kind of the first order. For large values of x {\textstyle x} , this can further be approximated by

ch ⁡ ( x ≫ 1 , π 2 ) ≈ π 2 x {\displaystyle \operatorname {ch} \left(x\gg 1,{\frac {\pi }{2}}\right)\approx {\sqrt {{\frac {\pi }{2}}x}}} .

For x → ∞ {\textstyle x\rightarrow \infty } and 0 ≤ z < π / 2 {\textstyle 0\leq z<\pi /2} , the Chapman function converges to the secant function:

lim x → ∞ ch ⁡ ( x , z ) = sec ⁡ z {\displaystyle \lim _{x\rightarrow \infty }\operatorname {ch} (x,z)=\sec z} .

In practical applications related to the terrestrial atmosphere, where x ∼ 1000 {\textstyle x\sim 1000} , ch ⁡ ( x , z ) ≈ sec ⁡ z {\textstyle \operatorname {ch} (x,z)\approx \sec z} is a good approximation for zenith angles up to 60° to 70°, depending on the accuracy required.

See also

References

  1. ^ 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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