Chapman function (original) (raw)

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Graph of ch(x, z)

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, sec ⁡ ( z ) {\displaystyle \sec(z)} $\sec(z)$ .

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]

In an isothermal model of the atmosphere, the density ϱ ( h ) {\textstyle \varrho (h)} ${\textstyle \varrho (h)}$ varies exponentially with altitude h {\textstyle 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)} $\varrho (h)=\varrho _{0}\exp \left(-{\frac {h}{H}}\right)$ ,

where ϱ 0 {\textstyle \varrho _{0}} ${\textstyle \varrho _{0}}$ denotes the density at sea level ( h = 0 {\textstyle h=0} ${\textstyle h=0}$ ) and H {\textstyle H} ${\textstyle H}$ the so-called scale height. The total amount of matter traversed by a vertical ray starting at altitude h {\textstyle 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)} $X_{0}(h)=\int _{h}^{\infty }\varrho (l)\,\mathrm {d} l=\varrho _{0}H\exp \left(-{\frac {h}{H}}\right)$ .

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

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

The Chapman function ch ⁡ ( x , z ) {\textstyle \operatorname {ch} (x,z)} ${\textstyle \operatorname {ch} (x,z)}$ is defined as the ratio between slant depth X z {\textstyle X_{z}} ${\textstyle X_{z}}$ and vertical column depth X 0 {\textstyle X_{0}} ${\textstyle X_{0}}$ . Defining x = s / H {\textstyle 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} $\operatorname {ch} (x,z)={\frac {X_{z}}{X_{0}}}=\mathrm {e} ^{x}\int _{0}^{\infty }\exp \left(-{\sqrt {x^{2}+u^{2}+2xu\cos z}}\right)\,\mathrm {d} u$ .

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

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

Huestis[3] 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 } $\operatorname {ch} (x,z)=1+x\sin z\int _{0}^{z}{\frac {\exp \left(x(1-\sin z/\sin \lambda )\right)}{1+\cos \lambda }}\,\mathrm {d} \lambda$ ,

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

For z = π / 2 {\textstyle z=\pi /2} ${\textstyle z=\pi /2}$ (horizontal incidence), the Chapman function reduces to[4]

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

Here, K 1 ( x ) {\textstyle 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} ${\textstyle x}$ , 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}}} $\operatorname {ch} \left(x\gg 1,{\frac {\pi }{2}}\right)\approx {\sqrt {{\frac {\pi }{2}}x}}$ .

For x → ∞ {\textstyle x\rightarrow \infty } ${\textstyle x\rightarrow \infty }$ and 0 ≤ z < π / 2 {\textstyle 0\leq z<\pi /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} $\lim _{x\rightarrow \infty }\operatorname {ch} (x,z)=\sec z$ .

In practical applications related to the terrestrial atmosphere, where x ∼ 1000 {\textstyle x\sim 1000} ${\textstyle x\sim 1000}$ , ch ⁡ ( x , z ) ≈ sec ⁡ z {\textstyle \operatorname {ch} (x,z)\approx \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.

^ 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.
^ Simple Comparative Ionospheres Using the Chapman Layer Model https://heliophysics.ucar.edu/sites/default/files/heliophysics/resources/presentations/2014_Lab_4.pdf
^ 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.
^ 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.

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. Bibcode:1972JGR....77.3592S. doi:10.1029/JA077i019p03592.