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Table 1 Analytical approximations of the Chapman function (in chronological order). The solar zenith angle \(\zeta\) is given in radian units

From: Accurate analytic approximation for the Chapman grazing incidence function

Index Reference Approximation of the Chapman function
(a) Green et al. (1964); Green and Barnum (1963) \({\text {Ch}}(X,\zeta ) \approx \exp \left[ \frac{\frac{1}{2}\zeta ^2}{1-0.115\zeta ^2-\alpha (X)\zeta ^4}\right] ,\)
\(\alpha (X)=\frac{16}{\pi ^4}\left\{ 1-(\pi /2)^2[0.115+ 1/\ln (X\pi /2)]\right\}\)
(b) Fitzmaurice (1964) \({\text {Ch}}(X,\zeta )\approx \sqrt{\frac{\pi X}{2}}{\text {erfc}}\left( \sqrt{\frac{X}{2}}\cos \zeta \right) \exp \left( \frac{X}{2}\cos ^2\zeta \right)\)
(c) Swider (1964) \({\text {Ch}}(X,\zeta )\approx -X\cos \zeta +\sqrt{1+X(1+\sin \zeta )}\Biggl \{\sqrt{X(1-\sin \zeta )}\)
\(+\frac{\sqrt{\pi }}{2}{\text {erfc}}\left[ \sqrt{X(1-\sin \zeta )}\right] \exp [X(1-\sin \zeta )]\Biggr \}\)
(d) Titheridge (1988, 2000) \({\text {Ch}}(X,\zeta ) \approx \sec (\zeta -d),\quad d = 3.88 X^{-1.143}\left[ \sec \left( A\zeta \right) -0.834\right]\),
\(A = 1.0123 -1.454 X^{-1/2}\)
(e) Kocifaj (1996); Schüler (2012) \({\text {Ch}}(X,\zeta ) \approx \frac{1}{2}\Biggl [\cos \zeta +\sqrt{\frac{\pi X}{2}}{\text {erfc}}\left( \sqrt{\frac{X}{2}}\cos \zeta \right) \exp \left( \frac{X}{2}\cos ^2\zeta \right) \Biggr .\)
\(\Biggl .\times \left( \frac{1}{X}+1+\sin ^2\zeta \right) \Biggr ]\)
(f) Huestis (2001) \({\text {Ch}}(X,\zeta ) \approx \sqrt{\frac{\pi X}{(1+\sin \zeta )}}{\text {erfc}}\left[ \sqrt{X(1-\sin \zeta )}\right] \exp \left[ X(1-\sin \zeta )\right]\)