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

(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)

\({\text {Ch}}(X,\zeta ) \approx \sqrt{\frac{\pi X}{(1+\sin \zeta )}}{\text {erfc}}\left[ \sqrt{X(1-\sin \zeta )}\right] \exp \left[ X(1-\sin \zeta )\right]\)