Comment on "Improved ray tracing air mass numbers model"

Siebren Y. van der Werf

doi:10.1364/AO.47.000153

Applied Optics
Vol. 47,
Issue 2,
pp. 153-156
(2008)
•https://doi.org/10.1364/AO.47.000153

Comment on “Improved ray tracing air mass numbers model”

Siebren Y. van der Werf

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Siebren Y. van der Werf, "Comment on "Improved ray tracing air mass numbers model"," Appl. Opt. 47, 153-156 (2008)

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Abstract

Air mass numbers have traditionally been obtained by techniques that use height as the integration variable. This introduces an inherent singularity at the horizon, and ad hoc solutions have been invented to cope with it. A survey of the possible options including integration by height, zenith angle, and horizontal distance or path length is presented. Ray tracing by path length is shown to avoid singularities both at the horizon and in the zenith. A fourth-order Runge–Kutta numerical integration scheme is presented, which treats refraction and air mass as path integrals. The latter may optionally be split out into separate contributions of the atmosphere's constituents.

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Integration Variable = h	Integration Variable = β	Integration Variable = ϕ	Integration Variable = s
$d β = \frac{[1 + \frac{(R_{0} + h)}{r \cos (β)}]}{(R_{0} + h) \tan (β)} d h$	$d h = \frac{(R_{0} + h) \tan (β)}{[1 + \frac{(R_{0} + h)}{r \cos (β)}]} d β$	$d h = (R_{0} + h) \tan (β) d ϕ$	$d h = \sin (β) d s$

$d ϕ = \frac{1}{(R_{0} + h) \tan (β)} d h$	$d ϕ = \frac{1}{[1 + \frac{(R_{0} + h)}{r \cos (β)}]} d β$	$d β = [1 + \frac{(R_{0} + h)}{r \cos (β)}] d ϕ$	$d β = [\frac{\cos (β)}{(R_{0} + h)} + \frac{1}{r}] d s$

$d s = \frac{1}{\sin (β)} d h$	$d s = \frac{\frac{(R_{0} + h)}{\cos (β)}}{[1 + \frac{(R_{0} + h)}{r \cos (β)}]} d β$	$d s = \frac{(R_{0} + h)}{\cos (β)} d ϕ$	$d ϕ = \frac{\cos (β)}{(R_{0} + h)} d s$

$d ξ = \frac{1}{r \sin (β)} d h$	$d ξ = \frac{\frac{(R_{0} + h)}{r \cos (β)}}{[1 + \frac{(R_{0} + h)}{r \cos (β)}]} d β$	$d ξ = \frac{(R_{0} + h)}{r \cos (β)} d ϕ$	$d ξ = d β - d ϕ = \frac{1}{r} d s$

$d M = \frac{ρ}{\sin (β)} d h$	$d M = \frac{ρ \frac{(R_{0} + h)}{\cos (β)}}{[1 + \frac{(R_{0} + h)}{r \cos (β)}]} d β$	$d M = ρ \frac{(R_{0} + h)}{\cos (β)} d ϕ$	$d M = ρ d s$

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