Question 5.1: Estimate the time difference between sunset and the beginnin...
Estimate the time difference between sunset and the beginning of civil twilight in Greenwich, England (51.5°N, 0.0°W) on 21 June (solar declination 23.5°).
Civil twilight is when ψ = 96° and sunset is when ψ = 90°. The small correction in the following calculations for the equation of time is ignored. To find the hour angle θ at civil twilight, solve the equation cos 96 = sin 51.5 sin 23.5 + cos 51.5 cos 23.5 cos θ, giving cos θ = −0.730 and θ = 136.9°. Hence astronomical twilight is 136.9/15 = 9.13 h after solar noon. Similarly (or using the simplification of Eq. 5.4)
\cos \theta _{s} = – \frac{\sin \varphi }{\cos \varphi } \frac{\sin \delta }{\cos \delta } = – \tan \varphi \tan \delta (5.4)
θ at sunset is 123.1°, so sunset is 123.1/15 = 8.21 h after solar noon. The difference is therefore 0.92 h, or 55 min.