Astronomy Principles and Practice Fourth Edition.pdf

106 The celestial sphere: timekeeping systems
Figure 9.11. The calculation for twilight.
below the horizon. Corresponding definitions hold for morning twilight: thus civil morning twilight,
for example, begins when the Sun’s centre is 6 ◦ below the horizon.
Twilight is a nuisance, astronomically speaking, often preventing the observation of very faint
celestial objects. We shall see later that in some latitudes during part of the year, twilight is indeed
continuous throughout the night, evening and morning twilight merging because the Sun’s centre at all
times of the night is less than 18 ◦ below the horizon.
An accurate enough approximation to the duration of twilight is to calculate twice the difference
between the hour angles of the Sun’s centre when it is 18 ◦ below the horizon and when it is on the
horizon.
In figure 9.11, let H 1 , H 2 be the values of the Sun’s hour angle when its centre is on the horizon
at X and 18 ◦ below the horizon at Y respectively, its declination having value δ ⊙ N. Let the observer’s
latitude be φ N. Then
PZ = 90 − φ PX = PY = 90 − δ ⊙ ZX = 90 ZY = 108 ◦ .
In △PZX, using the cosine formula, we have
cos 90 = cos(90 − φ)cos(90 − δ ⊙ ) + sin(90 − φ)sin(90 − δ ⊙ ) cos H 1
giving
cos H 1 =−tan φ tan δ ⊙ .
or
In △PZY, again using the cosine formula, we obtain
cos 108 = sin φ sin δ ⊙ + cos φ cos δ ⊙ cos H 2
cos H 2 = cos 108 − sin φ sin δ ⊙
cos φ cos δ ⊙
.
The duration of astronomical twilight is then given by H 2 −H 1 . It is seen that the duration depends
upon the Sun’s declination and the latitude of the observer.
The limiting case when evening twilight ends as morning twilight begins will occur when B in
figure 9.11 coincides with D, i.e. the Sun’s centre is exactly 18 ◦ below the horizon at apparent midnight.