Astronomy Principles and Practice Fourth Edition.pdf

70 The celestial sphere: coordinate systems
Figure 8.10. Example 8.3—conversion of hour angle and declination to azimuth and altitude.
Example 8.3. A star of declination 42 ◦ 21 ′ N is observed when its hour angle is 8 h 16 m 42 s . If the
observer’s latitude is 60 ◦ N, calculate the star’s azimuth and altitude at the time of observation.
It is often a great help to sketch as accurately as possible a celestial sphere diagram of the problem.
This provides a visual check on deductions about quadrants in which an angle lies.
Since PX = 90 − δ, we see that its value is 47 ◦ 39 ′ .
We convert the hour angle value of 8 h 16 m 42 s to angular measure by means of table 7.1 (see
page 48).
8 h 16 m 42 s = 8 h + 16 m + 42 s
= (8 × 15) ◦ + (16/4) ◦ + (40/4) ′ + (2 × 15) ′′
= 120 ◦ + 4 ◦ + 10 ′ + 30 ′′
= 124 ◦ 10·′5.
Hence, H = ZPX = 124 ◦ 10·′5, in figure 8.10 where X is the star.
PZ = 90 ◦ − φ = 90 ◦ − 60 ◦ = 30 ◦ .
Applying the cosine formula to △PZX, we may write
cos ZX = cos 30 ◦ cos 47 ◦ 39 ′ + sin 30 ◦ sin 47 ◦ 39 ′ cos 124 ◦ 10·′5
or
sin a = cos 30 ◦ cos 47 ◦ 39 ′ − sin 30 ◦ sin 47 ◦ 39 ′ cos 55 ◦ 49·′5.
The calculation then proceeds as follows:
sin a = cos 30 ◦ cos 47·◦6500 − sin 30 ◦ sin 47·◦6500 cos55·◦8253
giving, on reduction,
a = 22 ◦ 04·′6.
Applying the cosine formula once more to △PZX,wehave
cos 47 ◦ 39 ′ = cos 30 ◦ cos(90 − a) + sin 30 ◦ sin(90 − a) cos(360 − A)