Positional Calculations
Using the correctly labelled spherical triangle shown below, we can make detailed calculations of the position of a star, using the spherical trigonometry rules we have learnt.
Using the correctly labelled spherical triangle shown below, we can make detailed calculations of the position of a star, using the spherical trigonometry rules we have learnt.
Figure 4: The spherical triangle for positional calculations.
We can calculate the altitude and azimuth of the star shown above, by applying the cosine rule.
Applying the cosine rule to the triangle above we get two equations:
1

cos(90alt) = cos(90Φ) cos(90δ) + sin(90Φ) sin(90δ) cos HA


2

cos(90δ) = cos(90Φ) cos(90alt) + sin(90Φ) sin(90alt) cos(360Az)


These can be simplified enormously using the trigonometric identities
 cos(360x) = cos 360 cos x + sin 360 sin x = cos x
 cos(90x) = cos 90 cos x + sin 90 sin x = sin x
 sin(90x) = sin 90 cos x  cos 90 sin x = cos x
1

sin alt = sin Φ sin δ + cos Φ cos δ cos HA


2

sin δ = sin Φ sin alt + cos Φ cos alt cos Az


These equations, together with the material you have learnt in the previous lectures, are all you need to predict the position of a celestial object from any point on Earth, at any time of day or year. The problems in this session and the next session will give you plenty of practise doing just that. There is one special case which is worth mentioning; the rise and set times of stars.