Astronomy Principles and Practice Fourth Edition.pdf

Transformation of one coordinate system into another 69
Figure 8.9. The conversion of azimuth and altitude to hour angle and declination.
The required celestial sphere is shown in figure 8.9 where X is the body’s position.
In spherical triangle PZX, we see that we require to find arc PX and angle ZPX. We calculate
PX first of all, using the cosine formula because we know two sides PZ, ZX and the included angle
PZX.
Hence, we may write,
cos PX = cos PZ cos ZX + sin PZ sin ZXcos PZX
or
sin δ = sin φ sin a + cos φ cos a cos A.
This equation enables δ to be calculated.
A second application of the cosine formula gives
or
cos ZX = cos PZ cos PX + sin PZ sin PX cos ZPX
sin a = sin φ sin δ + cos φ cos δ cos H.
Re-arranging, we obtain
sin a − sin φ sin δ
cos H =
cos φ cos δ
giving H ,sinceδ is now known.
Alternatively, using the four-parts formula with ZX, PZX, PZ and ZPX, we obtain
or
cos PZ cos PZX = sin PZ cot ZX − sin PZX cot ZPX
sin φ cos A = cos φ tan a + sin A cot H
giving
sin A
tan H =
sin φ cos A − cos φ tan a .
We consider the problem in reverse by means of a numerical example.