Azimuth and Altitude and the Theory of Astro Navigation
So far, we have considered azimuth and altitude from a position on the surface of the Earth. To fully understand how these phenomena relate to the Local Hour Angle and the declination of a celestial body and hence, how they help us to establish our position, we need to consider them in relation to the celestial sphere.
In the following diagram, the celestial sphere is drawn in the plane of the observer’s meridian with the observer’s zenith (Z) at the top.
The Azimuth is the angle PZX (that is, the angle between the observer’s celestial meridian and the vertical circle through the celestial body). It is measured from 0o to 180o east or west from the observer’s meridian and depending on whether the observer is in the northern hemisphere or the southern hemisphere, it is named north or south.
The Altitude is the angle AOX, that is the angle from the celestial horizon to the celestial body measured along a vertical circle.
The Zenith Distance is the angular distance ZX measured along a vertical circle from the zenith to the celestial body; that is the angle XOZ.
Relationship between Altitude and Zenith Distance. Since the celestial meridian is a vertical circle and is therefore, perpendicular to the celestial horizon, it follows that angle AOZ is a right angle and that angles AOX and XOZ are complementary angles. From this we can deduce that:
Zenith Distance = 90o – Altitude and Altitude = 90o – Zenith Distance
Local Hour Angle (LHA). LHA is the angle ZPX; that is the angle between the observer’s celestial meridian and the meridian of the celestial body.
Relationship between LHA and Azimuth. Consider the next diagram:
This diagram is drawn in the plane of the celestial horizon. Imagine that you are looking down on the celestial sphere from a position directly above the observer’s zenith which is in the centre of a circle that represents the celestial horizon.
NZS represents the observer’s celestial meridian.
WQE represents the celestial equator,
P is the celestial pole,
X is the position of the celestial body,
PXR represents part of the meridian of the celestial body which cuts the Equator at R.
ZPX is the LHA.
PZX is the Azimuth.
When the LHA is less than 180o, the celestial body lies to the west of the observer’s meridian and when it is greater than 180o it lies to the east. (Remember LHA is measured westwards from the observer’s meridian).
It follows that if the celestial body is to the west of the observer’s meridian, the azimuth must be west and when to the east, the azimuth must be east.
So we have the rule:
LHA = 0o – 180o : Azimuth West
LHA = 180o – 360o : Azimuth East
The Theory of Astro navigation. The relationships discussed above illustrate the importance of azimuth and altitude in position finding at sea. The theory of astro navigation depends on the ability to solve the spherical triangle PZX and the azimuth and altitude give us the essential data we need to do this. With this data we are able to find the LHA, declination and zenith distance of a celestial body and armed with this information, we are able to establish our position on the Earth’s surface.
Links: Azimuth and Altitude part 1 Azimuth and Altitude part 3.
It is possible only to give a brief outline of this topic here but an in-depth treatment is given in my book Astro Navigation Demystified.
Astro Navigation Demystified 
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