To fully understand how the azimuth angle and the altitude of a celestial body help us to establish our position, we need to consider them in relation to the celestial sphere.

Consider the diagram below:

The celestial sphere is drawn in the plane of the observer’s meridian with the observer’s zenith (Z) at the top.

Point O represents both the observer and the Earth.

The arc PZQSP’ represents the observer’s celestial meridian.

The arc NAS is the celestial horizon and QRQ’ represents the celestial equator.

ZXAZ’ is a vertical circle running through the position of the celestial body (X). (A vertical circle is a great circle that passes through the observer’s zenith and is perpendicular to the celestial horizon).

** ****The Azimuth Angle** is the angle PZX (that is, the angle between the observer’s celestial meridian and the vertical circle through the celestial body).

**The Altitude **is the angle AOX (that is the angle from the celestial horizon to the celestial body measured along the vertical circle).

**The Zenith.** Point Z in the diagram represents the observer’s zenith which is an imaginary point on the celestial sphere directly above the observer. It is the point where a straight line drawn from the geocentric centre of the Earth, through the observer’s position and onwards, intersects with the celestial sphere.

** ****The Zenith Distance. **The Zenith Distance is the angular distance from the zenith to the celestial body measured from the Earth’s centre. In the diagram above, it is the angular distance ZX which is subtended by 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 angles AOX and XOZ are complementary angles. From this we can deduce that:

**Zenith Distance = 90 ^{o} – Altitude **and

**Altitude = 90**

^{o}– Zenith Distance**Calculating the Zenith Distance. **

Consider the next diagram.

The diagram shows that the angular distance AU on the Earth’s surface is equal to the angular distance ZX in the spherical triangle PZX.

X represents the position of a celestial body on the celestial sphere,

Z represents a point on the sphere which coincides with the zenith of the DR position (A),

P represents the projection of the North Pole onto the celestial sphere,

PX = NU = (90^{o} – the declination of the body),

PZ = NA = (90^{o} – the latitude of the DR position),

ZX = AU = (90^{o} – the altitude of the body).

We can see that the triangle NAU on the Earth’s surface can be solved, in effect, by solving the triangle PZX in the celestial sphere.

**Local Hour Angle (LHA)**

In the PZX triangle diagram, 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 Angle. **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 the circle.

The circle WANESW 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 angle.

When the LHA (ZPX) is less than 180** ^{o},** the celestial body lies to the west of the observer’s meridian and when the LHA is greater than 180

**it lies to the east. (Remember LHA is measured westwards from the observer’s meridian from 0**

^{o}**to 360**

^{o}**).**

^{o}It follows that if the celestial body is to the west of the observer’s meridian, the azimuth angle must be west and when to the east, the azimuth angle must be east.

So we have the rule:

**LHA 0 ^{o} to 180^{o }= Azimuth Angle West**

**LHA 180 ^{o }to 360^{o} = Azimuth Angle East**

In Astro navigation. It can be seen that by measuring the altitude of a celestial body, we are able to easily calculate the zenith distance which will give us the distance in nautical miles from the observer’s position to the geographical position of the body. The azimuth angle will give us the direction of the GP from the observer’s position. This explains why calculating the altitude and azimuth angle are the first steps in determining our position in celestial navigation.

**Relationship Between Azimuth Angle and Azimuth.**

**Azimuth** is a specific type of bearing which measures the direction of an object in relation to true north, in the horizontal plane, clockwise from 0^{o} to 360^{o}.

**Azimuth Angle.** In astro navigation, when we calculate the azimuth of a celestial body, what we actually calculate is the azimuth angle. Azimuth angle is measured from 0^{o} to 180^{o} either westwards or eastwards from either north or south. If the observer is in the northern hemisphere, the azimuth is measured from north and if in the southern hemisphere, it is measured from south.

For example, if the true azimuth of an object is 225^{o}, the azimuth angle for an observer in the northern hemisphere will be N135^{o}W but for an observer in the southern hemisphere, it will be S045^{o}W.

** ****Summary Of The Discussions Above. ** The relationships discussed above illustrate the importance of altitude and azimuth angle in position finding at sea. It can be seen that from the altitude of a celestial body, we are able to easily calculate the zenith distance which will give us the distance in nautical miles from the observer’s position to the geographical position of the body. From the calculated azimuth angle we can find the true azimuth and this will give us the direction of the GP from the observer’s position. This explains why determining the altitude and azimuth angle are the first steps in determining our position in astro navigation.

In the next post, we will discuss sight reduction which is the process of reducing the data gathered from an observation of a celestial body down to the information needed to establish an astronomical position line.

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Applying Mathematics to Astro Navigation

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