Calculating Azimuth And Altitude At The Assumed Position By Spherical Trigonometry.

There are several ways of calculating the azimuth and altitude at the assumed position; these include the use of sight reduction methods and software solutions. However, the traditional method is by the use of spherical trigonometry which is demonstrated below.

 

 

 

 

 

 

 

 

 

 

In the diagram above,

PZ is the angular distance from the Celestial North Pole to the zenith of the observer and is equal to 90^o – Lat.

PX is the angular distance from the Celestial North Pole to the celestial body and is equal to 90^o – Dec.

ZX is the Zenith Distance and is equal to 90^o – altitude.

Therefore, altitude is equal to 90^o – ZX

The angle ZPX is equal to the Local Hour Angle of the Celestial Body with respect to the observer’s meridian.

The angle PZX is the azimuth of the body with respect to the observer’s meridian.

Summary.

PX = 90^o – Dec.

PZ = 90^o – Lat.

ZX = 90^o – Alt.

Alt = 90^o – ZX

<PZX = Azimuth.

<ZPX = Hour angle.

In order to calculate the azimuth and altitude of a celestial body we must solve the triangle PZX in the diagram above.  Specifically, we must calculate the angular distance of side ZX so that we can find the altitude and we must calculate the angle PZX so that we can find the azimuth.

However, because the triangle PZX is on the surface of an imaginary sphere, we cannot solve this triangle by the use of ‘straight line trigonometry’; instead we must resort to the use of ‘spherical trigonometry’ which is explained here.

Examples of the use of spherical trigonometry to calculate the azimuth and altitude of celestial bodies.

Note.  Traditionally, the ‘half-haversine’ formula was used for this task but this formula does not lend itself well to solution by electronic calulator; therefore, the following solutions involve the cosine formula.

Example 1.  Star Sight.

Scenario:     Greenwich date: 30 June 18hrs 05 mins  33 secs

Assumed Position:  Lat. 30^oN    Long. 45^oW

Selected body: Alioth

SHA: 166

Declination: 56^oN

GHA Aries:  250

Step 1.  Calculate LHA

SHA Alioth:   166

Add GHA Aries: 166 + 250 = 416

Subtract Long(W) = 416 – 45 = 371

Subtract 360 = 11

Therefore,  LHA  =  11W

(all results in degrees)

Step 2. Calculate PZ/PX/ZPX

PZ = 90^o – 30^o = 60^o      ∴PZ = 60^o

PX=  90^o – 56^o = 34^o     ∴PX = 34^o

ZPX = LHA = 11^o west

Step 3.  Calculate Zenith Distance  (ZX).

As explained here, the formula for calculating side ZX is:

Cos (ZX) =  [Cos(PZ) . Cos(PX)] + [Sin(PZ) . Sin(PX) . Cos(ZPX)]

∴To calculate zenith distance of Alioth:

Cos (ZX) =  [Cos(PZ) . Cos(PX)] + [Sin(PZ) . Sin(PX) . Cos(ZPX)]

=  [Cos(60^o) . Cos(34^o)] + [Sin(60^o) . Sin(34^o) . Cos(11^o)]

=  [0.5 x 0.829} + [0.866 x 0.559 x 0.982]

=  0.415 + 0.475

Cos (ZX) =  0.89

∴ ZX      =  Cos^-1 (0.89)  =  27^o

Step 4.  Calculate Altitude.

Altitude  = 90^o – ZX   = 90^o – 27^o  = 63^o

Step 5.  Calculate Azimuth (PZX)

As explained here the formula for calculating angle PZX is:

Cos PZX = [Cos(PX) – Cos(ZX) . Cos(PZ)]  /  [Sin(ZX) . Sin(PZ)]

∴To calculate azimuth of Alioth:

Cos PZX   = [Cos(34) – Cos(27) . Cos(60)]  /  [Sin(27) . Sin(60)]

= [0.829 – (0.89 x 0.5)]  /  [0.454 x 0.866)]

=  (0.829 – 0.445) / 0.393

=  0.384   / 0.393   = 0.977

Cos(PZX) = 0.977

∴  PZX   =  Cos^-1(0.977)  = 12.31

∴  Azimuth = N12^oW (since LHA is west)

In terms of bearing, the azimuth is 348^o.

Step 6.  Summarize results.

LHA =  11^o west

Declination = 56^oN

Azimuth at assumed position = N12^oW

Altitude at assumed position = 63^o