UNIT 6 –  Calculating zenith distance and azimuth at assumed position.

We can use sight reduction tables to calculate the zenith distance and azimuth at the assumed position or else we can use the traditional method of making the calculations by spherical trigonometry.  The sight reduction method, as well as being less accurate, is quite involved and requires the provision of expensive sets of tables.  Alternatively, spherical trigonometry provides a more accurate and inexpensive method which requires only a calculator and little mathematical ability.

In this unit, we will focus on the spherical trigonometry method but an explanation of the sight reduction method can be found in ‘Astro Navigation Demystified’, the parent to this book.

Calculating zenith distance and azimuth by spherical Trigonometry. At first sight, the spherical trigonometry method might seem quite daunting and difficult but with the knowledge of just two formulas and with a little practice of the procedures explained below, it will be found to be quick and easy.

Essentially, we need to know only two formulas which are are explained below.

 To Calculate Zenith Distance (ZX).  The formula for calculating side ZX is:

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

To Calculate Azimuth (PZX)  The formula for calculating angle PZX is:

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

The use of these formulas will become clear if we study the example below which shows how the zenith distance and azimuth of a celestial body can be calculated by the use of the above formulas.

Example.   Assume that we have measured the altitude and azimuth of the star Alioth from our true position and have calculated that the intercept there is 1618.54 n.m. and that we can now wish to calculate what the intercept would be from our assumed position.

Celestial body:  The star Alioth

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

Data for Alioth taken from Nautical Almanac daily pages:

SHA: 166    Declination: 56^oN   GHA Aries:  250

Note.  Refer to the PZX diagram in unit 4 to identify PZX. PX, PZ and ZX in the calculations below.

 Step 1.  Calculate LHA

SHA Alioth +  GHA Aries = 166+  250 = 416

Apply Long :         416 -45 =  371

If greater that 360 then subtract 360:     371-360  = 11

∴ LHA  = 11

∴ ZPX = LHA = 11^o

Step 2. Calculate PZ/PX

PZ = 90^o – Lat.  = 90^o – 30^o = 60^o

PX = 90^o – Dec. = 90^o – 56^o = 34^o

Step 3.  Calculate Zenith Distance (ZX). Use the following formula to calculate ZX:

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.13   Therefore zenith distance at assumed position =  27^o.13

 = 1627.8′    = 1627.8 nm

 Step 4.  Calculate Altitude.

Altitude  = 90^o – ZX

= 90^o – 27^o.13          = 62^o.83

Step 5.  Calculate Azimuth (PZX)

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(34) – [Cos(27.13) . Cos(60]/ [Sin(27.13) . Sin(60)]

= 0.829 – [ 0.89 x 0.5] / 0.456 x 0.866

=  0.829 – 0.445 / 0.394

=  0.384 /  0.394

∴ Cos(PZX) = 0.975

∴  PZX   =  Cos^-1(0.975)  = 12.8  ≈ 13

Note. Azimuth is west if LHA is less than 180 ^o. Azimuth is east if LHA is greater than 180^o.

Therefore, azimuth is N13^oW.  In terms of bearing this is 347^o (360^o – Az.)

 Step 6.  Calculate Intercept.

Zenith Distance at assumed position  = 1627.8 nm

Zenith Distance at true position  = 1618.54 nm.

Therefore intercept = 1627.8 – 1618.54  = 9.26 nm.

Note. Since true position is closest to the GP, the intercept must be 9.26 nm away from the assumed position towards the azimuth (i.e. 347 ^o).

Therefore,  Alioth intercept = 1.46 nm from the assumed position towards 347 ^o.

A thorough treatment of this topic can be found in  ‘Astro Navigation Demystified‘

Where to buy books of the Astro Navigation Demystified series:

Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

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