Celestial Navigation Calculations

An imaginary sphere surrounding the Earth is used for calculations. This is known as the Celestial Sphere. Declination corresponds to Latitude and Hour Angles to Longitudes.

Solving a sight uses spherical trigonometry. The triangle is known as the PZX triangle.

P is the Pole, Z, the Observer’s Zenith and X is the body.

Altitude Vs Zenith Distance

An Altitude is a terrestrial measurement while calculations are carried out on the Celestial sphere.

The reason that the calculations do not allow for the shape of the Earth is because the calculations are performed on the Celestial Sphere. Gravity ensures that the horizon (and thus Altitude,) corresponds with the equivalents on the celestial Sphere.

Zenith Distance is the correct term and helps avoid confusion. 

Parts of the PZX Triangle

Q1 to Q2 is the Equator and the Vertical line from P to G is the Greenwich Meridian (0° GHA and Longitude.)

The compliment of an angle is 90° - the angle. This is also true of the sides in a spherical triangle.

The distance from P to Q1 is 90 ° therefore PX = 90° - Declination or

PX = co-Declination.

Compliments enable formulae to be simplified because the Sine of an angle equals the Co-Sine of the compliment of that angle. This also applies to Tangents and Cotangents, Secants and Cosecants.

Sin(60°) = Co-Sine(90° - 60°) = Cosine(30°)

The compliment of a compliment is the same as the original angle;

Cos(co-30°) = Sin(90° - (90° - 30°)) = Sin(30°)

Simplifying the diagram and adding some labels:-

SPHERICAL TRIGONOMETRY  FORMULAE

To Calculate a Side - The Cosine Formula

Cos(a) = Cos(b) x Cos(c) + Sin(b) x Sin(c) x Cos(A)

Applying this to the PZX triangle we get:-

Cos(Zenith Distance) = Cos(co-Lat) x Cos(co-Dec) + Sin(co-Lat) x Sin(Co-Dec)

         x Cos(LHA)

Because Sin(co-A) = Cos(A) and Cos(co-A) = Sin(A)

Cos(Zenith Distance) = Sin(Lat) x Sin(Dec) + Cos(Lat) x Cos(Dec) x Cos(LHA)

If Altitude is preferred; Zenith Distance = co-Altitude thus

Sin(Altitude) = Sin(Lat) x Sin(Dec) + Cos(Lat) x Cos(Dec) x Cos(LHA)

For an Angle

Tan(C) = Sin(A)/ [Sin(b)/ Tan(c) – Cos(b) x Cos(A)]

Inserting terms from the PZX triangle this becomes

Tan(Az) = Sin(LHA)/ [Sin(co-Lat)/ Tan(co-Dec) – Cos(co-Lat) x Cos(LHA)]

Or

Tan(Az) = Sin(LHA)/ (Cos(Lat) x Tan(Dec) – Sin(Lat) x Cos(LHA)) 

The Spherical Sine Formulae

Sin(a)/ Sin(A) = Sin(b)/ Sin(B) = Sin(c)/ Sin(C)

Napier’s Rules

These can be used when one of the sides or angles is 90°.

If angle A is 90° then a diagram is drawn with A above the circle and the sectors filled with the adjoining sides.

Notice that the three sectors of the lower half are marked “co-” In other words the compliment of these angles is used.

The two formulae are;

Sin(Mid Part) = Tan (Adjacent Parts)

e.g. Sin(c) = Tan(co-B) x Tan(b)

or Sin(c) =Cot(B) x Tan(b)

And

Sin (Mid Part) = Cos(Opposite Parts)

e.g. Sin(c) = Cos(co-a) x Cos(co-C)

or Sin(c) = Sin(a) x Sin(C)

Example using Napier’s  Rules

Assume that the True Altitude is 0° therefore the Zenith Distance is 90°.

Co-Dec is the mid-part therefore the two opposites are Azimuth and co-Lat.

Sin (Mid Part) = Cos(Opposites)

Sin(co-Dec) = Cos(Azimuth) x Cos(co-Lat)

Cos(Dec) = Cos(Azimuth) x Sin(Lat)

Cos(Azimuth) = Sin(Dec)/ Sin(Lat)

The Amplitude of a body is measured from East or West rather than North. In other words Amplitude = 90° - Azimuth = co-Azimuth.

Sin(Amplitude) = Sin(Dec)/ Sin(Lat)

This gives the formula that many readers will be familiar with of

Sin(Amplitude) = Sin(Dec) x Sec(Lat)

END OF TUTORIAL