Part 3 – Calculating Altitude and Azimuth at the DR Position by Spherical Trigonometry.
Links: Astro Navigation In A Nutshell Part One
Astro Navigation In A Nutshell Part Two
Astro Navigation In A Nutshell Part Four
There are several ways of calculating the azimuth and altitude at the DR 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,
X is the position of a celestial body in the Celestial Sphere and U is its Geographical Position on the surface of the Earth.
B is the position of an observer on the surface of the Earth and Z is the zenith of that position.
P is the celestial North Pole and N is the Earth’s North Pole.
PZ is the angular distance from the Celestial North Pole to the zenith of the observer and is equal to 90o – Lat.
PX is the angular distance from the Celestial North Pole to the celestial body and is equal to 90o – Dec.
ZX is the Zenith Distance and is equal to 90o – altitude.
Therefore, altitude is equal to 90o – 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 = 90o – Dec.
PZ = 90o – Lat.
ZX = 90o – Alt.
Alt = 90o – 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.
Example of the use of spherical trigonometry to calculate the azimuth and altitude of a celestial body.
Note. Traditionally, the ‘half-haversine’ formula was used for this task but this formula does not lend itself well to solution by electronic calculator; therefore, the following solutions which involve the cosine formula have been developed.
Example: Star Sight.
Scenario: Greenwich date: 30 June 18hrs 05 mins 33 secs
DR Position: Lat. 30oN Long. 45oW
Selected body: Alioth
SHA: 166
Declination: 56oN
GHA Aries: 250
- 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)
- Calculate PZ/PX/ZPX
PZ = 90o – 30o = 60o ∴PZ = 60o
PX= 90o – 56o = 34o ∴PX = 34o
ZPX = LHA = 11o west
- 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(60o) . Cos(34o)] + [Sin(60o) . Sin(34o) . Cos(11o)]
= [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) = 27o
- Calculate Altitude.
Altitude = 90o – ZX = 90o – 27o = 63o
- 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 = N12oW (since LHA is west)
In terms of bearing, the azimuth is 348o.
- Summarize results.
LHA = 11o west
Declination = 56oN
Azimuth at DR position = N12oW
Altitude at DR position = 63o
Please Note. This topic is explained in far greater depth in my books ‘Astro Navigation Demystified’ and ‘Celestial Navigation – The Ultimate Course’.
The next post in this series will give a full example of calculating an astronomical position line.
Where to buy books of the Astro Navigation Demystified series:
Celestial Navigation at Amazon.com
Celestial Navigation at Amazon.uk
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
web: http://www.astronavigationdemystified.com
e: astrodemystified@outlook.com
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