Part 4 – Full procedure for establishing an astronomical position line.
This post brings together all of the information from parts 1, 2 and 3 of this series to demonstrate the full procedure for establishing an astronomical position line.
.Links: Astro Navigation In A Nutshell Part One
Astro Navigation In A Nutshell Part Two
Astro Navigation In A Nutshell Part Three
Please note. There is not sufficient scope in this post to fully explain this topic; however, there are indepth expositions in my books ‘Astro Navigation Demystified’ and ‘Celestial Navigation – The Ultimate Course’.
Demonstration of the procedure:
Scenario:
Date: 18 July
D.R. Position at Zone Time: 16^{h} 44^{m}: 52^{o }N 21^{o }43.1’W.
Time Zone +1
Deck Watch Time (DWT): 17^{h} 50^{m} 28^{s}.
Deck Watch Error (DWE) 40^{s} fast (40^{s})
Body observed: Sun lower limb.
Sextant Altitude at true position: 32^{o} 10.’4 = 32^{o}.173
Compass Bearing at true position: 261^{o } (for rough check on azimuth)
Index error: +0’.54. Ht. of eye: 8m.
Temperature: 28^{o}C. Pressure: 991mb.
Step 1. Note Lat and Long of DR Position.  
Lat: 52^{o} N  
Long: 21^{o }43.1’W  
Step 2. Calculate PZ. (90 – Lat).
PZ = 90^{o} – 52^{o} = 38^{o} 

Step 3. Calculate Greenwich Date at time of observation.  
Date: 18 July  
Zone time: 16^{h} 44^{m}  
Zone correction: +1^{h}  
Universal Time (GMT): 17^{h} 44^{m}  
Deck watch time: 17^{h} 50^{m} 28^{s}  
Deck watch error: 40^{s}  
Greenwich date: 18^{d} 17^{h} 49^{m} 48^{s }July  
Step 4. Calculate Greenwich Hour Angle and Declination.  
Date: 18 July  
GHA Dec  
UT 17^{h} 73^{o} 26’.1 N20^{o} 54’.7 (d:0’.5 decreasing)  
Inc. 49^{m} 48^{s}: 12^{o} 27’.0 0’.4  
85^{o} 53’.1 N20^{o} 54’.3  
= 85^{o}.885 = N20^{o}.9  
Step 5. Determine if Lat and Dec are ‘Same’ or ‘Contrary’.  
Lat = N
Dec = N Therefore Lat and Dec are same. 

Step 6. Calculate PX
(Lat and Dec Same therefore PX = 90 – Dec). PX = 90^{o} – Dec. = 90^{o} – 20^{o}.9 = 69^{o}.1 

Step 7. Calculate the Local Hour Angle (LHA). (Longitude combined with GHA should equal LHA as a whole number of degrees).  
DR Long: 21^{o }43.1’W  
GHA: 85^{o}.885  
DR Long: 21^{o}.718 West ()  
LHA: 64^{o} .167  
Step 8. Determine Angle ZPX.
ZPX = LHA = 64^{o} .167 

Step 9. Calculate True Altitude at True Position (Observed altitude corrected for IE, Dip, Parallax and Refraction).  
Sextant Altitude = 32^{o} 10′.40
Index error (IE) = +0′.54 Observed Altitude = 32^{o} 10′.94 Dip (ht. 8m.) = 5′.00 (table 6a) Apparent Altitude = 32^{o} 05′.94 Altitude correction = +14′.50 (table 6d) Added refraction (28^{o}/991mb) = +0′.10 (table 6c) True Altitude = 32^{o} 20′.54 = 32^{o}.34 

Step 10. Calculate Zenith Distance at True Pos. (90^{o} – Altitude).  
Zenith Dist = 90^{o} – 32^{o}.34 = 57^{o}.66 = 3459.6′  
Step 11. Calculate Zenith Distance at DR Position. (ZX).  
Lat. = 52^{o}N
Declination = N20^{o}.9 (From Step 4) Lat and Dec Same (From Step 5) ZPX = 64^{o} .167 ^{ }(From Step 7) PZ = 38^{o} (From Step 2) PX = 69^{o}.1 (From Step 5) 

Cos (ZX) = [Cos(PZ) x Cos(PX)] + [Sin(PZ) x Sin(PX) x Cos(ZPX)]
= [Cos(38) x Cos(69.1)] + [Sin(38) x Sin(69.1) x Cos(64.167)] 

= 0.53  
ZX = Cos^{1} (0.53) = 57^{o}.99
Zenith Distance at DR position = 57^{o}.99 = 3479′.4 

Step 12. Calculate Azimuth at DR Position (PZX)  
Cos PZX = Cos(PX) – [Cos(ZX) x Cos(PZ]
[Sin(ZX) x Sin(PZ)] 

Cos PZX = Cos(69.1) – [Cos(57.99) x Cos(38]
[Sin(57.99) x Sin(38)] 

= 0.119  
PZX = Cos^{1}(0.119) = 96^{o}.8 ≈ 97^{o}
Azimuth at DR position = 97^{o} 

Step 13. Convert azimuth angle to true bearing (ZN):  


DR Lat. = 52^{o}.0N  
Azimuth (Z) = 97^{o} (from step 12)  
LHA = 64^{o} (from step 7)  
Therefore ZN = 360^{o} – 97^{o} = 263^{o}
Therefore true bearing of body at DR position = 263^{o} Compass Bearing at true position: 261^{o }(for rough check on azimuth) 

Step. 14. Note observed compass bearing at true position and compare with true bearing at DR position for rough check.  
Observed compass Bearing at true position: 261^{o}
True Bearing of body at DR position = 263^{o} 



Step 16. Plot the position line. (Reminder: Plot intercept from DR position along azimuth line).
Latitude: 52^{o }00’N Longitude: 21^{o} 53’W Intercept = 19.8 n.m. towards 263^{o }(From step 15) Note. Drawing not drawn to scale. 
INSERT PLOT
This topic is explained in far greater depth in my books ‘Astro Navigation Demystified’ and ‘Celestial Navigation – The Ultimate Course’.
Many thanks to Jeremy Parker for his help with this post.
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