Astro Navigation In A Nutshell Part Four

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 in-depth expositions in my books ‘Astro Navigation Demystified’ and ‘Celestial Navigation – The Ultimate Course’.

Demonstration of the procedure:


Date: 18 July

D.R. Position at Zone Time: 16h 44m:  52N   21o  43.1’W.

Time Zone +1

Deck Watch Time (DWT): 17h 50m 28s.

Deck Watch Error (DWE) 40s fast (-40s)

Body observed:  Sun lower limb.

Sextant Altitude at true position:  32o 10.’4  =  32o.173

Compass Bearing at true position: 261o  (for rough check on azimuth)

Index error:  +0’.54.     Ht. of eye:  8m.

Temperature:  28oC.  Pressure:  991mb.

Step 1. Note Lat and Long of DR Position.
Lat:   52o N
Long: 21o  43.1’W
Step 2. Calculate PZ. (90 – Lat).

PZ = 90o – 52o  =  38o

Step 3.  Calculate Greenwich Date at time of observation.
Date:  18 July
Zone time:                          16h 44m
Zone correction:                +1h
Universal Time (GMT):     17h 44m
Deck watch time:               17h 50m 28s
Deck watch error:                            -40s
Greenwich date:          18d 17h 49m 48s July
Step 4. Calculate Greenwich Hour Angle and Declination.
Date: 18 July
                             GHA                Dec
UT 17h                73o 26’.1          N20o 54’.7 (d:0’.5 decreasing)
Inc. 49m 48s:    12o 27’.0                   -0’.4
                             85o 53’.1          N20o 54’.3
                         = 85o.885        = N20o.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 = 90o – Dec. =  90o – 20o.9  =  69o.1

Step 7. Calculate the Local Hour Angle (LHA). (Longitude combined with GHA should equal LHA as a whole number of degrees).
DR Long: 2143.1’W
GHA:                     85o.885
DR Long:              21o.718 West (-)
LHA:                      64o .167
Step 8. Determine Angle ZPX.

ZPX = LHA = 64o .167

Step 9. Calculate True Altitude at True Position (Observed altitude corrected for IE, Dip, Parallax and Refraction).
Sextant Altitude     =              32o 10′.40

Index error (IE)  =                        +0′.54

Observed Altitude =                32o 10′.94

Dip (ht. 8m.)         =                        -5′.00   (table 6a)

Apparent Altitude =                 32o 05′.94

Altitude correction  =                   +14′.50   (table 6d)

Added refraction (28o/991mb) =    +0′.10   (table 6c)

True Altitude =                        32o 20′.54  = 32o.34

Step 10. Calculate Zenith Distance at True Pos. (90o – Altitude).
Zenith Dist = 90o – 32o.34  = 57o.66  = 3459.6′
Step 11. Calculate Zenith Distance at DR Position. (ZX).
Lat. = 52oN

Declination =  N20o.9   (From Step 4)

Lat and Dec Same (From Step 5)

ZPX = 64o .167     (From Step 7)

PZ = 38o    (From Step 2)

PX = 69o.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)  =  57o.99

Zenith Distance at DR position =  57o.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)  =  96o.8  ≈ 97o

Azimuth at DR position = 97o

Step 13. Convert azimuth angle to true bearing (ZN):
Rules for converting Azimuth (PZX) to True Bearing (Zn)
                Lat. North                              Lat. South
LHA>180o Zn = Z                Zn =  180o – Z
LHA<180o   Zn = 360o-Z      Zn =  180o + Z   
DR Lat.  =   52o.0N
Azimuth  (Z)   =   97o       (from step 12)
LHA                =   64o       (from step 7)
Therefore ZN = 360o – 97o  = 263o

Therefore true bearing of body at DR position = 263o

Compass Bearing at true position: 261(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: 261o

True Bearing of body at DR position = 263o

Step 15.  Calculate intercept.

Reminder: Subtract the ZD at the true position (a) from the ZD at the DR position (b).

  • If the result is positive, the intercept is towards the azimuth.
  • If the result is negative, the intercept is from the azimuth.
a. Zen. Dist. at DR Pos.        57o.99                                     (from step 11)
b. Zen. Dist. at True Pos.     57o.66                          (from step 10)
Intercept = a – b =                +0o.33                                 
Convert to minutes:               +19.8′
Azimuth:                                     263o
Intercept:                             19.8 nautical miles to 263o
Step 16.  Plot the position line.  (Reminder: Plot intercept from DR position along azimuth line).

Latitude:   52o 00’N

Longitude:  21o 53’W

Intercept  = 19.8 n.m. towards 263(From step 15)

Note. Drawing not drawn to scale.



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

Celestial Navigation at

Astro Navigation Demystified at

Astro Navigation Demystified at

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at

Astronomy for Astro Navigation at