Meridian Passage Long Method

Links:  Understanding Meridian Passage, Equation of Time,   Meridian Passage Short Method,  Short Distance Sailing Formulae,  What is the point of meridian passage?         Zone Time,    Local Hour Angle and Greenwich Hour Angle,   Converting GMT to GHA ,  Altitude Corrections

Outline Method

If a vessel is under way, its position at the time of meridian passage will not be known beforehand.  However, it is necessary to calculate the approximate time of meridian passage before it occurs so that the altitude measurements can begin in time.  This is achieved by calculating the time of meridian passage at the present D.R. position a few hours before noon and then estimating what the vessel’s new position will be at that time.  The approximate time of meridian passage can then be recalculated for the new position.

To calculate the new position, we use the Short distance sailing formulas which are fully explained in my book ‘Celestial Navigation – The Ultimate Course’. Short distance sailing is a term which is applied to sailing along a rhumb-line for distances less than 600 nautical miles.  Of course, we could simply extend our dead reckoning track for the required distance on a mercator chart.  However, mercator charts, which do not allow for the curvature of the Earth, are not very accurate for plotting tracks which cross high latitudes or which are predominately east-west.  Therefore, for the sake of accuracy, the meridian passage long method employs the short distance sailing formulas for calculating the new position.

Greenwich Mean Versus Universal Time.The nautical almanac lists the times of events such as mer. pas. in terms of universal time (UT); however, since we are dealing with the time that the Sun crosses the Greenwich Meridian, it is more helpful to refer to the time in terms of Greenwich Mean Time (GMT) instead. (Remember the terms Greenwich Mean Time and Universal Time are generally considered to be synonymous).

Refraction.  Since refraction is negligible when the Sun is at its zenith, additional altitude correction for non standard conditions is not necessary when calculating the true altitude at meridian passage.

 Relationship between Altitude and Zenith Distance.

Zenith Distance = 90o – Altitude

and Altitude = 90o – Zenith Distance

 Rules For Calculating Meridian Passage.

  • Latitude and declination same names but latitude greater than declination:

LAT  =  DEC + (90o – ALT)

  • Latitude and declination same names but declination greater than latitude:

LAT  =  DEC – (90o – ALT)

  • Latitude and declination opposite names:

LAT  =  (90o – ALT) – DEC

Nine-Step Proforma. This easy to the follow nine-step proforma has been devised provide a step by step method for calculating a vessel’s position by the Meridian Passage Long Method.  The method will become clear as you work your way through the guided examples below.  A blank copy of the proforma template can be found in appendix 10.

Outline of the 9-step Proforma.

Pre Planning.  At least one hour before noon, preferably two, calculate your vessel’s first DR position.  Busy navigators in ships travelling at speed will need to do this early so that they will have time to calculate what the ship’s  new position will be at the time of meridian passage.

  • Step 1. Using the nautical almanac daily page, find the time of meridian passage for the first DR position
  • Step 2.  Convert the time of meridian  passage from GMT to zone time.  (Remember, zone time will not correspond to the Sun’s apparent time so although meridian passage occurs at noon apparent time, the zone time is likely to be several minutes either side of this).
  • Step 3.  Calculate what the new position will be at the time of meridian passage as calculated at step 2.  (The short distance sailing formulas which are explained in note 15 should be used to calculate the new position).
  • Step 4.  Calculate the time of meridian passage at the new position.
  • Step 5.  Calculate the declination at the new time of local meridian passage.
  • Step 6.  Calculate the Meridian Altitude and note the deck-watch time.
  • Step 7.  Calculate the vessel’s latitude from the values of the meridian altitude calculated at step 6 and the declination calculated at step 5.
  • Step 8.  Calculate the vessel’s longitude from the deck watch time noted at step 6.
  • Step 9.  Summarise position at zone time of meridian altitude.

Example:    Use the Meridian Passage Long method to calculate the position of the vessel in the scenario below by following the nine-step proforma explained above.

Scenario.

Date: 17 December

(zone -9)

Mer. Pas.:  11h  56m

DR Position at 1000 (zone time): 41o 15’.0S. 134o 52’.0E.

Course: 030o  Speed: 15 knots.

Sextant Altitude at Mer. Pas. (Meridian Altitude):  72o 18’.2

Index error: +2’.1.

Height of eye: 12m

Deck Watch Time at Meridian Altitude:  02h 59m 10s

DWE 5s fast

Solution:

Pre Planning.

Date: 17 Dec.  Zone Time: 1000(-9).

DR Pos: 41o 15’.0S. 134o52’.0E.

Course: 030o  Speed: 15 knots.

Mer. Pas: 11 56
 
Step 1. Determine Time of Mer. Pas. at Greenwich.

From the Nautical Almanac Daily Page for 17 Dec,

Mer. Pas. at Greenwich =  1156 GMT.
Step 2. Estimate time of local mer.pas.at first D.R. Pos.

Starting Data: Long. 134o 52’.0E

 Calculations:

·      Convert Long. to time.

4 x 134o ÷ 60 =  8.93h               =  8h    55m    48s

4 x 52’.0 ÷ 60 = 3.46m                =  0h    03m    27.6s

= 8h   59m    15 .6s

·      Estimate zone time of local mer.pas.

Mer. Pas. Greenwich :                     11h   56m    00s

Long. (long east GMT least) :     –08h   59m    15.6s

Local Mer. Pas (GMT)    =               02h   56m   44.4s             

=               02h   57m (nearest minute)

Zone (-9) :                                        +09h             (- for GMT, + for ZT)

Zone time Mer. Pas.                        11h   57m
 
Step 3.  Calculate new position at estimated time of Mer. Pas.

Starting data:

Course = 030o Speed = 15 knots.

Zone time at first DR position = 1000

DR Position at 1000 = 41o 15’.0S. 134o 52’.0E.

Estimated zone time of Mer. Pas = 1157

Time elapsed since 1000 = 1h 57m  = 1.95h

Distance run at 15 knots in 1.95h = 15 x 1.95 = 29.25n.m.

Calculations:

Dep.            = Dist x Sin(course)

= 29.25 x Sin(30)  = 14’.6E

D.Lat.          = Dist x Cos(course)

= 29.25 x Cos(30)  = 25’.3N

New Lat.   = Lat – D.Lat

= 41o 15’.0S – 25’.3N  = 40o 49’.7S

Mid. Lat       = Lat – (D. Lat ÷ 2)

= 41o 15’.0S. – 12’.65N

= 41o 02’.35S.  = 41o.04S

D.Long.      = Dep. x Sec(M.Lat)

= 14’.62 x Sec(41o.04)

= 14′.62 x 1.3258 = 19’.4

New Long.  = 134o 52’.0E + 19’.4

= 135o 11’.4E

Summary:  New Position at 1157 = 40o 49’.7S. 135o 11’.4E.
Step 4.  Calculate time of Mer.Pas. at new position.

Starting data:

New Long. 135o 11’.4E (from step 3)

Calculations:

  • Convert Long. to time.

4 x 135o ÷ 60                       =  9h    00m    00.0s

4 x 11.4’ ÷ 60 = 0.76m           =  0h    00 m   45.6s

=   9h    00m   45.6s

  • Calculate new time of Mer. Pas.

Mer. Pas. Greenwich:       =   11h    56m   00s        (GMT)

Long (135o 11’.4E):             = –09h    00m   45.6s

Local Mer. Pas (GMT)       =  02h   55m   14.4s

Zone (-9)                               =+09h                   . 

Zone time Mer. Pas.            =  11h   55m    14.4s

=   11h 55m (nearest minute)
Step 5. Determine declination at new time of local Mer.Pas.

Starting data:

Date: 17 Dec.

Local Mer. Pas (GMT):      02h  55m 14.4s  » 02h 55m

Calculations:

Dec Sun (02h):                S23o 21’.2  (d = 0’.1 increasing)

d Correction (55m):                      + 0’.1

Dec Sun (02h 55m):         S23o 21’.3
Step 6.  Calculate Meridian Altitude.

Starting data:

Sextant Altitude: 72o 18’.2

Index error: +2’.1.

Height of eye: 12m

Calculations:

Sextant Altitude:            72o 18’.2

I.E.                                          + 2’.1

Observed Altitude:        72o 20’.3

Dip (12m):                             – 6’.1   

Apparent Altitude:        72o 14’.2

Altitude Correction:         + 15’.9

True Altitude  =             72o 30’.1

(Note deck watch time: 02h 59m 10s (DWE -5s)) 
Step 7. Calculate Latitude

Starting Data:

Estimated Lat = 40o 49’.7S (From step 3)

Estimated Dec = S23o 21’.3 (From step 5)

Altitude = 72o 30’.1 (From step 6)

Rule: Same hemisphere Lat > Dec = rule i

Calculations:

LAT  =  DEC + (90o – ALT) (rule i)

= 23o 21’.3 + (90o – 72o 30’.1)

= 23o.36 + (90o – 72o.5)

= 23o.36 + 17o.5

= 40o.86

Calculated Latitude = 40o.86 = 40o 51’.6S. 
Step 8.  Calculate Longitude From Deck Watch Time.

Starting Data:

Estimated Longitude = 135o 11’.4E (from step 3)

Deck Watch Time = 02h 59m 10s   (from step 6)

DWE = -05s

Mer. Pas.:  11h  56m

Calculations:

  • Calculate time difference.

Deck Watch Time:        02h 59m 10s

DWE:                                              -05s

GMT/UT:                        02h 59m 05s (Longitude East, GMT Least)

Mer. Pas:                        11h  56m 00s  

Time Diff:                       11h  56m 00s   -02h 59m 05s  = 08h  56m 55s

(Longitude East, GMT Least)

  • Convert Time to Arc.

9h  =  8 x 15                  =      120o 00’  00”

56m  = 56 ÷ 4                      =        14o 00’  00”

55s = 55 ÷ 4                   =          0o  13’  45”

=     134o  13’  45”  =  134o  13’.75E

Calculated Longitude at 02h 59m 05s GMT = 134o  13’.75E 
Step 9.  Summarise position at zone time of meridian altitude.

Starting Data:

GMT of meridian altitude:  02h 59m 05s  (from step 6)

Zone: -9

Calculated latitude:  40o 51’.6S.  (from step 7)

Calculated longitude: 134o  13’.75E (from step 8)

Calculate zone time of meridian altitude:

GMT of meridian altitude             =    02h 59m 05s

Zone correction                               =  +09h

Zone time of meridian altitude    =    11h 59m 05s  ≈ 11h 59m

 Summary:

Observed Position at 1159 (zone time)  = 40o 51’.6S. 134o 13’.75 E

 

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