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 rhumbline 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 eastwest. 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 = 90^{o} – Altitude
and Altitude = 90^{o} – Zenith Distance
Rules For Calculating Meridian Passage.
 Latitude and declination same names but latitude greater than declination:
LAT = DEC + (90^{o} – ALT)
 Latitude and declination same names but declination greater than latitude:
LAT = DEC – (90^{o} – ALT)
 Latitude and declination opposite names:
LAT = (90^{o} – ALT) – DEC
NineStep Proforma. This easy to the follow ninestep 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 9step 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 deckwatch 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 ninestep proforma explained above.
Scenario.
Date: 17 December
(zone 9)
Mer. Pas.: 11^{h } 56^{m}
DR Position at 1000 (zone time): 41^{o} 15’.0S. 134^{o} 52’.0E.
Course: 030^{o} Speed: 15 knots.
Sextant Altitude at Mer. Pas. (Meridian Altitude): 72^{o} 18’.2
Index error: +2’.1.
Height of eye: 12m
Deck Watch Time at Meridian Altitude: 02^{h} 59^{m} 10^{s}
DWE 5^{s} fast
Solution:
Pre Planning.
Date: 17 Dec. Zone Time: 1000(9). DR Pos: 41^{o} 15’.0S. 134^{o}52’.0E. Course: 030^{o} 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. 134^{o} 52’.0E Calculations: · Convert Long. to time. 4 x 134^{o} ÷ 60 = 8.93^{h} = 8^{h} 55^{m} 48^{s} 4 x 52’.0 ÷ 60 = 3.46^{m }= 0^{h} 03^{m} 27.6^{s} = 8^{h} 59^{m} 15 .6^{s} · Estimate zone time of local mer.pas. Mer. Pas. Greenwich : 11^{h} 56^{m} 00^{s} Long. (long east GMT least) : –08^{h} 59^{m} 15.6^{s} Local Mer. Pas (GMT) = 02^{h} 56^{m} 44.4^{s } = 02^{h} ^{ } 57^{m} (nearest minute) Zone (9) : +09^{h} ( for GMT, + for ZT) Zone time Mer. Pas. 11^{h} 57^{m} 
Step 3. Calculate new position at estimated time of Mer. Pas.
Starting data: Course = 030^{o} Speed = 15 knots. Zone time at first DR position = 1000 DR Position at 1000 = 41^{o} 15’.0S. 134^{o} 52’.0E. Estimated zone time of Mer. Pas = 1157 Time elapsed since 1000 = 1^{h} 57^{m }= 1.95^{h} Distance run at 15 knots in 1.95^{h} = 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 = 41^{o} 15’.0S – 25’.3N = 40^{o} 49’.7S Mid. Lat = Lat – (D. Lat ÷ 2) = 41^{o} 15’.0S. – 12’.65N = 41^{o} 02’.35S. = 41^{o}.04S D.Long. = Dep. x Sec(M.Lat) = 14’.62 x Sec(41^{o}.04) = 14′.62 x 1.3258 = 19’.4 New Long. = 134^{o} 52’.0E + 19’.4 = 135^{o} 11’.4E Summary: New Position at 1157 = 40^{o} 49’.7S. 135^{o} 11’.4E. 
Step 4. Calculate time of Mer.Pas. at new position.
Starting data: New Long. 135^{o} 11’.4E (from step 3) Calculations:
4 x 135^{o} ÷ 60 = 9^{h} 00^{m} 00.0^{s} 4 x 11.4’ ÷ 60 = 0.76^{m }= 0^{h} 00^{ m} 45.6^{s} = 9^{h} 00^{m} 45.6^{s}
Mer. Pas. Greenwich: = 11^{h } 56^{m} 00^{s} (GMT) Long (135^{o} 11’.4E): = –09^{h } 00^{m} 45.6^{s} Local Mer. Pas (GMT) = 02^{h } 55^{m} 14.4^{s} Zone (9) =+09^{h} . Zone time Mer. Pas. = 11^{h} 55^{m} 14.4^{s} = 11^{h} 55^{m }(nearest minute) 
Step 5. Determine declination at new time of local Mer.Pas.
Starting data: Date: 17 Dec. Local Mer. Pas (GMT): 02^{h } 55^{m }14.4^{s }» 02^{h} 55^{m} Calculations: Dec Sun (02^{h}): S23^{o} 21’.2 (d = 0’.1 increasing) d Correction (55^{m}):^{ }+ 0’.1 Dec Sun (02^{h }55^{m}): S23^{o} 21’.3 
Step 6. Calculate Meridian Altitude.
Starting data: Sextant Altitude: 72^{o} 18’.2 Index error: +2’.1. Height of eye: 12m Calculations: Sextant Altitude: 72^{o} 18’.2 I.E. + 2’.1 Observed Altitude: 72^{o} 20’.3 Dip (12m): – 6’.1 Apparent Altitude: 72^{o} 14’.2 Altitude Correction: + 15’.9 True Altitude = 72^{o} 30’.1 (Note deck watch time: 02^{h} 59^{m} 10^{s} (DWE 5^{s})) 
Step 7. Calculate Latitude
Starting Data: Estimated Lat = 40^{o} 49’.7S (From step 3) Estimated Dec = S23^{o} 21’.3 (From step 5) Altitude = 72^{o} 30’.1 (From step 6) Rule: Same hemisphere Lat > Dec = rule i Calculations: LAT = DEC + (90^{o} – ALT) (rule i) = 23^{o} 21’.3 + (90^{o} – 72^{o} 30’.1) = 23^{o}.36 + (90^{o} – 72^{o}.5) = 23^{o}.36 + 17^{o}.5 = 40^{o}.86 Calculated Latitude = 40^{o}.86 = 40^{o} 51’.6S. 
Step 8. Calculate Longitude From Deck Watch Time.
Starting Data: Estimated Longitude = 135^{o} 11’.4E (from step 3) Deck Watch Time = 02^{h} 59^{m} 10^{s} (from step 6) DWE = 05^{s} Mer. Pas.: 11^{h } 56^{m} Calculations:
Deck Watch Time: 02^{h} 59^{m} 10^{s} DWE: 05^{s} GMT/UT: 02^{h} 59^{m }05^{s (Longitude East, GMT Least)} Mer. Pas: 11^{h } 56^{m} 00^{s } Time Diff: 11^{h } 56^{m} 00^{s } 02^{h} 59^{m }05^{s = }08^{h} 56^{m} 55^{s} (Longitude East, GMT Least)
9^{h }= 8 x 15 = 120^{o }00’ 00” ^{56m = 56 ÷ 4 }= 14^{o} 00’ 00” 55^{s} = 55 ÷ 4 = 0^{o} 13’ 45” = 134^{o }13’ 45” = 134^{o }13’.75E Calculated Longitude at 02^{h} 59^{m }05^{s} GMT = 134^{o }13’.75E 
Step 9. Summarise position at zone time of meridian altitude.
Starting Data: GMT of meridian altitude: 02^{h} 59^{m }05^{s }(from step 6) Zone: 9 Calculated latitude: 40^{o} 51’.6S. (from step 7) Calculated longitude: 134^{o }13’.75E (from step 8) Calculate zone time of meridian altitude: GMT of meridian altitude = 02^{h} 59^{m }05^{s} Zone correction = +09^{h} Zone time of meridian altitude = 11^{h} 59^{m }05^{s } ≈ 11^{h} 59^{m} Summary: Observed Position at 1159 (zone time) = 40^{o} 51’.6S. 134^{o} 13’.75 E

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