Sailings

Sailings

 Plane (or Plain) Sailing

The relationships can be laid out as two triangles. The sides of the top triangle are Distance, difference in latitude (dLat) and Departure. Departure is the longitude distance in miles. The lower triangle relates Departure to the difference in longitude (dLong.) The angle labelled mLat stands for Mean Latitude.

 

From the top triangle:-

dLat = Distance x Cos (Course)    Eq 1

Departure = Distance x Sin (Course)    Eq 2

From the lower triangle:-

dLong = Departure/ Cos(Mean Latitude)   Eq 3

Tabulated values are found in Nautical Tables as “Traverse Tables.” However the simplicity of the formulae are ideal for calculators.

 

Example

Initial Position  45 N 30 W

Course   045 T

Distance   100’

 

Using the first formula (Eq 1):

dLat =    100 x Cos(45)

dLat =    70’.7  = 1 10’.7

Final Latitude   45 + 1 10’.7  = 46 10’.7 N

 

From the second formula (Eq 2);

Departure = 100 x Sin (045) = 70.71

 

From the third formula (Eq 3)

 dLong = 70.71/ Cos ( 45 + 70.7/ 2)

 dLong = 101’.04 = 1 41’.0

Therefore the final longitude = 30 + 1 41’.0 = 31 41’.0 W

 

For a Noon calculation, dLat is the difference between the calculated and observed latitudes. The "Course" becomes the direction of the Position Line.

 

Distance = dLat/ Cos(P/L Dirn)

Departure = Distance x Sin(P/L Dirn)

dLong = Departure/ Cos(Mean Latitude)

Plane Sailing is adequate for distances up to about 60’. If the example is worked using Mercator Sailing, there is a difference of 0’.3 in the final longitude.

 

Mercator Sailing.

Mercator Sailing allow for the oblate shape of the Earth. (A squashed sphere.)

 

The formulae are

dLat = Dist x Cos(Course)   As for Plane Sailing - Eq 1

and

dLong = Tan(Course) x DMP   Eq 4

 

DMP stands for Difference in Meridian Parts. Meridian Parts are the distance in nautical miles from the equator to the required latitude. These are tabulated in Nautical Tables.

 

Example

Initial Position  45 N   30 W

Course   045 T

Distance   100’

 

Using the first formula (Eq 1):

dLat =    100 x Cos(45)

dLat =    70’.7 =  1 10’.7 N  because the track is northerly

Final Latitude   45 + 1 10’.7  = 46 10’.7 N

 

Meridian Parts

for 45   3013.38

for 46 10’.7  3114.08

Difference in Meridian Parts    100.7

 

From the second formula (Eq 4);

 dLong = Tan (045) x 100.7   = 100’.7 =  1 40’.7

 

Original Longitude       30 00’.0 W

dLong                         1 40’.7 West because the course is westerly

Final Longitude          31 40’.7 W

 

The drawback with Mercator Sailing is the need to refer to tables. If you use a programmable calculator then this is the formula to calculate Meridian Parts. (The infinite series of terms in Bowditch et al is simply an expansion of this.)

 

A x Ln(Tan( 45 + Lat/ 2)/ ((1 + e x Sin(Lat))/ (1 – e x Sin(Lat)) ^ (e/ 2)

For WGS84; A = 3437.74677 and e = 0.08182

 

Note that most nautical tables are based on the Clarke 1880 spheroid that uses a different compression to WGS84. The difference is small but noticeable (0.27 at 45.) For Clarke 1880 use e = 0.08248.

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Celestial Navigation Calculations