Navigating beyond sight of land

Ranger Hope © 2016   (View as Pdf)

 

Contents

Introduction

Terrestrial position

Celestial position

The Sailings

Azimuth or Amplitude

Position without a chart

Answers Compass checks

Answers Traverse

 

Introduction

 

Traditional coastal navigation begins from a known point or a fix from land based features plotted on a suitable chart. During the passage, course steered from the ships compass and speed from the ships log provides dead reckoning plots of the ships progress, confirmed by additional fixes en route. Whenever the opportunity arises a confirmation of continuing compass accuracy is checked by land based transits. 

 

When on passage out of sight of land the navigator relies on the continuing accuracy of the dead reckoning or GPS for notifying of change in position. If she experiences compass error she is none the wiser and if she experiences power failure navigation position is lost. However, two basic computational processes using Nautical (navigational tables) are available. The first is to compare the predicted bearing of the sun at rising or setting with the observed compass reading (using azimuths or amplitudes tables) and the second to determine the changes in position without plotting on a chart (by traverse tables).

 

In order to use the tabulated data an underpinning knowledge of terrestrial and celestial navigational definitions is required.


Terrestrial position

 

Latitude and Longitude

Latitude and longitude are used as reference points for any geographical point (GP) on the earth’s surface. They are the angles at the centre of the earth between a GP that is North or South a base line of the equator (latitude) and a GP that is East or West of a base of the prime meridian of Greenwich (longitude).

 

 

Small circles and Great circles

The planes of circles drawn around the earth’s greatest girth cut the earth’s centre and are called great circles, being the shortest distance between two points on the spherical earth’s surface. The planes of circles drawn around the earth’s lesser girths do not cut the earth’s centre and are called small circles. All longitude lines follow great circles and are called meridians. All latitude lines apart from the Equator follow small circles and are called parallels.

 

 

 

Charts and Mercator projection

In order to represent the spherical earth on a flat surface the Mercator projection is commonly used for coastal charts in low latitude (less than 60º N/S). The projection allows rhumb lines with constant true bearing to be plotted over short distance with fair accuracy. Distance can also be measured using the parallels of latitude as a scale (1º = 60 nm). Measures cannot be taken using the Longitude scale as distortion occurs as the segments of longitude narrow to meet at the Poles. 

 

As the earth spins once every twenty four hours its longitude can be equated to time:

 

180ºE + 180ºW = 360º

 

360º ÷ 24 = 15º

 

A reference point is taken as the Greenwich Prime Meridian (0º). The time at that meridian was called Greenwich Mean Time (GMT) but is now called Universal Time Constant (UTC) or just Universal Time (UT). So each 15º of longitude is equivalent to one hour, or each 1º of longitude is equivalent to 4 minutes. For convenience the world is divided into 12 hourly time zones East of UT (+ before) of Greenwich Prime Meridian and 12 hourly time zones West of UT (-after) of Greenwich Prime Meridian.  

 

 

 

 

Charts and Gnomic projection

Straight lines drawn on gnomic charts are great circles of shortest distance from point to point but with constantly changing compass direction, particularly so between GP’s with similar latitudes. However, these charts are required in the high latitude zones for distance measuring where the Mercator projection is too distorted.

 

 

 

 

Gnomic & Mercator - Great Circle Sailing

A rhumb line drawn on a Mercator chart below represents a small circle of constant direction that is not the shorter great circle route.

 

 

mercator chart.jpg

 

 

The straight line drawn on a Gnomic chart below represents a great circle with constantly changing direction. This line can be transposed from the GP below as waypoint legs onto the Mercator chart above.

 

 

gnomonic chart.jpg

Drawings above are Courtesy of Bowditch-American Practical Navigator

 

Celestial position

 

The earth’s spin and its revolution

The earth spins (approx) once every 24 hours as it revolves around the sun in an elliptical orbit of (approx) 365¼ days. Its axis remains tilted with the consequence that the sun is overhead (in zenith) varying from 23.5ºN to 23.5ºS over the yearly orbit. When the sun’s rays are directly overhead the heat to earth’s surface is concentrated causing summer. This results in the opposing seasons experienced by Northern and Southern Hemispheres.

 

 

Geographic position (GP), Time and Longitude

The earth’s elliptical orbit causes it to accelerate on approach to the sun and decelerate in moving away. Our mechanical or atomic clocks maintain a constant beat referred to as mean time. As 15º of longitude is equivalent to 1 mean hour of time passing, or 1º is equivalent to, mean 4 minutes mean time and longitude can be equated. Local mean time is therefore unique at every GP’s latitude. However, for convenience the navigator divides the world into 12 hourly time zones East (+UT) and 12 hourly time West (-UT). 

 

Greenwich Hour Angle GHA & Local Hour Angle (LHA)

The earth is considered as at the centre of a celestial sphere on which stars and planets are projected. The position on the celestial dome directly overhead of any GP is called its Zenith. GP’s are described by the co-ordinates Lat. & Long – Celestial positions are described by Dec. and GHA. GHA increases from 0º to 360º measured clockwise (westward) following the apparent motion of the heavenly bodies (HB) overhead.

The angle between the observer’s longitude and a heavenly body GHA is called the Local Hour Angle (LHA) and is always measured clockwise (westward) from the observer. Hence if the observer is in:

East longitude the LHA is the GP longitude East + the HB’s GHA.

West longitude the LHA is the GP longitude West - the HB’s GHA.

The Sailings

 

Coastal passage requiring charted land based features are unavailable beyond sight of land. Other mathematical solutions of determining passage progress without a chart are available, collectively called the Sailings.

 

Plane sailing - These solutions provide sufficiently accurate single course/distance, difference in latitude and departure for passages of less than a few hundred miles. They assume the earth traversed (crossed over) is a flat surface (a plane) consequently difference in longitude cannot be calculated, that requiring spherical trigonometry.

 

Traverse sailing - Extends plane sailing to summing multiple rhumb lines of changes in course/distance over a passage.

 

Parallel sailing – The historic practice of maintaining an East or West heading along a parallel of latitude (by altitude of a heavenly body) until dead reckoning (departure) indicated that the final position sought was directly North or South.

 

Mid latitude sailing – A course that is not East/West will start and finish in different latitudes. In converting departure into difference in longitude the mean (average) of the two latitudes is applied.

 

Mercator sailing – A mathematical alternative to plotting on a Mercator chart that uses computational tables of meridional difference and difference in longitude rather than difference of latitude and departure.

 

Great circle sailing – Courses and distances that follow great circles, those being the shortest distance between two points of a sphere.

 

Composite sailing –Great circle sailing adapted for the purpose of routing convenience.

 

Azimuth or Amplitude

 

Checking for Compass error

Without terrestrial features a practical way for a navigator to check the accuracy of the ships steering compass is to compare its bearing of a heavenly body with that predicted. The predicted position of GHA and Declination of heavenly bodies are tabulated in Nautical Almanacs. The most accurate bearings can be observed from sun on rising and on setting. Due to refraction in the earth’s atmosphere the moment of sunrise and sunset is actually when the sun is half a diameter above the horizon, this normally being the tabulated time.

 

Bearings used by coastal navigators are measured from North in a clockwise fashion through 360º back to North – not so the tabulated quadrant notation of azimuths or amplitudes.

 

Azimuths are tabulated from North or South towards East or West, i.e.:

N 20 ºE  =  020º       N 20 ºW  =  340º    S 20 ºE  =  160º       S 20 ºW  =  200º

 

Amplitudes are tabulated from East or West towards North or South, i.e.:

E 20 ºN  =  060º       E 20 ºS  =  110º    W 20 ºN  =  290º       W 20 ºS  =  250º

 

The symbols AZI and AMP may help to memorise the different reference points of azimuths or amplitudes, being oriented as from the heavy lines of Z or M.

 

Three methods commonly used to find compass error from the sun:

 

1.  a scientific calculator, or

2.  azimuth or amplitude tables and form, or

3.  the ABC tables and form.

 

In all three methods the time, the ship’s approximate position, the local variation and ship’s compass bearing of a rising or setting sun must be known. The time of the risings or settings in local time (at the ship’s longitude) is tabulated in the current Nautical Almanac. The ship’s Mean Time of the observation must be converted into Universal Time (UT) so the sun’s declination can be extracted from the tables for method 1 & 2, and the sun’s declination and GHA can be extracted from the tables for method 3.

Finding time of Sunrise or Sunset, Declination and GHA

Nautical almanacs and online sources publish annual details of celestial positions. Follow the steps below to use the Nautical Almanac:

 

 

Tabulated range is N58º- 08:45  & N60º - 09:02.

The difference in time is      08:45 - 09:02 = +00:17.

           The difference in latitude is N58º  - N60º  =  N2º = +120.0.

The correction is time range ÷ latitude range x difference in latitude, eg:  (17 ÷ 120) x 90 = +12’.7. Time of local sunrise = 08:45 + 12’.7 = 08:57’.7

 

 

08:58’ local sunrise – 10:03hr position east = 22:55 UT on 1/1/11

 

          (0.2 ÷ 60) x 55 = 0.18 so declination = S22º 58’.2 – 0’.18 = S22º 58’.02

 

164º 05’.3 - 149º 05’.6 = 14º 59’.7    (say 15º ÷ 60) x 55 = 13.75º = 13º 45’

149º 05’.6 + 13º 45’  = GHA 162º 50’.6

 

 

 

1. Scientific calculator method:

Example - find the amplitude for sunrise with a tabulated Declination of 22º  30’S  in Latitude of 33º  45’S. Convert the degree/minutes to degrees/decimals and enter the formula Sin Amp=Sin Dec ÷ Cos Lat. Enter

the quantities to calculate Sin Amp; covert this to Amp using the invert sine key pads. Calculator key pads differ, but a common system is shown below.

 

Sin

22.50

÷

Cos

33.75

=

Inv

Sin

=

27.4

 

 

 

Next covert Amp to 360º notation, East if rising, West if setting and named as the declination, in the case above,    E 27.4º S   or  117.4º T then compare with ships compass for error.

Follow the worked example A1 below, then try the questions A2 and A3.

 

 

2.  Azimuths or amplitudes tables and form

After finding time of sunrise/sunset and declination from a current Nautical Almanac use nautical tables (Nories or Burtons) as per their instructions in the Explanation of the Tables. (click on link to see table extracts):

 

Follow the worked example A1 below, then try the questions A2 and A3.

 

 

3.  The ABC tables and form.

After finding time of sunrise/sunset, declination and GHA from the current Nautical Almanac calculate the LHA. Then use nautical tables (Nories or Burtons) as their instructions in the Explanation of the Tables.

(click on link to see tables):

 

Follow the worked example B1 below, then try the questions B2 and B3.

 

 

 

Worked Examples

 

Required:

Time, approx lat & long, variation and ship’s bearing of sun on rising or setting 

Nautical Almanac of the sun’s GHA and Declination

Nautical (Navigational) Tables – Nories, Burtons, etc

Azimuth or Amplitude form

ABC form

Amplitude tables (example only)

Traverse tables (example only)

 

 

 

 

 

 

Example A1 using the Azimuth/Amplitude form or calculator:

At sunrise on 11th January 2011 in latitude                                     45° 00’S

                                                 and longitude                                  75° 30’E

the sun was observed (half a diameter over the horizon) to bear 110° C

Find the time of sunrise, true amplitude and deviation if variation = 9° E

(Determine declination to nearest 1°)

 

From Almanac

 

Date…11/1/11.......…Heavenly body …Sun……Compass bearing…110° (C)

 

Latitude….45° 00’............ (N) (S)    ....... Longitude………75° 30’..…(E) (W)

 

Average local time of rising (E) or setting (W)..04h   27m …...

 

Time correction (if required to < 1 degree)

 

Time difference from UT (-E) or (+W)                     -5h      00m      hours

 

Time difference from UT (-E) or (+W)             -0h     02m      mins

                                                                                       -05h    02m       hours & mins

 

UT  (GMT)  (04h 27m -05h 02m)  Date 10/1/12     23h    25m       hours & mins

 

UT tabulated  Declination (hours)                      S21°  53.70m     hours

d correction (minutes)     + or -   0.4 x 25’/60 =             0.17m     mins

                                 

Heavenly body declination at chosen long.        S21° 53.53m      (N) or (S)

 

 

From amplitude or azimuth tables

Enter with declination and latitude (apply same name as declination)

 

Tabulated/calc. Amplitude   (E) (W)         E 32°S           (N) (S)

                           or Azimuth   (N)  (S)         S 58°E           (E) (W)

 

 

(T) True Azimuth       90 + E 32°S   = 122°T      (Amplitude tables)

                                    180 - S 58°E   = 122°T       (Azimuth tables)

 

 


          (T)   True bearing   122°T                        

W+     (V)   Variation            9°E

          (M)   Magnetic        113°M                        

E-       (D)   Deviation           3°E                        

          (C)   Compass        110°C

 

From calculator

Sin Amp=Sin Dec ÷ Cos Lat= 0.5272     Amp = 31.82  = E 31° 49’ S = 121.8°T     


Question A2 using the Azimuth/Amplitude form or calculator:

At sunset on 12th January 2011 in latitude                                      60° 00’N

                                                and longitude                                   20° 25’W

the sun was observed (half a diameter over the horizon) to bear 327.5° C

Find the time of sunset, true amplitude and deviation if variation=10° W

(Determine declination to nearest 1°)

 

From Almanac

 

Date……………Heavenly body ……………Compass bearing…….…….(C)

 

Latitude………………..(N) (S)     Longitude……………………...…(E) (W)

 

Average local time of rising (E) or setting (W)……..……………...

 

Time correction (if required to < 1 degree)

 

Time difference from UT (-E) or (+W)            ……..……………... hours

 

Time difference from UT (-E) or (+W)            _______________ mins

 

                                                                       _______________ hours & mins

 

UT  (GMT)   (Local +/- Long)  Date ________      _______________ hours & mins

 

UT tabulated declination (hours)                    ……..……………... hours

d correction (minutes)    + or -                        _______________ mins

                                

Heavenly body declination at chosen long.   _______________  (N) or (S)     

 

 

From amplitude/azimuth tables or calculator

Enter with declination and latitude (apply same name as declination)

 

 

Tabulated/calc. Amplitude  (E) (W..................................(N)  (S)

                        or Azimuth  (N) (S)..................................(E) (W)

 

(T) True Azimuth  ………………

 

 


           (T)   True bearing………………..

W+     (V)   Variation      ..………………

          (M)   Magnetic      ..………………

E-       (D)   Deviation     ..………………

          (C)   Compass      ..………………

 

 

From calculator

Sin Amp=Sin Dec ÷ Cos Lat=

See Answers

 

 

 

Question A3 using the Azimuth/Amplitude form or calculator:

At sunrise on 16th November 2011 in latitude                                    35° 00’S

                                                  and longitude                                  155° 30’E

the sun was observed (half a diameter over the horizon) to bear    116° C

Find the time of sunrise, true amplitude and deviation if variation  = 10° E

 (Determine declination to nearest 1°)

 

From Almanac

 

Date……………Heavenly body ……………Compass bearing…….…….(C)

 

Latitude………………..(N) (S)     Longitude……………………...…(E) (W)

 

Average local time of rising (E) or setting (W)……..……………...

 

Time correction (if required to < 1 degree)

 

Time difference from UT (-E) or (+W)            ……..……………... hours

 

Time difference from UT (-E) or (+W)            _______________ mins

 

                                                                       _______________ hours & mins

 

UT  (GMT)   (Local +/- Long)  Date ________      _______________ hours & mins

 

UT tabulated declination (hours)                    ……..……………... hours

d correction (minutes)    + or -                        _______________ mins

                                

Heavenly body declination at chosen long.   _______________  (N) or (S)     

 

 

From amplitude/azimuth tables or calculator

Enter with declination and latitude (apply same name as declination)

 

 

Tabulated/calc. Amplitude  (E) (W..................................(N)  (S)

                        or Azimuth  (N) (S)..................................(E) (W)

 

(T) True Azimuth  ………………

 

 


           (T)   True bearing………………..

W+     (V)   Variation      ..………………

          (M)   Magnetic      ..………………

E-       (D)   Deviation     ..………………

          (C)   Compass      ..………………

 

 

From calculator

Sin Amp=Sin Dec ÷ Cos Lat=

See Answers

 

 

Example B1 using the ABC tables:

At sunrise on 11th January 2011 in latitude                                     45° 00’S

                                                 and longitude                                       75° 30’E

the sun was observed (half a diameter over the horizon) to bear 110° C

Find the time of sunrise, true amplitude and deviation if variation = 9° E

(Determine declination to nearest 1°)

 

1

Date

Ship

11/1/11  

Smt

Ship mean time

04h   27m

HB

Heavenly body

Sun

 

2

DR Lat. CP

45° 00’

N / S

DR Long. CP

75° 30’

E / W

 

 

Variation Local

  9° E

E / W

Bearing of HB

 

 110°C

ºC

 

3

Convert SMT to UT (GMT)

4

Extract GHA (almanac) & Calculate LHA

Smt

 04h  

  27m

UT/GHA

hour

163°

06’.0

 

Zone Time

- east

+ west

  -5h

  -2m

UT/GHA

min & sec increment

    6°

15’.0

 

(75 ÷ 15)

(0.5 ÷ 15)x60

UT

23h  

25m

UT/GHA

169°

21’.0

 

UT Date

10/1/11  

  

DR long

+ E  -  W

  75°

30’.0

 

Tab Inc & corrections  6°21’.3

Calc. (178°05.7-163°06’.0) = (14° 59’.7 ÷ 60) =14’.99 x 25= 6°15’

LHA

244°

51’.3

 

 

5

Calculate Declination

 

Dec.

at hour

21° 53’.7

N / S 

 

‘d’  minute correction

Int. 0’.4 x 25/60

        0’.17

+ or

as table trends

 

Dec.

21° 53’.53

N / S  ‘d’

 

 

6

Enter ABC tables

10

Compass error check

LHA/Latitude

 

 

A=   0.47

N / S name opp. to Lat except LHAs 90-270

T

123°T

 

7

LHA/Dec

 

 

B=   0.45

N / S

name same as Dec

V

09°E

 

8

A+or-B

 

0.47 + 0.45

N / S

name same as greatest

M

114°M

 

A+or-B /lat

C=  0.92

 

D

  4°E

 

Quadrant notation =

S  57° E

W if LHA 000-180 

E if LHA 180-360

C

110°C

 

 

9

Convert to 360ºT notation

 

 

 

123°T

 

 

 

Question B2 using the ABC tables:

At sunset on 12th January 2011 in latitude                                      60° 00’N

                                                and longitude                                        20° 25’W

the sun was observed (half a diameter over the horizon) to bear 327.5° C

Find the time of sunset, true amplitude and deviation if variation=10° W

(Determine declination to nearest 1°)

 

1

Date

Ship

 

Smt

Ship mean time

 

HB

Heavenly body

 

 

2

DR Lat. CP

 

N / S

DR Long. CP

 

E / W

 

 

Variation Local

 

E / W

Bearing of HB

 

 

ºC

 

3

Convert SMT to UTC (GMT)

4

Extract GHA (almanac) & Calculate LHA

Smt

 

 

UT/GHA

hour

 

 

 

Zone Time

- east

+ west

 

 

UT/GHA

min & sec

 

 

 

UT

 

 

UT/GHA

 

 

 

UT Date

 

 

DR long

+ E  -  W

 

 

 

 

LHA

 

 

 

 

5

Calculate Declination

 

Dec.

at hour

 

N / S 

 

‘d’  minute correction

 

 

+ or –

as table trends

 

Dec.

 

N / S  ‘d’

 

 

6

Enter ABC tables

10

Compass error check

LHA/Latitude

A=

N / S

name opp. to Lat except LHAs 90-270

T

 

 

7

LHA/Dec

B=

N / S

name same as Dec

V

 

 

8

A+or-B

 

 

N / S

name same as greatest

M

 

 

A+or-B /lat

C=

 

D

 

 

Quadrant notation =

N / S       E / W

W

if LHA 000-180 

 E

if LHA 180-360

C

 

 

 

9

Convert to 360ºT notation

 

 

 

 

See Answers

 

 

 

 

Question B3 using the ABC tables:

At sunrise on 16th November 2011 in latitude                                    35° 00’S

                                                  and longitude                                       155° 30’E

the sun was observed (half a diameter over the horizon) to bear    116° C

Find the time of sunrise, true amplitude and deviation if variation  = 10° E

 (Determine declination to nearest 1°)

 

1

Date

Ship

 

Smt

Ship mean time

 

HB

Heavenly body

 

 

2

DR Lat. CP

 

N / S

DR Long. CP

 

E / W

 

 

Variation Local

 

E / W

Bearing of HB

 

 

ºC

 

3

Convert SMT to UT (GMT)

4

Extract GHA (almanac) & Calculate LHA

Smt

 

 

UT/GHA

hour

 

 

 

Zone Time

- east

+ west

 

 

UT/GHA

min & sec

 

 

 

UT

 

 

UT/GHA

 

 

 

UT Date

 

 

DR long

+ E  -  W

 

 

 

 

LHA

 

 

 

 

5

Calculate Declination

 

Dec.

at hour

 

N / S 

 

‘d’  minute correction

 

 

+ or –

as table trends

 

Dec.

 

N / S  ‘d’

 

 

6

Enter ABC tables

10

Compass error check

LHA/Latitude

A=

N / S

name opp. to Lat except LHAs 90-270

T

 

 

7

LHA/Dec

B=

N / S

name same as Dec

V

 

 

8

A+or-B

 

 

N / S

name same as greatest

M

 

 

A+or-B /lat

C=

 

D

 

 

Quadrant notation =

N / S       E / W

W

if LHA 000-180 

 E

if LHA 180-360

C

 

 

 

9

Convert to 360ºT notation

 

 

 

 

See Answers

 

Position without a chart

 

Charts for ocean areas are relatively blank. To remove the need to buy many blank charts to record a position, mathematical calculation rather than geometrical plotting is adopted using trigonometry and/or traverse tables.

 

 

Trigonometry

Trigonometry defines and provides formulas regarding relationships between sides and an angle within a right angle triangle. These definitions include:

 

The opposite side is opposite angle θ.

The adjacent side is adjacent (next to) to angle θ.

The hypotenuse is the longest of the sides.

 

 

 

These trigonometric formulas include:

 

 

Sine (sin) of  θ

=

Opposite

=

b

Hypotenuse

a

 

 

 

 

 

Cosine (cos) of  θ

=

Adjacent

=

c

Hypotenuse

a

 

 

 

 

 

Tangent (tan) of   θ

=

Opposite

=

b

Adjacent

c

 

These formulas can be remembered by the mnemonic of:

SOH-CAH-TOA

 

 

The sine, tangent or cosine of the angle θ can be found tabulated in published books of tables, or can be found using the function on a scientific calculator.

 

 

 

Also the reciprocal trigonometrically functions and formulas could be used. When using a calculator, the shift key is used to select reciprocal functions.

 

 

  1

θ

=

Secant (sec)      θ

=

Hypotenuse

=

a

Sin

Adjacent

c

 

 

 

 

 

 

 

 

   1

θ

=

Cosecant (cosec) θ

=

Hypotenuse

=

a

Cos

Opposite

b

 

 

 

 

 

 

 

 

   1

θ

=

Cotangent (cot) θ

=

Adjacent

=

c

Tan

Opposite

b

 

 

 

If any two of the components of the formula is known then the last can be calculated using substitution if required (rearranging the order of the formula).

 

 

 

 

Finding Difference in Latitude and Departure using trigonometry

This solution assumes that the earth surface is a flat plane, in other words it is a plane sailing solution that is accurate limited to a few hundred miles. A right angle triangle and (θ angle) ABC is used for solutions:

 

 

We start from position B steering 301ºT and log 348 nm travelled along line a (the hypotenuse) to reach a more northerly position of C. This course is the angle between ABC (301º - 270º) = 31º subtended north of the line c (the adjacent). The line b (the opposite) is equivalent to our change in latitude.

 

 

 

 

Sine (sin) of  θ

=

Opposite

=

b

Hypotenuse

a

 

 

We find the length of the opposite using the formula and sine function of a scientific calculator.  Sine 31º = 0.515.

 

Sine = Opposite ÷ Hypotenuse    or    Hypotenuse x Sine = Opposite  Therefore 348 nm x 0.515 = 179.23 nm, entered on a calculator as:

 

 

348

x

Sin

31

=

179.23             

 

 

 

 

 

As our initial latitude at B was 33º 00’.0 S then our difference in latitude having steamed north to C will be 179.23 nm ÷ 60  = North 2.9872º  =  -2º 59’

 

Our new latitude C is 33º 00’.0 S - 2º 59’ = 30º 01’.0 S.

 

As our initial longitude at B is 160º 00’.0 E the departure (our East/West shift in miles) will be the length of the adjacent (BA).

 

 

 

Cosine (cos) of  θ

=

Adjacent

=

c

Hypotenuse

a

 

 

 

We find the length of the adjacent using the formula and cosine function of a scientific calculator. Cos 31º = 0.978

 

Cos = Adjacent ÷ Hypotenuse or    Adjacent = Hypotenuse x Cosine

Therefore 348 nm x 0.978 = 298 nm, entered on a calculator as:

 

 

 

348

x

cos

31

=

298 nm

 

 

 

 

 

 

We have travelled 298 nm in a westerly direction (our departure). How this relates to our change in degrees of longitude depends on the latitudes we were travelling in. Remember, longitudes are meridians all meeting at a point at the poles while having a separation of 60 nm per degree at the equator. The varying equivalence of departure and difference in longitude dependant on latitude is best found by using the traverse tables, as explained below.

 

 

 

Distance and courses using Traverse tables

 

Trigonometric solutions are tabulated in the traverse tables using the definitions below:

 

 

 

Rhumb line - A course line of constant true bearing. It crosses all meridians at the same angle and appears on a Mercator chart as a straight course line. It is adequate for short distances of less than a few hundred miles.

 

Distance -A separation of between one point and another expressed in units of length, for navigational purposes these being divisions of a nautical mile (equal to 1852 metres).

 

Departure -The easting or westing of a vessel’s progress measured in nautical miles. Note this is only equivalent to Difference Longitude on the equator.

 

Difference in latitude -The angular distance from one parallel of latitude to another, being approximately 60 nm per degree or 1 nm per minute.

 

Difference in longitude - The angular distance from one meridian of longitude to another, being approximately between 60 nm per degree at the equator to 0 nm at the poles.

 

 

 

Traverse course is tabulated from North or South towards East or West, i.e.:

 

N 68 ºE  =  068º       N 68 ºW  =  292º    S 68 ºE  =  112º       S 68 ºW  =  248º

 

 

 

 

Enter or extract your required course on the page with the quadrant notation of your course. Read the table from the top down or the bottom up dependant on the 360º notation of your course. See example below:

 

 

 

 

 

 

 

 

 

 

 

Example C1. 

At 09:00 in latitude 40° 00’S and longitude 150° 00’E a ship changes course to 112° T and steams at 8.4 knots until 19:00. Assuming no drift and steady course and speed, calculate the 19:00 position.

 

Finding Distance and Quadrant notation to enter tables.

  19:00

-09:00

  10 hours x 8.4 = Distance 84 nm steamed.   Course 112° = S 68° E

 

Finding Difference in Latitude and Departure   

From Page 68º reading from bottom up          D. Lat = 31.5         Dep  = 77.9 miles

 

Old Latitude            40° 00’.0 S

Change in Latitude      +31.5 S

New Latitude           40° 31.5 S

 

Finding Difference in Longitude at Mid Latitude

Mid Lat = (40° 31.5 - 40° 00’.0) ÷ 2 + old lat = 40° 15’S

 

From Page top 40º use D’Long and Dep columns for nearest to Dep 77.9

Find    D’Long 101     Dep 77.4

          D’Long 102     Dep 78.1

 

Interpolate for D’Long correction for 77.9.

 (D’Long increases 101 to 102 = 1 between 77.4 and 78.1)

 

                    78.1                           77’.9

 Interp =    -77.4                           77’.4

                  0’.7                            0’.5

 

0’.7 Dep increase is equivalent to 1’.0 D’Long increase.

0’.5 Dep increase is equivalent to  0’.5 ÷ 0’.7 = + 0’.71

 

D long     101’.0

Corr.      + 0’.71

          101’.71 ÷ 60 = 1° 41.71’E

 

Old Longitude         150° 00’.00E

Change in longitude   +1° 41’.71E

New longitude         151° 41’.71E

 

 

 

 

 

 

 

Question C2.

On 1/2/11 at 12:00 in latitude 25° 00’S and longitude 100° 00’E a ship changes course to 220° T and steams at 14 knots until noon on 2/2/11.

Assuming no drift and steady course and speed, calculate the last position.

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

____________________________________________________________

 

 

Question C3.

Work up the change in position of the yacht that beats into the wind from position 31° 30’S and longitude 153° 30’E.

Course

Distance run

Diff of Lat

Departure

Notes

 

 

North

South

East

West

 

035°T

20 nm

 

 

 

 

 

 

260°T

35 nm

 

 

 

 

 

 

125°T

8 nm

 

 

 

 

 

 

290°T

15 nm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Compass checks - Answers

Answer A2 using the Azimuth/Amplitude form or calculator:

At sunset on 12th January 2011 in latitude                                      60° 00’N

                                                     and longitude                                   20° 25’W

the sun was observed (half a diameter over the horizon) to bear  230° C

Find the time of sunset, true amplitude and deviation if variation= 10° W

(Determine declination to nearest 1°)

                                                                              

From Almanac

 

Date…12/1/11.......…Heavenly body …Sun……Compass bearing…230° (C)

 

Latitude….60° 00’............ (N) (S)    ....... Longitude………20° 25’..…(E) (W)

 

Average local time of rising (E) or setting (W)..15h   22m …...

 

Time correction (if required to < 1 degree)

 

Time difference from UT (-E) or (+W)                  +1h      20m      hours

 

Time difference from UT (-E) or (+W)             +0h       1.7m    mins

 

UT  (GMT)   (15h 22m + 1h  22m)      12/1/12           16h    44m       hrs & mins

 

UT tabulated  Declination (hours)                       S21°  37.5m       hours

d correction (minutes)  + or -   (0.4 x 44/60) =               0.3m       mins

                                

Heavenly body declination at chosen long.         S21° 37.2m (N) or (S)     

 

 

From amplitude or azimuth tables

Enter with declination and latitude (apply same name as declination)

Tabulated/calc. Amplitude   (E) (W)         47.1° (N)  (S)

                           or Azimuth   (N)  (S)         42.9° (E) (W)

 

 

(T) True Azimuth      270 -  W 47.1° S    = 222.9° (Amplitude tables)

                                  180 +  S 42.9° W   = 222.9° (Azimuth tables)