To be able to calculate the great circle distance between any two points on the Earth s surface.

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Cameron Mason
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1 17. Great Circles Objective To be able to calculate the great circle distance between any two points on the Earth s surface. Introduction The shortest distance between any two points on the Earth s surface is part of a great circle. When planning ocean voyages of several hundreds of miles it is typically a great circle route that is chosen providing of course it does not cross any land masses or dangerous areas. For short voyages a rhumb line course is acceptable. A rhumb line course is one which crosses each line of longitude at the same angle. The rhumb line course distance and the great circle course distance are the same in only two special cases: a) a course along the equator or b) a course which is either due north or due south. Neither of these happens often. The further one is from the equator i.e. the higher the latitude the greater the difference between the rhumb and great circle courses. For example imagine your voyage is an East-West passage of 90 degrees of longitude along the equator, for which the great-circle and rhumb-line distances are the same at 5,404 nautical miles. At latitude 20 N the great-circle distance for the same voyage is 4,997 nautical miles while the rhumb-line distance is 5,074 miles, about 1½ percent further. At 60 N the great circle distance is 2,485 nautical miles while the rhumb-line is 2,700 nautical miles, a difference of 8½ percent. Gnonomic chart projections can be used to plot great circle courses but the method of determining distance is not easy or intuitive. Using the exact same process that you have learned to perform sight reductions you will be able to calculate the distance between any two points and the initial course to steer. The process is repeated at regular intervals during the voyage to update the course to be steered. The sight reduction process gives solutions to spherical triangles and therefore can be used to determine the great circle distances between any two points on the Earth s surface. In all our work up to now we have used a figure of 1 of latitude as being equivalent to 60 nautical miles (1 = 1nm). For short distances that is very adequate. However a more accurate figure would be 1 = 60.04nm. However in the examples below I have used the usual 1 = 1nm. Page 17-1

2 Method As you already know the entry arguments to the sight reduction tables and formulae are Lat, LHA, & Dec. They are used exactly as they would be for a sight reduction using the following substitutions. Use the Latitude of the point of departure (rounded in the tabular method) as Lat. Use the Longitude of the Destination as GHA. Calculate LHA in the normal manner using the Longitude of the departure as the AP longitude for the tabular method. Use the Latitude of the destination as the Dec. The outputs from the tables and the formulae are HC and Z. Great Circle Distance (nm) = (90 - Hc) x 60 Zn = Initial course to steer. Example Calculate the great circle distance and the initial course to steer from Boston (42 21 N, W) to La Rochelle (46 10 N, W) Tabular Method Use the following values in the NAO or HO 249 tables: Lat = 42 N; LHA = 290 ; Dec = N (GHA = ; LHA = = 290 ) NAO Table results: Hc = ; Zn = 60 The distance between the two points is actually the zenith distance, which you will remember is 90-Hc, converted from an angle to nautical miles. The distance calculation can be done in a couple of ways: 1. Hc as a decimal. a. Convert the minutes of Hc to decimal degrees; 12/60 = 0.2; b. Hc = 41.2 ; Zenith Distance = = 48.8 c. D= 60 x 48.8 = 2928nm 2. Split the Zenith Distance into a degrees part and a minutes part. a. Zenith Distance = = b. D= (60 x 48).+ 48 = = 2928nm Page 17-2

3 However we are not finished. What we have so far is the distance between the AP and the destination. So a small correction is required to allow for the distance from the point of departure. A plot is the easiest way to determine the required correction. Set up a plotting sheet for the latitude of the point of departure (42 ) Plot the AP (L 42 N Lo W) Plot Zn (060 ) from the AP Plot the point of departure (42 21 N, W) Plot the initial course (060 ) from the point of departure Drop a perpendicular line from the departure point on the course line to the azimuth line (vice versa if necessary) and measure the distance between the AP and where the perpendicular intersects the azimuth line. Add or subtract this distance as appropriate. (In this case we need to subtract 14 nm) The distance between Boston and La Rochelle = = 2914nm and the initial course is 060 true. Page 17-3

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5 Calculator Method Another way to do this is to use a scientific calculator. In this case we can use the actual co-ordinates of the point of departure and arrival and LHA does not have to be a whole number. Lat = N, Dec = N, LHA = = Here is the formula for Hc Hc = Sin-1 [(Sin lat * Sin dec) + (Cos lat * Cos dec * Cos LHA)] Hc = Sin-1 [(Sin * Sin ) + (Cos * Cos * Cos )] Hc = Sin-1 [(0.674 * 0.721) + (0.739 * * 0.343)] Hc = Sin-1 [(0.486) + (0.176)] Hc = Sin-1 [(0.662)] Hc = Zenith Distance = (90 - Hc) = = Distance = 60 * = This correlates well with the tabular answer. Here is the formula for Z Z = Cos -1 [(Sin dec (Sin lat * Sin Hc)) / (Cos lat * Cos Hc)] Z = Cos-1 [(0.721 (0.674 * 0.662)) / (0.739 * 0.75)] Z = Cos-1 [( ) / (0.554)] Z = Cos-1 [0.275 / 0.554] Z = Cos-1 [0.496] Z = which of course would be rounded to 60. This correlates well with the tabular answer. Here s a comparison of the answers: Source Distance nm Initial Course NAO Calculator Web Javascript Calculator Page 17-5

6 17.1. Exercise a) Calculate the great circle distance and the initial course to steer from Halifax (44 40 N, W) to Gibraltar (36 7 N, 5 20 W). Use a tabular method and then verify your answers using the formulae and a scientific calculator. Your answers should be close to 2655nm and 080 b) Calculate the distance and initial course to steer from Gibraltar to Halifax. Your answers should be close to 2655nm and 300 c) Why are the courses not reciprocals? Page 17-6

CHAPTER 24 THE SAILINGS

CHAPTER 24 THE SAILINGS INTRODUCTION 2400. Introduction Dead reckoning involves the determination of one s present or future position by projecting the ship s course and distance run from a known position.