Question 4 gives us two points: J at 58°12'N 004°00'W, and K at 58°12'N… — Page 256, Lesson 231

I want to walk you through a set of practical navigation questions that bring together convergency, rhumb line tracks, and great circle tracks. These are the kind of calculations you'll be doing as a professional pilot, so let's take them step by step. Question 4 gives us two points: J at 58°12'N 004°00'W, and K at 58°12'N 006°00'E. Notice they share the same latitude — both are at 58°12'N — so they lie on the same parallel. Part (a) asks for the value of convergency between J and K. Convergency is the angle at which meridians approach each other as you move toward the pole. The formula for convergency is: change of longitude multiplied by the sine of the mean latitude. Here, the change of longitude from 004°00'W to 006°00'E is 10 degrees — that's 4 degrees from the prime meridian to 004°W, plus 6 degrees from the prime meridian to 006°E. The mean latitude is 58°12'N. The sine of 58°12' is approximately 0.85. So convergency equals 10° × 0.85, which gives us 8.5°. That's the answer to part (a). Part (b) asks for the rhumb line track from J to K. Since both points are on the same latitude, the rhumb line track is simply along the parallel — due east or due west. From J at 004°W to K at 006°E, you're going eastward, so the rhumb line track is 090°(T). The answer key confirms that: on the same latitude, a rhumb line track is 090°(T). Part (c) asks for the initial great circle track from K to J. Now, the great circle track is not the same as the rhumb line track when you're not on the equator or a meridian. The initial great circle track from K to J will be different from the reciprocal of the rhumb line track from J to K. The answer given is 274.25°(T). That's the initial true great circle track when departing K heading toward J. Notice that the reciprocal of 090° would be 270°, but the great circle initial track is 274.25° — that's 4.25° greater than 270°, which relates to the convergency we calculated. Question 5 describes an aircraft flying a great circle track via three waypoints: 53°N 030°W, then 53°N 020°W, then 53°N 010°W. All three are on the same latitude of 53°N, but the aircraft is being steered along great circle tracks between them, not rhumb lines. The question asks: the track change on passing the second turning point will be approximately what? Let me explain what's happening here. The rhumb line track all the way from waypoint 1 to waypoint 3 is 090°(T) because they're on the same latitude. However, the INS — the Inertial Navigation System — will steer the aircraft along the great circle track from waypoint 1 to waypoint 2, then at waypoint 2 it changes over to a continuous great circle track from waypoint 2 to waypoint 3. On each leg, the great circle track follows what we call the D-I-I-D rules. That stands for: in the Northern hemisphere, going easterly, the track angle increases. So on the first leg from 030°W to 020°W, the great circle track starts at less than 090°(T) and ends at more than 090°(T). Same on the second leg from 020°W to 010°W. At waypoint 2, when the INS switches from the first great circle leg to the second, there will be a left turn at the waypoint changeover. To find the approximate track change, we calculate the conversion angle. Conversion angle is half the change of longitude multiplied by the sine of the mean latitude. For each leg, the change of longitude is 10 degrees — from 030°W to 020°W, or from 020°W to 010°W. The mean latitude is 53°N. The sine of 53° is approximately 0.8. So conversion angle equals ½ × 10 × 0.8, which is 5 × 0.8, giving 4 degrees. That means on each leg, the great circle track starts 4 degrees less than the rhumb line track and ends 4 degrees greater. At waypoint 2, the aircraft finishes the first leg at a track of about 094°(T) — that's 090° plus 4° — and then starts the second leg at about 086°(T) — that's 090° minus 4°. So the track change at the waypoint is a decrease of approximately 8 degrees. The answer is an 8° decrease, which corresponds to option (d). Question 6 gives us point A at N55° E/W000° — that's 55°N on the prime meridian — and point B at N54° E010°. The initial true great circle track from A to B is given as 100°(T). The question asks: what is the true rhumb line track at A? Here we need to use the relationship between great circle track and rhumb line track. The conversion angle tells us the difference between them. Conversion angle equals half the change of longitude times the sine of the mean latitude. The change of longitude from A to B is 10 degrees — from 000° to 010°E. The mean latitude is the average of 55°N and 54°N, which is 54.5°N. The sine of 54.5° is approximately 0.815. So conversion angle equals ½ × 10 × 0.815, which is 5 × 0.815, giving about 4.075°, or roughly 4°. Now, in the Northern hemisphere, when flying easterly, the great circle track is initially less than the rhumb line track. The initial great circle track from A to B is 100°(T). Since the great circle track starts off less than the rhumb line track, the rhumb line track at A must be greater than 100°(T) by the conversion angle. So the true rhumb line track at A is 100° plus 4°, which is 104°(T). That corresponds to option (c). Let me recap the key principles we've used. Convergency is change of longitude times sine of mean latitude. Conversion angle is half of convergency. In the Northern hemisphere, going easterly, the great circle track starts less than the rhumb line track and ends greater — that's the D-I-I-D rule. And when points share the same latitude, the rhumb line track is simply 090° or 270° depending on direction.