The conversion angle will be 42 degrees, so this makes the initial great… — Page 560, Lesson 558

Let's pick this up with the conversion angle problem, because that's the heart of great-circle navigation. We have a conversion angle of 42 degrees, and that gives us an initial great circle track of 132 degrees true. So the question is: where does that 42 degrees come from, and why does it turn into 132(T)? The conversion angle is the angular difference between the great circle track and the rhumb line track at any point. For a great circle, the track changes continuously as you fly, because you're following the shortest path over a sphere. The rhumb line, by contrast, cuts every meridian at the same angle — it's a constant track. The conversion angle tells you how much the great circle track differs from that constant rhumb line track at a given point. Now, the formula for conversion angle is half the change of longitude times the sine of the mean latitude. So if you're flying between two points, you take the total change of longitude, halve it, and multiply by the sine of the mean latitude — the average of the two latitudes. That gives you the conversion angle in degrees. In this case, the conversion angle comes out to 42 degrees. So the initial great circle track is 132 degrees true. That means at the start of your great circle, your track is 132(T), and as you fly, that track will gradually change because you're on a great circle, not a rhumb line. Now, let's move to a different concept — the relationship between latitude and change of longitude. Here's the key idea: 6 degrees south is a greater value of latitude than 4 degrees north. What does that mean? Latitude is measured from the equator, so 6°S is further from the equator than 4°N. For a given departure — that is, for a given east-west distance travelled — a greater latitude means you get a greater change of longitude eastwards. Why? Because at higher latitudes, the parallels of latitude are closer together. The same distance covers more degrees of longitude when you're further from the equator. So if you depart from 4°N, fly to 6°S, and then come back to 4°N, here's what happens. Going south, you cover 600 NM and that takes you a certain change of longitude eastwards. But when you come back north to 4°N again, that same 600 NM does not take you so far westwards, because at 4°N the parallels are wider apart. So you finish up east of where you started. That's the classic result — a closed loop at different latitudes leaves you displaced east or west. Now let's look at a fuel or weight conversion problem. This is about converting imperial gallons to kilograms. The key relationships: there are 5 imperial gallons to 6 US gallons, and the imperial gallon to litres conversion is 4.55. So if you have 380 imperial gallons, you convert to US gallons by multiplying by 5/6, then to litres by multiplying by 4.55, then to kilograms by multiplying by 0.78 — the density of the fuel. So the calculation is 380 × 5/6 × 4.55 × 0.78, which gives 1123.85 kilograms. You could do this on the CRP-5 flight computer, which is probably easier, but the arithmetic works out the same. Next, a time and distance problem. The distance still remaining is 475 minus 190, which is 285 nautical miles. The time to go is 1130 minus 1040, which is 50 minutes. So you have 285 NM to cover in 50 minutes. Now, a plotting question — these are solved by measurement on the chart. There's a note about both DME distances decreasing, and that explanation is given in the Plotting chapter. DME, or Distance Measuring Equipment, gives you slant range to a beacon. If both DME distances are decreasing, that tells you something about your position relative to the two beacons — but the full explanation is in the Plotting chapter. Next, a question about Jeppesen conventions. The Jeppesen conventions differ slightly from the ICAO ones. The key is given in the introduction to the Jeppesen Student Pilots' Manual. So if you're using Jeppesen charts, you need to be aware of those differences. Now, a great circle versus rhumb line question. The mean great circle is the same as the rhumb line track. The question is asking which pairs of latitudes will give the greatest difference between great circle and rhumb line track — that is, which will give the greatest conversion angle. The conversion angle formula is half the change of longitude times the sine of the mean latitude. So to get the greatest conversion angle, you want the greatest change of longitude and the mean latitude where the sine is largest — which is at 90 degrees, where sine is 1. Now, a temperature and Mach number point. Temperature normally decreases with increasing altitude. This means the speed of sound will decrease. So for a given TAS — true airspeed — the Mach number will increase. That's an additional effect. Mach number is the ratio of true airspeed to the speed of sound. If the speed of sound drops because it's colder, then for the same TAS, your Mach number goes up. Now, a variation problem. Apply 17 degrees west variation to 120 degrees magnetic to get 103 degrees true heading. Variation is the difference between true north and magnetic north. West variation means magnetic north is west of true north, so you subtract it from magnetic to get true. Then the island is 15 degrees true to the left, which makes the true bearing TO the island 088 degrees true. So 103 minus 15 is 88. Now, a departure problem. In one hour, the aircraft covers 360 NM. The departure formula is: departure equals change of longitude in minutes times cosine latitude. So 360 NM equals change of longitude in minutes times cosine 60, which is 0.5. So change of longitude equals 720 minutes. At the Equator, 720 minutes equals 720 NM, which also has to be covered in one hour. Finally, a standard time problem. Use a table. Standard Time at Kuwait: today at 0700 ST, STD — that's standard time — with longitude east, UTC is least, so you subtract 3 hours to get UTC. That gives 0400 UTC today. Then for Algeria, standard time is UTC plus 1 hour, so 0500 ST today. The table helps you keep track of the day and the hour. So that's the full set of answers and explanations. Each one is a different type of navigation problem — great circle tracks, latitude and longitude relationships, fuel conversions, time-speed-distance, plotting, variation, departure, and time zones.