Departure and Rhumb-Line Track on a High-Latitude Sector
A CPL candidate plans a VFR sector from an airfield at 58°20'N 004°10'W to another at 62°50'N 003°30'E. The aircraft cruises at 115 KT TAS. The forecast wind is 250°(T)/35 KT throughout. The pilot intends to fly a rhumb line because the GNSS database route is not available and only a Lambert conformal chart is carried. Chart convergence for the sheet is stated as 0.8. The pilot computes a constant heading from the mercator-style mid-latitude track and notes that the magnetic variation along the route changes from 4°W at the departure airfield to 2°E at the destination. Before flight the pilot must decide whether the rhumb-line assumption and the single heading are acceptable for the fuel carried, which allows 3 hours 10 minutes of endurance with no reserve margin to spare.
1. Explain why the rhumb line and the great circle differ on this sector, and quantify roughly how large the difference in track is at the midpoint.
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The rhumb line crosses every meridian at the same angle, so on a Lambert conformal chart it plots as a straight line. The great circle plots as a curve concave toward the pole, and its track changes continuously along the route.
The convergence angle between the two is approximately the change of longitude multiplied by the sine of the mean latitude. The change of longitude here is from 004°10'W to 003°30'E, which is 7°40', or about 7.67°. The mean latitude is about 60.6°N, and sin 60.6° is about 0.872. The convergence is therefore roughly 7.67 × 0.872 ≈ 6.7°.
This is the total angle between the initial great-circle track and the rhumb-line track at the departure meridian, and the great circle's track rotates by this amount across the sector. At the midpoint the great-circle track differs from the rhumb-line track by about half that, roughly 3.3°. The rhumb line is the longer path but the difference over this short sector is small.
For a 115 KT aircraft the practical consequence is a lateral displacement of a few miles at most, so the rhumb-line assumption is acceptable provided the pilot allows for the track change and does not treat the single heading as exact for the whole leg.
2. The pilot proposes one constant magnetic heading for the whole leg. Assess this against the wind and variation data given.
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A single heading cannot be correct for the whole leg because both the true track and the variation change.
First, the track. If the pilot flies the rhumb line, the true track is constant and equals the rhumb-line track. If the pilot flies the great circle, the true track rotates by about 6.7° across the sector, so a constant heading would produce an increasing cross-track error.
Second, the variation. It runs from 4°W to 2°E, a swing of 6°. Magnetic heading equals true heading minus easterly variation (or plus westerly). At departure, with 4°W, the magnetic heading is the true heading plus 4°. At destination, with 2°E, it is the true heading minus 2°. A single magnetic heading therefore misstates the required heading by up to 6° at one end of the leg.
Combined with the wind, the drift correction also changes as the along-track and cross-track wind components vary with the changing track. The pilot should split the leg into two or three segments, recompute heading, groundspeed and ETA for each, and update variation at each turning point rather than fly one heading for the whole sector.
3. Given the 3 h 10 min endurance and the wind, decide whether the flight is safe as planned and state what the pilot should verify.
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The distance along the rhumb line is approximately the change of longitude times the cosine of the mean latitude. The change of longitude is 7.67° and cos 60.6° is about 0.491, so the departure is about 7.67 × 60 × 0.491 ≈ 226 NM. The great circle is slightly shorter, perhaps 224 NM.
The wind is 250°(T)/35 KT. The rhumb-line true track is roughly 070°(T) (northeast). The wind is therefore almost directly across the track, with a small headwind component. With 115 KT TAS, the groundspeed is close to 110 KT, perhaps 108 KT. At 108 KT, 226 NM takes about 2 h 05 min.
That leaves about an hour of endurance margin, which is adequate for the cruise but not generous once climb, descent, and any holding are counted. The pilot should verify the actual groundspeed against the forecast within the first 20 minutes, confirm the wind at altitude, and check that the fuel plan includes climb and descent allowances. If groundspeed falls below about 90 KT the margin erodes quickly, so a diversion or refuel decision point should be set.