Why planes fly over Greenland
Tonight's idea
Pull a string tight between two points on a ball. It marks the shortest path, and it always lies along a great circle: a circle whose centre is the centre of the sphere. The equator is one. So is any meridian (a line of longitude) with the one opposite it.
A circle of latitude such as 60°N is not. It is smaller, and its centre is north of the Earth's centre. Following it is a detour. The great circle between two northern cities swings toward the pole.
On a flat map that looks wrong, because the map stretched the north (Why Greenland looks as big as Africa). The orange route is straight there, yet nearly 1,000 km longer than the blue one over southern Greenland.
Sphere of radius 6,371 km; London 51.5°N, 0.1°W; Los Angeles 34.1°N, 118.2°W. Both distances are approximate.
The Earth is close to 40,000 km round, so \(\theta\) degrees of a great circle is \(\tfrac{\theta}{360} \times 40{,}000\) km.
Watch one: two cities at 60°N, on opposite sides of the Earth
Along the 60°N circle. Its radius is half the orange line in the side view, from the axis to a city: the short side of a 30°, 60°, 90° triangle with hypotenuse \(R\), so \(\tfrac12 R\) (Two triangles worth knowing by heart). Half the radius gives half the circumference, 20,000 km. Half way round that is 10,000 km.
Over the pole. Each city is 30° from the pole, so the blue arc is 60° (A slice of a circle): \(\tfrac{60}{360} \times 40{,}000 \approx 6{,}667\) km. A third shorter.
Your turn
- Take the Earth as 40,000 km round. (a) How far is the equator from the North Pole? (b) How long is one degree of a great circle? (c) Two towns on the same meridian are at 20°N and 65°N. How far apart are they?
- Two ships are 90° of longitude apart. How far apart are they (a) along the equator, if both are on it; (b) along the 60°N circle, if both are on that? (c) In which case is that the shortest route, and why?
- Two cities are at 45°N on opposite sides of the Earth. The 45°N circle has radius \(R/\sqrt2\). How far is it (a) half way round that circle; (b) over the pole? (c) How many times longer is route (a)?
- Sydney is at 34°S and Santiago, in Chile, is at 33°S. (a) On a flat map with north at the top, does the shortest route between them bend up or down? Explain. (b) Describe two pairs of places for which the straight line on the map really is the shortest route.
Stretch: north-east from the equator
A plane leaves a point on the equator heading exactly north-east and flies straight ahead along a great circle. What is the farthest north it gets? Where does it next cross the equator, and how far has it flown by then? (Hint: a great circle is a tilted copy of the equator.)
Answers
- (a) A quarter of a great circle: 10,000 km. (b) \(40{,}000 \div 360 \approx 111\) km. (c) They are 45° apart on a great circle: \(\tfrac{45}{360} \times 40{,}000 = 5{,}000\) km.
- (a) \(\tfrac{90}{360} \times 40{,}000 = 10{,}000\) km. (b) The 60°N circle is 20,000 km round, so a quarter of it is 5,000 km. (c) Only on the equator, because the equator is a great circle. At 60°N the great circle between the ships bends north of the 60°N circle and is shorter, about 4,600 km (this needs trigonometry you have not met yet).
- (a) The circle is \(40{,}000 \div \sqrt2 \approx 28{,}280\) km round, so half is about 14,140 km. (b) Each city is 45° from the pole, so the arc is 90°: 10,000 km. (c) \(\sqrt2 \approx 1.41\) times.
- (a) Down. Neither the 33°S nor the 34°S circle is a great circle, and the great circle between two southern cities swings toward the South Pole, just as northern ones swing toward the North Pole. This one goes down to about 62°S, far south of both cities. (b) Two places on the equator, and two places on the same meridian (one due north of the other). The equator and the meridians are great circles.
Stretch. Take the equator and tilt it by 45° about the line joining the starting point to the centre of the Earth. That is the plane's great circle, because at the start it crosses the equator at 45°. A circle tilted by 45° reaches 45°N at its highest point and 45°S at its lowest. It crosses the equator again at the point exactly opposite the start, half a great circle later: 20,000 km.