Convergency and Conversion Angle — Page 247, Lesson 221

I want to walk you through the concept of convergency — also called Earth convergence — which is absolutely central to understanding how direction works on a sphere, and why your great circle track changes as you fly. Let’s start with the definition. Convergency is defined as the angle of inclination between two selected meridians, measured at a given latitude. In plain language: if you pick any two lines of longitude on the Earth, they don’t stay parallel — they get closer together as you move toward the poles. The angle between them at a particular latitude is the convergency. Now, let’s look at the extremes to see the pattern. At the Equator, where latitude is 0°, the meridians are parallel to each other — they don’t converge at all. So the ratio of convergency to change in longitude is zero. At the Poles, where latitude is 90°, the meridians all meet. The angle between them at the Pole is exactly equal to the change in longitude between them — so the ratio of convergency to change in longitude is one, or a factor of 1.0. Take a look at Figure 14.5, which shows this as a 2-D drawing of the spherical Earth problem. In that figure, you can see a table. For a position at the Equator, latitude 0°, with a change of longitude of 60°, the convergency is 0°. For a position at the Pole, latitude 90°, with the same change of longitude of 60°, the convergency is 60° — exactly equal to the change in longitude. So the relationship between convergency and latitude is a function of the sine of the latitude. That gives us the formula: Convergency = Change in Longitude × Sine Latitude Let’s see that formula in action with Figure 14.6. In this figure, the latitude chosen is 30° North. Tangents have been drawn to each meridian at this latitude. The angle between the slope of those two tangents is the convergency. In this example, the change of longitude is 40°. The sine of 30° is 0.5. So convergency equals 40° times 0.5, which gives us 20°. So now we have a formula for calculating convergency between any two meridians at the same latitude. But you might be wondering — why does a pilot need to know this? Here’s the key: at any point on the Earth, True North is defined with reference to the direction of the local meridian. If you fly to a different meridian than where you started, your local direction of True North has changed. This is exactly why a rhumb line — a line of constant true bearing — is a continuously curved line on the Earth’s surface. If it didn’t constantly curve, it couldn’t cut all the meridians, which are at different angles to each other, at the same angle. Now, if you are flying a great circle track — and all automatic computing systems like Inertial Navigation Systems, Global Positioning Systems, and Flight Management Computers compute steering signals along great circle tracks — then your great circle track will alter over time with respect to True North by exactly the amount of convergency. The track direction is defined by reference to the meridians, and if the direction of the meridians changes, then the track direction changes. So the change in great circle track direction is the angle of inclination of the meridians at the two points where the track is measured. That angle is the convergency.