Convergency and Conversion Angle — Page 247, Lesson 223

I want to walk you through a key idea in navigation: convergency and how we calculate it when a great circle crosses two meridians at different latitudes. Let’s start with Figure 14.7. It shows a great circle cutting two meridians, which we’ll call meridian X and meridian Y. At the points where the great circle cuts each meridian, we’ve drawn tangents to those meridians. The convergency is shown in the same way we’ve seen before — as the angle between those two tangents. Now, the great circle direction at meridian X is represented by the angle ‘a’. The great circle direction at meridian Y is represented by the angle ‘b’. The difference between these two directions — that is, ‘b’ minus ‘a’ — equals the convergency. So convergency is simply the change in great circle direction as you move from one meridian to the other. But here’s the important twist: in this case, the great circle does not cross the two meridians at the same latitude. In fact, there are only two mean track directions where that happens — due East or due West. On any other mean track, by definition, there will be some element of north or south latitude change between the two meridians. Because the latitudes are different, we cannot use the simple convergency formula we might have used before. Why? Because that simple formula requires a single term for ‘sine latitude’. Since we have two different latitudes, we need the sine of the mean latitude between the two points. However, determining the true mean latitude is a complex process. The mean latitude will actually be closer to the nearer pole than the mid-latitude is — with two exceptions. So, although the formula technically requires mean latitude, for simplicity we use mid-latitude instead. The difference between the sine of mid-latitude and the sine of mean latitude is so small that it is virtually insignificant — unless a large change of longitude is involved, which is unlikely in typical navigation scenarios. So let me summarize what convergency is: it is the angle of inclination between two selected meridians, measured at a given latitude. It is equal to the difference between the great circle directions measured at each meridian. And its value may be calculated from the formula: Convergency = Change in Longitude × Sine Mean Latitude In practice, we substitute mid-latitude for mean latitude in that formula, and the result is accurate enough for our purposes.