Departure — Page 266, Lesson 235

I want to walk you through the concept of departure, which is a fundamental idea in general navigation. Let's start with the definition. Departure is the distance between two meridians along a specified parallel of latitude, and it's usually expressed in nautical miles. Now, let's think about what that really means. Look at Figure 15.2 in your materials. You can see that two meridians represent the same change of longitude, no matter what latitude you're at. But here's the key point: because meridians converge from the Equator toward the poles, the same change of longitude does not represent the same east-west distance in nautical miles at different latitudes. If we take the Equator as our specified parallel of latitude, the distance between two meridians will be greater than at some higher latitude. So departure varies with latitude. Let me give you the two extreme cases. Departure is maximum at the Equator, where one degree of change of longitude equals 60 minutes of arc of a Great Circle. At the other extreme, departure is zero at both poles because the meridians converge and meet at those two points. So departure varies as the cosine of the latitude. That gives us the formula you'll use: Departure in nautical miles equals change of longitude in minutes multiplied by the cosine of the latitude. In symbols: Departure (NM) = ch.long (in minutes) × cos lat Since this distance is always measured along a parallel of latitude, it represents a Rhumb Line distance — that's a line of constant bearing, which follows a parallel of latitude. Let's work through a calculation example to make this concrete. Consider two meridians on the Earth joined by a parallel of latitude. In the example in your materials, the change of longitude is 20 degrees. First, we multiply by 60 to convert that into minutes — so 20 × 60 gives us 1200 minutes. Then we multiply by the cosine of 52 degrees. Departure = 20 × 60 × cos 52° = 738.8 nautical miles. So the east-west distance along that parallel of latitude at 52 degrees is 738.8 nautical miles. Now, there are two main types of departure questions you'll encounter. The first type involves variations on the basic departure formula. The second type gives you departure at one latitude and asks you to calculate it at another. Let me walk through the first type with two examples. Example 1: An aircraft at position 60°00'N, 005°22'W flies 165 kilometers due east. What is the new position? First, we convert 165 kilometers to nautical miles. 165 divided by 1.852 equals 89 nautical miles. Now substitute into the departure formula: 89 = ch.long × cos 60°. Cosine of 60 degrees is 0.5, so 89 = ch.long × 0.5. Therefore, ch.long = 89 divided by 0.5, which is 178 minutes of longitude. 178 minutes of longitude is 2 degrees and 58 minutes. Since we're flying east from 005°22'W, we add that 2°58' eastward. The new longitude becomes 002°24'W. So the new position is 60°00'N, 002°24'W. Example 2: In which latitude is a difference of longitude of 44°11' equivalent to a departure of 2000 nautical miles? First, convert 44°11' into minutes of arc. 44 degrees times 60 gives 2640 minutes, plus 11 minutes gives 2651 minutes. Now use the formula: 2000 = 2651 × cos lat. Rearranging, cos lat = departure divided by ch.long, so cos lat = 2000 divided by 2651, which equals 0.7544. The latitude whose cosine is 0.7544 is 41 degrees. And since the cosine is the same for north and south, the latitude is 41° North or South. That's the core of departure — the east-west distance between meridians, how it varies with latitude, and how to calculate it using the formula.