Right, let's pick this up with the actual mechanics of twilight — Page 432, Lesson 392

Right, let's pick this up with the actual mechanics of twilight. We've established the basic idea of the Sun's path, but now we're going to look at how that path dictates the length of twilight, and we start with a very specific, and very important, case: the Equator. There's a common romantic notion that at the Equator, the moment the Sun sets, it's pitch black. That is completely false, and I want to show you why. Let's consider an observer standing exactly on the Equator on or about March 21st or September 21st — these are the Equinoxes. On those days, the Sun's path is beautifully simple. Looking east, the Sun rises and travels apparently vertically upwards, until it is directly overhead the observer. Then it descends vertically down to the western horizon, where it sets. Now, let's look at the geometry of that sunset, because this is where the precise numbers come in. At the moment of sunset, at position A, the top edge of the Sun has just passed below the visual horizon. But here's the key: the Sun's centre is not on the horizon. Because of the Sun's apparent diameter, at that exact moment the centre of the Sun is 50 minutes of arc below the sensible horizon. That's the first number to lock in: 50' of arc. Now, twilight doesn't end at sunset. It ends when the Sun's centre reaches a specific depression below the horizon. For evening civil twilight, that point is when the centre of the Sun is 6 degrees below the sensible horizon. So, we have a start point at 50 minutes of arc, and an end point at 6 degrees. The duration of twilight is the angular arc the Sun's centre must travel between those two points. That's 6 degrees minus 50 minutes, which gives us 5 degrees and 10 minutes of angular arc. And here's the beautiful part: at the Equator, because the Sun descends vertically, that angular distance converts directly into a time. 5 degrees 10 minutes of arc corresponds to 21 minutes of time. So the minimum period of twilight at the Equator is 21 minutes. That's our baseline, and it's a critical number to remember. Now, let's move away from the Equator and see how this duration changes with latitude. We're going to use that equatorial case, our 21 minutes, as our baseline. Let's call that case (i). Now consider case (ii): an observer whose latitude is farther north than the declination of the Sun. In this example, that's an observer in the Northern hemisphere. For this observer, the Sun will rise in the East, and it will travel westward. At 1200 LMT — that's Local Mean Time, noon — the Sun will be due south of the observer. And it will set in the West, but following a general path that is inclined, not vertical. The key result here is that the duration of twilight will be longer than 21 minutes. Now case (iii): an observer whose latitude is farther south than the declination of the Sun — in this example, an observer in the Southern hemisphere. The Sun will rise in the East, travel westward, and at 1200 LMT, noon, the Sun will be due north of the observer. It sets in the West, following the general path indicated. And again, the duration of twilight will be longer than 21 minutes. So the core principle is this: the minimum twilight of 21 minutes only happens at the Equator at the Equinoxes, where the Sun's path is vertical. Anywhere else, where the Sun's path is inclined to the horizon, the Sun takes longer to descend through that 6-degree band, so twilight lasts longer. And for both of those cases, the duration of twilight can be calculated using the Air Almanac — that's the professional reference you'd use to get the precise figure for your specific latitude and date.