Gyroscopes — Page 148, Lesson 172

Let’s pick this up right where the earth’s rotation starts to matter for your gyro. We’ve already seen that at the equator, the gyro’s axis stays fixed in space while the earth’s surface moves underneath it, so the apparent wander is zero. Now I want to walk you through what happens at the poles, and then everywhere in between, because this is where the numbers and the signs come in. At the pole, the gyro remains aligned in space, but the meridian direction—that is, the direction of local true north—changes as the earth rotates. So we get the full earth rate of 15° per hour. Why 15? Because the earth rotates one full revolution, 360°, in one day, 24 hours. 360 divided by 24 is exactly 15 degrees per hour. That’s the earth’s spin rate. Now, for anywhere between the equator and the pole, the earth rate is not the full 15. It’s given by this formula: Earth Rate = 15 × sine latitude, in degrees per hour. Let me unpack that. Latitude at the pole is 90°, and sine 90° equals 1, so you get the full 15. Latitude at the equator is 0°, and sine 0° equals 0, so you get zero. That matches what we said about the equator. So the sine of the latitude scales the earth rate from zero at the equator up to the full 15 at the pole. Now here’s the part that trips people up: whether earth rate is positive or negative depends on the hemisphere. Let’s look at the northern hemisphere first. In Figure 11.11, at position 1, the aircraft and the gyro are both pointing north. At position 2, the gyro remains orientated to the original ‘gyro north’, but the local earth direction of the parked aircraft is up the local meridian, i.e. true north. So if you take the gyro as the datum, the aircraft is now on a heading of about 280°. The gyro heading has appeared to decrease. We call this a negative earth rate, and so it has a value of -15 × sine latitude, degrees per hour. Now flip to the southern hemisphere, as in Figure 11.12. The earth rotates eastwards. As seen from above the north pole, looking down—Figure 11.11—that rotation is anticlockwise. But when seen from the south pole looking upwards, as in Figure 11.12, it’s a clockwise rotation. So if the aircraft is parked pointing northwards, that’s 12 o’clock in position 1, when the earth rotates, the aircraft is now still pointing in a direction of true north—outwards from the south pole—pointing at about 3 o’clock in the diagram. But if the gyro is taken as the datum, the gyro heading is about 080°. The gyro heading has appeared to increase. So the rule is: earth rate is negative in the northern hemisphere, and earth rate is positive in the southern hemisphere. That sign convention is what you’ll carry into the wander calculations later, so make sure you can see why—north hemisphere, the heading appears to decrease; south hemisphere, it appears to increase.