Another kind of gravity anomaly perturbation is nicely illustrated by considering the motion of a spacecraft in geostationary Earth orbit (GEO). As we are now aware, the Earth's shape is predominantly that of a squashed sphere, but there is another aspect of the Earth's shape that is surprising. If you were to slice the Earth through the equatorial plane, the shape of the resulting cross section is not circular but approximately elliptical. The Earth's shape can be represented, in an exaggerated way, by the outline shown in Figure 3.4. To get this shape, we take a sphere and give it a good squeeze at the poles to make it oblate, and then give it a small squeeze at the equator, to make the equatorial cross section slightly elliptical. We end up with a form defined by three perpendicular axes, a, b, and c, each having different lengths. From our previous discussions, we know that the equatorial radius b is greater than the polar radius a by about 21 km (13 miles). But now we are suggesting that b and
c are different also, but this time by a small amount—less than a kilometer. If we recall that the equatorial radius of Earth b = 6378 km (3963 miles), then this difference between b and c means that the degree of ellipticity of the equatorial cross section is small indeed. However, we now focus attention on this, and discuss the effect it has on the motion of a GEO spacecraft.
But how can such a small deviation in the shape of the equatorial cross section have any effect on an orbiting spacecraft? The answer to this question comes from the fact that although the perturbing forces are small, they tend to act on the spacecraft in the same way on each orbit revolution, so that lots of small changes accumulate to cause an effect that is sizable.
To explain this, in Figure 3.5 we are looking down on Earth and the GEO. The elliptical cross-section of the equator is shown in a rather exaggerated way, with the Greenwich Meridian drawn in to relate the ellipse to Earth's geography. In Figure 3.5a, the points A and B represent the "bulges" in the equatorial cross section that occur at around 160 and 350 degrees longitude east, respectively (it may be helpful to have a globe of the world handy while reading this chapter to help with the geography). This places them in the region of the western Pacific Ocean on the one hand, and the west African coast on the other. A spacecraft is also shown in GEO at an arbitrary position, which happens to be at a longitude of about 40 degrees east. Note that the longitudinal position of the GEO spacecraft would depend on what region it serves; in this case it just happens to be stationed above east Africa and the Middle East.
An important thing to note about Figure 3.5a is that, for an ideal GEO, the geometry is unchanging. What I mean is that the Earth rotates once per day, in the same time as the spacecraft takes to complete one orbit. The situation is equivalent to rotating the whole of Figure 3.5a about the North Pole axis once per day. In this process the relative positions of Earth and spacecraft do not change. If we now focus on the spacecraft, the gravity force acting on it is modified by the elliptical mass distribution around Earth's equator. Because the spacecraft is closer to the "bulge'' at B than that at A, the direction of the gravity force, indicated by the arrow labeled g, is deflected slightly toward B, rather than pointing precisely at Earth's center. The deflection is exaggerated in the diagram for clarity; in reality it is tiny. The resulting force acting on the spacecraft can now be resolved into two components, the main one labeled g1 directed toward Earth's center, and a tiny force component g2 acting in the local horizontal direction at the spacecraft.
To illustrate the concept of resolving forces into components, a good example is the use of a heavy roller to flatten the bumps in a lawn. If we look at Figure 3.6, there are basically two ways of doing this: we can either push the roller or pull it. Does it make any difference which way we choose? Well, if we resolve the forces into components as shown, we can see that it does. When we push the roller (Fig. 3.6a), the force we use (the solid arrow) can be resolved into two force components (the broken arrows)—one directed horizontally to move the roller along, and another vertical component directed downward. When we pull the roller (Fig. 3.6b), there is the same horizontal component, but now the vertical component is directed upward.
If we push the roller, we produce a vertical force component that tends to increase its effective weight and therefore increase friction with the ground, making it harder to move. On the other hand, if we pull the roller, we tend to reduce the ground friction, making it easier to move. The trick of resolving forces into components is often used by engineers, and in the case of the force of gravity acting on a GEO spacecraft, it is just convenient for the argument to resolve it into the vertical and horizontal directions.
Equipped with this intuitive notion of force components, let's return to our discussion about GEO perturbations. Having components of gravity acting in the local horizontal direction is something that we are not very familiar with! In our normal experience, gravity forces always act down the local vertical direction. But in this case the elliptical mass distribution at the equator is producing this rather strange occurrence. The consequences of this for the spacecraft motion are significant. In Figure 3.5b, at point 1 the small horizontal component of gravity force acts in a direction that is opposed to the spacecraft's motion around the GEO; this direction is referred to as retrograde. This small retrograde force causes a decrease in the energy of the orbit, with a corresponding decrease in orbit height. What happens if the height of a GEO decreases? The spacecraft orbit speed increases, and it goes round the orbit in slightly less than 1 day; it is no longer synchronous with Earth's rotation. As well as losing height, it also drifts away from its on-station longitude. This situation is depicted by the broken curve from point 1 to point 2. It should be noted that the height change illustrated at point 2 is again exaggerated to make it clear.
At point 2, the spacecraft is at the same distance from the bulges at A and B, and all the gravity force is now directed toward Earth's center. However, because the orbit height is still low, the spacecraft continues to drift, and once it passes point 2 the small horizontal gravity force recurs. But now its direction is reversed, as the spacecraft is now closer to bulge A. The small horizontal gravity force is now in the prograde direction—in the same direction as the spacecraft's motion. This tends to increase the orbit energy, causing the height to increase again, as indicated by the broken curve from point 2 to point 3. At point 3, the spacecraft has regained GEO height, and so becomes synchronous again, halting the drift in longitude. However, the force continues to act in a prograde direction, causing the orbit height to increase above the GEO. Now the orbit speed becomes less than the GEO speed, and the synchronism is again lost as the spacecraft drifts in the opposite direction, represented by the broken curve from point 3 to point 4. Beyond point 4, the force reverses, acting in a retrograde direction once again. This causes a reduction in orbit height until the spacecraft finally returns to its on-station position at point 1.
This rather interesting circuitous journey takes a typical spacecraft quite a long time—of the order of hundreds of days—and is generally a bit of a nuisance for the spacecraft operators, who would prefer the spacecraft to stay at the required on-station position! The operators have to plan and execute orbit control maneuvers to combat the effects of these gravity anomaly perturbations. This means firing small rocket thrusters on the spacecraft (see Chapter 9) to ensure that the spacecraft stays in position. Without this, the spacecraft would oscillate indefinitely in longitude about the stable point S, which is where the minor axis of the equatorial ellipse cuts the GEO arc. In the example above, this would mean that an uncontrolled spacecraft would wander off from its on-station position at 40 degrees to around 120 degrees longitude east (East Africa to Indonesia), and back again on a regular basis. I always find it amazing how such a small variation in Earth's shape can cause such a large change in the spacecraft's orbit!
Note that the orbital positions above the bulges in the equator are unstable—labeled US in Figure 3.5b. Uncontrolled spacecraft positioned near these would move off toward the nearest stable point, labeled S.
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