If the Earth were perfectly spherical, and its internal density distribution had a particularly simple form, then the gravity field of Earth would be a perfect inverse square law, as described by Isaac Newton. However, Earth is not perfectly spherical, nor does it have a simple internal mass distribution. There are topographical features that spoil that perfection, such as mountains that are 8 km (5 miles) high and ocean trenches that are 11 km (7 miles) deep. In addition, Earth's shape is basically that of an oblate spheroid (Fig. 3.1)—a squashed sphere—such that if you were to stand on the poles, you would be 21 km (13 miles) closer to Earth's center than if you stood on the equator.

This is not too surprising given that Earth rotates on its axis once per day, causing the equator to bulge and the poles to flatten. This degree of flattening of Earth—21 km in a radius on the order of 6400 km (4000 miles)—is not too noticeable, however. For example, if you were to stand on the moon and look back at Earth, this degree of oblateness would probably not distract you from the apparent spherical perfection of the brilliant blue orb in the black sky. However, if you think of it in terms of the possible perturbing effects on a spacecraft in a LEO, there is an extra 21 km of crust—of gravitational mass—at the equator, which will have a significant

effect on the spacecraft's motion as it flies over the equatorial region. The gravity field of Earth deviates from that proposed by Newton, causing gravity anomaly orbit perturbations.

What effect does the oblate shape of Earth have on the motion of orbiting satellites? The answer to this is a fairly complex affair, and one that has always challenged me in my career as a teacher in space engineering, even when I am allowed to use mathematics! There are two main effects on the orbit, perigee precession and nodal regression, both of which produce major changes in the orbital motion.

Perigee precession is a gravity anomaly perturbation that affects elliptic orbits. In an ideal elliptic orbit, the major axis of the ellipse, that is, the line from perigee to apogee, remains fixed in direction. If you decided you wanted your spacecraft to be in an orbit with the perigee point above the North Pole, then you would launch into a polar orbit in such a way as to achieve this. In the absence of gravity anomalies, the perigee would remain frozen above the North Pole as required. However, the presence of the extra gravitational mass around the equator, due to Earth's oblateness, causes a greater acceleration on the spacecraft in the perigee region, which in turn causes the trajectory to curve a little more acutely. As a result, when the spacecraft climbs to its apogee, the apogee point has moved, causing the line of the major axis to rotate in the plane of the orbit. This is illustrated in

Figure 3.2, where the major axis of the orbit has rotated through an angle a (from 1 to 2) over four orbit revolutions.

In fact to move the major axis through this angle may require many orbit revolutions, particularly for the size of orbit shown, but the diagram has exaggerated the effect to aid clarity. Coming back to our orbit with the perigee above the North Pole, due to the effects of perigee precession, the perigee point will not stay fixed above the Pole but will move steadily around the orbit.

The rate at which the perigee moves is dependent on the size, shape, and orbital inclination of the orbit, so it is difficult to generalize. One thing that can be said, however, is that low orbits are affected more than high ones, since the influence of the extra mass due to the equatorial bulge is more strongly felt when a spacecraft makes a low pass over the equator. To give you an idea of the numbers, for a low elliptical orbit with a perigee height of 300 km (185 miles), an apogee height of 500 km (310 miles) and an orbital inclination of 30 degrees, the major axis of the orbit will rotate at a rate of about 11 degrees per day. To get a feel for altitude dependence, if we stick with the same orbital inclination but increase the perigee altitude to 1000 km (620 miles) and the apogee altitude to 10,000 km (6200 miles), then the perigee precesses at a rate of about 2 degrees per day. In terms of magnitude, this is a large perturbation, even for the higher orbit. Other changes to the orbit that mission analysts get excited about, due to other types of perturbation, are typically of the order of fractions of a degree per day!

Nodal regression is a gravity anomaly perturbation that affects both circular and elliptic orbits. The first question is, What is a node? You may remember the answer from Chapter 2. This is just the point on the orbit where the spacecraft crosses the equator. Clearly it does this twice on each orbit revolution, once when traveling from south to north, and once on the other side of the orbit when traveling from north to south. When the spacecraft motion is "ascending" from south to north, the equator crossing is called the ascending node, and when descending from north to south we have a descending node. The line between the nodes—the intersection of the orbit plane and the equatorial plane—is called the line of nodes. For an ideal orbit, the line of nodes remains fixed in direction with respect to the distant stars, but in a real orbit the gravity anomaly perturbation causes it to move around the equator. This nodal movement—or nodal regression—is illustrated in Figure 3.3a, and is due to the extra mass associated with Earth's equatorial bulge.

For orbital inclinations less than 90 degrees, the node moves west around the equator (as shown). When the inclination is 90 degrees—a polar orbit— the node remains stationary, and when the inclination is greater than 90 degrees the node moves east. As the node moves, the orbit plane rotates, while the orbital inclination remains constant. This nodal movement, and plane rotation, will continue indefinitely, as shown in Figure 3.3b. The tip of the arrow describes a circle, while the arrow always remains at right angles to the orbit plane. How nodal regression is produced by Earth's oblate shape is a little difficult to explain, but it is related to something called gyroscopic precession.

Figure 3.3: (a) In an ideal orbit the node remains stationary, but Earth oblateness causes the node to move so that the spacecraft crosses the equator at a slightly different position on each orbit, (b) Another way of describing the effect is to imagine an arrow that always remains perpendicular to the orbit plane, Over time, Earth's oblateness causes the tip of the arrow to describe a circle,

Figure 3.3: (a) In an ideal orbit the node remains stationary, but Earth oblateness causes the node to move so that the spacecraft crosses the equator at a slightly different position on each orbit, (b) Another way of describing the effect is to imagine an arrow that always remains perpendicular to the orbit plane, Over time, Earth's oblateness causes the tip of the arrow to describe a circle,

If you are interested to know what this is about, and perhaps have a bit of a technical background, then read the Nodal Regression box. If not, then you can skip it without compromising your understanding of what follows.

The effect of nodal regression on an orbit is similar to the effect of the precession of a gyroscope, when subjected to a torque. A torque is a rotational force, like the force you have to apply to remove a bolt from a wheel when you get a flat tire. You apply a force in a rotational sense by pushing at the end of the handle of the wrench. The size of the applied torque is not just to do with the amount of force exerted but also the length of the handle of the wrench you are using. The longer the handle, the greater the "moment arm" and the more torque there is. The axis of the torque is parallel to the direction about which the rotation takes place—in this case the long axis of the bolt.

Now, if we refer to the diagram, Earth is depicted as a rather exaggerated oblate sphere. The orbital motion of the spacecraft about Earth produces a spin axis 5, called an orbital angular momentum vector, which is perpendicular to the orbit plane. At the northernmost position on the orbit, point 1, the gravity force on the spacecraft is deflected slightly downward to the extra mass around the equator, producing a small out-of-plane component of gravity as shown. 5imilarly, at point 2 a small out-of-plane force is produced, but this time directed upward. The combination of these small forces produces a torque on the orbit shown as the arrow T, the direction of which is aligned with the orbit's line of nodes N — N'. As with a gyroscope, if the orbit plane is torqued in this way, its spin axis will tend to align itself with the torque axis; that is, the angular momentum vector Swill precess toward the torque vector T. Put more simply, the spin axis Swill tilt toward the torque axis T. Since the spin axis S is always perpendicular to the orbit plane, as S tilts so does the orbit plane, causing the node N to move westward along the equator. Also, given that the torque axis T remains parallel to the line of nodes, its direction also rotates westward in the orbit plane. The result is a precessional motion of the orbit spin axis as shown in Figure 3.3b.

How big an effect is nodal regression? If we take a circular orbit typical of a space shuttle, for example 300 km (185 miles) altitude with an orbital inclination of 30 degrees, then the orbit node will move at a rate of about 7 degrees per day in a westward sense. This is again a huge effect compared to other types of perturbation. The effect is less for higher orbits, however, since the spacecraft is further away from the extra mass associated with the equatorial bulge of Earth. For example, a circular orbit at 10,000 km (6200 miles) altitude, with the same inclination, has a modest nodal regression rate of about 0.3 degrees per day.

The bottom line of this discussion about the effects of Earth oblateness is that they are really, really important in low Earth orbit spacecraft operations. If they are neglected, then the position of the spacecraft over time will be in error by many thousands of kilometers!

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