In earlier chapters, we described a variety of orbits around Earth, but in this section we discover that it is possible for a spacecraft to orbit around a point in space where there is no mass! This rather fascinating state of affairs requires some explanation.

The story starts in the 1770s with an Italian-French mathematician named Joseph Lagrange. Note that this is about 100 years after Newton gave the world his revelations about how objects move in gravity fields, giving the world's scientists and mathematicians potentially centuries of research work

to do to reveal their full significance. Lagrange was using Newton's laws to study something called the three-body problem, which, as the name suggests, is the investigation of how three massive bodies move around each other under gravity. Unfortunately, he found the problem to be complex and unsolvable, which it remains to this day. However, his work was not entirely fruitless. In his attempts to make the problem more amenable to solution, Lagrange examined simplified versions of the full three-body problem and in the process discovered Lagrangian points, which, as we will see, are relevant to modern spacecraft mission design.

The simplified version that Lagrange looked at is shown in Figure 4.9, and involves two massive bodies, M1 and M2, in circular orbits around each other, and a third much smaller body M3 moving along a trajectory influenced by the gravity fields of its two larger neighbors. This setup is referred to as the circular restricted three-body problem (CRTBP). The important thing to note here is that the third body is so small in mass that it has negligible effect on the motion of the two larger bodies. So how do two massive bodies rotate around each other in circular orbits? If they are of equal mass, then they rotate about a point that is halfway between their centers (Fig. 4.10a). If their masses are dissimilar, they rotate about a point that is closer to the larger object (Fig. 4.10b). This point about which the rotation takes place is referred to as the barycenter of the system. For example, in the Earth-Moon system, the mass of Earth is about 81 times

larger than that of the Moon, so the barycenter about which Earth and the Moon orbit is only about 5000 km (3100 miles) from Earth's center— actually beneath Earth's surface.

If we return to Lagrange's simplified problem, illustrated in Figure 4.9, there are a number of good examples of this type of system that are relevant to modern spacecraft mission design. The most obvious of these, from the 1960s, is an Apollo spacecraft on its way to the moon. In this example, Earth and the Moon represent the larger bodies in (nearly) circular orbits around each other, and the spacecraft represents the third body of negligible mass. Another example of a CRTBP with wide applications is the Sun-Earth-spacecraft system.

In his mathematical exploration of the CRTBP, Lagrange discovered five points in the rotating system where the third body (of negligible mass) could remain stationary relative to the two larger bodies. These equilibrium points are illustrated in Figure 4.11, and are referred to as Lagrangian points L1, L2, L3, L4 and L5 in Lagrange's honor. When looking at Figure 4.11, it is important to realize that the relative geometry between the large bodies and the Lagrangian points remains fixed, and rotates about the system's barycenter.

What is the relevance of all this to spacecraft mission design? To focus the discussion, let's look at the situation where the two larger bodies are the Sun and Earth, and the smaller body is a spacecraft. Each Lagrangian point is a place where the forces of gravity and those due to the rotation of the system add up to nothing (we'll return in a moment to discuss what we mean by

"forces due to rotation''), which gives it the characteristic of being an equilibrium point. If you locate a spacecraft at any one of these points, it will remain there. To further focus the discussion, let's consider the L1 and L2 points in the Sun-Earth system, as these have attracted most interest in terms of spacecraft applications. In this system, the L1 point is located approximately 1,500,000 km (930,000 miles) away from Earth, in the direction of the Sun, and the L2 point is a similar distance from Earth in the opposite direction (Fig. 4.11). Positioning a spacecraft at the L1 and L2 points seems like a straightforward affair, apart from one detail. A more detailed look at the mathematics tells us that these are points of unstable equilibrium, which means that if the spacecraft is disturbed by the slightest perturbation, it will move away from the Lagrangian points. This state of unstable equilibrium is a bit like trying to balance a small metal ball, like a ball bearing, on top of a smooth dome-shaped surface. With enough care, you may be able to balance the ball at the summit of the dome, but the slightest disturbance will cause it to roll away down the slope. And so it is with a spacecraft. However, with the help of the spacecraft's propulsion system, it is possible to regain stability, and furthermore to control the spacecraft in an orbit around the Lagrange point. How the spacecraft orbits a massless point is most easily explained by considering the motion about the L1 point in the Sun-Earth-spacecraft system.

Strictly speaking, the L1 point is a location in the rotating system where the gravitational and rotational forces sum to zero. I think we are fairly happy thinking about the forces of gravity of the Sun and Earth acting upon the spacecraft, but what do we mean when we talk about rotational forces? The Sun-Earth system rotates only slowly, about 1 degree per day due to the Earth orbiting the Sun with a period of 1 year, so whatever forces there are due to rotation, they must be small. But in this instance, they do nevertheless play an important role.

Perhaps the best way to think about rotational forces is to imagine yourself on a small merry-go-round, the kind you see in a child's playground. Maybe as a child you stood on one of these, holding on to the safety rails, while a friend spun it up to perhaps an uncomfortably high speed. In this situation, you certainly get a good impression of a rotational force, as this is the force you feel tending to throw you off the merry-go-round. This outward directed force you experience in a rotating system—the merry-go-round in this case—is referred to as centrifugal force. To prevent yourself from being hurled off the merry-go-round, you have to hold on tightly to the safety railings. You are able to remain standing on the same spot on the merry-go-round because the force in your arms pulling toward the center of the merry-go-round balances the centrifugal force tending to throw you off.

As it happens, this is a remarkably good analogy to describe the manner in which the spacecraft can remain "standing on the same spot'' at the L1 point in the rotating Sun-Earth system. The force tending to pull the spacecraft toward the Sun is the Sun's gravity, and this is analogous to the force in your arms as you hold on tightly to the merry-go-round's safety rails. The force tending to pull the spacecraft outward is predominantly centrifugal, generated by the rotation of the system, but there is also a small contribution from Earth's gravity. There is a balance of forces on the spacecraft—solar gravity inward, and centrifugal force plus Earth gravity outward—allowing the spacecraft to remain stationary at the L1 point. To be precise, 97% of the outward force balancing solar gravity is centrifugal, and only 3% is Earth's gravity.

But we still have not addressed how a spacecraft can orbit the L1 point. In Figure 4.12 we see that if the spacecraft is displaced along the line joining the Sun and Earth to point 1 or point 2, then the sum of gravity and rotational forces is no longer zero, and the vehicle will tend to move away from the Lagrangian point. This is an expression of the instability of the L1 point that we referred to earlier.

However, there is a surface of stability upon which the spacecraft can orbit the L1 point. This is the surface at right angles to the Sun-Earth line, which passes through the L1 point, as shown in Figure 4.12. It is slightly curved, as shown in a rather exaggerated way in the figure, but it can be thought of as a plane surface in which the orbital motion takes place. Now we can see that if

the spacecraft is located at point 3 or point 4, the sum of the Sun's gravity force in one direction, and centrifugal force (with a little bit of Earth gravity) in the other, produces a resultant force directed toward the L1 point. In fact, it is easy to see that this L1 directed force occurs at any point on the illustrated halo orbit in the figure, thus allowing an orbital motion around the massless L1 point. The shape of the orbit around L1 is certainly not a conic section, in general, and can in fact be a weird variety of looping curves referred to as a Lissajous orbit. Also, as we have mentioned already, the spacecraft has to use its propulsion system to tweak the motion to ensure long-term stability of the orbit.

A similar explanation of the orbital motion around the L2 point can be argued, with a balance between an inward directed force composed of Sun and Earth gravity and a outward directed centrifugal force. The latter force is slightly larger at the L2 point since it is further away from the center of rotation.

As for the idea of a resultant force, the configuration of force vectors at point 3 in the figure is similar to the forces acting on an arrow when it is fired from a bow. In Figure 4.13 the force vectors actually acting on the arrow are the tension forces in the string on either side of the arrow. But the sum of these—the resultant force—is actually directed along the arrow, and produces the acceleration that makes it fly.

Why are we interested in this concept? The idea of using Lagrangian point orbits for spacecraft is not a new one. A halo orbit around the Li point, between Earth and the Sun, is an ideal location for a spacecraft with a Sun-

viewing payload, as such a spacecraft has an uninterrupted view of the Sun. Examples of solar observatory spacecraft that have resided at L1 are the SOlar and Heliospheric Observatory (SOHO) and the Advanced Composition Explorer (ACE). Conversely, the L2 point is further away from the Sun than the Earth, above Earth's night side, and this point is a good location for space telescopes. The sky is not obscured by Earth as it is for the Hubble Space Telescope in its low Earth orbit; Earth subtends an angle of only about half a degree from the L2 point. The Wilkinson Microwave Anisotropy Probe (WMAP) spacecraft is an example of a space observatory that has used a L2 point orbit. Looking to the next generation of space telescope beyond Hubble, the James Webb Space Telescope (JWST) is destined for a L2 halo orbit around the year 2013.

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