SwingBy Trajectories

If a spacecraft passes close by a planet, as it journeys through interplanetary space, then the path it takes is described as a swing-by trajectory. It may be worth recalling some of the background we discussed in Chapter 1 about this type of trajectory, and indeed you may wish to reread the text associated with Figures 1.9 and 1.10 to refresh your memory. The shape of the curve describing the spacecraft's path is called a hyperbola, and it is one of the four basic conic section shapes found by Isaac Newton in his equations describing motion in an inverse square law gravity field. As mentioned in Chapter 1, the shape can be seen all over the place once you start looking for it. You may even have your very own hyperbola in the room where you are reading this,

G. Swinerd, How Spacecraft Fly: Spaceflight Without Formulae, DOI: 10.1007/978-0-387-76572-3_4, © Praxis Publishing, Ltd. 2008

if you have a table or standard lamp placed adjacent to a wall (see Figure 1.10).

The shape of the hyperbolic swing-by trajectory is shown in Figure 4.1. When the spacecraft is a long way away from the planet, before it even gets to point A in the diagram, the gravity force of the planet is so feeble that the spacecraft effectively travels in a straight line. This line is called an asymptote of the hyperbola. However, as the spacecraft closes in on the planet, the gravity force steadily increases, and the spacecraft's path describes the classic hyperbolic shape depicted in Figure 4.1 from point A through point B to point C. Beyond point C, the gravity force decreases rapidly, and the trajectory tends to the straight line given by the asymptote once again. During this process, the planet has changed the direction of travel of the spacecraft, and this change is given by the deflection angle, which is effectively the angle between the incoming and outgoing asymptotes, as shown. The amount by which the trajectory is deflected is

Figure 4.1: The classic hyperbolic shape of a swing-by maneuver past a planet.

Asymptotes

Figure 4.1: The classic hyperbolic shape of a swing-by maneuver past a planet.

Asymptotes dependent on three things: how massive the planet is, how close point B is to the planet, and how fast the spacecraft is traveling on approach.

This description echoes much of what was said in Chapter 1, but one thing that was not discussed was the speed of the spacecraft on the hyperbola. As the spacecraft falls toward the planet on the hyperbola, the gravity force increases, causing a corresponding increase in the vehicle's speed. Conversely, after the point of closest approach, the speed decreases again as the spacecraft climbs away against the force of gravity. The important thing to remember for what follows is that as speed and height are traded, and no energy is lost in the encounter, the speed of approach at A is identical to the speed of departure at C, relative to the planet. This is analogous to a cyclist negotiating a road that falls into a valley between two hilltops. The cyclist leaves the first hill at, say, 10 mph, and accelerates on the downhill section, reaching a maximum speed at the bottom of the valley. The cycle then slows down steadily on the uphill gradient on the other side of the valley, finally reaching the second hilltop at 10 mph again. To make the comparison with the spacecraft's motion complete, we have to assume that no energy is lost by the cyclist due to things like friction with the road or wind resistance. But nevertheless it is quite a good way of thinking about how the speed of the spacecraft varies on the hyperbola.

Gravity Assists

However, this is not the end of the story of hyperbolic swing-bys, because swing-bys can be used to increase the speed of spacecraft in their travels around the Sun without firing a rocket motor. (Swing-bys can also be used to decrease spacecraft speed, but we will not discuss this aspect.) Mission analysts get excited at the prospect of gaining spacecraft energy without having to use rocket fuel! It means that their spacecraft can achieve the journey to a distant planet more quickly, and do it without having to exchange payload mass for propellant mass. This type of maneuver is often referred to as a gravity assist, and it has been used many times by mission designers.

Perhaps the most famous example of its use is in the Voyager spacecraft program to explore the outer solar system. If we look back on the history of astronautics in the 20th century, the events that stand out for me are Sputnik 1, the first manned orbital flight by Yuri Gagarin, the Apollo moon landings, and the exploration of the outer solar system by the Voyager 1 and 2 spacecraft. Voyager 2 was launched in 1977, and took advantage of a rare alignment of the planets to visit Jupiter, Saturn, Uranus, and Neptune before finally leaving the solar system. The scientific return was huge! And it was achievable through the use of gravity-assist maneuvers at each planetary encounter, which increased the spacecraft's speed relative to the Sun so that the Neptune fly-by could be achieved after just 12 years from launch. Without this technique, and using the same launch energy, the transfer to Neptune would have taken around 30 years, and we would only now be learning about Neptune's mysterious moon Triton as I write this in 2006 (assuming that the spacecraft would still be operating after 30 years!).

Gravity-assist maneuvers clearly have great benefit, but how exactly do they work? Let's consider gravity-assist maneuvers using the planet Jupiter as an example. Given that Jupiter is the largest planet in the solar system, it has the strongest gravity field, and so the effect it has on the dynamics of a passing spacecraft is very significant. Let's imagine our spacecraft wandering through interplanetary space, traveling out from Earth toward Jupiter. We also suppose that, in this interplanetary cruise, the spacecraft is in an elliptical orbit around the Sun, so that it is the Sun's gravity that governs its motion. However, when the spacecraft is about 40 million kilometers (25 million miles) from Jupiter, the gravity field of Jupiter begins to be about the same as that of the Sun. Beyond this point, as the spacecraft closes in on Jupiter, the force of Jupiter's gravity increases, and once it is within, say, 10 million kilometers (6 million miles) of Jupiter, the Sun's influence is negligible. At this position, point A in Figure 4.2, the spacecraft is effectively falling toward Jupiter, moving in a (more or less) straight line with a speed given by Vin relative to Jupiter. In this part of the flight, while governed by Jupiter's gravity field, the spacecraft performs the classic hyperbolic trajectory as described above. It reaches a maximum speed at closest approach (point B), and then climbs away arriving at point C traveling effectively in a straight line once again with a speed Vout relative to Jupiter. From our discussion above about the spacecraft's speed on the hyperbola, we know that Vin and Vout are the same, so we don't seem to have gained anything from the maneuver! Remember, however, that these are speeds relative to Jupiter. The thing we have forgotten is that Jupiter itself is moving along its orbit around the Sun at about 13 km/sec (8 miles/sec) relative to the Sun, and this makes all the difference.

In Figure 4.2, the speeds of objects are represented by arrows. The orientation of the arrow represents the direction in which the object is moving, and the length of the arrow indicates how fast it is going: long arrows denote fast objects, and short arrows denote slow objects. These arrows are called velocity vectors, which are useful tools in the analysis of this type of problem. Incidentally, in Chapter 3 we used force vectors, although we did not describe them as such, in the discussion of orbit perturbations, where the direction of an arrow indicated the direction of a force, and the length of the arrow indicated how much force was applied. There are lots of

Spacecraft's path

Spacecraft's path

Figure 4.2: A schematic of a Jupiter gravity-assist maneuver.

objects in physics and dynamics that require two pieces of information— direction and magnitude—to describe them fully, and these are all described as vectors by scientists and engineers. For example, there are position vectors, velocity vectors, force vectors, torque vectors, and others, but I'm getting away from the point here.

Getting back to our gravity-assist maneuver, what defines the orbit around the Sun before the swing-by is the position and velocity of the spacecraft relative to the Sun before it enters Jupiter's gravitational influence. Let's suppose that point A in Figure 4.2 is now right at the edge of Jupiter's sphere of influence, approximately 40 million kilometers (25 million miles) out from the planet. If Vin again represents the incoming velocity of the spacecraft relative to Jupiter at A, and Vjup is the velocity of Jupiter on its orbit around the Sun, then we can calculate the velocity of the spacecraft relative to the Sun before the encounter. This is denoted by Vbefore in the figure, and it is the sum of the velocity of the spacecraft relative to Jupiter and the velocity of Jupiter relative to the Sun. Given that the arrows, or vectors, representing these velocities are not parallel, we have to use a velocity vector diagram to do this sum, and this is shown as the triangle in the lower right of the figure. The important thing to note is that Vbefore represents the velocity of the spacecraft on its elliptical orbit around the Sun before meeting Jupiter.

Working out the corresponding velocity of the spacecraft relative to the Sun afterward is a little easier, as in this case the velocities Vout and Vjup are parallel to each other, so Vafter is just a simple arithmetical sum. This is shown in the lower left in Figure 4.2, opposite the vector triangle. V^ut and Vjup need not necessarily be parallel to each other, but I have devised the gravity-assist geometry so that they are, to make things a little easier to visualize. Remember that Vafter represents the spacecraft's velocity relative to the Sun after the gravity assist, and so defines the subsequent orbit around the Sun after the encounter with Jupiter. In the bottom left of Figure 4.2, the magnitudes of the Sun-relative speeds before and after the encounter— Vbefore and Vafter—are compared, and it is easy to see that the spacecraft has gained speed.

The next question to ask is, Where has that speed gain come from? The answer is that the spacecraft's gain is Jupiter's loss. The planet has tugged on the spacecraft, to give it a significant boost in speed, while at the same time the spacecraft has exerted an opposite tug on Jupiter, causing it to lose speed. However, given the mass of the spacecraft compared to the huge mass of Jupiter, the effect on Jupiter is immeasurably small, although it is measurable if you wait long enough; in the words of a National Aeronautics and Space Administration (NASA) press release about the Voyager Jupiter swing-by, "The position of Jupiter will change by about 1 foot every trillion years''!

Figure 4.3 shows a remarkable plot of the Sun-relative speed of Voyager 2 during its interplanetary cruise to the outer planets. The "blips'' in the curve at around 5,10,20 and 30 astronomical units (AU) correspond to the gravity-assist maneuvers at Jupiter, Saturn, Uranus, and Neptune, respectively. You may recall from Chapter 1 that 1 AU is equal to the mean Earth-Sun distance. These maneuvers maintained the speed of the spacecraft well above that needed to escape the Sun, which is indicated by the broken line.

We can get a feel for the way gravity-assist maneuvers work by proposing a useful analogy, in the form of a rather unusual experiment involving a double-decker bus. Firstly, I would definitely recommend that you do not attempt to perform the experiment, as I would not wish to be responsible for any resulting injuries—and secondly, it allows the rather unusual notion of having a picture (see Figure 4.4a) of a double-decker bus in a book about spaceflight! Those of you who are familiar with the old-style double-decker bus know that the entrance is via a step-up, open platform at the back of the bus as shown in Figure 4.4, with a vertical handrail for passengers to hang onto to prevent them from falling out while the bus is moving. The vertical handrail plays the role of the planet Jupiter in the experiment!

Distance from Sun (AU)

Figure 4.3: The speed of the Voyager 2 spacecraft relative to the Sun during its 12-year mission to the outer planets. (Figure compiled from data courtesy of Steve Matousek, National Aeronautics and Space Administration [NASA]/Jet Propulsion Laboratory [JPL]—Caltech.)

The layout of the experiment is illustrated in Figure 4.4b, which shows a view of the bus looking down from above. The vertical handrail is positioned at the rear corner of the bus, where the open platform entrance is located. Note that the entrance has been switched to the right-hand side of the bus, to allow us to compare the geometry more easily with Figure 4.2. A person, shown rather unimaginatively as a blob with an extended arm and hand, is shown standing on the road, swinging on the handrail. Surprisingly, this rather unlikely arrangement gives a good depiction of what happens in a gravity-assist maneuver, as discussed above. Each of the items shown in Figure 4.4b represents some part of the real thing. For example, as mentioned above, the vertical handrail represents the planet Jupiter, where its movement over the ground, as the bus moves forward, represents the movement of the planet in its orbit around the Sun. The person swinging on the rail represents the spacecraft, and the force in the person's arm as he swings on the rail plays the role of the gravitational attraction between the planet and the spacecraft. Now we can play the experiment two ways—first with the bus stationary and then with it moving.

The first task for our willing helper is fairly simple then, as we would just like them to run up to the rear of the stationary bus at, say, 10 mph (4.5 m/ sec)—see plan view in Figure 4.5—grab hold of the vertical hand rail at point 1, and then release his grip at point 2. This simply has the effect of changing the direction of his run from the vector Vin to the vector Vout as

person arm and hand

Front vertica hand

Rear

Figure 4.4: (a) The back of a bus! Note the vertical handrail in the open, step-up platform entrance, (b) Plan view of bus, showing relevant features of the thought experiment described in text,

Figure 4.5: Depiction of a hyperbolic swing-by, using the bus analogy.

shown, without effectively changing his speed. This parallels a hyperbolic swing-by trajectory, where the incoming and outgoing speeds relative to the planet are the same but the direction of travel of the spacecraft is changed.

To go any further with the experiment, we have to suppose that our helper is fairly athletic, and has good hand-eye coordination, as we will now expect them to grab the handrail while the bus is moving. Not wishing to make this too difficult, we'll constrain the speed of the bus to 10 mph. This situation is now shown in plan view in Figure 4.6. Again, the runner grabs the rail at point 1 and releases it at point 2. To make a proper parallel with the

Figure 4.6: Depiction of a gravity-assist maneuver. The speed of the runner over the ground is increased by the movement of the bus. Compare with Figure 4.2.

spacecraft gravity assist, the runner has to perform his swinging movement on the rail in such a way that it looks the same as the stationary bus swing from the perspective of someone standing on the moving bus platform. But then, perhaps this is one technical detail too far? The important thing to take away from the moving bus analogy is an intuitive feel that once the runner grabs the rail, he acquires additional speed over the ground from the movement of the bus. If you imagine grabbing the moving handrail yourself, you can almost feel the stress in your arm, tending to pull the socket, as your body mass is accelerated by the bus! The force in your arm will also act on the bus in the opposite sense, tending to slow it down a little, in the same way as we saw Voyager taking a tiny amount of orbital speed from Jupiter. The speeds over the ground of the runner Vbefore and Vafter are worked out at the bottom of Figure 4.6, in an analogous way to that done in the swing-by case in Figure 4.2, confirming an increase in speed. Note that the speed over the ground in this analogy corresponds to the speed of a spacecraft relative to the Sun in the gravity-assist maneuver.

This analogy presents a situation in which you can at least imagine experiencing a boost in speed in a way that is similar to what happens in a gravity assist.

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