Slingshots and Space ShotsCassini, the spacecraft that flew through Saturn's rings not too long ago, started out from Earth with far less speed than it needed to get to its goal, and picked up what energy it needed by a succession of energy thefts. Bill Casselman
IntroductionFor a long time now, NASA has been using a technique that they call gravity assist and that almost all the rest of the world calls the slingshot in order to send space payloads to the far solar system using as little fuel as possible. The effects might seem miraculous to the uninitiated. Pioneer 10 was launched with enough speed to get not much further than Jupiter, but by swinging it around Jupiter it was brought to a speed sufficient to take it out of the solar system. NASA at one of its web sites devoted to the subject says
Cassini, the spacecraft that flew through Saturn's rings not too long ago, started out from Earth with far less speed than it needed to get to its goal, and picked up what energy it needed by a succession of energy thefts. Here is one of the NASA images of what the trajectory looked like: This image has been copied from NASA's site The early techniques used were mathematically rather simple, depending essentially only on the kinematics of two isolated gravitational bodies. I'll explain this here. More recent techniques have involved a clever application of the theory of "chaos," or more prosaically the sensitive dependence of trajectories on initial conditions in the neighbourhood of an unstable equilibrium point. This is explained in a recent book by Edward Belbruno. I'll not say anything about that topic here, but hope to deal with it in a future feature. In the last section I'll illustrate a very curious example of something like the slingshot effect that was encountered on theoretical grounds a long time before space flight. Kepler dynamics of two isolated gravitational bodiesAs a spacecraft approaches a planet, the forces on it become appreciably strong in the context where the two objects are considered, to a close approximation, as isolated from all other objects in space. Since the spacecraft comes from a great distance, it will not do too much damage to assume it comes from infinitely far away, which means that it is following a hyperbolic orbit with respect to the center of gravity of the planet and itself. I'll first recall here the situation without providing details. (Danby's book does a good job of filling them in.) Suppose we consider more generally two objects of masses where we can track the objects themselves. If we set is also constant. Here where
If
The geometry of hyperbolasIf a conic section is an ellipse, its eccentricity Given two points is an hyperbola with foci where Its asymptotes ("limiting" lines) are the lines Therefore the eccentricity is related to the direction of the asymptotes by the equation and comparison with an earlier formula tells us that
Here are a few hyperbolas with eccentricity indicated. They approach a parabola as
Hyperbolic trajectories In this section, suppose we are working with two bodies, one of them much, much larger than the other, or more precisely suppose that and since the inital velocity and position are in effect perpendicular we can calculate the eccentricity This tells us the asymptotes of the orbit, which is all that will be important. The smaller body thus swings around the origin and passes off to infinity again in a direction obtained from the original direction by rotating it through the deflection angle
The slingshotThe paths of the objects are conic sections if the origin is taken to be their center of gravity, or more generally if the center of gravity is fixed in the coordinate system. But if the center of gravity is moving, the paths are stranger looking. For example, here is a trace of the paths if
If v_{f} = Rv_{i} + V  RV v_{f} = R(v_{i} + R^{1}V  V) The vector
One conclusion to be drawn from the basic formula is that the acceleration or retardation will always be smaller in magnitude than
Burrau's problemThe slingshot effect is a good example of slightly nonintuitive behaviour with a simple explanation. It is especially striking since it involves only two bodies, a configuration that is completely understood. If we start to look at the behaviour of systems of three or more objects, which is what more recent NASA techniques involve, the variety of surprises may be unlimited. One of the earliest surprises involved a variation on the slingshot. For reasons not entirely clear to me, the astronomer Carl Burrau suggested trying to determine the motion of a system of three bodies initially at rest, of masses
The gravitational constant is taken to be
When the system starts out, it proceeds rather sedately, then gets progressively more tangled up, with the heaviest of the three bodies weaving back and forth between the lighter two. The resemblance of the traces of the motion to a plate of overcooked spaghetti is presumably accidental. Occasionally there are some very close encounters between the two heavy bodies. But then close to I do not know to what extent the empirical behavior exhibited here is assured on theoretical grounds. Remarks on programmingThe diagrams above are based on those of Szebehely. They occur often in the literature. The programming underlying them is tricky, because the step size has to be reduced drastically near close encounters. I used Störmer's method with polynomial extrapolation and a primitive step adjustment procedure. ReferencesJ. M. A. Danby, Fundamentals of celestial mechanics, WillmannBell, 1988. Edward Belbruno, Capture dynamics and chaotic motions in celestial mechanics, Princeton Press, 2004. Victor Szebehely, Burrau's problem of three bodies, P. N. A. S. 58 (1967), 6065 A list of missions using gravity assist A tutorial on gravity assist, by James van Allen Johnson's home page containing a link to useful notes by Johnson NASA's primer on gravity assists NASA's page on Cassini and gravity assists Bill Casselman NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services. 
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