The article by Van Allen quoted in the Reference recalled my attention to the subject of encounters between orbiting bodies and the interesting effects on their orbits. Apparently, this is a subject of some mystery to physics students, though it is really quite simple in principle. The reader is strongly encouraged to read Dr Van Allen's excellent article, in which the encounter of Pioneer 10 with Jupiter in December 1973 is discussed as an example. I won't reproduce here the details in this article, except to mention the broad picture for those who may not have immediate access to it. The article, despite its clarity, still has the aura of mystery, or at least complicated physics, here and there, and I hope to give a simpler explanation here.
One source says that the first use of gravitational assist was with the Mariner 10 probe of 1973-1975. An encounter with Venus on 10 March 1974 was used to decelerate the vehicle and send it towards Mercury. Mariner 10 made three passes around Mercury, giving us most of the information on the surface of that planet that we have to date. This date is later than the December, 1973 date for the Pioneer 10 encounter with Jupiter, so perhaps Pioneer 10 was actually the first.
Jupiter is about 5 AU (the AU is the radius of the earth's orbit) from the sun, and revolves at an orbital speed of about 13.1 km/s on the average, completing its orbit in 11.857 years. Pioneer 10 approached Jupiter on an elliptical orbit at a speed of 9.8 km/s, not far from aphelion. After its closest approach of 2.027 x 105 km, it swung away, approximately tangent to Jupiter's orbit, at a speed of 22.4 km/s, which was sufficient to put it into hyperbolic orbit about the Sun. Since then, Pioneer 10 has left the solar system, and just recently its final radio transmission was received. This is an example of gravitational assist, which has been frequently used in planetary probe trajectories, performing feats that would be impossible, or very costly, if done directly.
Although we do deal with three bodies here, the sun, Jupiter and the spacecraft, none of the complicated results of the celebrated "problem of three bodies" is required for an understanding. There is a massive body, Jupiter, and a body of negligible mass, the spacecraft, in independent orbits about the even more massive Sun. A close encounter between Jupiter and the spacecraft, which is an elastic collision in which the bodies interact gravitationally, takes place in a short time and limited space to change the instantaneous velocities of the participants, sending them into new orbits. The new orbit of Jupiter is indistinguishable from the old one; though slightly different, the difference is extremely small. The new orbit of the spacecraft, however, can be very different, and its kinetic energy can change by a considerable fraction. This energy comes from Jupiter, of course, which does not notice the theft.
In the elastic collision, the total kinetic energy and total momentum of the two bodies is conserved. However, it may be distributed differently before and after the collision, as is very well known. Let us reduce the problem to its simplest state, that of a one-dimensional collision, as on an air track. An actual collision in three dimensions may approach this more or less closely when the impact parameter (distance of closest approach if the bodies did not interact) is small. First, let's assume that the large mass M is moving with velocity w to the right, and the small mass m is moving with velocity v to the left, as shown in the figure. The magnitudes of the vectors are shown above the arrows giving the directions. The relative velocity is v + w, and in the elastic collision is reversed in the center of mass system in which the center of gravity is fixed. This system moves with velocity w to the right, very closely, since M >> m. Therefore, after the collision, the mass m is moving to the right with a velocity v + 2w. In a head-on collision, the light body can pick up twice the velocity of the heavy body. In the figure, momentum and energy are apparently not conserved, but this is only because the very small decrease in w is not shown. An accurate figure would differ imperceptibly.
Now suppose the large mass is still moving with velocity w, but the small mass comes up behind it with velocity v > w, approaching with relative velocity v - w. In the center of mass system this velocity is reversed after the collision, and the final velocity of the small mass is v - 2w. In a rear-end collision, the small mass can lose twice the velocity of the heavy body. We have now found the limits of the velocity of the light body after collision, and how these limits depend on the velocity of the heavy body. In a three-dimensional collision, the velocity change will be smaller, and can easily be calculated explicitly for any case. In the case of Jupiter, the speed of Pioneer 10 after the encounter could be as high as 9.8 + 26.2 = 36 km/s, or as low as zero. The actual result of 22.4 km/s is well within the possible range. An unlucky encounter could send the spacecraft plummeting rapidly into the Sun.
Comets have long been known to interact with Jupiter in this way. If they come in behind Jupiter, they lose energy and are put into short-period elliptical orbits, many of which do indeed come perilously close to the sun. If they come in ahead, they are flung well out of the solar system, like Pioneer 10. Saturn exerts a very similar effect. Saturn's orbital velocity is 9.67 km/s, so the changes it can make are not as large as those of Jupiter. If Pioneer 10 got a kick in the apse from Saturn, it could only with difficulty attain the required escape velocity. Uranus and Neptune move even more slowly, but still could perturb a parabolic comet into an elliptical one. These massive planets can create a tight orbit in an encounter that leads to orbit modification.
Mars, with 24.1 km/s; Earth, with 29.8 km/s; and Venus, with 35.0 km/s orbital velocities could transfer large velocities in an encounter, but the smaller masses do not reach as far. Any planet is sufficiently massive relative to a comet, spacecraft or meteoroid to modify its orbit with no change to its own. It should be noted that a planet cannot capture a body in this way, in a two-body encounter. It is the velocity relative to the Sun that is changed. An exception is when the collision is not elastic, and the body encounters some atmosphere or even the solid surface of the planet. This is, of course, of frequent occurrence on Earth, and has occurred elsewhere as well, as the pock-marked surface of the Moon attests. We also note that encounters in which kinetic energy is lost generally give rather elliptical orbits; circular orbits are not to be expected.
The gain or loss of energy in an encounter of a light body with a heavy one is only a natural consequence of the conservation of momentum and energy, and the transformation from one inertial system to another, with no mystery. The infinite possiblities introduced with three-dimensional orbits adds complexity, but no essential difference. The effect could easily be seen with an air track in the laboratory, or indeed with pendulums of elastic balls of very different mass, which is the basis of the argument in this article.
Composed by J. B. Calvert
Created 23 April 2003
Last revised 7 May 2003