The trajectory of the Mercury MESSENGER, which flew by Mercury in 2008 and 2009, and orbited the planet beginning in 2011. (A long discussion of the complexities of the planetary flybys and gravitational assists required to achieve this goal, which required a considerable amount of time to write, was inadvertantly lost in saving this file; and as a result of the frustration involved in that loss, will be put off until another day. In the meantime, this page serves as a placeholder and a prod to repost the discussion sooner rather than later.) (Johns Hopkins, Carnegie Institution, NASA).

(placeholder for at least two pages -- Orbital Perturbations, which will be more about the basic physics, and Gravitational Assists, which will be more about its application to spacecraft, such as Mariner 10 and Mercury MESSENGER)
     Gravitational assists (sometimes referred to as "gravitational slingshots") involve the use of a planet's gravity to change the orbit of a spacecraft. Similar changes occur when celestial bodies, such as comets or asteroids, pass near a planet; but in those cases, the orbital changes are referred to as perturbations. Gravitational assists, although fundamentally the same, involve the deliberate use of planetary perturbations to change spacecraft orbits, to reduce the amount of fuel required to achieve a desired orbit.
     The Mercury MESSENGER spacecraft's orbital path(s) represent one of the best examples of the use of gravitational assists, both because of the complexity of the orbital changes (half a dozen gravitational assists were used), and because the assists allowed an otherwise impossible result -- namely, insertion into orbit around Mercury.
     (keeping in mind that this is a very quick and dirty introduction, to be replaced by a clearer and more detailed discussion, complete with diagrams:)
     To send a spacecraft to another planet, we must first get it off the ground and away from the Earth. This requires huge amounts of energy, and enormous rockets, in comparison to the size of the payload (the scientific instruments, power and control systems, and radio transmitters and antennas). But even if we expend such energies, all we basically do is put the spacecraft into an orbit around the Sun essentially identical to the orbit of the Earth (it is in essentially the same place, relative to the Sun, moving at essentially the same velocity as the Earth, if all we do is get it off the Earth; having the same motion in the same place as the Earth, the Sun's gravity will give it the same orbit). To send it to another planet, we have to speed it up relative to the Sun (if we want to send it to an outer planet) or slow it down (if we want to send it to an inner planet). As an example, if we launch a spacecraft into nearly-Earth-orbital flight, then change its motion relative to the Earth to send it to another planet, we need to speed it up by about 3 miles per second to send it to Mars, and 9 miles per second to send it to Jupiter; whereas to reach Venus, we need to slow it down by about 3 miles per second, and to reach Mercury, we need to slow it down by about 9 miles per second. (The energy required to do this goes as the square of the speed change, so it takes nearly 10 times as much energy per pound to send a spacecraft to Mercury or Jupiter as it does to send one to Mars or Venus).
     The change in motion described above can be accomplished by firing the spacecraft's rockets while it is in Earth orbit, headed forward relative to the Earth (to increase its speed and send it outward), or backward relative to the Earth (to decrease its speed and send it inward). Part of the rocket power changes the orbit from one around the Earth to one around the Sun (taking it away from the Earth on a more or less permanent basis), and the rest changes the orbital velocity (as specified above, for the four planets listed there).
     Now suppose we follow a spacecraft thrown/fired backward relative to our orbit, at 9 miles per second, so that instead of moving forward (with the Earth) at 18 miles per second, it is only moving forward at 9 miles per second (both speeds relative to the Sun). If the spacecraft still had 18 miles per second speed, it would follow the same orbit as the Earth; but reducing its speed to half that value would cause it to fall inward, toward the Sun, and eventually, to the orbit or Mercury. So if the relative positions of the Earth and Mercury are correct, the spacecraft can sweep past Mercury, as it falls to its orbit. The trouble is, that is exactly what it would do -- just sweep past Mercury, take a few pictures, then continue on its orbit, and to a first approximation, return to the origin of its motion; namely, the place in our orbit where we launched it.
     As it happens, that's essentially what was done with the first spacecraft to fly by Mercury -- Mariner 10 -- in 1974. It was thrown backwards, fell towards Mercury, and snapped pictures of the sunlit side of the planet as it swept past it. And if its orbit had remained as it started off, that would have been just about it. However, by going past Mercury, the spacecraft experienced a change in its orbital motion which reduced its speed, and the size of its orbit, so instead of returning to our orbit, it moved only partway outward, then fell back to Mercury's orbit. Fortunately, it was realized beforehand that the time required for it to follow the new, smaller orbit was approximately two Mercury years, so that when the spacecraft returned to the point where it passed Mercury in 1974, Mercury wouldn't be far from returning to the same point. By carefully choosing the distance and orientation of the spacecraft as it passed Mercury, the orbital period of the spacecraft was made exactly equal to two Mercury years, so that it was able to take additional photos of the planet, about six months later (unfortunately, Mercury rotates exactly once every two "years", so exactly the same side of Mercury was facing the Sun; but at least this did allow for better and more complete photographic coverage of the sunward side of the planet). In fact, it proved possible to arrange the second flyby to allow for a third and final flyby, under the same circumstances, another two Mercury years or six months later, in 1975. (Hence the three flyby dates shown on the Mercury page, in March and September of 1974, and March of 1975).
(to follow later -- aside from a grammatically and pedadogically superior discussion, and diagrams of the Mariner 10 flyby -- drawings showing how MESSENGER passed by the Earth a year after its initial launch, and was slowed down enough to fall toward Venus; a discussion of how the half dozen gravitational assists allows the spacecraft to lose enough speed to attain orbit (in 2011); an introductory discussion of the physics of planetary perturbations; and examples of similar interactions involving asteroids and comets)