Gravity Assist - Explanation

Explanation

A gravity assist or slingshot maneuver around a planet changes a spacecraft's velocity relative to the Sun, though the spacecraft's speed relative to the planet on effectively entering and leaving its gravitational field, will remain the same (as it must according to the law of conservation of energy). To a first approximation, from a large distance, the spacecraft appears to have bounced off the planet. Physicists call this an elastic collision even though no actual contact occurs. A slingshot maneuver can therefore be used to change the spaceship's trajectory and speed relative to the Sun.

A close terrestrial analogy is provided by a tennis ball bouncing off a moving train. In the cartoon at right, a boy throws a ball at 30 mph toward a train approaching at 50 mph. The engineer of the train sees the ball approaching at 80 mph and then departing at 80 mph after the ball bounces elastically off the front of the train. Because of the train's motion, however, that departure is at 130 mph relative to the station.

Translating this analogy into space, then, a "stationary" observer sees a planet moving left at speed U and a spaceship moving right at speed v. If the spaceship has the proper trajectory, it will pass close to the planet, moving at speed U + v relative to the planet's surface because the planet is moving in the opposite direction at speed U. When the spaceship leaves orbit, it is still moving at U + v relative to the planet's surface but in the opposite direction (to the left). Since the planet is moving left at speed U, the total velocity of the rocket relative to the observer will be the velocity of the moving planet plus the velocity of the rocket with respect to the planet. So the velocity will be U + ( U + v ), that is 2U + v.

This oversimplified example is impossible to refine without additional details regarding the orbit, but if the spaceship travels in a path which forms a parabola, it can leave the planet in the opposite direction without firing its engine, the speed gain at large distance is indeed 2U once it has left the gravity of the planet far behind.

This explanation might seem to violate the conservation of energy and momentum, but the spacecraft's effects on the planet have not been considered. The linear momentum gained by the spaceship is equal in magnitude to that lost by the planet, though the planet's enormous mass compared to the spacecraft makes the resulting change in its speed negligibly small. These effects on the planet are so slight (because planets are so much more massive than spacecraft) that they can be ignored in the calculation.

Realistic portrayals of encounters in space require the consideration of three dimensions. The same principles apply, only adding the planet's velocity to that of the spacecraft requires vector addition, as shown below.

Due to the reversibility of orbits, gravitational slingshots can also be used to decelerate a spacecraft. Both Mariner 10 and MESSENGER performed this maneuver to reach Mercury.

If even more speed is needed than available from gravity assist alone, the most economical way to utilize a rocket burn is to do it near the periapsis (closest approach). A given rocket burn always provides the same change in velocity (Δv), but the change in kinetic energy is proportional to the vehicle's velocity at the time of the burn. So to get the most kinetic energy from the burn, the burn must occur at the vehicle's maximum velocity, at periapsis. Powered slingshots describes this technique in more detail.