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The closer it gets, the greater the deflection. The amount of deflection can be controlled by adjusting how close the spacecraft comes to the planet. The direction of the spacecraft changes during the encounter, however, so typically it leaves the planet heading in a different direction. After the encounter it climbs back out of the gravity well and loses whatever kinetic energy it gained during the approach, ending up with the same final speed it started with. speed) and loses gravitational potential energy, trading one for the other just like a ball rolling downhill. During the approach, as the spacecraft falls into the gravity well of the planet, it gains kinetic energy (i.e. In the planet frame, then, the spacecraft indeed speeds up on approach and slows down by the same amount while departing, just like my colleague thought.
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This means the spacecraft's total energy, made up of kinetic energy (energy of motion) plus potential energy (energy due to proximity to a massive object), is conserved throughout the encounter in this frame. For example, Jupiter is about 10 to the 24th power times more massive than the Voyager spacecraft, so Jupiter ignores an encounter to an extremely high degree of precision. More importantly, since the planet is so much more massive than the spacecraft, the planet sits almost exactly at the center of mass of the two objects and does not react by any measurable amount as a result of the encounter. In the planet frame, the planet sits still (by definition!). For economy of language I'll call them the "planet frame" and the "sun frame." It's convenient to think about reference frames for both the planet and for the sun (or the solar system). The key to understanding how a gravity assist works is to consider the problem from two different points of view, or reference frames. Recently I was talking with a colleague, an excellent plasma physicist who knew the phrase "gravity assist" but thought it must be marketing hyperbole because he didn't believe it could actually work. Since energy is conserved, you reason, how can a spacecraft obtain a net velocity boost by passing by a planet? Energy conservation suggests the spacecraft should speed up while approaching the planet, but then lose the same speed while departing. This feeling can persist even if you know some physics.
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Gravity assists seem a bit mysterious, like one is getting something for nothing. Furthermore, the extra speed gained by gravity assists dramatically reduces the duration of a mission to the outer planets. Lifting extra fuel into orbit, just so it can be used later, is exponentially expensive.
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Mission planners use gravity assists because they allow the objective to be accomplished with much less fuel (and hence with a much smaller, cheaper rocket) than would otherwise be required.