A gravity assist is free energy for a spacecraft.

David Shortt identifies why the gravity assist can be confusing, theoretically, and then, clears up the confusion:

Gravity assists seem a bit mysterious, like one is getting something for nothing. This feeling can persist even if you know some physics. 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. 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. The mystery begs to be explained.

The “Shortt” answer: It all depends on point of reference, or “reference frames.”

The key to understanding how a gravity assist works is to consider the problem from two different points of view, or reference frames. It’s convenient to think about reference frames for both the planet and for the sun (or the solar system). For economy of language I’ll call them the “planet frame” and the “sun frame.”

If the Planet is stationary, energy gained on entry is lost on exit from gravitational force.

In the planet frame, the planet sits still (by definition!). 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. 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. 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.

There’s a static part of the planet frame, where energy is conserved.

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. During the approach, as the spacecraft falls into the gravity well of the planet, it gains kinetic energy (i.e. speed) and loses gravitational potential energy, trading one for the other just like a ball rolling downhill.

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.

Although energy is conserved in the planet frame, a directional change is caused by the gravity assist.

The direction of the spacecraft changes during the (gravitational) encounter, however, so typically it leaves the planet heading in a different direction.

The amount of deflection can be controlled by adjusting how close the spacecraft comes to the planet. The closer it gets, the greater the deflection. It’s possible to have a very small deflection, near zero degrees, by arranging a wide miss.

The maximum deflection is 180 degrees, sending the spacecraft back where it came from, obtained by arranging an extremely close approach. Mathematically the spacecraft’s path is a hyperbola, so we say the spacecraft follows a hyperbolic trajectory in the planet frame.

Visit the next page to consider the Sun frame, where the planet is not stationary.