Gravity assist

Fuel is 12 km/s. A Hohmann transfer out to Jupiter’s orbit takes about 9 km/s of that. Passing behind Jupiter can raise the heliocentric speed by about 10 km/s. The extra speed is not burned. It is borrowed from Jupiter’s orbital motion.

In Jupiter’s frame the probe only turns, and its speed relative to Jupiter stays the same. In the Sun’s frame Jupiter is already moving at about 13 km/s, so the exit velocity is Jupiter’s velocity plus that turned relative velocity. A sharper turn behind the planet adds more heliocentric speed than a head-on pass.

The model is a point mass in the Sun’s gravity, with Jupiter on a circle. It leaves out eccentricity, inclination, and the other planets. A 0.1 km/s change in the burn can move the closest approach from 0.024 AU to 0.005 AU and change the final speed by about 5 km/s. The interactive controls are on the Chinese demo.