How much speed increase can a probe get by using each planet for gravity assist?

How much speed increase can a probe get by using each planet for gravity assist? I'm specifically wondering about Jupiter since it has the most gravity so I would assume it could provide the most gravity assist.|||The *theoretical* maximum speed increase is twice the velocity of the planet around the Sun. The mass of the planet is not relevant.





Here is why: the planet is moving left at speed U and the spaceship moving right at speed v. If the spaceship is on the right path, 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.





For Jupiter U is approximately 13 km/sec.





In practice the velocity change is less than 2U. The spaceship's direction is not changed by 180掳, but by an angle T between 0 and 180掳. The velocity change is then: U(3-cosT) /2