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You wrote, FWIW, I looked up slingshot maneuver, and I now understand it. The spacecraft steals momentum from the planet/star to impart the sideways delta-vee.
Still don't quite grasp why doing an impulse acceleration at the deepest part (closest approach) gives a multiplicative boost. Gonna have to read and re-read that a few times. It's been a long time since I messed with integrals.

The part you are missing is called the Obert Effect.

It explains why apoapsis (and periapsis) is the most efficient point in an orbit to apply trust.

- Joel
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