It seems too me there’s an analogy between the two, but I understand we have a loss of kinetic energy in the former and we can have a gain in the latter. What am I missing?
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Bremsstrahlung radiation comes from charged particles quickly changing direction. A charged particle swinging past a planet would likely produce this as well, but most flybys of planets don’t happen with lots of electrical charge and the acceleration is too low.
Gravity slingshots work because the planet is moving and can donate some of its movement (a very small amount) to the rocket.
There is probably some gravitational analogue to bremsstrahlung that applies, generating gravitational waves, but it would be incredibly weak.
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