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I recently read the discussion in the question, Gravity Assist Braking that momentum degradation used while entering in the opposite direction to the body's rotation in the Hill's Sphere of that body cannot be used to make the spacecraft orbit around it.

I couldn't help but wonder about the low energy transfers (Complete low energy transfer introduced into practice during JAXA's Hiten mission and consequently in NASA's Grail and Genesis missions which utilizes Lissajous trajectories through manifolds and ideally assumes 0 $\Delta v$ for ballistic capture around the body and completion of at least 1 orbit thereafter.) Can that be related as an exception to gravity assist braking or is it a totally different case?

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When working with the patched conics approximation, it is not possible to brake into orbit using gravity alone. The rule used is that vinf in equals vinf out. But this is only an approximation used to divide a n-body problem into a more easily solved two-body problem. It assumes a distance, a sphere of influence, where an object stops orbiting one body (for example the Sun), and starts to orbit another body (for example the Earth).

In reality, this transition is more smooth, giving a spacecraft some wiggling room when it orbits very close to edge of the planets influence. Ballistic capture is thus not an exception from the usual rules, it is just a phenomenon that the traditional approximations fails to handle properly. They are limited though, saving only a little bit of delta-v, and have been criticised for giving people an impression of a new revolutionary concept in space travel.

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  • $\begingroup$ +1. Though I don't totally dis Belbruno, Shane Ross, et al. If the mass parameter is large, 3 body mechanics is of great interest. Some systems with big mass parameters: Pluto-Charon, Earth-Moon, Jupiter-Galilean Moon. Also Sun-Gas giant. $\endgroup$ – HopDavid Jan 4 '16 at 15:56
  • $\begingroup$ Of interest is the transition region where patched conics don't work. David Hammen wrote a nice explanation of the Hill Sphere as well as what most of us call the Sphere of Influence -- the Lagrange/Tisserand Sphere: space.stackexchange.com/questions/3015/… $\endgroup$ – HopDavid Jan 4 '16 at 16:00

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