One of the types of gravitational scattering in the Solar System within the framework of the circular restricted three-body problem (CR3BP) are the gravity assist maneuvers of "particles of insignificant mass" (spacecraft, asteroids, comets, etc.). For their description, a physical analogy with the scattering of beams of charged alpha-particles in the Coulomb field is useful. However, unlike the scattering of charged particles, there are external restrictions on the ability to perform gravity assists associated with the limited size of the spheres of influence of the planet. At the same time, internal limitations on the possibility of performing gravity assists are known from the literature on CR3BP, estimated by the effective radii of planets (including gravitational capture by a planet falling into it). They depend on the asymptotic velocity of the particle relative to the planet. For obvious reasons, their influence makes it impossible to effectively use gravity assist maneuvers. The paper presents generalized estimates of the sizes of near-planetary regions (flat "perturbation rings" or "perturbation hollow spheres" rotating synchronously with a small body in the three-dimensional case), falling into which is a necessary condition for the implementation of gravity assists. A detailed analysis shows that Neptune and Saturn have characteristic of perturbation hollow spheres of the largest size in the Solar System, and Jupiter occupies only the fourth place in this list
Боровин Г.К., Голубев Ю.Ф., Грушевский А.В., Тучин А.Г. Моделирование пертурбационных оболочек для гравитационных маневров в Солнечной системе.Математическое моделирование и численные методы, 2023, № 4, с. 64–73.