Restricted three-body problem in units GM = 1, moon on a circular orbit of radius a = 1 (period 2*pi). Each test particle feels the planet, -GM r/r^3, and the moon, +mu (r_m - r)/|r_m - r|^3 (softened slightly), integrated with a symplectic leapfrog. The Hill radius is r_H = a (m/3M)^(1/3); co-orbital horseshoe motion occupies a radial half-width of roughly 2*sqrt(3) r_H (Lissauer 1985).

The mass ratio is exaggerated (1e-6 to 1e-4) so the effect is visible in seconds; real shepherds such as Prometheus, Pandora or Daphnis have m/M below 1e-8. Particles do not collide or self-gravitate here, so real ring-edge waves and gap sharpness are simplified.