Hill Three-Body Problem

Author: CislunarSpace

Website: https://cislunarspace.cn

Definition

The Hill Three-Body Problem is a further simplification of the Circular Restricted Three-Body Problem (CR3BP) introduced by G. W. Hill. It translates the rotating coordinate origin to the second primary body and linearizes the gravitational terms under the assumption R/R₁ ≪ 1, where R is the distance between the two primaries and R₁ is the distance from the massless body to the second primary. This model was developed to study lunar motion in the Sun-Earth-Moon system, later refined by Brown into the Hill-Brown theory. It can also approximately describe spacecraft motion near the second primary in systems with small mass ratios.

Key Properties

  • The coordinate origin is placed at the second primary (e.g., the Moon), simplifying analysis of orbits near it.
  • Gravitational terms from the first primary are linearized, reducing the equations to a tractable form.
  • The model is valid when the distance to the second primary is much smaller than the distance between the two primaries.
  • Brown's refinement improved accuracy for lunar theory and satellite motion studies.

References

  • Hill, G. W. "Researches in the Lunar Theory." American Journal of Mathematics, 1(1):5–26, 1878.
  • Brown, E. W. An Introductory Treatise on the Lunar Theory. Cambridge University Press, 1896.