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5. Restricted three-body problem

The restricted three-body problem is a simplified version of the three-body problem where one of the masses m₃ is assumed much smaller than the primaries m₁ and m₂. Thus, m₁ and m₂ move in Keplerian orbits which are not affected by m₃. The Sun-Earth-Moon system provides an example where we further have m₂ ≪ m₁. In the planar circular restricted three-body problem, the primaries move in fixed circular orbits around their common CM with angular speed Ω given by Kepler’s third law, and m₃ moves in the same plane as m₁ and m₂. Here, d is the separation between primaries. This system has 2 degrees of freedom associated to the planar motion of m₃, and therefore a 4-dimensional phase space just like the planar Kepler problem for the reduced mass. However, unlike the latter, which has three conserved quantities and is exactly solvable, the planar restricted three-body problem has only one known conserved quantity, the ‘Jacobi integral’, which is the energy of m₃ in the co-rotating frame of the primaries.

E = ½m₃ṙ² + ½m₃r²φ̇² − Gm₃(m₁/r₁ + m₂/r₂) ≡ T + V_eff   (15)