Regularization of the Triple Collision Manifold in the Planar Three-Body Problem

Abstract

In his work about the triple-collision manifold of the rectilinear three-body problem, McGehee used Lundman’s method to regularize the binary collisions. We apply a similar method to the three binary collisions of the planar three-body problem.

The positions M₁, M₂, M₃, can be represented by the radius of inertia r, and three angles s₁, s₂, s₃, so that cos s₁ is vanishing with the distance ∥M₁ M₂ M₃∥, (s₁ - s̄₁) with ∥M₂ M₃∥, and $\left( {{s_1} - {{\bar s}_1}} \right)$ with ∥M₂ M₃∥, and $\left( {{s_1} - {{\mathop s\limits^ = }_1}} \right)$ with ∥M₃ M₁∥ (where $\bar s$ and $\mathop s\limits^ =$ are constants which depend on the masses m₁ only). Thus, we propose the Lundman-McGehee transformation, defined in particular by the change of time τ-> τ’: dτ = η(s₁) · dτ’, where: $\left( {{s_1}} \right) = \cos {s_1} \cdot si{n^\alpha }\left( {{s_1} - {{\bar s}_1}} \right) \cdot si{n^\alpha }\left( {{s_1} - {{\mathop s\limits^ = }_1}} \right)$. The exponent α must be chosen so that the new velocities and the new potential are regular and the new time τ’ admits a finite limit, at each binary collision. Then, one shows that α must be an irreducible fraction p/q (where p has to be even) such that: 1≤α≤3/2.

By such a transformation, the differential system of the problem is regularized simultaneously at each of the three binary collisions, and for any value of the integrals h and C.

The regularized triple collision manifold can be described as follows: to each configuration-point (s₁, s₃)∈S², it associates an ellipsoid in the space of velocities; this ellipsoid degenerates to a cylinder, at each of the six points of S² representing the three binary Collision-configurations.