Sufficient conditions for hyperbolic-elliptic escape and for ‘ejection without escape’ in the three-body problem

Celestial Mechanics and Dynamical Astronomy - The three-body problem is decomposed into the motion of the binarym 1,m 2 and the motion of the third massm 3 with respect to the center of masses ofm...     

Sufficient conditions for return in the three-body problem

Sufficient conditions are given, which, if fulfilled, enable one to determine, rigorously, that an ejected particle will not escape from a three-body system.     

Close triple approaches and escape in the three-body problem

We have studied a total of 5000 close triple approaches resulting in escape, for equal-mass systems with zero initial velocities. Escape is shown to take place in the majority of the cases after a fly-by close triple approach when the escaper passes near the centre of mass along an almost straight-line orbit. A number of configurational and kinematical parameters are introduced in order to characterize the triple approach. The distributions of these parameters are investigated. A comparison with 831 examples in the vicinity of the so-called Pythagorean problem is carried out. We find that the general features of close triple approaches which result in escape are the same for both types of systems.     

Free time minimizers for the three-body problem

Free time minimizers of the action (called “semi-static” solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120–131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton–Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton’s three-body problem which is asymptotic to Lagrange’s parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange’s solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209–227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler’s central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.     

The three-body problem

Book review

The three-body problemby Christian Marchal, Elsevier, Amsterdam, 1990, (ISBN 0444874402)     

Non-integrability of the three-body problem

We consider the planar problem of three bodies which attract mutually with the force proportional to a certain negative integer power of the distance between the bodies. We show that such generalisation of the gravitational three-body problem is not integrable in the Liouville sense.     

High-order resonances in the restricted three-body problem

This paper consists in analyzing very simple resonance models for the j+i/j (i=2, 3, 4) resonance cases by averaging, truncating and scaling the restricted three body problem. The phase space, the equilibria, the critical areas and the probability of capture are analytically calculated for each case.     

Retrograde resonance in the planar three-body problem

We continue the investigation of the dynamics of retrograde resonances initiated in Morais and Giuppone (Mon Notices R Astron Soc 424:52–64, doi:10.1111/j.1365-2966.2012.21151.x, 2012). After deriving a procedure to deduce the retrograde resonance terms from the standard expansion of the three-dimensional disturbing function, we concentrate on the planar problem and construct surfaces of section that explore phase-space in the vicinity of the main retrograde resonances (2/ $-$ 1, 1/ $-$ 1 and 1/ $-$ 2). In the case of the 1/ $-$ 1 resonance for which the standard expansion is not adequate to describe the dynamics, we develop a semi-analytic model based on numerical averaging of the unexpanded disturbing function, and show that the predicted libration modes are in agreement with the behavior seen in the surfaces of section.     

Motion in the generalized restricted three-body problem

This paper investigates the motion of an infinitesimal body in the generalized restricted three-body problem. It is generalized in the sense that both primaries are radiating, oblate bodies, together with the effect of gravitational potential from a belt. It derives equations of the motion, locates positions of the equilibrium points and examines their linear stability. It has been found that, in addition to the usual five equilibrium points, there appear two new collinear points L _n1, L _n2 due to the potential from the belt, and in the presence of all these perturbations, the equilibrium points L ₁, L ₃ come nearer to the primaries; while L ₂, L ₄, L ₅, L _n1 move towards the less massive primary and L _n2 moves away from it. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ _c and unstable for $\mu_{c} \le\mu\le\frac{1}{2}$ , where μ _c is the critical mass ratio influenced by the oblateness and radiation of the primaries and potential from the belt, all of which have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near the oblate, radiating binary stars systems surrounded by a belt.     

Periodic orbits in the general three-body problem

Stability regions are identified in the neighborhood of periodic orbits. Features of motion in these regions are investigated. The structure of stability regions in the neighborhood of the Schubart, Moore, and Broucke orbits, the S-orbit, and the Ducati orbit is studied. The following features of motion are identified near these periodic orbits: libration, precession, symmetrization, centralization, bounce (a transition between types of trajectories), ejections, etc.     

Triple collisions in the one-dimensional three-body problem

We consider the motions of particles in the one-dimensional Newtonian three-body problem as a function of initial values. Using a mapping of orbits to symbol sequences we locate the initial values leading to triple collisions. These turn out to form curves which give clear structure to the region in which the motions depend sensitively on initial conditions. In addition to finding the triple collision orbits we also locate orbits which end up to a triple collision in both directions of time, that is, orbits which are finite both in space and time. This revised version was published online in July 2006 with corrections to the Cover Date.     

Escape or retention in the three-body problem

Sufficient conditions for escape and for retention in the three-body problem are derived which, for escape, are more stringent than those previously derived and, for retention, avoid the difficulty of containing the inverse of the mass of the escaping particle.     

Approach-ejection correlation in the general three-body problem

The relations between parameters of triple approaches and the lengths of subsequent ejections are analyzed for the general three-body problem with components of equal masses and zero initial velocities. A statistically significant correlation is shown to exist between the closeness of approaches and the lengths of subsequent ejections: closer approaches generally result in longer ejections. We have found several systems that evolve to a temporary quasi-stable chain-like configuration.     

Periodic solutions in the commensurable three-body problem

Various families of periodic solutions are shown to exist in the three body problem, in which two of the bodies are close to a commensurability in mean motions about the third body, the primary, which is considerably more massive than the other two. The cases considered are

1. The non-planar circular restricted problem (in which one of the secondary bodies has zero mass, and the other moves in a fixed circular orbit about the primary).
2. The planar non-restricted problem (in which the three bodies move in a plane, and both secondaries have finite mass).
3. The planar elliptical restricted problem (in which the three bodies move in a plane, one of the secondary bodies has zero mass, and the other moves in a fixed elliptical orbit about the primary).

The method used is to eliminate all short period terms from the Hamiltonian of the motion by means of a von Zeipel transformation, leaving only the long period terms which are due to the commensurability. Hence only the long period part of the motion is considered, and the variables used differ from the variables describing the full motion by a series of short-period trigonometric terms of the order of the ratio of the mass of the secondaries to that of the primary body. It is shown that solutions of the long-period problem in which the variables remain constant are equivalent to solutions in the full motion in which the bodies periodically return to the same configuration, and these are the types of periodic solution that are shown to exist. The form of the disturbing function, and hence of the equations of motion, is found up to the fourth powers of the eccentricities and inclination by considering the d'Alembert property. The coefficients of the terms appearing in this expansion are functions of the semi-major axes of the orbits of the secondary bodies. Expressions for these coefficients are not worked out as they are not required. Lete, n, m be the orbital eccentricity, mean motion and mass of one of the secondary bodies, and lete′, n′, m′ be the corresponding quantities for the other. (The mass of the primary is taken as unity). In cases (a) and (c) we will havem=0. In case (a)e′ will be zero, and in case (c) it will be a constant. Leti be the mutual inclination of the orbits of the secondary bodies. Suppose the commensurability is of the form(p+q) n =pn′, wherep andq are relatively prime integers, and put γ=(p+q) n/n′?p. The families of periodic solutions shown to exist are as follows. For q=1 No periodic solutions are found withi≠0 in case (a), and none withe′≠0, in case (c). In case (b) periodic solutions are found in whiche=0 (m′/γ),e′=0 (m/γ) for values of γ away from the exact commensurability. As γ approaches zero thene ande′ become 0 (1). For q≠1 Case (a). Families of periodic solutions bifurcating from the family withe=0, i=0 are shown to exist. Families in whichi=0 ande becomes non-zero exist for all values ofq. Families in whiche=0 andi becomes non-zero exist for even values ofq. Families in whiche andi become non-zero simultaneously exist for odd values ofq. Case (b). No families are found other than those withe=e′=0. Case (c). Families are found bifurcating from the familye=e′=0 in whiche ande′ become non-zero simultaneously. For all these solutions existence is only demonstrated close to the point of bifurcation, where all the variables are small, as the method uses series expansions ine, e′ andi. From the form of the solutions it is clear that the non-zero variables will become large for values of γ away from the bifurcation point.     

Global chaoticity in the Pythagorean three-body problem

The effects of small changes in the initial conditions of the Pythagorean three-body problem are investigated by computer simulations. This problem consists of three interacting bodies with masses 3, 4 and 5 placed with zero velocities at the apices of a triangle with sides 3, 4 and 5. The final outcome of this motion is that two bodies form a binary and the third body escapes. We attempt to establish regions of the initial positions which give regular and chaotic motions. The vicinity of a small neighbourhood around the standard initial position of each body defines a regular region. Other regular regions also exist. Inside these regions the parameters of the triple systems describing the final outcome change continuously with the initial positions. Outside the regular regions the variations of the parameters are abrupt when the initial conditions change smoothly. Escape takes place after a close triple approach which is very sensitive to the initial conditions. Time-reversed solutions are employed to ensure reliable numerical results and distinguish between predictable and non-predictable motions. Close triple approaches often result in non-predictability, even when using regularization; this introduces fundamental difficulties in establishing chaotic regions.     

The stellar three-body problem

Two applications of von Zeipel's method to the stellar three-body problem eliminate the short period terms and establish two new integrals of the motion beyond the classical integrals. The remaining time averaged problem with only the second order Hamiltonian has one additional integral and can be solved. The motion with the third order averaged Hamiltonian included is more complex, in that there may be additional resonances, and the additional integral does not exist in all cases.