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Families of relative periodic orbits in the planar three-body problem via consecutive alignments
Joan Gimeno, Marc Jorba-Cuscó, Begoña Nicolás
TL;DR
The paper addresses how to construct coherent relative periodic motions for planar three-body dynamics, which are needed for realistic restricted four-body formulations. It develops an alignment-based numerical continuation method and applies it to multiple orbit families, revealing broad stable structures and transitions between satellite, circumstellar, planetary, and binary configurations.
Problem
The paper seeks a systematic way to construct coherent three-body motions for restricted four-body models and to understand the dynamical limits of planetary and lunar prescribed motions.
Method
The method integrates from one syzygy to the next, matches positions and momenta to impose relative periodicity, continues the resulting one-parameter family, and evaluates reduced monodromy eigenvalues for stability.
Results
The procedure produces Poincaré, Hill, and binary-type families, including highly eccentric continuations and Hill transitions from satellite to circumstellar motion.
Takeaways & Limitations
Stable solutions occupy wide portions of continuation curves, providing a robust structure that can include nearby motions with small inclinations, imperfect alignments, or slightly different parameters.
Takeaways & Limitations
The analysis uses approximations for gravitational-boundary estimates, and some unstable circumbinary solutions could not be computed exactly at θ = 2π.
Abstract
from arXiv · showhide
Relative periodic orbits (RPOs) are solutions of the three-body problem that are periodic in a uniformly rotating reference frame and, in general, quasi-periodic in inertial coordinates. We present a numerical procedure for computing and continuing one-parameter families of RPOs of the planar Newtonian three-body problem. The method exploits consecutive syzygies, understood here as configurations in which the three bodies are aligned and their velocities satisfy the corresponding symmetry conditions. Matching the positions and momenta at two consecutive alignments reduces the computation of RPOs to a low-dimensional nonlinear problem. Its solutions are then numerically continued, and linear stability is determined from the nontrivial eigenvalues of the rotated monodromy matrix after removing the neutral directions associated with conserved quantities and continuous symmetries. The procedure is applied to several mass distributions and initial configurations, producing families of Poincaré, Hill, and binary-type solutions. These families exhibit transitions from nearly circular to highly eccentric motion, changes of stability near resonances and turning points, and absolute periodic solutions when the rotation angle is a rational multiple of 2π. In the Hill families, the continuation connects satellite configurations with circumstellar motion as the smallest body loses its gravitational binding to the intermediate body. Circumbinary and circumstellar configurations are also obtained in the binary regime. The results illustrate the dynamical diversity of RPOs and provide coherent three-body motions that can be used as prescribed trajectories in restricted four-body models.
1. Introduction
The paper develops a systematic numerical route from the general three-body problem to coherent relative periodic orbits, motivated by restricted four-body applications. It continues planetary, satellite, and binary-type families across nearly circular and highly eccentric regimes.
- Background: Quasi-Bicircular solutions are special relative periodic orbits belonging to one-parameter families that describe prescribed planetary and lunar motions.The paper asks about the dynamical limits and stability of these motions for given masses.
- Motivation: The study targets coherent restricted four-body models by constructing relative periodic solutions of the general planar three-body problem.This addresses the limitation of non-coherent models whose prescribed primary motion is not itself a three-body solution.
- Contribution: The proposed method systematically constructs three-body solutions from alignments, which constitute relative periodic solutions.The approach is intended to broaden the motions available for restricted four-body formulations.
- Contribution: Continued families extend nearly circular solutions to highly eccentric relative periodic solutions for all three primaries.The paper distinguishes the nearly circular cases as Quasi-Bicircular solutions and the extended eccentric families as relative periodic solutions.
- Results: Hill families connect satellite motion around a planet to circumstellar motion when the satellite becomes unbound from the planet.The same continuation also represents a small body that begins around a star and is later trapped by a planet.
- Results: The study also includes binary solutions that may represent a binary star and a planet, alongside Poincaré and Hill families.The stated broader objective is to use these coherent motions in restricted four-body problems with a massless body.
2. The Three-Body Problem
The planar three-body problem is formulated as a Hamiltonian gravitational system with conserved quantities, symmetries, collisions, and distinct absolute and relative periodic motions. These structures support dimensional reduction and motivate the study of RPO families.
- Model: The planar three-body problem models three point masses interacting through universal gravitation in a twelve-dimensional phase space.The state contains two positions and two velocities for each body.
- Global behavior: The flow can encounter double or triple collisions, with collision initial conditions forming a null-measure, first-category set.Triple collision requires the angular momentum to vanish according to Sundman’s theorem.
- Structure: The Hamiltonian, center of mass, linear momentum, and angular momentum provide conserved quantities that reduce the system to three degrees of freedom.The reduction removes translational and additional energy-angular-momentum degrees of freedom.
- Periodic motion: Absolute periodic orbits return positions and velocities to their initial values, whereas relative periodic orbits repeat in a uniformly rotating frame.RPOs become absolute periodic when θ/(2π) is rational and are otherwise quasi-periodic in inertial coordinates.
- Periodic motion: Absolute periodic orbits are isolated, while relative periodic orbits lie in one-parameter families.The paper studies these families as continuations of three-body and restricted-problem periodic motions.
- Orbit classes: Poincaré, Hill, and third-kind orbits classify planetary relative periodic motions by eccentricity and inclination, with first-kind orbits numerically continuable to second-kind orbits.Poincaré type denotes planetary solutions, while Hill type denotes lunar or satellite solutions.
3. Numerical computation of RPO
The computation uses consecutive syzygies as a Poincaré section, matches states at successive alignments to impose relative periodicity, and continues the resulting low-dimensional solution curve while assessing stability.
- Method overview: The method searches for two consecutive alignments of the three bodies and uses them to compute and continue relative periodic orbits.Syzygies provide the phase-space section on which the continuation scheme operates.
- Stability: Stability is computed from the nontrivial eigenvalues of a reduced rotated monodromy matrix after projecting out symmetry-associated directions.The algorithm forms an orthogonal complement with SVD before evaluating the reduced matrix.
- Syzygies: A syzygy is characterized by collinear positions, with the paper requiring simultaneous alignment of positions and velocities for its target solutions.The positional condition can be expressed through a vanishing determinant or signed area.
- RPO conditions: The residual system matches distances from the barycenter and individual momenta across consecutive alignments while enforcing velocity-position orthogonality.These conditions imply that the second alignment is a rotated copy of the first.
- Dimensional reduction: After fixing the initial alignment and rescaling x1 to 1, the RPO determination requires the variables x2, ẏ1, and ẏ2.The resulting one-parameter family is parameterized essentially by the distance between bodies 1 and 2.
- Continuation: The continuation algorithm combines predictor and Gauss-Newton corrector steps with flow integration to the next syzygy and residual evaluation.Pseudo-arclength updates maintain the solution curve through successive corrected points.
4. Dynamical diversity of RPO families in the Three-Body Problem
The paper examines diverse relative-periodic-orbit families across mass distributions and initial configurations, emphasizing shared dynamical features while retaining family-specific behavior. Some solutions model astronomical situations, while others serve as academic three-body examples.
- Scope: Family behavior is sensitive to initial relative positions and velocities but appears less sensitive to exact masses and other parameters.The authors therefore examine a variety of academic examples rather than claiming a complete generic analysis.
- Case studies: The case studies cover Poincaré planetary families and Hill satellite families, with the first case presented in greater detail.Later sections focus on particular dynamical features while avoiding repetition of shared analysis steps.
- Interpretation: Many computed families reproduce characteristic situations in real astronomical systems, whereas others do not correspond to known systems and are retained for academic interest.The latter are treated as solutions of the three-body problem rather than direct astronomical models.
4.1 Poincar´e solutions
The Poincaré families are continued from stable, nearly circular planetary configurations toward eccentric motions, with resonances and turning points organizing stability changes and transitions between orbit species.
- Configurations: The study considers prograde Poincaré configurations with planets ordered either m0 ≥ m1 ≥ m2 or m0 ≥ m2 ≥ m1.The second case places the intermediate body as the smaller one.
- Continuation from circular motion: Initially, all four mass cases yield stable, nearly circular solutions, with B1's velocity near unity and its Keplerian period near 2π.These properties correspond to e1 ≈ 0 while B1 and B2 remain far apart.
- Resonances and stability: Crossing θ = π produces unstable solutions when m2 is comparable to m1, whereas the same resonance is unstable for every tested mass ordering when m2 > m1.At θ = π, the computed periods satisfy Ts = T2/2 and T2 = 3T1, with x2 ≈ 2.08 and small eccentricities.
- Turning points: Near θ = 2π, continuation curves develop turning points around x2 ≈ 1.59, where B2's velocity is maximal and total angular momentum approaches a minimum.The minimum angular momentum is close to zero, a condition under which binary or triple collision can become possible.
- Turning points: Passing a turning point changes stability and drives increasing velocities and eccentricities, so the families connect first-species to second-species Poincaré orbits.The resulting unstable intervals can later return to stable solutions as the bodies separate; at θ = 2π, the orbits are highly eccentric.
- Methodological comparison: The alignment-based continuation computes the θ = 2π resonance for highly eccentric orbits and reveals post-turning-point instability that earlier continuation from circular CRTBP orbits did not capture.RPOs with θ/(2π) rational, including θ = π and θ = 2π, are absolute periodic orbits, but the θ = 2π solutions are not classical Euler collinear solutions.
- Continuation boundary: Continuation ends when the bodies are far apart, significantly eccentric, and total energy approaches zero, where the three-body motion ceases to be bounded.
4.2 Hill solutions
Hill solutions begin as hierarchical star–planet–satellite configurations and continue across regimes as the satellite becomes unbound and circumstellar. The families are mostly stable and are tracked through energy, angular, eccentricity, resonance, and geometric diagnostics.
- Initial configuration: Hill solutions model body 2 orbiting body 1 while body 1 orbits body 0 in a hierarchical star–planet–satellite configuration.The initial approximation treats the system as two decoupled two-body problems when m0 ≫ m1 ≫ m2 and bodies 1 and 2 are sufficiently close.
- Continuation procedure: The continuation starts with bodies 1 and 2 close together and follows RPO families using curves colored by stability, velocity, alignment angle, relative energy, and eccentricity.The mass cases use m1 = 10^-4 and m2 ∈ {10^-7, 10^-6, 10^-5, 10^-4}, with m0 = 1 − m1 − m2.
- Stability and mass dependence: For most computed configurations, the families are stable, with no real eigenvalues, although their shapes and orbital regimes vary with the satellite-to-planet mass ratio.When the satellite mass is comparable to the planet mass, satellite-type solutions persist, but body 1 has velocity greater than 1.
- Satellite escape: As x2 increases, the relative energy h1,2 changes from negative to positive while body 2’s orbit around body 1 becomes increasingly eccentric.The sign change marks the loss of gravitational capture; body 2 then begins orbiting body 0, with its circumstellar orbit initially low-eccentricity and later more eccentric.
- Binding boundary: The approximate Hill radius is 0.032, while the first unbounded solutions occur at distances from body 1 of approximately 0.04–0.05.The Hill radius and relative-energy construction are explicitly described as useful approximations rather than exact three-body invariants.
- Configuration transition: A Hill-family continuation can connect a typical satellite regime to a central body with two circumstellar companions, rather than a planetary Poincaré configuration.The paper states that continuation from Hill-type to Poincaré-type solutions requires appropriate regularization.
4.3 Binary solutions
Binary families describe configurations with two massive bodies and a comparatively small third body, spanning circumbinary and more varied binary-system motions. Continuation reveals circular-to-eccentric transitions, turning points, resonance behavior, stability loss, and eventual changes in the small body's rotation.
- Binary configuration: Binary solutions use bodies 0 and 1 as the massive pair, while body 2 has comparatively small mass, modeling systems such as binary stars with a planet.The considered masses are m2 ∈ {10^-6, 10^-5, 10^-4, 10^-3}, with m0 = m1 = (1 − m2)/2.
- Circumbinary planet: When body 2 has velocity comparable to body 1, it follows a circumbinary orbit around the orbit defined by the two stars.For the configuration with ẏ2 = 0.7, the solutions describe a binary star system and a circumbinary planet.
- Continuation behavior: Far from the binary, bodies 0 and 1 follow circular stable orbits, including the resonance θ = π, while body 2 develops eccentricity as the bodies approach.After the turning point, the binary orbits decouple; the minimum of body 2's eccentricity occurs near a selected continuation solution.
- Continuation behavior: The continuation has a turning point where the two relevant bodies approach closely, followed by growing eccentricity and velocity and slower variation of the alignment angle.The continuation curves retain H < 0, so the three-body motion remains bounded.
- Stability and limits: Stability is lost before the turning point, where body 2 reaches maximum velocity; some unstable solutions have four real eigenvalues, and θ = 2π was not computed.The last computed orbit lies at an angle extremely close to 2π rather than exactly at that alignment.
- Configuration diversity: As body 2 approaches body 1, its velocity eventually becomes negative, indicating clockwise rotation instead of the previously observed counter-clockwise motion.The family contains substantially different configurations, including solutions not always corresponding to known astronomical systems.
5. Conclusions and further work
The paper introduces a systematic numerical methodology for constructing planar three-body relative periodic orbits from consecutive alignments. The computed families show diverse, robust, and potentially useful coherent motions for future restricted four-body studies.
- Conclusions: The paper introduces a systematic numerical method for computing solutions of the general planar Three-Body Problem.The method targets coherent three-body solutions suitable for restricted four-body formulations.
- Method: The method matches positions and velocities at two consecutive alignments, producing Relative Periodic Orbits and reducing numerical costs in stability analysis.The approach also reduces error accumulation when analyzing stability.
- Results: Illustrative examples produce three-body solutions with diverse configurations and phenomena compatible with known real systems.The examples demonstrate the potential of the methodology across multiple solution types.
- Results: Stable solutions occupy wide continuation-curve ranges rather than appearing as isolated points, forming a robust structure that includes nearby perturbed configurations.The authors connect this structure to small inclinations, imperfect alignments, and slightly different masses or orbital parameters.
- Further work: The longer-term goal is to study small bodies in restricted four-body problems using the computed relative periodic orbits.Planned work includes reproducing real systems and adding a massless particle to analyze third-body effects on triangular-point stability.