New-family orbit search
A search for planar periodic three-body orbits of new topology at unequal masses produced 42 distinct primitive topology candidates. Each passed independent periodicity and zero-angular-momentum checks, is absent from the catalogues checked, and satisfies a topological exclusion of Euler-half-twist ancestry. Parameter repeats and points of known families are not counted. They are candidate families relative to the catalogues checked; worldwide publication priority is not established.
All 42 are unstable. One further orbit of the search, golden pancake drama, is linearly stable; it sits on an isolated mass loop described below and in its own note.
In the atlas. The 42 candidates form the family New unequal-mass families, each named by its gallery label where one exists and otherwise by its free-group word, and qualified by its masses. The atlas’s own tool re-converged every one in double-double arithmetic (closure below 10⁻²⁴) and reproduces the periods and multipliers below. Golden pancake drama has its own family, Golden pancake drama, together with six samples of its isola.
| Name | Masses | Period at E = −1 | Largest nontrivial multiplier | Status | Atlas |
|---|---|---|---|---|---|
| Long-lobed flower | 1.000000, 1.076717, 1.000000 | 20.500200 | 64.1091 | unstable | open |
| Interlaced petals | 1.250000, 1.111899, 0.750000 | 22.918608 | 31.3818 | unstable | open |
| Asymmetric loops | 1.120000, 1.030420, 0.880000 | 17.708147 | 1452.85 | unstable | open |
| Woven loops | 1.000000, 0.562707, 1.000000 | 17.777741 | 834.081 | unstable | open |
| Nested crossings | 1.250000, 1.133505, 0.750000 | 16.337180 | 121.969 | unstable | open |
| Weak-instability lattice | 1.000000, 1.077141, 1.000000 | 68.271204 | 1.29126 | unstable | open |
| Near-equal-mass weave | 1.000000, 1.023100, 1.000000 | 83.841446 | 1.57752 | unstable | open |
| Golden pancake drama | 1.000000, 0.612356, 1.000000 | 16.958477 | 1 | linearly elliptic | open |
Numerical evidence
The force is the unsoftened Newtonian inverse-square force with G = 1 and positive masses. Candidates were corrected in 160-bit arithmetic, checked in an independent 12-dimensional Cartesian formulation with the full variational equations, and audited with 192-bit native event detection. The full labelled inertial state must close, not just a rotating-frame shape. The selected orbits also have separate SciPy DOP853 and dense high-precision trajectory checks. Closest approaches are numerically refined estimates, not interval-certified lower bounds.
Family comparisons
The comparison covered the published 695 and 1349 word catalogues, 971 stable itineraries, 12,431 Euler itineraries, the figure-eight and BHH root classes, the 135,445-member Moth-I continuation, Rose’s 221 collisionless entries and the earlier results of this site. Four independently reconstructed Rose trajectories validate the symbolic decoding; Rose’s 142 collision entries have no collision-free topology to compare. The full 421,562-row Euler initial-condition list is available for direct section matching, but its entire word list was not verified. A symmetry-derived homology obstruction excludes the Euler family class for the off-line candidates. Euler catalogue study, stable-orbit study, recent relative-family study.
Collision-free continuation preserves the shape word, and a non-power shape word rules out a repeated traversal of a shorter orbit, conditional on the validated numerical loop. Similar shapes or a shared word do not by themselves prove common ancestry. Connections through collisions, infinite-period limits or unindexed literature remain outside the exclusion arguments.
Golden pancake drama
The stable orbit belongs to a numerically closed, isolated mass-continuation loop, an isola, with masses (1, μ, 1) and μ between about 0.4689 and 0.6346; the branch never reaches three equal masses. Golden pancake drama sits at μ = 0.612356088827994835… with the primitive shape word AABAbbaabaBB, a genuine full labelled inertial period and zero angular momentum. Independent 160-bit Cartesian validation gives normalised closure 1.52×10⁻³⁶, and the four nontrivial Floquet multipliers lie on the unit circle to about 35 decimal places: numerical planar linear stability, not nonlinear or KAM stability.
The loop has two topological segments: one with golden pancake drama’s non-power word, one with (ABab)³, the topology of a threefold figure-eight satellite. The latter is genuine longer-period motion, not a shorter orbit run three times: return tests including T/3 found no shorter return. The topology changes at binary-collision limits near μ ≈ 0.50655 and μ ≈ 0.62944, where the closest pair’s relative angular momentum changes sign; separations were resolved down to about 1.9×10⁻⁷ with the third body more than 1.1 units away. The two directions of continuation meet at μ = 0.55 with initial conditions and period agreeing within 1.90×10⁻²⁹ and opposite tangents, the main evidence for a closed loop.
No published counterpart or matching word was identified in the audited catalogues, including the 821 figure-eight satellites and the 462 choreographies, whose tables contain no power-three examples. The supported description is an isolated unequal-mass family not represented in the audited sources, with a linearly stable segment of distinct primitive topology; worldwide priority is not proved. The typeset note golden pancake drama, with the family map and the exact initial conditions, is linked from that orbit’s page.
Files
- Selected orbits: exact initial conditions, masses, periods, energies, multipliers and proof hashes for the orbits above. The Cartesian state holds six positions followed by six canonical momenta; velocities are momentum divided by each mass.
- Candidate inventory: the 42 candidates with their words, exclusions and validation records.
- The numerical sources, proofs and literature audits are available on request from info@threebodyorbits.com.