Three Body Orbits

Families of periodic three-body orbits

Periodic three-body orbits come in families: sequences of solutions that share a shape and a topology and differ in how many times the bodies wind around each other. The atlas has 3,866 orbits in 37 families, each traced back to the paper that found it.

Belgrade classes

Figure-8 V.1.A, an example of the Figure-8 familyFigure-8 orbits28 orbitsThree equal masses chase each other along a single figure-eight curve, the most famous choreography.Butterfly I.2.B, an example of the Butterfly I familyButterfly I orbits19 orbitsTwo wings of tight loops, with the bodies crossing through the centre at high speed.Dragonfly II.11.A, an example of the Dragonfly II familyDragonfly II orbits16 orbitsSlim, elongated wings that stretch far from the centre of mass.Yin-Yang III.12.Aα, an example of the Yin-Yang III familyYin-Yang III orbits20 orbitsTwo interlocking halves whose lobes wrap around each other.Moth IVa.2.A, an example of the Moth IVa familyMoth IVa orbits36 orbitsBroad, fringed wings built from many overlapping passes.Butterfly IVb.3.A, an example of the Butterfly IVb familyButterfly IVb orbits32 orbitsA second butterfly sequence with a different topology from class I.Moth IVc.12.B, an example of the Moth IVc familyMoth IVc orbits22 orbitsMoth-like shapes from a third topological class.Yarn VI.4.A, an example of the Yarn VI familyYarn VI orbits6 orbitsDense skeins of thread wound around a hollow core.Moth VIIa.11.A, an example of the Moth VIIa familyMoth VIIa orbits8 orbitsSymmetric moths whose wings meet along a luminous seam.Moth VIIb.7.A, an example of the Moth VIIb familyMoth VIIb orbits12 orbitsRelated moths with a different winding of the outer lobes.VIII.8.A, an example of the Sequence VIII familySequence VIII orbits3 orbitsThe eighth sequence of the Belgrade gallery, recovered from its archived pages; three orbits that appear in no other catalogue.Split figure-8 SE.87, an example of the Split figure-eights familySplit figure-eights2 orbitsTitov’s continuations of the figure-eight to a lighter third body, where the curve splits into two lobes.Moth satellite M13, an example of the Certified satellite familyCertified satellite orbits1 orbitA satellite of moth I whose existence at masses 1, 1, 1.001 is established by a computer-assisted proof.Bestiary t0(2,4), an example of the Rose’s Bestiary familyRose’s Bestiary13 orbitsOrbits from Rose’s survey of the symmetry-reduced problem, which reaches parts of phase space the other searches do not.Loop-ended triangles, an example of the Sheen’s orbits familySheen’s orbits12 orbitsA dozen orbits Matthew Sheen found in 2016 with a root finder and hand-picked starting guesses while a teaching assistant on Cornell’s dynamics course, named for what they look like; six hang in the Belgrade gallery. Most of them carry angular momentum.

Broucke

Broucke A1, an example of the Broucke A familyBroucke A orbits16 orbitsBroucke’s 1975 family of orbits with a distant third body circling a close binary; several are linearly stable.Broucke R3, an example of the Broucke R familyBroucke R orbits13 orbitsRetrograde relatives of the A family, with the third body circling the other way.Broucke–Boggs S3, an example of the Broucke–Boggs familyBroucke–Boggs orbits23 orbitsBroucke and Boggs’s 1975 survey of periodic orbits at many mass ratios, most of them reported linearly stable.Rosette 4:5, an example of the Broucke-like rosettes familyBroucke-like rosettes14 orbitsZero-angular-momentum relatives of Broucke’s rosettes: two equal bodies in a close binary and a lighter third body circling them the other way, its mass tuned so that the precession closes the pattern after a whole number of turns and the inner paths cross the outer one. Fourteen representatives from the site’s own September 2026 search, all linearly stable; points on a known continuous family rather than a new one. Named by the inner and outer winding counts.

Li–Liao classes

I.A.1, an example of the I.A familyI.A orbits513 orbitsClass I.A orbits found with Liao’s clean numerical simulation method, for a third body of varying mass.I.B.1, an example of the I.B familyI.B orbits540 orbitsClass I.B orbits, for equal and unequal masses.I.C.1, an example of the I.C familyI.C orbits3 orbitsThe rare class I.C orbits.II.A.2, an example of the II.A familyII.A orbits20 orbitsClass II.A orbits with an unequal third mass.II.B.1, an example of the II.B familyII.B orbits35 orbitsClass II.B orbits, mostly long-period.II.C.1, an example of the II.C familyII.C orbits844 orbitsThe largest class in the atlas: hundreds of II.C orbits for seven mass ratios.II.D.1, an example of the II.D familyII.D orbits2 orbitsTwo class II.D orbits with a heavy third body.Moth IVa.2.A, an example of the Moth IVa.2.A, unequal masses familyMoth IVa.2.A, unequal masses7 orbitsThe stable moth IVa.2.A continued to unequal masses: one body a little lighter or heavier than the other two, and the orbit stays stable.

Free-fall

Free-fall F1, an example of the Free-fall familyFree-fall orbits311 orbitsThree bodies released from rest fall together, swing past each other and return to rest exactly where they started.

Sofia searches

Figure-8 satellite HS.10, an example of the Figure-8 satellites familyFigure-8 satellites196 orbitsStable orbits found by an exhaustive search of the figure-eight’s neighbourhood: each traces the eight with extra loops and twists, and every one is linearly stable to 40 digits.Moth region HS.44, an example of the Moth region familyMoth region103 orbitsStable orbits from the moth-like part of the search domain, between the figure-eight and the Schubart regions.Schubart region HS.75, an example of the Schubart region familySchubart region213 orbitsStable orbits that pass extremely close to a binary collision, descended from Schubart’s rectilinear orbit. They needed 100-digit arithmetic to find.Broucke region HS.47, an example of the Broucke region familyBroucke region74 orbitsStable orbits related to Broucke’s isosceles orbit, with one body swinging through the other two.Family NF.1, an example of the New families familyNew families123 orbitsOrbits from topological families that no earlier catalogue contained; all of them unstable.BH-818, an example of the New Broucke-region orbits familyNew Broucke-region orbits3 orbitsThree linearly stable orbits found here in September 2026 by pushing the Broucke region past the published catalogue’s limit of T* = 800: two continue known lap sequences, one is a seventeen-fold resonant satellite of the Broucke-region orbit B1. Verified independently to 160 bits and again by this atlas’s own tool; not published elsewhere.

Choreographies

Choreography CH.203, an example of the Choreographies familyChoreographies76 orbitsAll three bodies follow one closed curve, each a third of a period behind the next. Every one here is linearly stable.Choreography NC.1, an example of the Nontrivial choreographies familyNontrivial choreographies164 orbitsChoreographies whose knot is not a power of the figure-eight’s: three bodies on one closed curve with a topology of its own. Computed to 180 digits; all unstable.Choreography SC.2, an example of the Simó’s choreographies familySimó’s choreographies343 orbitsCarles Simó’s catalogue of three-body choreographies with nonzero angular momentum, the curves that made choreographies famous. All unstable except the figure-eight.