Central Configurations

Gravitational molecules at minimum shape complexity

Aaron Lax — March 2026

Critical points of VS = √Icm × W on the pre-shape sphere. Heavy particles red, light particles blue. 3D positions projected via PCA. N = 100 to 10,000.

Highlights

Click any image to explore.

Bond angles
Bond angle peaks at 107.9° — water is 104.5°
N=1000 1:2000
N=1,000, 1:2000 — mass shell structure
Oblate
Oblate shape — Jupiter-like flattening
V_EM
VEM — magnetic vorticity breaks symmetry

Mass Ratio Progression — N=100, 2D

Equal
1:2
1:10
1:80
1:2000

Scaling — Equal Masses

N=100
N=500
N=1K
N=2K
N=5K

Scaling — Mass Ratio 1:2000

Shell structure strengthens with N.

N=100
N=500
N=1K
N=2K
N=5K

All Configurations

N=100

2D + 3D, 5 mass ratios

N=500, 3D

5 mass ratios

N=1,000, 3D

5 mass ratios

N=2,000, 3D

5 mass ratios

N=5,000, 3D

5 mass ratios

Molecule Search

500 starts, angle distributions, recurring motifs

Oblate Shapes

Jupiter/Saturn-like flattening

VEM

Maria’s 3-body triangle vorticity

Shape Complexity Values

ConfigEqual1:21:101:801:2000
N=100, 2D52,231142,7843.6×1065.1×1081.6×1012
N=100, 3D43,325118,8353.0×1064.3×1081.3×1012
N=5002.5×1067.0×1061.8×1082.6×10107.9×1013
N=1,0001.4×1074.0×1071.0×1091.5×10114.5×1014
N=2,0008.2×1072.3×1085.8×1098.6×10112.6×1015
N=5,0008.2×1082.3×1095.8×10108.5×10122.6×1016
N=10,0004.6×109———1.5×1017
Method. Minimise VS = √Icm × W on {∑qi=0, ∑|qi|²=1}. Gradient descent with mass preconditioning, cosine-annealed learning rate, 8K–40K iterations, 12–100 random restarts. N≥10,000 uses tiled pairwise computation (O(N×T) memory). JAX float64 on NVIDIA H100 GPUs via Modal.
Data and source code available on request.

N=100

2D results reproduce Maria’s methodology. 3D results projected via PCA.

2D — Maria Comparison

Equal
1:2
1:10
1:80
1:2000

3D

Equal
1:2
1:10
1:80
1:2000

N=500, 3D

Molecular segregation persists at scale.

Equal
1:2
1:10
1:80
1:2000

N=1,000, 3D

Mass-dependent shell structure clearly visible.

Equal
1:2
1:10
1:80
1:2000

N=2,000, 3D

Shell structure increasingly pronounced.

Equal
1:2
1:10
1:80
1:2000

N=5,000, 3D

Five thousand particles. Clear mass-dependent spatial organisation.

Equal
1:2
1:10
1:80
1:2000

Oblate Configurations

z-variance penalty breaks spherical symmetry → Jupiter/Saturn-like shapes. N=200, 3D, aligned to principal axes.

Equal Masses

λ=0 — spherical (z/xy=1.99)
λ=0.5 — mildly oblate (z/xy=0.92)
λ=2.0 — strongly oblate (z/xy=0.39)
λ=10 — pancake (z/xy=0.006)

Mass Ratio 1:80

λ=0 (z/xy=2.00)
λ=0.5 (z/xy=0.91)
λ=2.0 (z/xy=0.40)
λ=10 (z/xy=0.04)

Maria’s VEM

3-body oriented triangle terms: VEM = √I × (α Welec + β Wmag). The magnetic term couples triangle areas to a global vorticity axis — oblate shapes from pure geometry.

N=100, varying β

β=0 (spherical)
β=0.1 (oblate, 0.70)
β=0.5 (0.70)
β=2.0 (0.90)
β=10 (0.85)

N=200

β=0 (spherical)
β=1.0 (0.79)
β=5.0 (0.85)

Molecule Search

500 random starts per mass ratio, keeping ALL local minima. Bond angles, asphericity, and cluster counts classify each configuration.

Equal Masses — 107.9°

All 500 starts converge to one minimum. Bond angle peaks at water’s 104.5° line.

Angle distribution. Red dashed = 105° (water).
Gallery — same minimum, different PCA views.

1:10 — 98.1°

Still one minimum. Two light-particle clusters.

Angles
Gallery

1:80 — 86.6°

3 distinct minima. Dominant (434/500) has 7 light clusters.

Angles
Gallery

1:2000 — 71.4°

2 distinct minima. Dominant (447/500) with pronounced segregation.

Angles
Gallery

Angle Trend

Mass RatioDistinct MinimaDominant AngleLight Clusters
Equal1107.9°—
1:10198.1°2
1:80386.6°7
1:2000271.4°—

Bond angle decreases monotonically: 108° → 98° → 87° → 71°.