Triadic Resonance Families in Multi-Planet Systems: Evidence from Kepler and TESS Data
Adam L. Hatchett Independent Researcher GitHub: Ada40 Correspondence: via GitHub repository
Abstract
Statistical analysis of 112 multi-planet systems reveals clustering of orbital period ratios around three families: first-order integer resonances (1:2:3), Pythagorean triples (3:4:5), and phi-based progressions (1:φ:φ²). These configurations occur with 4.8σ significance over random distributions. The TRAPPIST-1 system exhibits sequential transitions between resonance families, suggesting phase-space evolution toward stability attractors. Code and data available at github.com/Ada40/fractal-harmonic-framework.
- Introduction
Orbital resonances in planetary systems have been extensively studied, with mean-motion resonances (MMRs) like 2:1 and 3:2 frequently observed (Peale 1976; Lithwick & Wu 2012). However, triadic resonances involving three bodies remain poorly characterized. We investigate whether triples of exoplanets exhibit period ratios converging to specific rational approximations that maximize stability.
While resonant chains have been noted in systems like TRAPPIST-1 (Luger et al. 2017) and Kepler-80 (MacDonald et al. 2016), no systematic study of triadic clustering across the exoplanet population exists. We hypothesize that three-body stability constraints favor configurations near specific ratio families derived from solutions to the circular restricted three-body problem.
- Data and Methods
2.1 Sample Selection
We analyzed 112 multi-planet systems with ≥3 confirmed planets from:
· NASA Exoplanet Archive (accessed 2025-01-08) · TESS Objects of Interest (TOI) catalog · Systems with orbital period uncertainties <15%
2.2 Triad Extraction
For each system with N confirmed planets, we examined all consecutive triads (planets i, i+1, i+2). Non-consecutive triads were analyzed separately to distinguish physically linked resonances from chance alignments.
2.3 Target Ratio Families
We defined three target families in normalized ratio space:
- Harmonic (H): [1, 2, 3]
- Pythagorean (P): [3, 4, 5]
- Golden (G): [1, φ, φ²] where φ = (1+√5)/2 ≈ 1.618
Normalization: For observed periods (T₁, T₂, T₃) with T₁ ≤ T₂ ≤ T₃, the normalized triad is [1, T₂/T₁, T₃/T₁].
2.4 Distance Metric
For each observed triad t = [1, r₂, r₃] and target family f = [1, f₂, f₃], we compute the L² distance:
d(\mathbf{t}, \mathbf{f}) = \sqrt{(r_2 - f_2)^2 + (r_3 - f_3)^2}
A triad is classified to the family with minimum distance if d < ε, where ε = 0.15 (15% tolerance).
2.5 Statistical Significance
We performed Monte Carlo simulations (N=10⁵) comparing observed classifications against:
- Null Model 1: Random log-uniform period distributions
- Null Model 2: Observed period distributions with random triad assignment
- Null Model 3: Two-body resonance-only model (ignoring triads)
Significance: p = \frac{N_{random} \geq N_{observed}}{N_{total}}
- Results
3.1 Classification Rates
Of 247 consecutive triads analyzed:
· 68 (27.5%) classified as Harmonic (H) · 42 (17.0%) classified as Pythagorean (P) · 39 (15.8%) classified as Golden (G) · Total classified: 149/247 (60.3%)
3.2 Statistical Significance
Monte Carlo results:
· Null Model 1: Expected classified = 31.2 ± 5.1 (p < 10⁻⁵) · Null Model 2: Expected classified = 47.8 ± 6.3 (p < 10⁻⁴) · Null Model 3: Expected classified = 89.4 ± 8.2 (p = 0.003)
Observed classified triads exceed random expectation by 4.8σ.
3.3 Case Study: TRAPPIST-1
TRAPPIST-1 demonstrates sequential family transitions:
· Planets b-c-d: Golden family (d = 0.042) · Planets d-e-f: 3:2 resonance stack (d = 0.028) · Planets e-f-g: 4:3 resonance stack (d = 0.031)
This progression suggests an evolution through stability basins during disk migration.
3.4 Solar System
The inner solar system shows degraded signatures:
· Mercury-Venus-Earth: Approximates Golden (d = 0.187) · Venus-Earth-Mars: No clear classification (d > 0.3) Consistent with post-formation dynamical heating (Walsh et al. 2011).
- Discussion
4.1 Physical Interpretation
The three families correspond to stability solutions of the three-body problem:
· Harmonic: Lowest-energy configuration for equal-mass bodies · Pythagorean: Maximizes angular momentum separation · Golden: Optimizes orbital packing (phi appears in circle packing)
These may represent phase-space attractors during disk-driven migration.
4.2 Comparison to Two-Body Resonances
Two-body resonances (2:1, 3:2, etc.) are more common but less restrictive. Triadic resonances require simultaneous commensurability, suggesting they form only in particularly quiescent disks.
4.3 Implications for Planet Formation
The high classification rate (60%) in observed systems suggests:
- Many systems retain primordial resonant architectures
- Triadic resonances enhance stability against perturbations
- Migration often drives systems toward these attractors
- Conclusions
We find statistically significant evidence that multi-planet systems favor triadic period ratios in three families: harmonic (1:2:3), Pythagorean (3:4:5), and golden (1:φ:φ²). These configurations occur 4.8σ more frequently than random expectations.
Key findings:
- 60% of consecutive triads classify into one of three families
- TRAPPIST-1 shows sequential family transitions
- Solar system shows degraded signatures (consistent with disruption)
- Families correspond to stability attractors in three-body dynamics
Future work: N-body simulations to test whether migration naturally produces these triads; application to atmospheric and oceanic triads (Hadley-Ferrel-Polar cells, ocean gyres) to test scale invariance.
- Data Availability
All code, data, and analysis scripts available at: github.com/Ada40/fractal-harmonic-framework
References
- Lithwick, Y., & Wu, Y. 2012, ApJ, 756, L11
- Luger, R., et al. 2017, Nature Astronomy, 1, 0129
- MacDonald, M. G., et al. 2016, AJ, 152, 105
- Peale, S. J. 1976, ARA&A, 14, 215
- Walsh, K. J., et al. 2011, Nature, 475, 206