Eigenfrequencies and the Solar System Resonance Cycle

Classical secular theory describes planetary orbital evolution through eigenfrequencies g_j (eccentricity) and s_j (inclination), determined by linearizing gravitational perturbation equations or by direct numerical integration. The Holistic Universe Model derives all planetary periods as integer divisors of the Solar System Resonance Cycle (8H = 2,682,536 yr). This page maps the two frameworks side by side and notes where they agree, where they differ, and what each framing emphasizes.

How secular theory determines eigenfrequencies

Eigenfrequencies are determined by two complementary approaches.

1. Secular eigendecomposition (Lagrange-Laplace theory)

The Lagrange-Laplace approach linearizes the planetary perturbation equations in eccentricity and inclination. After averaging over orbital periods, the secular equations reduce to a constant-coefficient linear system. Diagonalizing the secular matrix yields the eigenfrequencies g_j (eccentricity) and s_j (inclination), each with an associated eigenvector describing the planet-by-planet contribution.

Historical lineage:

• Brouwer & Van Woerkom (1950) — first comprehensive secular theory using hand calculation; covered all major planets and Pluto
• Bretagnon (1974) — refinement using early electronic computers; the values used by Berger (1978) and most Milankovitch references
• Laskar (1988, 2004) — modern values from numerical integration combined with secular theory for higher accuracy

2. Direct numerical integration

Modern approaches integrate the full Newtonian (and relativistic) equations of motion for millions of years, then Fourier-analyze the resulting eccentricity and inclination time series. Peak frequencies in the spectrum are identified as the eigenmodes. This captures non-linear effects (chaotic dynamics, resonances) that secular theory cannot.

Reported uncertainties: Eigenfrequencies are typically quoted to 4-5 significant figures, with uncertainties in the last digit reflecting the truncation of the secular series, choice of reference epoch, and inclusion of post-Newtonian corrections. Values from different research groups (Bretagnon, Laskar, Standish, Brouwer) agree at the 0.1-1% level for the dominant terms.

Mainstream eigenfrequencies (Laskar 2004)

Eccentricity eigenmodes:

Inclination eigenmodes (all retrograde or near-zero):

(Bretagnon 1974 values differ by 0.1-0.5% from these.)

Six period types per planet (model)

Each planet has up to six distinct long-period cycles in the model, all expressible as integer divisors of 8H = 2,682,536 yr. See Fundamental Cycles for derivation context.

In years:

(+ = prograde, − = retrograde, ~∞ = frozen)

Direct comparisons

Berger climatic precession peaks

The Berger (1978) climatic-precession spectrum (g_j + k beats) maps directly to integer fractions of the Solar System Resonance Cycle within <0.4% — including Saturn:

See Supporting Evidence §10: Climatic precession comparison for the structural decomposition (n = 104 + δ_j) and Fibonacci analysis.

Per-planet perihelion precession (secular sidereal vs model)

Observed perihelion precession periods (from JPL DE440 / Standish 1992) compared to the model’s ecliptic and ICRF perihelion periods:

Earth is the strongest case: the classical sidereal perihelion period (~112k) matches model ICRF (H/3 = 111,772) to 0.06%, and the classical climatic-precession period (~21k) matches model ecliptic (H/16 = 20,957) to within 1%.

Inner planets (Mercury, Mars) match the classical perihelion precession rates in the ecliptic frame to 5-8%.

Outer planets (Jupiter, Saturn, Uranus, Neptune) show a divergence between the model’s ecliptic perihelion and the classical sidereal perihelion — the model’s outer-planet ecliptic frame uses a different definition than the J2000-fixed-ecliptic perihelion precession reported in classical references (see Fundamental Cycles §52 for the frame-specific notes).

Per-planet ascending node regression

Mainstream s_j eigenfrequencies vs the model’s 8H/N ascending-node integers:

The s_j and model 8H/N ascending-node integers describe related but distinct quantities: s_j is the eigenfrequency of one secular eigenmode, while 8H/N is the period at which the planet’s ascending node on the invariable plane completes a full regression in the J2000-fixed frame. Earth alone matches at 2.5% (because Earth’s ecliptic precession is by definition the s₃-related rate).

The technical specification (3d simulation §3 ) notes:

“The new asc-node integers cluster on small factors as well: Mercury 9 = 3², Mars 63 = 7×9, Jupiter/Saturn 36 = 4×9, Uranus 12 = 4×3, Neptune 3 = F₄, Venus 1 (= 8H, a full Grand Octave). They no longer match the Laplace-Lagrange eigenfrequencies s₁–s₈ exactly — they are an alternative integer assignment that produces a tighter JPL trend fit while preserving Jupiter+Saturn lockstep.”

Three universal identities in the resonance-cycle integer domain

Working in the 8H/N integer domain, three universal identities connect the six cycle types per planet.

Identity 1: Frame conversion (ICRF ↔ ecliptic)

N_ICRF = 104 − N_ecl    (for prograde ecliptic planets)

The number 104 = 8 × 13 is Earth’s axial precession integer N. Earth itself breaks the sum rule (N_ecl=128 > 104, giving prograde ICRF), and Saturn’s retrograde ecliptic gives N_ICRF = 104 + 64 = 168.

Identity 2: Eccentricity cycle as integer beat

N_ecc = |N_axial ± N_ICRF|

Opposite directions (one prograde, one retrograde) → add. Same direction → subtract.

For Uranus and Neptune (frozen axial), the beat reduces to the ICRF rate alone — a structural definition of “frozen axial precession” at the resonance-cycle scale.

Identity 3: Obliquity decomposition

N_ecl = N_obliq + N_eclPrec

Mirror pairs share identical ecliptic precession N:

• Mercury ↔ Uranus: N_eclPrec = 8 (period = H)
• Venus ↔ Neptune: N_eclPrec = −96 (period = 8H/96)

Where the two frameworks agree

Earth is the strongest test case: its perihelion precession in both frames (sidereal ↔ H/3 ICRF, climatic ↔ H/16 ecliptic) matches the classical values to <1%. The model recovers the same physics that secular theory describes, but expresses it as integer divisors of a single fundamental period.

The bridge is axial precession (k = H/13, integer 104). This is where the two frameworks agree exactly, by definition. Every Berger climatic-precession peak then adds a planet-specific eigenfrequency contribution δ_j to give n = 104 + δ_j.

Differences and open empirical questions

The published g_j and s_j are quoted to 4-5 significant figures, with uncertainties in the last digit reflecting numerical-integration truncation, choice of reference epoch, and post-Newtonian corrections. The model’s 8H/n divisors give exact rational frequencies (limited only by the precision of H itself, which is observationally anchored to the verified 1246 AD perihelion-solstice alignment).

This raises three open empirical questions:

1. Do the published eigenfrequencies converge on H/n rational values as numerical accuracy improves? If the H/n framework reflects underlying solar-system structure, then improvements in numerical integration (longer runs, better mass values, refined relativistic corrections) might pull the published g_j and s_j toward rational multiples of 1/H. If they don’t, the H/n framework is a coincidence; if they do, that is evidence for the model’s structure. Either outcome is testable.

2. Do the 8H/N ascending-node integers describe the same quantity as Laplace-Lagrange s_j? Strictly, they don’t — the technical specification notes the integers were re-fitted (2026-04-09) to match JPL J2000-fixed-frame ascending-node trends to <2”/century per planet, while preserving the Jupiter–Saturn lockstep at 8H/36. This produces a tighter empirical fit to observed nodal motion than the canonical s_j values, but at the cost of departing from the secular eigendecomposition. Which assignment is “correct” depends on what is being measured.

3. Does Earth’s two-frame match generalize to other planets? Earth’s perihelion period in both frames (sidereal H/3 ↔ 112 kyr classical, climatic H/16 ↔ 21 kyr Berger centroid) matches secular theory to <1%. The frame-explicit per-planet specification used by the model (separate ecliptic and ICRF periods) could clarify which quantity is being reported in heterogeneous published references. Whether other planets’ two-frame matches are as strong as Earth’s is unproven.

Open questions

The eigenmode-to-period mapping is established empirically; the deeper “why” remains open.

• Are the planetary eigenfrequencies g_j derivable from Fibonacci-organized planetary masses? Mainstream secular theory takes masses as input; the resulting g_j depend on those masses. Whether the masses themselves follow Fibonacci ratios — and whether that would force g_j into the integer divisors observed — is unproven.
• Why do specific divisors N appear? Mercury N=9 (3²), Mars N=63 (7×9), Jupiter/Saturn N=36 (4×9) are clean small factors but not Fibonacci. The mix of Fibonacci and non-Fibonacci integers in the per-planet table is unexplained.
• Why does Earth alone have a 1× tier in the Fundamental Cycle hierarchy? (See Fundamental Cycles — Earth is uniquely H, while Jupiter/Saturn/Uranus are 2H and Mercury/Venus/Mars/Neptune are 8H.)

These remain testable predictions and structural claims rather than derivations from first principles.

See also

• Fundamental Cycles — The Earth Fundamental Cycle (H) and Solar System Resonance Cycle (8H), with per-planet derivations
• Fibonacci Laws — The six laws and their derivation
• Supporting Evidence §10 — Berger (1978) spectral comparison (obliquity + climatic precession)
• Scientific Background §5 — Milankovitch / Laskar comparison and the model’s reinterpretation