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.

The eigenmodes are real; only the planet attribution differs

Before showing the tables: the eigenmodes g_j, s_j are mathematical objects — eigenvalues of the Laplace-Lagrange secular perturbation matrix that captures gravitational coupling between all eight planets. Their numerical values (g₁ ≈ 5.5965 ″/yr, g₂ ≈ 7.4555 ″/yr, …) come from the planetary masses and semi-major axes. They exist in any framework that solves the same physics. Both Berger 1978 and the Holistic model accept the eigenmodes. What differs between frameworks is physical attribution:

Notation on this page. The “Planet association” / “(Jupiter)” / “(Mercury)” labels in the tables below are Berger’s convention — included because that’s what the literature uses. The Holistic model does not equate these to its own planet-specific cycles. When the model needs a planet-specific period, it uses the six-period-types table below (per Solar System Resonance Cycle period table on GitHub ).

For empirical climate evidence that the eigenmodes are real but their beats and combinations — not single-planet attributions — are what shows up in paleoclimate spectra, see the Orbital Forcing Formula page (companion empirical document with the 26-component 8H formula and per-planet contribution analysis).

Published eigenfrequencies (Laskar 2004)

Eccentricity eigenmodes:

Inclination eigenmodes (all retrograde or near-zero):

*The “Berger planet label” column shows the standard literature convention of naming each eigenmode after the planet whose contribution dominates that mode. The Holistic model does not endorse this single-planet attribution — see the framework comparison above.

(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.

(+ = prograde, − = retrograde, ~∞ = frozen. Obliquity and eccentricity cycles are oscillation periods and shown unsigned.

In years:

(+ = prograde, − = retrograde, ~∞ = frozen). Venus is the sole planet with a prograde axial precession — its obliquity is ~177° (essentially upside-down rotation), so the standard cos(obliquity) geometric factor flips sign and the axial precession appears prograde (Cottereau & Souchay 2009 ). Uranus would also be prograde (obliquity ~98°) if its axial precession were not effectively 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. Parenthetical planet names below are Berger’s labels (see note on attribution above):

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

g_j eigenmode periods vs model H/n divisors

A direct comparison of each eigenfrequency’s period (= 1,296,000 / g_j) against the model’s nearest H/n divisor (parenthetical planet names are Berger’s labels):

Observation: Mercury (g₁), Mars (g₄), and Saturn (g₆) — the three planets where the model’s ecliptic divisor falls within 10% of the bare eigenfrequency period — are also the three planets where the dominant g_j is the strongest contributor to the planet’s perihelion precession with limited mode mixing. For the others, the bare g_j and the model’s divisor describe different things:

• Earth: the bare g₃ period (75 kyr) does not match any model H/n divisor directly. Earth’s actual full perihelion precession (~112k) does match H/3 cleanly — because Earth’s perihelion is a sum of g₃ plus substantial mixing from g₅ (Jupiter) and other modes, and the model’s H/n is calibrated to the full sum, not to g₃ alone.
• Outer planets (Jupiter, Uranus, Neptune): their bare g_j periods are 300-2000 kyr, but their model ecliptic divisors fall in the 67-670 kyr range. The model’s outer-planet ecliptic frame is defined differently than the bare g_j eigenmode (see Fundamental Cycles §52 for the frame-specific notes).

This indicates the model’s H/n divisors are calibrated to full per-planet perihelion rates with mode-mixing built in, not to bare eigenfrequency periods. The Berger spectrum match (above) reflects this: the g_j + k beats correspond to integer fractions of the resonance cycle, while bare g_j periods do not.

Per-planet theoretical perihelion precession (vs model)

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

Inner planets (Mercury, Mars) match the classical theoretical 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 theoretical 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

The published s_j eigenfrequencies vs the model’s ascending-node integers (where each period equals Resonance Cycle / N). The “Dominant s_j” column reflects Berger’s planet-association convention; see note above:

The s_j and the model’s ascending-node integer divisors describe related but distinct quantities: s_j is the eigenfrequency of one secular eigenmode, while the model’s integer N gives a period (Resonance Cycle / N) 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 Solar System Resonance Cycle). 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

Working with the Resonance Cycle integer divisors, 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. Three special cases:

• Earth breaks the sum rule (N_ecl = 128 > 104, giving a prograde ICRF perihelion — see Why Earth Is Special).
• Saturn is retrograde ecliptic, so N_ICRF = 104 + |N_ecl| = 104 + 64 = 168.
• Venus is also retrograde ecliptic, so N_ICRF = 104 + |N_ecl| = 104 + 6 = 110 (matching the master table).

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

(The “Period at 8H/N_eclPrec” column is the period associated with the residual integer N_eclPrec — the second factor in the obliquity decomposition. For Earth specifically this happens to equal the obliquity period (since 128 = 64 + 64), but in general it is a derived quantity, not the canonical “ecliptic precession” period reported elsewhere.)

Mirror pair: Mercury ↔ Uranus share N_eclPrec = 8 (period = H = 335,317 yr).

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.

Conclusions

Reading the four comparisons together, six substantive findings emerge:

1. The correspondence is real, not coincidental — Berger spectrum matches integer 8H/n within <0.4%; Earth’s perihelion in both frames within <1%; inner-planet theoretical rates within 5-8%; Saturn’s g₆+k ≈ ICRF perihelion within 3%. Independent matches at meaningful precision.

2. The match is empirical, not derivational — H/n divisors are calibrated to full perihelion rates (with mode mixing built in). Bare g_j eigenfrequencies don’t match cleanly for half the planets. The model captures observables, not the secular eigenstructure beneath them.

3. The three structural identities are the strongest non-numerical claim — N_ICRF + N_ecl = 104, N_ecc = |N_axial − N_ICRF|, and N_ecl = N_obliq + N_eclPrec hold across all planets via simple integer arithmetic. They tie together quantities that secular theory treats as independent eigenvalues, and are the kind of pattern that fitting alone cannot produce.

4. Earth is the validation case — Earth’s perihelion matches secular theory + GR in both frames simultaneously (sidereal ↔ H/3, climatic ↔ H/16) to <1%. No other planet shows this two-frame consistency, empirically validating Earth’s role as the model’s reference point (H = Earth Fundamental Cycle).

5. Divergences are honest, not failures — outer planets show 50%+ divergence on theoretical perihelion, but this reflects different frame definitions in the model (see Fundamental Cycles §52), not an error. The Berger spectrum match works because it uses the moving-equinox frame where definitions align.

6. The framework is falsifiable, not just descriptive — if the H/n hierarchy reflects real underlying structure, refined numerical integrations should pull published g_j toward rational multiples of 1/H. If they don’t, the framework is empirical-only. Either outcome is testable.

In one sentence: classical secular theory and the Holistic model describe the same orbital observables to <1% in the strongest cases, but they disagree on what is fundamental — eigenmodes versus integer divisors of a single fundamental period — and the model’s most distinctive claim is three integer identities that secular eigendecomposition does not expose.

Open questions and differences

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 integer divisors of the Resonance Cycle give exact rational frequencies (limited only by the precision of H itself, which is observationally anchored to the verified 1246 AD perihelion-solstice alignment).

The eigenmode-to-period mapping is established empirically; several questions remain open.

Empirical convergence:

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 model’s 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 (both at integer N = 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.

Structural questions:

4. Are the planetary eigenfrequencies g_j derivable from Fibonacci-organized planetary masses? 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.

5. 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.

6. Why does Earth alone have a 1× tier in the Fundamental Cycle hierarchy? (See Fundamental Cycles — Earth’s Fundamental Cycle is uniquely H, while Jupiter / Saturn / Uranus sit at 2H and Mercury / Venus / Mars / Neptune at the full Resonance Cycle.)

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
• Orbital Forcing Formula — companion empirical document with the 26-component 8H Orbital Forcing Formula fit on LR04, per-planet contribution analysis, and the frame-transformation explanation of how eigenmodes reach climate (full technical record at Milankovitch Evidence (doc 17) on GitHub )
• Solar System Resonance Cycle period table (3d repo doc 55)  — full 8H/n period table per planet × cycle type, the model-frame counterpart to the eigenmodes above