Formula Derivation and Analysis

This page documents how the planetary precession formulas were derived — the physical reasoning, mathematical relationships, and coefficient breakdowns. For the practical “cookbook” formulas, see Formulas. For the model explanation, see Scientific Background.

Quick Reference

Formula Types

Contents

1. Fibonacci Hierarchy in Orbital Periods
2. Saturn-Jupiter-Earth Resonance Loop
3. Mercury Formula: Key Combination Periods
4. Mercury Formula: Coefficient Breakdown (Predictive)
5. Venus Formula: Coefficient Breakdown (Observed)
6. Mars Formula: Coefficient Breakdown
7. Jupiter Formula: Coefficient Breakdown
8. Saturn Formula: Coefficient Breakdown
9. Uranus Formula: Coefficient Breakdown
10. Neptune Formula: Coefficient Breakdown
11. Time-Varying Fluctuation
12. Planetary Physical Comparison
13. Uncertainties and Limitations
14. Observed-Angle Formulas (Using Observational Data) — includes ERD definition & formulas

1. Fibonacci Hierarchy in Orbital Periods

A remarkable pattern emerges when dividing the Holistic-Year by Fibonacci numbers. The resulting periods correspond to major planetary cycles:

Beat Frequency Rule: Just as Fibonacci numbers add (F(n) + F(n+1) = F(n+2)), the corresponding beat frequencies follow the same pattern:

1/H(n) + 1/H(n+1) = 1/H(n+2)

For example:

• 1/H(3) + 1/H(5) = 1/111,296 + 1/66,778 = 1/41,736 = 1/H(8) ✓
• 1/H(5) + 1/H(8) = 1/66,778 + 1/41,736 = 1/25,684 = 1/H(13) ✓
• 1/H(8) + 1/H(13) = 1/41,736 + 1/25,684 = 1/15,899 = 1/H(21) ✓

2. Saturn-Jupiter-Earth Resonance Loop

A remarkable discovery emerges when analyzing the planetary precession periods: Saturn’s retrograde precession creates a closed resonance loop with Jupiter and Earth that explains why certain periods appear in the Mercury fluctuation formula.

Saturn’s Unique Motion: Saturn is the only planet whose perihelion precesses retrograde (opposite to orbital motion) with a period of ~41,736 years. All other planets precess prograde. This creates beat frequencies when combined with prograde periods.

The Resonance Loop:

This is a closed loop: Starting from any one period, you can derive the others through beat frequency relationships. The three major solar system cycles are mathematically interlinked:

Jupiter (66,778) ←──────────────────────→ Saturn (41,736)
        ↑                                       │
        │     beat frequencies                  │
        │                                       ↓
Earth true perihelion (111,296) ←───── Axial precession (25,684)

Physical Interpretation: When Saturn’s retrograde wobble interacts with Jupiter’s prograde wobble:

• Their sum frequency matches Earth’s axial precession
• Their difference frequency matches Earth’s true perihelion period
• This suggests the solar system’s major cycles are coupled through gravitational resonances

3. Mercury Formula: Key Combination Periods

Mercury’s formula is more complex than other planets because three movements interact to create frequency mixing:

1. Earth’s effective perihelion: 20,868 years (from axial + true perihelion)
2. Earth’s true perihelion: 111,296 years
3. Mercury’s perihelion: 242,828 years

When these angular rates combine, they create new “sideband” frequencies through amplitude modulation — similar to how radio signals mix frequencies.

Mixing Frequencies

Where:

• φ_E = effective Earth angle = 360°/20,868 × t
• φ_M = Mercury angle = 360°/242,828 × t
• t = YEAR + 301340

Derived Shorter Periods

All periods in the formula have physical derivations:

Note: The periods are all physically derived from Mercury, Jupiter, Saturn, or Fibonacci harmonics.

4. Mercury Formula: Coefficient Breakdown (Predictive Formula)

The Mercury predictive fluctuation formula achieves R² = 0.9986 using 106 non-zero coefficients organized into categories:

4.1 Geometric Terms (6 terms)

Where δ = θ_E - θ_M (relative angle) and σ = θ_E + θ_M (sum angle).

4.2 Phase Terms (26 terms)

4.3 Auxiliary Terms (2 terms)

Note: The eccentricity mean has been updated to $e_0 = \sqrt{0.015321^2 + 0.0014226^2} = 0.015387$. The Python observed formula coefficients have been retrained against this value. The coefficient +478,089 shown here is from the legacy Excel approximation.

4.4 ERD Basic Terms (7 terms)

4.5 Higher Harmonics (7 terms)

4.6 ERD × Periodic Terms (12 terms)

4.7 ERD² × Periodic Terms (10 terms)

These quadratic rate terms were the key breakthrough for reaching 99.8% accuracy:

4.8 Triple Interactions (ERD × Periodic × Angle, 24 terms)

4.9 Mercury Period Terms (2 terms)

4.10 Periodic × Angle (no ERD, 11 terms)

Summary (Predictive Formula)

Constant: +22

Total: 106 non-zero coefficients (predictive formula, year-only input)

Accuracy: R² = 0.9986 (explains 99.86% of variance across full 333,888-year cycle) RMSE: 2.83 arcsec/century

Note: The observed formula (Section 14) achieves R² = 1.0000 with 225 terms by using actual perihelion data from CSV.

5. Venus Formula: Coefficient Breakdown (Observed Formula)

Venus presents a fundamentally different challenge than Mercury. With an eccentricity of only 0.00678 (compared to Mercury’s 0.20564), Venus has a nearly circular orbit where geometric modulation effects are minimal. Instead, Venus’s fluctuation is dominated by variations in Earth’s axial precession rate.

5.1 The Earth Rate Deviation Model

The Venus formula is based on the physical principle that:

1. Perihelion points move at eccentricity distance from orbit center

   • Venus: e = 0.00678, so perihelion is only 0.0049 AU from center (poorly defined)
   • Earth: e = 0.01671, so perihelion is 0.0167 AU from center

2. Earth’s reference frame rotates due to axial precession

   • Mean period: 25,684 years
   • Current period: ~25,772 years (varying with obliquity)

3. Earth Rate Deviation (ERD) captures this variation:

   ERD = (Earth perihelion rate) - (expected rate)
       = (Earth perihelion rate) - (360° / 20,868 years)

4. The observed fluctuation depends on:

   • The relative angle between Venus perihelion and Earth’s reference (θE - θV)
   • Earth’s instantaneous axial precession rate variation (ERD)
   • Interactions between these two factors

5.2 Relative Angle Terms (δ = θE - θV)

5.3 Earth Rate Deviation Terms

5.4 Individual Angle Terms

5.5 Periodic Terms

5.6 ERD × Periodic Terms

5.7 Periodic × Angle Terms (amplitude modulation)

5.8 ERD × Periodic × Angle Terms (triple interaction, KEY terms)

5.9 ERD² × Periodic Terms (quadratic rate modulation)

5.10 Higher Harmonic Terms

Summary

Constant: +984

Total: 328 non-zero coefficients (organized into relative angle, ERD, individual angle, periodic, triple interaction, ERD³, and 4δ harmonic terms)

Accuracy: R² = 1.0000 (explains 100% of variance) RMSE: 0.27 arcsec/century (using observed perihelion from CSV)

6. Mars Formula: Coefficient Breakdown

Mars presents a unique challenge among the terrestrial planets. With an eccentricity of 0.09339 (between Venus’s near-circular orbit and Mercury’s highly elliptical one), Mars shows moderate geometric modulation effects combined with strong coupling to Earth’s orbital dynamics.

6.1 Physical Driver

Mars’s precession fluctuation is driven by:

1. Relative geometry: The angle between Mars’s perihelion and Earth’s perihelion (δ = θE - θMars)
2. Earth Rate Deviation (ERD): Variations in Earth’s axial precession rate
3. Mars’s own perihelion precession: Period of ~77,051 years (H×3/13)
4. Jupiter-Mars resonance: Mars is strongly influenced by Jupiter’s gravitational perturbations

6.2 Formula Summary

6.3 Key Term Categories

The 225 terms include:

• Angle terms (δ harmonics through 4δ)
• Obliquity & eccentricity coupling terms
• ERD terms (linear, quadratic, cubic)
• Periodic terms from H, H/3, H/5, H/8, H/13, H/16, and Mars period
• Cross-products: ERD × periodic, periodic × angle, ERD × periodic × angle
• Beat frequency terms between Mars and Earth periods

For full implementation details, see the Python reference in /docs/mars_coeffs.py.

7. Jupiter Formula: Coefficient Breakdown

Jupiter, as the largest planet, dominates the outer solar system’s gravitational dynamics. Its precession fluctuation shows strong coupling with Saturn through the Saturn-Jupiter-Earth resonance loop described in Section 2.

7.1 Physical Driver

Jupiter’s precession fluctuation is driven by:

1. Saturn resonance: Jupiter and Saturn are locked in gravitational resonance
2. Earth’s reference frame motion: ERD effects still contribute
3. Fibonacci hierarchy: Jupiter’s period (H/5 = 66,778 years) is a key Fibonacci division
4. Long-term stability: Jupiter’s massive orbit shows slow, predictable precession

7.2 Formula Summary

7.3 Key Periods in Jupiter Formula

For full implementation details, see the Python reference in /docs/jupiter_coeffs.py.

8. Saturn Formula: Coefficient Breakdown

Saturn is unique in the solar system: it is the only planet with retrograde perihelion precession. While all other planets precess prograde (in the direction of orbital motion), Saturn’s perihelion precesses opposite to its orbital motion with a period of ~41,736 years.

8.1 Physical Driver

Saturn’s precession fluctuation is driven by:

1. Retrograde precession: Creates beat frequencies when combined with prograde periods
2. Jupiter resonance: Saturn and Jupiter form a closed resonance loop
3. Obliquity coupling: Saturn’s period (H/8) matches Earth’s obliquity cycle
4. Ring dynamics: Saturn’s rings influence its precession behavior

8.2 Formula Summary

8.3 The Saturn-Jupiter-Earth Loop

Saturn’s role in the resonance loop (from Section 2):

1/Jupiter + 1/Saturn = 1/66,778 + 1/41,736 = 1/25,684 ≈ Axial precession
1/Jupiter − 1/Saturn = 1/66,778 − 1/41,736 = 1/111,296 = Earth true perihelion

This closed loop means Saturn’s coefficients include strong coupling to Jupiter and Earth periods.

8.4 Predictive Formula Enhancement

The predictive formula for Saturn uses the unified 273-term matrix enhanced with time-varying obliquity and eccentricity (GROUP 15 terms):

This accounts for the resonance between Saturn’s perihelion period (41,736 years) and Earth’s obliquity cycle. The predictive formula achieves R² = 1.0000, RMSE = 0.29″/century.

For implementation details, see docs/predictive_formula.py (GROUP 15 terms in build_features).

9. Uranus Formula: Coefficient Breakdown

Uranus’s perihelion precession period (H/3 = 111,296 years) matches Earth’s inclination precession cycle. This places Uranus in a key resonance position within the Fibonacci hierarchy, sharing a period with one of Earth’s fundamental orbital oscillations.

9.1 Physical Driver

Uranus’s precession fluctuation is driven by:

1. Inclination cycle resonance: Uranus’s period matches H/3 (Earth’s inclination precession)
2. Extreme axial tilt: Uranus’s 98° tilt may influence its precession dynamics
3. Outer planet coupling: Interactions with Neptune, Saturn, and Jupiter
4. Ice giant dynamics: Different internal structure than gas giants

9.2 Formula Summary

9.3 Uranus-Earth Resonance

Uranus’s period (111,296 years) exactly matches Earth’s inclination precession cycle (H/3). This creates:

• Direct coupling between Uranus’s perihelion and Earth’s orbital plane oscillation
• Strong resonance terms in the formula
• Excellent predictability over historical timescales

For full implementation details, see the Python reference in /docs/uranus_coeffs.py.

10. Neptune Formula: Coefficient Breakdown

Neptune, the outermost major planet, has the longest precession period in the Fibonacci hierarchy: H×2 = 667,776 years. This ultra-slow precession, combined with Neptune’s nearly circular orbit, makes it the most challenging planet to model precisely.

10.1 Physical Driver

Neptune’s precession fluctuation is driven by:

1. Outer solar system dynamics: Dominated by interactions with Uranus
2. Venus period resonance: Period (H×2 = 667,776 years) matches Venus’s precession period
3. Long orbital period: 164.8 years means slow accumulation of precession
4. Kuiper Belt interactions: Possible perturbations from trans-Neptunian objects

10.2 Formula Summary

10.3 Neptune-Venus Period Match

Neptune’s precession period (H×2 = 667,776 years) exactly matches Venus’s precession period. Both planets have nearly circular orbits, which may explain why they share this ultra-long timescale in the Fibonacci hierarchy.

For full implementation details, see the Python reference in /docs/neptune_coeffs.py.

10.4 Predictive Formula Enhancement

Neptune now uses the unified 273-term predictive matrix, the same system used by all other planets. Despite Neptune and Venus sharing the same precession period (H×2 = 667,776 years), the unified matrix handles this through its ridge regression regularization (α=0.01), which prevents term interference between the two planets.

11. Time-Varying Fluctuation

The Mercury fluctuation is not constant — it varies over Mercury’s 242,828-year perihelion cycle. The formula’s many terms (geometric, periodic, ERD) combine to produce a time-dependent value.

At year 2000, these terms combine to give approximately +38.8 arcsec/century. The historical “43 arcsec anomaly” corresponds to Einstein’s era (~1900, when the model gives ~42.9″). The value is decreasing as Earth’s precession cycles progress.

Base Amplitude

The geometric coefficients scale with Mercury’s orbital properties:

A = Baseline × e_Mercury = 533.7 × 0.20564 ≈ 110 arcsec/century

Where:

• Baseline = 533.7 arcsec/century (Mercury’s Newtonian precession rate = 1,296,000″ ÷ 242,828 × 100)
• e_Mercury = 0.20564 (Mercury’s orbital eccentricity)

The actual coefficients in the formula are optimized jointly with ERD terms, resulting in values that differ from simple geometric predictions. The dominant cos(2θM) term (-441) and the large eccentricity coefficient (+478,089) reflect the complex interplay between geometric effects and Earth Rate Deviation.

12. Planetary Physical Comparison

The table below shows two sets of formula accuracy values:

• Observed: Uses actual perihelion positions from observational data (CSV)
• Predictive: Calculates all values from year only (standalone formulas)

Observed Formula Accuracy (using CSV data)

Predictive Formula Accuracy (year-only input, unified 273-term system)

Key Observations

Inner Planets (Mercury, Venus):

• Mercury’s high eccentricity (0.21) creates strong, predictable geometric modulation
• Venus’s near-circular orbit (e = 0.007) means precession fluctuation is dominated by ERD² effects
• Venus requires 328 features (including ERD³, 4δ harmonics, and obliquity/eccentricity coupling) to achieve 0.27 arcsec accuracy

Mars (Transition):

• Intermediate eccentricity (0.09) shows both geometric and ERD effects
• Achieves perfect fit (R² = 1.0000) when using observed perihelion data
• Acts as a bridge between inner planet and outer planet dynamics

Outer Planets (Jupiter, Saturn, Uranus, Neptune):

• All achieve perfect fits (R² = 1.0000) with 225 features
• Uranus and Neptune achieve the best fits (RMSE = 0.01 arcsec/century)
• Saturn is unique with retrograde precession, creating the resonance loop
• Neptune and Venus share the same period (H×2 = 667,776 years) despite being at opposite ends of the solar system

13. Uncertainties and Limitations

The formula is a provisional approximation — a best-fit model with physically-motivated terms. The predictive Mercury RMSE of ~2.83 arcsec/century (106-term legacy formula) arises from several sources of uncertainty:

13.1 Base Period Parameters

The fundamental periods are model parameters, not independently derived values:

Note: Saturn is the only planet with retrograde perihelion precession (opposite to orbital motion). Venus and Neptune share the same period (H×2). Uranus shares its period with Earth’s inclination precession (H/3).

If future research refines H (e.g., to 333,900 or 333,850), all beat frequencies and coefficients would need recalculation.

13.2 Beat Frequency Sensitivity

The periodic terms depend on precise period ratios. Small changes propagate:

• If Mercury = 242,828 ± 100 years → the 7,163-year term shifts by ~3 years
• If H = 333,888 ± 50 years → the 111,296-year term (H/3) shifts by ~17 years

Over 300,000+ years, even small period errors accumulate into phase drift.

13.3 Simplified Geometry

The formula assumes:

• Idealized two-body interactions (Earth-Mercury)
• Constant orbital eccentricities (in reality, they vary slightly)
• No higher-order gravitational perturbations

13.4 Coefficient Rounding

All coefficients are rounded to integers for simplicity. The optimal least-squares values are non-integer (e.g., 42.8 → 43, -89.6 → -90), introducing small systematic errors.

14. Observed-Angle Formulas (Using Observational Data)

The formulas in this section require observed orbital parameters as inputs — Earth perihelion position, planetary perihelion position, obliquity, eccentricity, and Earth Rate Deviation (ERD). These formulas were used during model development to fit against ice-core chronological data. For predictive formulas (year-only input), see Formulas.

Earth Rate Deviation (ERD) — Shared Helper

Both Mercury and Venus formulas use Earth Rate Deviation (ERD) to account for variations in Earth’s axial precession rate.

Scientific Notation:

Where:

• $\frac{d\theta_E}{dt}$ = Instantaneous Earth perihelion rate (°/year)
• $\omega_0 = \frac{360°}{20,868} = 0.01725°$/year — Expected (mean) rate

The derivative is computed as:

With angle wraparound correction:

Calculate ERD once in column DR:

=IF(AI2738-AI2737<-180,(AI2738-AI2737+360)/(A2738-A2737),IF(AI2738-AI2737>180,(AI2738-AI2737-360)/(A2738-A2737),(AI2738-AI2737)/(A2738-A2737)))-360/20868

Where: A = Year, AI = Earth Perihelion. This calculates the rate from the previous row, handling angle wraparound at ±180°.

Shared Column References

Mercury Fluctuation Formula (Using Observed Angles)

Scientific Notation:

Geometric Terms:

Where $\delta = \theta_E - \theta_M$ (relative angle) and $\sigma = \theta_E + \theta_M$ (sum angle).

With coefficients: $a_1 = -27$, $a_2 = -7$, $a_3 = +7$, $a_4 = -441$, $a_5 = +173$, $a_6 = +7$

Phase (Periodic) Terms:

Where $t$ = Year + 301,340 and periods $T_i$ are: 7,163; 19,206; 121,414; 28,185; 111,296; 66,778; 11,038; 12,521; 9,820; 20,868; 242,828 years.

ERD (Earth Rate Deviation) Terms:

With coefficients: $d_1 = +21,322$, $d_2 = +81,832$, $d_3 = +2,702$, $d_4 = -620$, $d_5 = +786$, $d_6 = +853,292$, $d_7 = -403$

ERD × Periodic, ERD² × Periodic, and Triple Interactions:

Key ERD² × Periodic coefficients: +1,784,775 · ERD² cos(t/20868), −1,134,935 · ERD² sin(t/20868), +1,123,268 · ERD² sin(t/121414), +823,617 · ERD² cos(t/111296)

Higher Harmonics:

Auxiliary Terms:

With coefficients: $e_1 = +12$ (obliquity), $e_2 = +478,089$ (eccentricity), $\varepsilon_0 = 23.414°$, $e_0 = 0.015387$ (these are legacy Excel coefficients; Python observed formula uses retrained coefficients)

Result units: arcseconds per century (″/century)

=-27*ABS(SIN((AI2738-DH2738)*PI()/180))*COS((AI2738+DH2738)*PI()/180)-7*COS((AI2738+DH2738)*PI()/180)+7*SIN((AI2738+DH2738)*PI()/180)-441*COS(2*DH2738*PI()/180)+173*SIN(2*DH2738*PI()/180)+7*COS(2*AI2738*PI()/180)+16*SIN(2*PI()*(A2738+301340)/7163)+4*COS(2*PI()*(A2738+301340)/7163)+18*SIN(2*PI()*(A2738+301340)/19206)-56*COS(2*PI()*(A2738+301340)/19206)+1*SIN(2*PI()*(A2738+301340)/121414)+431*COS(2*PI()*(A2738+301340)/121414)+24*SIN(2*PI()*(A2738+301340)/28185)+17*COS(2*PI()*(A2738+301340)/28185)-1*SIN(2*PI()*(A2738+301340)/111296)+4*COS(2*PI()*(A2738+301340)/111296)-2*SIN(2*PI()*(A2738+301340)/66778)+1*COS(2*PI()*(A2738+301340)/66778)+3*SIN(2*PI()*(A2738+301340)/11038)+1*COS(2*PI()*(A2738+301340)/11038)+1*SIN(2*PI()*(A2738+301340)/12521)-8*COS(2*PI()*(A2738+301340)/12521)-1*SIN(2*PI()*(A2738+301340)/9820)+3*COS(2*PI()*(A2738+301340)/9820)+12*(U2738-23.414)+478089*(F2738-0.015354)+21322*DR2738+81832*DR2738*COS((AI2738-DH2738)*PI()/180)+2702*DR2738*SIN((AI2738-DH2738)*PI()/180)-620*DR2738*COS(2*(AI2738-DH2738)*PI()/180)+786*DR2738*SIN(2*(AI2738-DH2738)*PI()/180)+853292*DR2738*DR2738-403*(U2738-23.414)*DR2738+4*COS(3*DH2738*PI()/180)-15*COS(4*DH2738*PI()/180)+9*SIN(4*DH2738*PI()/180)+3*COS(3*(AI2738-DH2738)*PI()/180)+2*SIN(3*(AI2738-DH2738)*PI()/180)+895*DR2738*COS(3*(AI2738-DH2738)*PI()/180)+232*DR2738*SIN(3*(AI2738-DH2738)*PI()/180)+603*DR2738*SIN(2*PI()*(A2738+301340)/7163)-176*DR2738*COS(2*PI()*(A2738+301340)/7163)-1677*DR2738*SIN(2*PI()*(A2738+301340)/20868)-10336*DR2738*COS(2*PI()*(A2738+301340)/20868)-7327*DR2738*SIN(2*PI()*(A2738+301340)/121414)+395*DR2738*COS(2*PI()*(A2738+301340)/121414)+20103*DR2738*SIN(2*PI()*(A2738+301340)/19206)+62079*DR2738*COS(2*PI()*(A2738+301340)/19206)-4451*DR2738*SIN(2*PI()*(A2738+301340)/28185)+3083*DR2738*COS(2*PI()*(A2738+301340)/28185)+3607*DR2738*SIN(2*PI()*(A2738+301340)/111296)-5291*DR2738*COS(2*PI()*(A2738+301340)/111296)-132402*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/7163)+36800*DR2738*DR2738*COS(2*PI()*(A2738+301340)/7163)-1134935*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/20868)+1784775*DR2738*DR2738*COS(2*PI()*(A2738+301340)/20868)+1123268*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/121414)+416835*DR2738*DR2738*COS(2*PI()*(A2738+301340)/121414)-623746*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/111296)+823617*DR2738*DR2738*COS(2*PI()*(A2738+301340)/111296)+197457*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/19206)+152907*DR2738*DR2738*COS(2*PI()*(A2738+301340)/19206)+7828*DR2738*SIN(2*PI()*(A2738+301340)/7163)*COS((AI2738-DH2738)*PI()/180)+4336*DR2738*SIN(2*PI()*(A2738+301340)/7163)*SIN((AI2738-DH2738)*PI()/180)-6657*DR2738*SIN(2*PI()*(A2738+301340)/7163)*COS(2*(AI2738-DH2738)*PI()/180)+22805*DR2738*SIN(2*PI()*(A2738+301340)/7163)*SIN(2*(AI2738-DH2738)*PI()/180)+5325*DR2738*COS(2*PI()*(A2738+301340)/7163)*COS((AI2738-DH2738)*PI()/180)-7226*DR2738*COS(2*PI()*(A2738+301340)/7163)*SIN((AI2738-DH2738)*PI()/180)+22722*DR2738*COS(2*PI()*(A2738+301340)/7163)*COS(2*(AI2738-DH2738)*PI()/180)+6210*DR2738*COS(2*PI()*(A2738+301340)/7163)*SIN(2*(AI2738-DH2738)*PI()/180)+40984*DR2738*SIN(2*PI()*(A2738+301340)/20868)*COS((AI2738-DH2738)*PI()/180)-7570*DR2738*SIN(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DH2738)*PI()/180)+314*DR2738*SIN(2*PI()*(A2738+301340)/20868)*COS(2*(AI2738-DH2738)*PI()/180)+1977*DR2738*SIN(2*PI()*(A2738+301340)/20868)*SIN(2*(AI2738-DH2738)*PI()/180)-12433*DR2738*COS(2*PI()*(A2738+301340)/20868)*COS((AI2738-DH2738)*PI()/180)-38981*DR2738*COS(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DH2738)*PI()/180)+1015*DR2738*COS(2*PI()*(A2738+301340)/20868)*COS(2*(AI2738-DH2738)*PI()/180)-139*DR2738*COS(2*PI()*(A2738+301340)/20868)*SIN(2*(AI2738-DH2738)*PI()/180)+31407*DR2738*SIN(2*PI()*(A2738+301340)/121414)*COS((AI2738-DH2738)*PI()/180)+25110*DR2738*SIN(2*PI()*(A2738+301340)/121414)*SIN((AI2738-DH2738)*PI()/180)-21*DR2738*SIN(2*PI()*(A2738+301340)/121414)*COS(2*(AI2738-DH2738)*PI()/180)-13622*DR2738*SIN(2*PI()*(A2738+301340)/121414)*SIN(2*(AI2738-DH2738)*PI()/180)-16012*DR2738*COS(2*PI()*(A2738+301340)/121414)*COS((AI2738-DH2738)*PI()/180)+40263*DR2738*COS(2*PI()*(A2738+301340)/121414)*SIN((AI2738-DH2738)*PI()/180)+12136*DR2738*COS(2*PI()*(A2738+301340)/121414)*COS(2*(AI2738-DH2738)*PI()/180)+260*DR2738*COS(2*PI()*(A2738+301340)/121414)*SIN(2*(AI2738-DH2738)*PI()/180)-15*SIN(2*PI()*(A2738+301340)/242828)-135*COS(2*PI()*(A2738+301340)/242828)-14713*DR2738*SIN(2*PI()*(A2738+301340)/242828)-35146*DR2738*COS(2*PI()*(A2738+301340)/242828)-3*SIN(2*PI()*(A2738+301340)/20868)+650*COS(2*PI()*(A2738+301340)/20868)+2*SIN(2*PI()*(A2738+301340)/7163)*COS((AI2738-DH2738)*PI()/180)-4*SIN(2*PI()*(A2738+301340)/7163)*SIN((AI2738-DH2738)*PI()/180)-2*COS(2*PI()*(A2738+301340)/7163)*COS((AI2738-DH2738)*PI()/180)+137*SIN(2*PI()*(A2738+301340)/20868)*COS((AI2738-DH2738)*PI()/180)+13*SIN(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DH2738)*PI()/180)-122*COS(2*PI()*(A2738+301340)/20868)*COS((AI2738-DH2738)*PI()/180)-134*COS(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DH2738)*PI()/180)-29*SIN(2*PI()*(A2738+301340)/121414)*COS((AI2738-DH2738)*PI()/180)-148*SIN(2*PI()*(A2738+301340)/121414)*SIN((AI2738-DH2738)*PI()/180)+114*COS(2*PI()*(A2738+301340)/121414)*COS((AI2738-DH2738)*PI()/180)-14*COS(2*PI()*(A2738+301340)/121414)*SIN((AI2738-DH2738)*PI()/180)+22

Excel Formula Notes (Legacy):

• This Excel formula is a legacy approximation (~104 terms) with manually rounded coefficients
• Period constants may differ slightly from current Python implementation
• The eccentricity mean in this formula (0.015354) has been updated to 0.015387 in the Python code; this Excel formula uses legacy coefficients
• For accurate calculations: Use /docs/observed_formula.py (R² = 1.0000, RMSE = 0.08″/cy, 225 terms)
• Predictive formula: R² = 0.9986, RMSE = 2.83″/cy (106 terms) — see Section 4

Predicted Values

Venus Fluctuation Formula (Using Observed Angles)

Venus presents a fundamentally different challenge than Mercury. With an eccentricity of only 0.00678 (compared to Mercury’s 0.20564), Venus has a nearly circular orbit where geometric modulation effects are minimal. Instead, Venus’s fluctuation is dominated by variations in Earth’s axial precession rate (ERD² terms).

Formula Summary:

• R² = 1.0000 (explains 100% of variance)
• RMSE = 0.27 arcsec/century
• Features: 328 terms (V3_VENUS optimized matrix)
• Key drivers: ERD² × periodic terms, ERD³ terms, ERD × obliquity/eccentricity coupling, 3δ and 4δ harmonics

For the complete coefficient breakdown and physical explanation, see Section 5.

Uses the shared ERD helper from column DR (see above for ERD calculation).

Scientific Notation:

Relative Angle Terms ($\delta = \theta_E - \theta_V$):

With coefficients: $a_1 = +1,037$, $a_2 = -1,312$, $a_3 = +80$, $a_4 = +860$, $a_5 = +33$, $a_6 = -44$, $a_7 = -14$, $a_8 = +10$

ERD Terms:

With coefficients: $b_1 = -358,356$, $b_2 = -234,829$, $b_3 = -11,488$, $b_4 = -1,038,655$, $b_5 = +570,929$, $b_6 = -112,855,487$

Individual Angle Terms:

With coefficients: $c_1 = -2,884$, $c_2 = +1,205$, $c_3 = -284$, $c_4 = +26$, $c_5 = +4,084$, $c_6 = -7,418$, $c_7 = -1,670$, $c_8 = +931$

Phase (Periodic) Terms:

Where $t$ = Year + 301,340 and periods $T_i$ are: 667,776; 333,888; 111,296; 41,736; 20,868; 6,956 years.

ERD × Periodic Terms:

Where periods $P_j$ are: 20,868; 667,776; 333,888; 111,296; 41,736; 6,956 years. See component table for coefficients.

Periodic × Angle Terms (amplitude modulation when ERD is small):

Key terms for $T = 20,868$: $p = -5,921$, $q = +7,926$, $r = -19,742$, $s = +22,347$

ERD × Periodic × Angle Terms (triple interaction):

Key terms for $P = 20,868$: $u = +125,840$, $v = -5,657,660$, $w = -8,035,742$, $x = +1,630,420$

ERD² × Periodic Terms (quadratic rate modulation):

Key terms: $f_{20868} = +99,207,877$, $g_{20868} = +111,052,242$, $g_{667776} = +286,162,990$

Higher Harmonic Terms (3δ):

Key terms for $P = 20,868$: $a = -426,328$, $b = -246,344$, $c = -490,205$, $d = +493,871$

Constant: $C = +984$

Result units: arcseconds per century (″/century)

Note (Legacy): This Excel formula is a legacy approximation simplified to fit Excel’s 8,192 character limit. Some period terms have been removed and coefficients are manually rounded. For accurate calculations: Use /docs/observed_formula.py (R² = 1.0000, RMSE = 0.27″/cy, 328 terms).

=1037*COS((AI2738-DX2738)*PI()/180)-1312*SIN((AI2738-DX2738)*PI()/180)+80*COS(2*(AI2738-DX2738)*PI()/180)+860*SIN(2*(AI2738-DX2738)*PI()/180)+33*COS(3*(AI2738-DX2738)*PI()/180)-44*SIN(3*(AI2738-DX2738)*PI()/180)-14*COS(4*(AI2738-DX2738)*PI()/180)+10*SIN(4*(AI2738-DX2738)*PI()/180)-2884*COS(AI2738*PI()/180)+1205*SIN(AI2738*PI()/180)-284*COS(2*AI2738*PI()/180)+26*SIN(2*AI2738*PI()/180)+4084*COS(DX2738*PI()/180)-7418*SIN(DX2738*PI()/180)-1670*COS(2*DX2738*PI()/180)+931*SIN(2*DX2738*PI()/180)-358356*DR2738-234829*DR2738*COS((AI2738-DX2738)*PI()/180)-11488*DR2738*SIN((AI2738-DX2738)*PI()/180)-1038655*DR2738*COS(2*(AI2738-DX2738)*PI()/180)+570929*DR2738*SIN(2*(AI2738-DX2738)*PI()/180)-112855487*DR2738*DR2738+5859*SIN(2*PI()*(A2738+301340)/667776)+8260*COS(2*PI()*(A2738+301340)/667776)+395*SIN(2*PI()*(A2738+301340)/333888)+746*COS(2*PI()*(A2738+301340)/333888)+39*SIN(2*PI()*(A2738+301340)/41736)-7833*SIN(2*PI()*(A2738+301340)/20868)-2925*COS(2*PI()*(A2738+301340)/20868)-135*SIN(2*PI()*(A2738+301340)/6956)+229*COS(2*PI()*(A2738+301340)/6956)-4863030*DR2738*SIN(2*PI()*(A2738+301340)/667776)+4374726*DR2738*COS(2*PI()*(A2738+301340)/667776)+322475*DR2738*SIN(2*PI()*(A2738+301340)/333888)-172362*DR2738*COS(2*PI()*(A2738+301340)/333888)+8125*DR2738*SIN(2*PI()*(A2738+301340)/41736)+17121*DR2738*COS(2*PI()*(A2738+301340)/41736)+3346580*DR2738*SIN(2*PI()*(A2738+301340)/20868)+1163334*DR2738*COS(2*PI()*(A2738+301340)/20868)-46310*DR2738*SIN(2*PI()*(A2738+301340)/6956)-61608*DR2738*COS(2*PI()*(A2738+301340)/6956)-5921*SIN(2*PI()*(A2738+301340)/20868)*COS((AI2738-DX2738)*PI()/180)+7926*SIN(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DX2738)*PI()/180)-19742*COS(2*PI()*(A2738+301340)/20868)*COS((AI2738-DX2738)*PI()/180)+22347*COS(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DX2738)*PI()/180)-2837*SIN(2*PI()*(A2738+301340)/20868)*COS(2*(AI2738-DX2738)*PI()/180)+4176*SIN(2*PI()*(A2738+301340)/20868)*SIN(2*(AI2738-DX2738)*PI()/180)+5363*COS(2*PI()*(A2738+301340)/20868)*COS(2*(AI2738-DX2738)*PI()/180)+2331*COS(2*PI()*(A2738+301340)/20868)*SIN(2*(AI2738-DX2738)*PI()/180)+7621*SIN(2*PI()*(A2738+301340)/667776)*COS((AI2738-DX2738)*PI()/180)-12107*SIN(2*PI()*(A2738+301340)/667776)*SIN((AI2738-DX2738)*PI()/180)+12298*COS(2*PI()*(A2738+301340)/667776)*COS((AI2738-DX2738)*PI()/180)-2692*COS(2*PI()*(A2738+301340)/667776)*SIN((AI2738-DX2738)*PI()/180)-17370*SIN(2*PI()*(A2738+301340)/667776)*COS(2*(AI2738-DX2738)*PI()/180)-3035*SIN(2*PI()*(A2738+301340)/667776)*SIN(2*(AI2738-DX2738)*PI()/180)+2123*COS(2*PI()*(A2738+301340)/667776)*COS(2*(AI2738-DX2738)*PI()/180)-17304*COS(2*PI()*(A2738+301340)/667776)*SIN(2*(AI2738-DX2738)*PI()/180)-707*SIN(2*PI()*(A2738+301340)/333888)*COS((AI2738-DX2738)*PI()/180)+665*SIN(2*PI()*(A2738+301340)/333888)*SIN((AI2738-DX2738)*PI()/180)-833*COS(2*PI()*(A2738+301340)/333888)*COS((AI2738-DX2738)*PI()/180)-414*COS(2*PI()*(A2738+301340)/333888)*SIN((AI2738-DX2738)*PI()/180)-918*SIN(2*PI()*(A2738+301340)/333888)*COS(2*(AI2738-DX2738)*PI()/180)+1582*SIN(2*PI()*(A2738+301340)/333888)*SIN(2*(AI2738-DX2738)*PI()/180)-1728*COS(2*PI()*(A2738+301340)/333888)*COS(2*(AI2738-DX2738)*PI()/180)-1133*COS(2*PI()*(A2738+301340)/333888)*SIN(2*(AI2738-DX2738)*PI()/180)+125840*DR2738*SIN(2*PI()*(A2738+301340)/20868)*COS((AI2738-DX2738)*PI()/180)-5657660*DR2738*SIN(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DX2738)*PI()/180)-8035742*DR2738*COS(2*PI()*(A2738+301340)/20868)*COS((AI2738-DX2738)*PI()/180)+1630420*DR2738*COS(2*PI()*(A2738+301340)/20868)*SIN((AI2738-DX2738)*PI()/180)-1775121*DR2738*SIN(2*PI()*(A2738+301340)/20868)*COS(2*(AI2738-DX2738)*PI()/180)-3139592*DR2738*SIN(2*PI()*(A2738+301340)/20868)*SIN(2*(AI2738-DX2738)*PI()/180)-2702485*DR2738*COS(2*PI()*(A2738+301340)/20868)*COS(2*(AI2738-DX2738)*PI()/180)+986428*DR2738*COS(2*PI()*(A2738+301340)/20868)*SIN(2*(AI2738-DX2738)*PI()/180)-2166058*DR2738*SIN(2*PI()*(A2738+301340)/667776)*COS((AI2738-DX2738)*PI()/180)+6117614*DR2738*SIN(2*PI()*(A2738+301340)/667776)*SIN((AI2738-DX2738)*PI()/180)-171831*DR2738*COS(2*PI()*(A2738+301340)/667776)*COS((AI2738-DX2738)*PI()/180)+1334347*DR2738*COS(2*PI()*(A2738+301340)/667776)*SIN((AI2738-DX2738)*PI()/180)-371905*DR2738*SIN(2*PI()*(A2738+301340)/667776)*COS(2*(AI2738-DX2738)*PI()/180)-289032*DR2738*SIN(2*PI()*(A2738+301340)/667776)*SIN(2*(AI2738-DX2738)*PI()/180)+1323251*DR2738*COS(2*PI()*(A2738+301340)/667776)*COS(2*(AI2738-DX2738)*PI()/180)-786120*DR2738*COS(2*PI()*(A2738+301340)/667776)*SIN(2*(AI2738-DX2738)*PI()/180)+306909*DR2738*SIN(2*PI()*(A2738+301340)/333888)*COS((AI2738-DX2738)*PI()/180)-86179*DR2738*SIN(2*PI()*(A2738+301340)/333888)*SIN((AI2738-DX2738)*PI()/180)-427267*DR2738*COS(2*PI()*(A2738+301340)/333888)*COS((AI2738-DX2738)*PI()/180)+507766*DR2738*COS(2*PI()*(A2738+301340)/333888)*SIN((AI2738-DX2738)*PI()/180)-376674*DR2738*SIN(2*PI()*(A2738+301340)/333888)*COS(2*(AI2738-DX2738)*PI()/180)-180509*DR2738*SIN(2*PI()*(A2738+301340)/333888)*SIN(2*(AI2738-DX2738)*PI()/180)+226744*DR2738*COS(2*PI()*(A2738+301340)/333888)*COS(2*(AI2738-DX2738)*PI()/180)-375559*DR2738*COS(2*PI()*(A2738+301340)/333888)*SIN(2*(AI2738-DX2738)*PI()/180)-1187*SIN(2*PI()*(A2738+301340)/6956)*COS((AI2738-DX2738)*PI()/180)+205*SIN(2*PI()*(A2738+301340)/6956)*SIN((AI2738-DX2738)*PI()/180)+342*COS(2*PI()*(A2738+301340)/6956)*COS((AI2738-DX2738)*PI()/180)+339*COS(2*PI()*(A2738+301340)/6956)*SIN((AI2738-DX2738)*PI()/180)-128955*DR2738*SIN(2*PI()*(A2738+301340)/6956)*COS((AI2738-DX2738)*PI()/180)+77955*DR2738*SIN(2*PI()*(A2738+301340)/6956)*SIN((AI2738-DX2738)*PI()/180)-47034*DR2738*COS(2*PI()*(A2738+301340)/6956)*COS((AI2738-DX2738)*PI()/180)+36966*DR2738*COS(2*PI()*(A2738+301340)/6956)*SIN((AI2738-DX2738)*PI()/180)-37979*DR2738*SIN(2*PI()*(A2738+301340)/41736)*COS((AI2738-DX2738)*PI()/180)-5593*DR2738*SIN(2*PI()*(A2738+301340)/41736)*SIN((AI2738-DX2738)*PI()/180)+9982*DR2738*COS(2*PI()*(A2738+301340)/41736)*COS((AI2738-DX2738)*PI()/180)+6997*DR2738*COS(2*PI()*(A2738+301340)/41736)*SIN((AI2738-DX2738)*PI()/180)+50061905*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/667776)+286162990*DR2738*DR2738*COS(2*PI()*(A2738+301340)/667776)-52734896*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/333888)+11348620*DR2738*DR2738*COS(2*PI()*(A2738+301340)/333888)+99207877*DR2738*DR2738*SIN(2*PI()*(A2738+301340)/20868)+111052242*DR2738*DR2738*COS(2*PI()*(A2738+301340)/20868)-426328*DR2738*SIN(2*PI()*(A2738+301340)/20868)*COS(3*(AI2738-DX2738)*PI()/180)-246344*DR2738*SIN(2*PI()*(A2738+301340)/20868)*SIN(3*(AI2738-DX2738)*PI()/180)-490205*DR2738*COS(2*PI()*(A2738+301340)/20868)*COS(3*(AI2738-DX2738)*PI()/180)+493871*DR2738*COS(2*PI()*(A2738+301340)/20868)*SIN(3*(AI2738-DX2738)*PI()/180)+120*SIN(2*PI()*(A2738+301340)/6956)*COS(2*(AI2738-DX2738)*PI()/180)+466*SIN(2*PI()*(A2738+301340)/6956)*SIN(2*(AI2738-DX2738)*PI()/180)+308*COS(2*PI()*(A2738+301340)/6956)*COS(2*(AI2738-DX2738)*PI()/180)-39764*DR2738*SIN(2*PI()*(A2738+301340)/6956)*COS(2*(AI2738-DX2738)*PI()/180)-45035*DR2738*SIN(2*PI()*(A2738+301340)/6956)*SIN(2*(AI2738-DX2738)*PI()/180)-32422*DR2738*COS(2*PI()*(A2738+301340)/6956)*COS(2*(AI2738-DX2738)*PI()/180)+72034*DR2738*COS(2*PI()*(A2738+301340)/6956)*SIN(2*(AI2738-DX2738)*PI()/180)+141083*DR2738*SIN(2*PI()*(A2738+301340)/667776)*COS(3*(AI2738-DX2738)*PI()/180)-80015*DR2738*SIN(2*PI()*(A2738+301340)/667776)*SIN(3*(AI2738-DX2738)*PI()/180)+69508*DR2738*COS(2*PI()*(A2738+301340)/667776)*COS(3*(AI2738-DX2738)*PI()/180)+25816*DR2738*COS(2*PI()*(A2738+301340)/667776)*SIN(3*(AI2738-DX2738)*PI()/180)+984