Sun-side Δa: Kepler Periods Across the Solar System

This page is the Sun-side companion to Mass from Moon Orbits. The Moon-side derivation extracts GM_planet from a moon’s orbit using a re-parameterized Δa correction. The Sun-side picture, documented here, is the mirror: extract any planet’s heliocentric period T from its semi-major axis using a closed-form Δa, and recover the same simple Kepler result that the model’s code actually uses.

Both pictures use the same algebraic machinery, but with one important asymmetry: the Sun-side Δa is an exact identity, not an approximation.

1. The Formula

For any planet b orbiting the Sun, the exact Sun-side Δa is:

Δa_b  =  a_b  ·  ( 1  −  ( (μ_S + μ_E − μ_b) / (μ_S + μ_E) )^(1/3) )

where:

Plugged into the elaborate two-body period formula:

T_b  =  2π · √( (a_b − Δa_b)³ / (μ_S + μ_E − μ_b) )

— and the (a−Δa)³ numerator exactly cancels the (μ_S + μ_E − μ_b) denominator, leaving the simple system-mass form:

T_b  =  2π · √( a_b³ / (μ_S + μ_E) )           ← algebraically identical

Same period, same precision, same answer.

2. The Algebraic Identity (Why Both Forms Agree Exactly)

Start from the symmetric Δa:

Δa_b / a_b      =   1 − ((μ_S + μ_E − μ_b)/(μ_S + μ_E))^(1/3)
(a_b − Δa_b)    =   a_b · ((μ_S + μ_E − μ_b)/(μ_S + μ_E))^(1/3)
(a_b − Δa_b)³   =   a_b³ · (μ_S + μ_E − μ_b) / (μ_S + μ_E)

Divide both sides by (μ_S + μ_E − μ_b):

(a_b − Δa_b)³ / (μ_S + μ_E − μ_b)   =   a_b³ / (μ_S + μ_E)

This is an algebraic identity, not an approximation — exact to machine precision. The body’s own gravitational parameter μ_b appears in both the numerator (via Δa) and the denominator (via the system-mass-minus-body), and they cancel in lockstep.

The same is not true for the Moon-side Δa = a · μ · m formula on the Mass from Moon Orbits page. That form is a re-parameterization of Hill-Brown’s ½·m² term that holds to ~3% in our solar system because μ ≈ m/6 for Earth-Moon-Sun. The Sun-side identity has no such “approximate” qualifier.

3. Per-Planet Δa Table

Computed with the model’s mass-ratio constants and standard semi-major axes:

The Δa scales roughly as a · μ_b / (3·μ_S) — i.e., the planet’s distance times one-third of its own mass-ratio. Jupiter dominates by being both far and heavy: its Δa is ~1,600× Earth’s, even though both are O(10⁻⁶) of their semi-major axes.

Notably, this formula works for all 8 planets plus Pluto (9 bodies in total), including Mercury and Venus — which the Moon-side derivation cannot reach because they have no moons.

4. Earth as the Anchor — Why Δa ≠ 0 by Construction

A subtle but important point: Earth’s Δa is not zero even in this symmetric form. Setting μ_b = μ_E:

Δa_E  =  a_E · ( 1 − (μ_S / (μ_S + μ_E))^(1/3) )  ≈  149.77 km

The numerator becomes μ_S + μ_E − μ_E = μ_S, recovering exactly the asymmetric Sun-side formula from the Mass-from-Moon page §3. So for Earth specifically, the symmetric and asymmetric formulas coincide:

Δa_E (symmetric)   =  a_E · (1 − (μ_S/(μ_S + μ_E))^(1/3))   ≈ 149.77 km
Δa_E (asymmetric)  =  a_E · (1 − (μ_S/(μ_S + μ_E))^(1/3))   ≈ 149.77 km  (same)

This is because Earth is the AU-anchor body. The formula’s reference frame (μ_S + μ_E in the denominator) and Earth’s own μ_b are the same μ_E, so the two forms reduce to the same expression. For any other planet, they diverge — see §5 below.

5. Symmetric vs Asymmetric Form

The asymmetric form (the Sun-side analog presented on the Moon-side page) uses just the planet’s own mass:

Δa_b (asymmetric)  =  a_b · ( 1 − (μ_S / (μ_S + μ_b))^(1/3) )

(The Earth-specific expression on the Moon-side page — Δa = AU / (3 × M_Sun / M_Earth) — is the small-Δa linearization of this cube-root form. Both yield 149.77 km for Earth; the cube-root form generalizes cleanly to other planets.)

The symmetric form (this page) uses the system mass minus the body:

Δa_b (symmetric)   =  a_b · ( 1 − ((μ_S + μ_E − μ_b) / (μ_S + μ_E))^(1/3) )

The asymmetric form matches the symmetric form to under a kilometer for all terrestrial planets and Pluto, but is incomplete for the gas giants. Only the symmetric form makes the algebraic identity exact. The 235 km Jupiter discrepancy with the asymmetric form is precisely the source of the ~0.5 second residual that appears if the asymmetric Δa is fed into the elaborate two-body formula. The simple form T = 2π·√(a³/(μ_S + μ_E)) — what the model’s code actually uses — has no residual either way, because it doesn’t compute Δa at all.

6. Mirror Diagram: Moon-side vs Sun-side

The Moon-side derivation and the Sun-side formula on this page are mirror images of the same machinery:

Both formulas extract a Kepler-effective semi-major axis from a geometric one to make a single-line Kepler formula match physics:

• Moon-side: a_M_geometric (LLR) → a_M_Kepler_eff (the textbook 384,748 km)
• Sun-side: a_b_geometric (in the Keplerian system formula) → a_b − Δa_b (in the bare body formula)

The asymmetry in direction (add vs subtract) and in exactness (re-parameterization vs identity) reflects the different roles the two formulas play. The Moon-side bridges raw observations to JPL DE440 reference values across the 3-body Earth-Moon-Sun system, with an honest few-ppm precision floor. The Sun-side is a pure two-body identity once μ_S + μ_E is the canonical denominator.

7. Practical Use in Code

For period computation in software or spreadsheets, the simple form is strictly preferable:

T_b  =  2π · √( a_b³ / (μ_S + μ_E) )

No Δa needed. No body-specific term. Same denominator for every planet. Exact to machine precision.

The elaborate Δa form is useful as a conceptual lens:

• It explains why the Earth-specific 149.77 km correction (subtract from AU) is the structural mirror of the 349 km Moon correction (add to a_M)
• It generalizes that picture to every planet (including Mercury and Venus)
• It makes the body-mass dependence visible in the formula (otherwise hidden in the AU anchor)

In the code, the simple form is used with GM_SUN_PLUS_EARTH as the canonical denominator. The Δa machinery is documented but not computed at runtime.

8. Spreadsheet Reference

In Excel-style notation, with cells:

• Q_col = planet’s semi-major axis (m)
• $G$11 = GM_Sun_alone (m³/s²)
• $G$15 = GM_Earth_alone (m³/s²)
• G_col = planet’s own GM (m³/s²)
• $A$28 = day length (s)
• $B$29 = H/13 frame factor = meanSolarYear / meanSiderealYear

The exact symmetric Δa:

M_col = Q_col * ( 1 - ( ($G$11 + $G$15 - G_col) / ($G$11 + $G$15) )^(1/3) )

The elaborate Kepler period (with Δa):

T_col = ( 2*PI() * SQRT( (Q_col - M_col)^3 / ($G$11 + $G$15 - G_col) ) / $A$28 ) * $B$29

The simple Kepler period (no Δa, identical result to machine precision):

T_col = ( 2*PI() * SQRT( Q_col^3 / ($G$11 + $G$15) ) / $A$28 ) * $B$29

For every planet, both T_col forms produce the same value:

Summary

• The symmetric Sun-side Δa, Δa_b = a_b · (1 − ((μ_S + μ_E − μ_b)/(μ_S + μ_E))^(1/3)), makes the elaborate two-body period formula T = 2π·√((a−Δa)³/(μ_S+μ_E−μ_b)) algebraically identical to the simple T = 2π·√(a³/(μ_S+μ_E)) — exact, not approximate.
• For Earth the symmetric and asymmetric forms coincide at Δa = 149.77 km — Earth is the AU-anchor body, so its μ_b equals the formula’s reference μ_E.
• For Jupiter–Neptune, only the symmetric form fully closes the residual to zero; the asymmetric form leaves a ~235 km Jupiter discrepancy.
• In the model’s code the simple form is preferred (a³/(GM_Sun + GM_Earth)); the Δa machinery lives here and on the Mass from Moon Orbits page as the conceptual explanation.

This is the Sun-side mirror of the Moon-side mass derivation. Together they form a single coherent picture: every planet’s GM and every planet’s period can be expressed as a single-line Kepler formula plus a single-line Δa shift — three classical effects (two-body Kepler, three-body solar perturbation, body-mass split) collapsed into one structural template.

See also

• Mass from Moon Orbits — the Moon-side derivation; this page is its Sun-side companion
• Formulas — model’s analytical formulas for obliquity, eccentricity, precession
• Glossary — definitions for GM, mass ratios, sidereal year
• JPL DE440 SPICE kernel: gm_de440.tpc  — authoritative system GMs