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. The Sun-side picture documented here goes in the opposite direction: given a planet’s heliocentric semi-major axis a, what is its orbital period T?

1. The question

Kepler’s third law relates a planet’s semi-major axis a to its orbital period T:

T  =  2π · √( a³ / GM )

The exact two-body answer is the system mass: GM_Sun + GM_body. Earth uses GM_Sun + GM_Earth, Jupiter uses GM_Sun + GM_Jupiter, and so on. The denominator is body-specific. Suppose instead you want one formula that works for every planet with a single fixed denominator. The Sun-side Δa makes that possible: shift a by a small amount, then plug into Kepler with the fixed denominator GM_Sun + GM_Earth. Same period, every planet, no body-specific term.

This page answers two questions: (1) what is Δa? and (2) why does it give exactly the right period — not approximately?

2. The formula

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

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

Plug the shifted axis into Kepler’s third law with μ_S + μ_E − μ_b in the denominator:

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

This produces a result algebraically identical to the simple Kepler formula with a fixed denominator:

T_b  =  2π · √( a_b³ / (μ_S + μ_E) )           ← same T, no Δa needed

The next section works through this for Earth, then §4 shows why the two forms must agree.

3. A worked example: Earth

Inputs:

a_E   =  149,597,870.70 km          (one astronomical unit, by definition)
μ_S   =  1.327 × 10¹¹  km³/s²       (GM_Sun alone)
μ_E   =  3.986 × 10⁵   km³/s²       (GM_Earth alone)
T_E   =  365.25636 days             (sidereal year — target answer)

For Earth, the planet’s own μ_b equals μ_E, so the numerator μ_S + μ_E − μ_b simplifies to μ_S:

μ_S / (μ_S + μ_E)  =  0.999996997
( ... )^(1/3)      =  0.999998999
Δa_E               =  149,597,870.70 × (1 − 0.999998999)  ≈  149.77 km

The AU shrinks by 149.77 km to a Kepler-effective AU of 149,597,720.93 km. Both Kepler forms give the same period:

T_E  =  2π · √( (149,597,720.93)³ / μ_S )           =  365.256 days   ✓
T_E  =  2π · √( (149,597,870.70)³ / (μ_S + μ_E) )   =  365.256 days   ✓

To machine precision, not coincidentally — that match is the exact algebraic identity the next section explains.

4. Why both forms agree (the identity)

Start from the Δa formula in §2 and rearrange:

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

Cube both sides, then multiply by a_b³ / (μ_S + μ_E − μ_b):

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

The left side is what the Kepler formula computes when the body-specific denominator is used; the right side is what the simple Kepler formula computes when the fixed denominator is used. The equation says they are equal as an algebraic identity — exact to machine precision. The planet’s own mass μ_b cancels: it appears in the numerator (via Δa) and in the denominator (via μ_S + μ_E − μ_b), and both cancellations happen in lockstep.

This is not the case for the Moon-side Δa = a · μ · m formula on the Mass from Moon Orbits page. That form is a re-parameterisation 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.

5. 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) — distance times one-third of the body’s mass-ratio. Jupiter dominates by being both far and heavy: its Δa ≈ 248,000 km is ~1,600× Earth’s. This formula works for all 8 planets plus Pluto (9 bodies), including Mercury and Venus — which the Moon-side derivation cannot reach because they have no moons.

6. Why Earth gets the special form

The worked example in §3 has a subtle simplification: for Earth, μ_b = μ_E, so the numerator μ_S + μ_E − μ_b collapses to just μ_S. The Δa formula reduces to:

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

For every other planet the numerator still contains a μ_E term and the formula keeps the full three-term form. Earth gets this clean form because Earth is the AU-anchor body: the astronomical unit is defined relative to Earth’s orbit, and the canonical Kepler denominator μ_S + μ_E is the system mass of Earth’s orbit specifically. When the formula’s reference (μ_S + μ_E) and the body’s own pair (μ_S + μ_E) coincide, the body’s mass cancels into a simpler expression. This is the structural reason Earth’s Δa = 149.77 km gets a clean closed form, while Jupiter’s keeps the full expression.

7. Physical interpretation: Δa as 1/3 of the Sun’s barycentric pull

Beyond being an algebraic identity, the leading-order Δa has a direct geometric reading. Each planet pulls the Sun toward itself by an amount set by the centre-of-mass definition of the Sun-planet system:

(Sun-SSB offset due to that planet)  =  a_b · M_b / (M_S + M_b)  ≈  a_b · M_b / M_S

where SSB is the Solar System Barycenter. The leading-order Δa from §2 reduces to:

Δa_b  ≈  a_b · M_b / (3 · M_Sun)

So each planet’s Δa is exactly one-third of its contribution to the Sun’s barycentric displacement. The ratio is precisely 3.00 for every planet — the factor of 3 traces to the cube root in Kepler’s third law (the derivative of a³ produces a 3, which propagates to the linear approximation of Δa).

Consequence: Σ(3·Δa) = maximum solar inertial motion

Summing each planet’s barycentric pull (3·Δa = a·M/M_Sun) when planets align in the same direction gives the maximum Sun-SSB excursion:

3 · Σ Δa  ≈  1,510,000 km  ≈  2.17 R☉

This is the maximum solar inertial motion (SIM) amplitude, documented since Jose (1965)  and Charvátová & Střeštík (1991). The Sun’s actual position relative to the solar-system barycenter sweeps from ~0 (planets scattered) to ~2.17 R☉ (alignment), on a Jupiter-Saturn-dominated cycle of roughly 178.7 years — the “Jose cycle”.

The Δa values aren’t merely abstract calibration constants — they are exactly one-third of each planet’s pull on the Sun, and their sum (multiplied by 3) is the radius of the solar inertial motion  trajectory the Sun traces around the SSB over centuries.

8. Mirror to the Moon-side page

The Mass from Moon Orbits page and this page are mirror images of the same machinery:

Both formulas extract a Kepler-effective semi-major axis from a geometric one so that a single-line Kepler formula returns the right answer. The Moon-side bridges raw observations to JPL DE440 reference values across the 3-body Earth-Moon-Sun system, with a few-ppm precision floor. The Sun-side is a pure two-body identity once μ_S + μ_E is the canonical denominator.

9. An alternative form (note)

The expression in §2 is the exact form. A simpler-looking variant exists that uses just the planet’s own mass:

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

This is what the Mass from Moon Orbits page shows for Earth (in its linearised form, AU / (3 × M_Sun/M_Earth) ≈ 149.77 km). For Earth specifically the two forms are literally identical. For the other terrestrials and Pluto they agree to under a kilometre. For the gas giants they diverge — most dramatically for Jupiter:

The two forms agree exactly for terrestrials and Pluto, and diverge only for the gas giants. The 235 km Jupiter discrepancy with the alternative form is precisely the source of the ~0.5 second residual that appears if the alternative is fed into the elaborate Kepler equation. The exact §2 form closes that gap. In practice this barely matters because the model’s code uses neither Δa form — see §10.

10. 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 Δa form is useful only 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 generalises that picture to every planet (including Mercury and Venus); and 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.

Summary

For any planet orbiting the Sun, the Sun-side Δa = a_b · (1 − ((μ_S + μ_E − μ_b)/(μ_S + μ_E))^(1/3)) shifts the geometric semi-major axis to a Kepler-effective value, so that one formula with the same denominator (μ_S + μ_E) computes the right period for every planet — exact, not approximate. For Earth specifically, Δa = 149.77 km (Earth is the AU-anchor body, so the formula reduces to a particularly clean form). For Jupiter through Neptune the full expression is required; for Mercury, Mars, Venus, and Pluto the body-mass term is tiny.

Each Δa equals exactly 1/3 of that planet’s contribution to the Sun’s barycentric displacement. Summing 3·Δa across all planets gives ~1.5 million km ≈ 2.17 R☉ — the maximum solar inertial motion amplitude (Jose 1965, Charvátová 1991).

The model’s code uses the simple Kepler form; the Δa machinery lives here as the conceptual explanation. The in-model simulation includes a live Sun-SSB trajectory chart in the Sun’s panel that visualises this in real time. Together with the Moon-side derivation, the picture is unified: every planet’s GM and every planet’s period can be expressed as a single-line Kepler formula plus a single-line Δa shift.

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
• Wikipedia — Barycenter (astronomy)  — the canonical reference for the Sun-SSB picture
• JPL DE440 SPICE kernel: gm_de440.tpc  — authoritative system GMs