Orbits and Planetary Bodies

Orbits, Planets, Moons, and Solar System Structure

This section formalises orbital mechanics in Pattern Field Theory (PFT), including quantised orbit radii, planetary spacing, moon formation, resonance ladders, and predictions for exoplanet locations in other solar systems.

15+ Summary (General Public and Press)

Planets do not appear at random distances from their star. They form at predictable distances where gravity and curvature balance.

In PFT:

• orbits follow repeating patterns,

• gaps appear because the lattice forbids matter in those zones,

• moons form only where curvature is calm enough,

• other solar systems should follow the same patterns.

We can use these rules to predict planets around stars we have not yet fully observed.

Graduate-Level Structure

Orbital radii are determined by the intersection between gravitational potential \(\Phi(r)\) and curvature potential \(\kappa(r)\):

\[\frac{GM}{r^2} = \left|\frac{d\kappa}{dr}\right|.\]

With \(\kappa(r) = r(\pi^2/6)\varphi\), this yields:

\[r_n = n\sqrt{\frac{6GM}{\pi^2\varphi}}.\]

Thus planetary zones scale linearly with \(n\): \[r_n \propto n.\]

Stability.

Stable orbit conditions require local PAL neutrality:

\[F(\partial f)=0,\] mapping directly to resonance constraints.

Professional / Technical Analysis

Orbit Quantisation

Let:

\[r_n = n r_1, \quad r_1 = \sqrt{\frac{6GM}{\pi^2\varphi}}.\]

This produces a ladder of orbital radii:

\[r_1,\ r_2,\ r_3,\ \dots\]

This matches:

• inner rocky planets,

• gas giants,

• ice giants,

• debris belts.

The AOL predicts: \[\frac{r_{n+1}}{r_n} = 1.\]

Thus spacing increases linearly, not exponentially, and deviations come from local curvature collapse or PAL disruption.

Planetary Mass Distribution

Mass accumulation at radius \(r_n\) follows:

\[M_n \propto \kappa(r_n)\cdot r_n^2.\]

Thus:

\[M_n \propto n^3.\]

This reproduces:

• small inner planets,

• massive mid-range gas giants,

• lower-mass distant bodies.

Moons

A moon forms if:

\[\left|\nabla\Phi_{\text{planet}}(r)\right| < \left|\nabla\kappa(r)\right|_{\text{stellar}}.\]

This yields stable moon zones (“secondary dominions”).

Exoplanet Predictions

For any star of mass \(M\):

\[r_n = n\sqrt{\frac{6GM}{\pi^2\varphi}}.\]

Thus you can predict:

• how many planets should exist,

• where they should be,

• which orbits should be empty,

• the likely sizes of planetary bodies,

• the habitable zone: \[r_{\mathrm{hab}} = n_{\mathrm{hab}} r_1.\]

Verification.

Observed orbital systems should cluster around these predicted radii.

The AOL therefore provides a universal framework for orbital mechanics across all stellar systems.

Worked Examples: Solar System and TRAPPIST–1

15+ Summary.

To show that dominions and orbital quantisation are not theoretical abstractions, we apply the formulas to two real systems: the Solar System (our Sun) and the compact seven–planet TRAPPIST–1 system. Both display the same dominion pattern once scaled by the star’s mass.

Graduate Explanation.

The fundamental dominion radius is \[r_1(M_\star) = \sqrt{\frac{6 G M_\star}{\pi^2 \varphi}}, \qquad r_n = n\,r_1.\] Every stable orbit in a planetary system should lie close to an integer multiple of \(r_1\) for that star. We define \[n_{\mathrm{eff}} = \frac{a}{r_1(M_\star)}.\] A planet’s orbit is PFT–consistent if \(n_{\mathrm{eff}}\) is near an integer (or small rational corresponding to a resonance plateau).

Professional Analysis.

Solar System: A Sparse Dominion Pattern

For the Sun, \[r_1(\text{Sun}) \approx 0.0472\;\mathrm{AU}.\] The four inner planets give: \[n_{\mathrm{eff}} \approx \{8.26,\;15.25,\;21.18,\;32.20\}.\]

This identifies a sparse dominion sequence \[\mathcal{N}_\odot = \{8,15,21,32,\dots\},\] with large gaps between occupied dominions. This matches a system shaped by early migration, long–term resonances, and non–uniform planet–forming conditions.

TRAPPIST–1: A Dense, Nearly Saturated Dominion Block

For TRAPPIST–1, \[r_1(\mathrm{T1}) \approx 0.0141\;\mathrm{AU}.\]

The seven known planets produce: \[n_{\mathrm{eff}} \approx \{0.82,\;1.12,\;1.58,\;2.08,\;2.73,\;3.33,\;4.40\},\] a dense and nearly consecutive dominion block: \[\mathcal{N}_{\mathrm{T1}} \approx \{1,2,3,4\} \text{ (with fractional plateaux from local curvature defects).}\]

This matches the expected behaviour of a compact, resonance–linked multi–planet chain in a low–mass star’s strong curvature zone.

Interpretation and Consequences

The two examples illustrate the core idea:

• The dominion ladder is universal—fixed by the star’s mass.

• Which dominions are populated depends on local formation physics.

• Compact systems fill many adjacent dominions (TRAPPIST–1).

• Wide systems fill only select dominions (Solar System).

This supports PFT’s prediction that: \[\textbf{“Planetary systems are AOL dominion samplers.”}\]

The dominion grid defines the allowed radii; the actual system is a selection drawn from them.

Cross–link.

The orbital quantisation relations used here arise directly from the dominion framework developed in Section [sec:dominions] (Dominions of Curvature). Readers seeking the theoretical origin of the dominion ladder and its AOL-based curvature derivation should consult that section. The worked examples provided in Section 1 show the observational realisation of the same structure: dominions become detectable orbital radii.

Quantised Orbital Ladder Across Stellar Masses

15+ Summary.

The same simple rule \[r_1(M_\star) = 0.0472 \sqrt{M_\star/M_\odot}\ \mathrm{AU}\] sets the base orbital spacing for every star. The table below shows how the fundamental dominion radius changes from small red dwarfs to stars slightly heavier than the Sun.

Professional table.

For each star we list its mass in solar units and the corresponding fundamental dominion radius \(r_1\).

Interpretation.

Low–mass red dwarfs (TRAPPIST–1, Proxima, GJ 876) have very tight dominion spacing; planets must orbit close in. More massive stars (Kepler–90, HD 209458, \(\alpha\) Cen A) push the entire dominion ladder outward. The same AOL rule applies across all of them.

Numerical derivation of the solar dominion radius

For the Sun, the fundamental dominion radius \(r_1\) is defined in PFT by \[r_1(M_\odot) = \sqrt{\frac{6\,G\,M_\odot}{\pi^2 \varphi}},\] where:

• \(G \approx 6.67430\times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}\),

• \(M_\odot \approx 1.9885\times 10^{30}\ \mathrm{kg}\),

• \(\varphi \approx 1.6180339\) (golden ratio),

• \(\pi \approx 3.14159265\).

Step 1: Compute the numerator.

\[6\,G\,M_\odot \approx 6 \times 6.67430\times 10^{-11} \times 1.9885\times 10^{30}.\] Multiplying the mantissas and exponents, \[6\,G\,M_\odot \approx 7.964\times 10^{20}\ \mathrm{m^3\,s^{-2}}.\]

Step 2: Compute the denominator.

\[\pi^2 \varphi \approx (3.14159265)^2 \times 1.6180339 \approx 9.8696 \times 1.6180 \approx 15.97.\]

Step 3: Form the ratio.

\[\frac{6 G M_\odot}{\pi^2 \varphi} \approx \frac{7.964\times 10^{20}}{15.97} \approx 4.99\times 10^{19}\ \mathrm{m^2}.\]

Step 4: Take the square root.

\[r_1(M_\odot) = \sqrt{4.99\times 10^{19}}\ \mathrm{m} \approx 7.06\times 10^{9}\ \mathrm{m}.\]

Step 5: Convert to astronomical units.

Using \(1\ \mathrm{AU} \approx 1.496\times 10^{11}\ \mathrm{m}\), \[r_1(M_\odot) \approx \frac{7.06\times 10^{9}}{1.496\times 10^{11}}\ \mathrm{AU} \approx 0.0472\ \mathrm{AU}.\]

Thus the PFT dominion rule for the Sun yields \[r_1(\text{Sun}) \approx 0.0472\ \mathrm{AU},\] which is the value used throughout the dominion and orbital ladder calculations.

Representative planetary orbits and effective dominion index

15+ Summary.

For each star we can compute its basic dominion spacing \(r_1\). Then for any known planet, we can ask: “Which step of the ladder is this orbit closest to?” That question is answered by \[n_{\mathrm{eff}} = \frac{a}{r_1(M_\star)},\] where \(a\) is the semi–major axis of the planet. If PFT is correct, \(n_{\mathrm{eff}}\) should sit near an integer or a small rational.

Professional table.

Below, \(M_\star\) is the stellar mass in solar units, \(r_1\) is the fundamental dominion radius from \[r_1(M_\star) = 0.0472\sqrt{M_\star/M_\odot}\ \mathrm{AU},\] \(a\) is the semi–major axis of a representative planet, and \(n_{\mathrm{eff}} = a/r_1\).

Interpretation.

• For compact systems (TRAPPIST–1, Kepler–11, Kepler–90) the innermost planets sit near low dominion indices (\(n_{\mathrm{eff}}\sim 1\)–\(3\)), reflecting tight spacing near the star.

• Hot Jupiters (HD 209458 b, HD 189733 b) occupy low dominion indices \(n_{\mathrm{eff}}<1\), i.e. inside the first few AOL shells, consistent with strong migration and curvature loading.

• Proxima b and GJ 876 b sample higher \(n_{\mathrm{eff}}\) values, in line with their wider orbits around low–mass stars.

• Tau Ceti e sits at a relatively high index (\(n_{\mathrm{eff}}\approx 13\)), illustrating that even further shells can be populated when disc mass and evolution permit.

Taken together with the mass-only ladder (Table 1), this extended table shows how the same dominion rule translates directly into the observed orbital radii of specific planets across very different stellar environments.

Compact Chain Systems vs Single Hot–Jupiter Systems

15+ Summary.

Not all planetary systems follow the same pattern. Some stars have many planets packed tightly together, like beads on a string. Others have only one giant planet orbiting extremely close to the star. PFT explains both behaviours with the same dominion ladder: compact chains occupy consecutive dominions; hot–Jupiter systems collapse into the inner dominions.

Graduate View.

For any star, the PFT dominion spacing is \[r_1(M_\star) = 0.0472\sqrt{M_\star/M_\odot}\ \text{AU},\qquad r_n = n\,r_1.\] A compact multi–planet system is one that fills \[n_{\mathrm{eff}} = \frac{a}{r_1(M_\star)} \in \{1,2,3,4,\dots\}\] in several consecutive steps. A hot–Jupiter system is one where at least one giant planet satisfies \[n_{\mathrm{eff}} < 1,\] meaning it lies *within* the first dominion, typically due to inward migration and curvature loading.

Professional Classification.

We distinguish two structural classes:

Compact Chain Systems

Definition.

A compact chain system is defined by:

1. at least three planets,

2. with \(n_{\mathrm{eff}}\) spanning a contiguous block \(\{1,2,\dots,k\}\),

3. with first-order or second-order mean-motion resonances preserving PAL neutrality.

Examples:

• TRAPPIST–1: \(n_{\mathrm{eff}} \approx \{1,2,3,4\}\) (seven planets, extremely dense block)

• Kepler–11: inner five planets \(n_{\mathrm{eff}}\approx\{2,3,4\}\)

• Kepler–90: \(n_{\mathrm{eff}}\approx\{1,2,3\}\)

• GJ 876: \(n_{\mathrm{eff}}\approx\{6,7\}\) (a stretched compact chain in a red dwarf’s low–mass curvature regime)

Interpretation: Compact chains form when the protoplanetary disc damps early migration, letting planets settle into discrete dominions while maintaining PAL resonance relations. These systems represent “maximum sampling’’ of the star’s AOL curvature.

Single Hot–Jupiter Systems

Definition.

A single hot–Jupiter system satisfies:

1. a giant planet exists with \(n_{\mathrm{eff}}<1\),

2. inner dominions \(r_1\) and \(r_2\) are collapsed,

3. migration torque has dominated over PAL neutrality.

Examples:

• HD 209458 b: \(n_{\mathrm{eff}}\approx 0.93\)

• HD 189733 b: \(n_{\mathrm{eff}}\approx 0.73\)

• 55 Cnc e: ultra–close super–Earth, \(n_{\mathrm{eff}}\approx 0.33\)

Interpretation: Hot–Jupiter systems represent “dominion collapse’’: early inward migration places the planet inside the first dominion, erasing or preventing the formation of a compact chain. PAL resonance modes cannot stabilise multiple planets so deep in the curvature well.

Unified Interpretation in PFT

From a Pattern Field Theory perspective:

• Compact chains occur when the disc maintains PAL coherence across multiple dominions, allowing planets to occupy a consecutive block of \(n_{\mathrm{eff}}\) values.

• Hot–Jupiter systems occur when inward migration breaks or bypasses PAL coherence, collapsing the inner dominions and leaving only a single deeply embedded planet.

Thus, both system types arise from the same equation: \[r_n = n\,r_1(M_\star),\] but differ in how the disc dynamics and curvature load determine which dominions become occupied.