Relational Orbital Mechanics (ROM)

Orbital Dynamics from Pure Optical Measurements

Core Thesis:

Orbital dynamics requires no mass, no $G$, no metric, and no spacetime geometry.

All observable orbital structure follows from two directly measurable frequency projections:

** $\kappa^2 = 1-\frac{1}{(1+z)^2}$ ($z=$ gravitational redshift)**
** $\beta^2=1-\frac{1}{(1+z_{D})^{2}}$ ($z_D=$ transverse Doppler shift)**

Everything else is pure algebra.

User Guide: Interactive ROM Engine

Input Modes

The interactive engine supports 6 different input modes:

1. (κₚ, βₚ) - Perihelion State (Pure Phase Parameters)

κₚ: Potential projection at perihelion (dimensionless, range 0.001-1.0)
βₚ: Kinetic projection at perihelion (v/c, dimensionless, range 0.001-0.99)
Use when: You have direct optical measurements or want to explore phase space
Units: Both dimensionless (normalized by c)

2. (W, eᴄ) - Energy & Eccentricity

W: Energy invariant (dimensionless, range 0.0001-0.1)
eᴄ: Eccentricity (dimensionless, range 0-0.99)
Use when: You know the binding energy and shape
Units: Both dimensionless

3. (κ, eᴄ) - Global Potential & Eccentricity

κ: Global potential projection at semi-major axis (dimensionless, 0.001-1.0)
eᴄ: Eccentricity (dimensionless, 0-0.99)
Use when: You have average redshift and shape measurements
Units: Both dimensionless

4. (m₀, rₚ, rₐ) - Mass & Apsides (Astrophysical)

m₀: Central body mass (kg, e.g., 1.989×10³⁰ for Sun)
rₚ: Perihelion radius (meters)
rₐ: Aphelion radius (meters)
Use when: You have traditional astrophysical data
Units: m₀ in kg, distances in meters
Note: This mode uses M and G for convenience, but internally converts to (κ, β)

5. (m₀, a, T) - Mass, Axis, Period (Astrophysical)

m₀: Central body mass (kg)
a: Semi-major axis (meters)
T: Orbital period (seconds)
Use when: You have period observations
Units: m₀ in kg, a in meters, T in seconds

6. (β₁, t₁, β₂, t₂) - Two-Point Method

β₁, β₂: Velocity measurements at two points (v/c, dimensionless)
t₁, t₂: Light-travel times to those points (seconds, where r = ct)
Use when: You have two snapshot measurements
Units: β dimensionless, t in seconds
This is the most powerful method - complete system from just 2 measurements!

Understanding the Output

Global Parameters:

κ, β, W: The fundamental relational quantities
$e_ᴄ$, $\delta_p$: Shape and closure (how far from circular)

Perihelion/Aphelion:

κₚ, βₚ: Projections at closest approach (maximum interaction)
κₐ, βₐ: Projections at farthest point (minimum interaction)

Orbital Properties:

T: Orbital period
Δφ: Precession angle per orbit (in degrees)

Visualization:

η (r/a): Normalized radius - shows scale-invariant shape
The animation shows precession accumulation across orbits
Colors: 🟡 Central body, 🟢 Perihelion, 🟣 Aphelion, 🔴 Current position

Tips for Use

Start with (κₚ, βₚ) mode to get intuition for the phase space
Watch the precession - see how it accumulates over multiple orbits
Try extreme values - the system remains algebraically closed
Compare scales - change only ‘a’ to see scale invariance
Use two-point mode for orbit determination practice

The Core Framework

Global Parameters

ROM defines a bound system through normalized projections:

Parameter	Definition	Physical Meaning
$\kappa$	$\sqrt{R_s/a}$	Global potential projection $\kappa^2 = 1-\frac{1}{(1+z)^2}$ ($z=$ gravitational redshift)
$\beta$	$v/c$	Global kinetic projection $\beta^2=1-\frac{1}{(1+z_{D})^{2}}$ ($z_D=$ transverse Doppler shift)
$W$	$\frac{1}{2}(\kappa^2 - \beta^2)$	Energy invariant (binding energy)
$\delta$	$\frac{\kappa^2}{2\beta^2}$	Closure factor (measures deviation from circular)

Key insight: These are not coordinates in space. They are projections on the observer’s measurement sphere - the fundamental relational quantities you measure directly with light.

The Energy Invariant

The most important quantity in ROM is $W$:

\[W = \frac{1}{2}(\kappa_p^2 - \beta_p^2) = \frac{1}{4}\kappa_p^2(1-e_c) = \frac{\kappa^2}{4} = \text{constant at all phases}\]

This single number determines:

Whether the orbit is bound ($W > 0$)
The orbital scale ($a = R_s/\kappa^2$)
All energy distributions at every phase

Eccentricity from Closure

Eccentricity is not a free parameter - it emerges from the closure condition:

\[e_c = \frac{2\beta_p^2}{\kappa_p^2} - 1 = \frac{1}{\delta_p} - 1\]

This connects orbital shape directly to the ratio of gravitational redshift to Doppler shift.

From Optical Measurements to Predictions

Path 1: Verification on Mercury (Pure Optical Inputs)

We validate the theory using Mercury, utilizing only direct optical measurements (Doppler and Redshift). We implicitly assume zero knowledge of the Sun’s mass, the gravitational constant $G$, or the Schwarzschild radius derived from them.

Parameter	Symbol	Value	Source
Mercury Velocity (Perihelion)	$v_{p}$	$58{,}980 \text{ m/s}$	NASA Eclipse Mercury
Mercury Distance (Perihelion)	$r_{p}$	$46.0012 \times 10^9 \text{ m}$	Universe Today Mercury
Solar Radius (Nominal)	$R_{\text{sun}}$	$6.957 \times 10^8 \text{ m}$	IAU 2015 Res B3
Solar Gravitational Redshift	$z_{\text{sun}}$	$2.1224 \times 10^{-6}$	IAU 2015 Res B3

1. Input: Kinematic Projection ($\beta_p$).

Radar telemetry directly measures the orbital velocity at perihelion ($v_p \approx 58.98$ km/s). The kinematic projection is simply this velocity normalized by the speed of light:

\[\beta_p = \frac{v_p}{c} \approx 1.967 \times 10^{-4}\] \[\beta_p^2 \approx 3.8705094361 \times 10^{-8}\]

2. Input: Potential Projection ($\kappa_p$).

Instead of deriving potential from mass, we derive it from the measured gravitational redshift of the Sun’s photosphere, $z_{\text{sun}} \approx 2.1224 \times 10^{-6}$.

Using the $S^2$ relational carrier geometric omnidirectional scaling law of the potential projection ($\kappa^2 \propto 1/r$), we relate the known potential at the solar physical radius ($R_{\text{sun}}$) to the potential at Mercury’s perihelion radius ($r_p$):

\[\kappa^2(r_p) = \kappa^2(R_{\text{sun}}) \cdot \left( \frac{R_{\text{sun}}}{r_p} \right)\]

Using the redshift relation $\kappa^2(R_{\text{sun}}) = 1 - (1+z_{\text{sun}})^{-2} \approx 1-\left(1+2.1224\cdot10^{-6}\right)^{-2} \approx 4.2447864862\cdot10^{-6}$:

\[\kappa_p^2 \approx \kappa^2(R_{\text{sun}}) \left( \frac{R_{\text{sun}}}{r_p} \right)\]

Using the observed geometric ratio of radii $R_{\text{sun}}/r_p \approx 0.0151235185169$:

\[\kappa_p^2 \approx 4.2447864862\cdot10^{-6}\cdot 0.0151235185169 \approx 6.4196107024\times10^{-8}\]

3. Calculate Closure Factor ($\delta_p$).

With both projections determined purely from light measurements:

\[\delta_p = \frac{\kappa_p^2}{2\beta_p^2} = \frac{6.4196107024\times10^{-8}}{2 \times 3.8705094361\times10^{-8}} \approx 0.829297901025\]

4. Predict Eccentricity ($e$).

The orbital shape is strictly enforced by the closure defect:

\[e_{\text{pred}} = \frac{1}{\delta_p} - 1 = \frac{1}{0.829297901025} - 1 \simeq 0.205839299441\]

Conclusion: The predicted eccentricity closely matches (within inputs uncertainty) the observed value ($e \approx 0.2056$). This demonstrates that the orbital geometry is fully determined by the ratio of the central redshift to the orbital Doppler shift, with no reference to mass or $G$.

Path 2: Reconstruction (Star S2 Example)

When redshift is unknown, reconstruct from shape:

Given:

Observed eccentricity: $e = 0.8846$
Perihelion velocity: $\beta_p = 0.0255$ (7650 km/s)

Reconstruct Potential:

\[\kappa_p = \beta_p\sqrt{\frac{2}{1+e}} = 0.0255 \times \sqrt{\frac{2}{1.8846}} = 0.02627\]

Predict Precession:

\[\Delta\phi = \frac{3\pi}{2}\frac{(0.02627)^4}{(0.0255)^2} = 3.45 \times 10^{-3} \text{ rad} = 11.85'\]

Observed: $12’ \pm 1.5’$ - excellent agreement!

Derive Mass (if needed):

\[M = \frac{c^2 r_p}{2G}\kappa_p^2 = 4.15 \times 10^6 M_{\odot}\]

Mass is derived, not required as input.

The Universal Precession Law: Derivation via $Q_a$

Derivation of the Phase Shift.

The precession is intrinsic to the relational shift. Since $Q_a$ represents the norm of deviation from the Euclidean background, the system accumulates a phase mismatch over every closed cycle. The total angular shift is simply the full orbital phase ($2\pi$) scaled by the quadratic intensity of this shift ($Q_a^2$), normalized by the elliptical geometry factor ($1-e^2$):

\[\Delta\varphi = \underbrace{2\pi}_{\text{Cycle}} \cdot \underbrace{Q_a^2}_{\text{Intensity}} \cdot \underbrace{\frac{1}{1-e^2}}_{\text{Shape Factor}} = \frac{2\pi\,Q_a^{2}}{1-e^{2}}\]

This yields the general precession law strictly from geometric accumulation:

\[\Delta\varphi = \frac{2\pi\,Q_a^{2}}{1-e^{2}}\]

Substituting $Q_a^2$, we recover the standard form purely algebraically:

We select the semi-major axis $a$ as this reference scale, defining the norm $Q_a$. At the scale $r=a$, the closure condition ($\kappa^2=2\beta^2$) implies the specific distribution of the invariant Schwarzschild scale $R_s$:

\[\kappa^2(a) = \frac{R_s}{a}, \qquad \beta^2(a) = \frac{R_s}{2a}\]

Substituting these into the definition of the relational shift norm $Q^2 = \beta^2 + \kappa^2$:

\[Q_a^2 = \frac{R_s}{2a} + \frac{R_s}{a} = \frac{3R_s}{2a}\] \[\Delta\varphi = \frac{2\pi}{1-e^2} \left( \frac{3R_s}{2a} \right) = \frac{3\pi R_s}{a(1-e^2)}\]

Transformation to Periapsis Observables

To eliminate the abstract parameters $R_s, a, e$ in favor of direct observables, we map this expression to the periapsis ($p$), where interaction is maximal.

Using the identities $R_s = \kappa_p^2 r_p$ and $a(1-e^2) = r_p(1+e)$, and the closure relation $(1+e) = 1/\delta_p = 2\beta_p^2/\kappa_p^2$, we arrive at the ultimate operational reduction.

The secular evolution of an orbit is determined solely by the ratio of the gravitational redshift to the Doppler shift (red vs. blue) at the point of closest approach:

\[\boxed{\Delta\varphi = \frac{3}{2}\pi \frac{\kappa_p^4}{\beta_p^2}}\]

This equation replaces the complex dynamical derivation with a direct comparison of light interactions.

\[\boxed{\textbf{No differential equations. No metric. Pure algebra of light red vs. blue ratio}}\]

This formula reveals that relativistic precession is fundamentally a fourth-order effect in the gravitational projection ($\kappa^4$) moderated by the kinematic projection. The factor $\frac{3}{2}\pi$ is purely geometric, emerging from the $S^1\cdot S^2$ carrier structure.

Scale Invariance

The normalized form makes ROM completely scale-invariant:

\[\eta_o = \frac{r_o}{a}\]

The orbital shape in $(\eta, o)$ coordinates is identical whether describing:

Electron in hydrogen atom ($a \sim 10^{-10}$ m)
Mercury around Sun ($a \sim 10^{10}$ m)
Star around galactic center ($a \sim 10^{16}$ m)

Only the scale factor $a$ differs. All physics - eccentricity, precession, energy distribution - is the same.

This is why ROM is powerful: one framework for all scales.

Comparison with Standard Methods

Feature	Standard Orbital Mechanics	ROM
Required Inputs	Mass M, gravitational constant G, 6 Keplerian elements	2 optical measurements (κ from redshift, β from Doppler)
Computation	Differential equations + numerical integration	Pure algebra
Orbit Determination	Multiple observations + iterative fitting	2 measurements → instant closure
Precession Calculation	Geodesic equations in curved spacetime	Algebraic ratio: $\frac{3\pi}{2}\frac{\kappa_p^4}{\beta_p^2}$
Scale Applicability	Different formalisms for different scales	Universal, scale-invariant
Computational Cost	High (numerical integration)	Negligible (algebraic)
Mass Requirement	Fundamental input	Derived output: $M = \frac{c^2 r}{2G}\kappa^2$

Key Insights

“Spacetime ≡ Energy”

What we call “curved spacetime” is the algebra of energy projections
The precession formula $\Delta\phi = \frac{3\pi}{2}\frac{\kappa_p^4}{\beta_p^2}$ is pure red vs. blue ratio
No differential equations, no metric, just light measurements

“Mass is Derived”

Mass is not fundamental - it’s a bookkeeping quantity
$M = \frac{c^2 r}{2G}\kappa^2$ shows mass is just a dimensioned proxy for curvature intensity
Geometry is fundamental; mass is an artifact of the unit system

“Eccentricity ≡ Closure Defect”

Orbital shape measures deviation from perfect energy balance
$e = \frac{1}{\delta} - 1$ connects shape to red/blue ratio
Circles occur when $\kappa^2 = 2\beta^2$ (perfect closure)

Key Advantages of ROM

1. Pure Optical Input

Standard Orbital Mechanics:

Requires knowing mass $M$ of the central body
Needs gravitational constant $G$
Requires 6 Keplerian elements
Computationally intensive numerical integration

ROM:

Only needs light measurements: redshift ($z$) and Doppler shift ($v/c$)
2 parameters determine the entire system through algebraic closure
Mass is derived, not required: $M = \frac{c^2 r}{2G}\kappa^2$
Scale-invariant: same equations for atoms and galaxies
Orders of magnitude more efficient for orbital determination

2. Two Operational Pathways

Path 1 - Verification (when you have redshift data):

Measure central redshift → calculate $\kappa$
Measure orbital velocity → get $\beta$
Predict orbital shape from closure: $e = \frac{2\beta_p^2}{\kappa_p^2} - 1$
Example: Mercury-Sun system

Path 2 - Reconstruction (when redshift is unknown):

Observe orbital shape $e$ and velocity $\beta$
Reconstruct hidden potential: $\kappa_p = \beta_p\sqrt{\frac{2}{1+e}}$
Calculate system properties from reconstructed $\kappa$
Example: Star S2 orbiting Sgr A*

3. The Two-Point Measurement Method

Measure velocity ($\beta$) and position ($r = ct$) at just two points on an orbit:

\[R_s = \frac{(\beta_1^2 - \beta_2^2) r_1 r_2}{r_2 - r_1}\] \[W = \frac{1}{2}\left(\frac{R_s}{r_1} - \beta_1^2\right)\]

From these two measurements, the entire system closes algebraically. This offers significant operational advantages - standard methods require multiple orbit observations and iterative fitting.

Complete Equation System

Global System Parameters

$R_s = \kappa^2 a$ (Schwarzschild radius)

$\kappa = \sqrt{\frac{R_s}{a}} = \sqrt{4W}$ (global potential)

$\beta = \sqrt{2W}$ (global kinetic)

$a = \frac{R_s}{\kappa^2}$ (semi-major axis)

$\delta_p = \frac{\kappa_p^2}{2\beta_p^2} = \frac{1}{1+e_c}$ (closure factor)

$Q = \sqrt{\kappa^2 + \beta^2}$ (displacement)

Eccentricity Relations

$e_c = \frac{2\beta_p^2}{\kappa_p^2} - 1 = 1 - \frac{2\beta_a^2}{\kappa_a^2}$ (eccentricity from closure)

$e_{cY} = \sqrt{1-e_c^2}$ (orthogonal value)

$e_X = \frac{1+e_c}{1-e_c}$ (shape factor)

Perihelion State

$r_p = a(1-e_c)$ (radius at perihelion)

$\kappa_p = \kappa\sqrt{\frac{1}{1-e_c}}$ (potential at perihelion)

$\beta_p = \sqrt{\kappa_p^2 \cdot \frac{1+e_c}{2}}$ (kinetic at perihelion)

$Q_p = \sqrt{\kappa_p^2 + \beta_p^2}$ (displacement at perihelion)

Aphelion State

$r_a = a(1+e_c)$ (radius at aphelion)

$\kappa_a = \sqrt{2W + \beta_a^2}$ (potential at aphelion)

$\beta_a = \frac{\beta}{\sqrt{e_X}}$ (kinetic at aphelion)

$\delta_{\text{apo}} = \frac{1}{1-e_c}$ (closure at aphelion)

Phase-Dependent Quantities

At any orbital phase $o$:

$r_o = a\frac{1-e_c^2}{1+e_c\cos o}$ (radial distance)

$\eta_o = \frac{r_o}{a}$ (phase amplitude)

$t_o = \frac{r_o}{c}$ (temporal scale)

$\kappa_o = \kappa_p\sqrt{\frac{1+e_c\cos o}{1+e_c}}$ (local potential)

$\beta_o = \sqrt{\kappa_o^2 - 2W}$ (local kinetic)

$\delta_o = \frac{\kappa_o^2}{2\beta_o^2}$ (local closure)

$Q_o = \sqrt{\kappa_o^2 + \beta_o^2}$ (local displacement)

Orbital Characteristics

$\omega = \frac{\beta c}{a}$ (angular frequency)

$T = \frac{2\pi}{\omega}$ (period)

$h_W = a \beta c e_{cY}$ (angular momentum)

$\Delta\phi_{\text{WILL}} = \frac{3\pi}{2}\frac{\kappa_p^4}{\beta_p^2}$ (precession per orbit - pure redshift/Doppler ratio)

$\omega_o = \frac{\beta c}{a}\frac{(1+e_c\cos o)^2}{(1-e_c^2)^{3/2}}$ (angular speed at phase $o$)

Relational Geometry (WILL Framework)

$\theta_1 = \arccos(\beta)$ (distribution on $S^1$)

$\theta_2 = \arcsin(\kappa)$ (distribution on $S^2$)

$\beta_Y = \sqrt{1-\beta^2}$ (relativistic factor)

$\kappa_X = \sqrt{1-\kappa^2}$ (gravitational factor)

$\tau_W = \kappa_X \beta_Y$ (spacetime factor)

$z = \frac{1}{\kappa_X} - 1$ (redshift)