Moon Formation via DRUMS Superfluid–Substrate Dynamics
1. Failure of Impact Composition Constraint
The Moon exhibits near-identical isotopic composition to Earth's mantle:
\[
\delta^{17}O_{\oplus} \approx \delta^{17}O_{\text{Moon}}
\]
Any viable model must produce material sourced primarily from Earth's mantle, not an external body.
2. DRUMS Rotating Superfluid Earth
Model early Earth as a rotating superfluid mass coupled to a cubic substrate. The velocity field is:
\[
\mathbf{v} = \frac{\hbar}{m} \nabla \theta + \mathbf{\Omega} \times \mathbf{r}
\]
Where \(\Omega\) is planetary rotation. This produces quantized vortex structures.
3. Vortex-Induced Mass Ejection
Energy density in rotating superfluid:
\[
E = \int \left( \frac{1}{2} \rho |\mathbf{v}|^2 + \gamma |\nabla^2 h|^2 \right) dV
\]
At critical rotation, centrifugal and surface tension terms produce instability:
\[
\rho \Omega^2 r \sim \frac{\gamma}{r^3}
\]
Solving for critical radius:
\[
r_c \sim \left( \frac{\gamma}{\rho \Omega^2} \right)^{1/4}
\]
This defines the radius at which material is preferentially ejected.
4. Harmonic Orbital Stabilization
Once ejected, material settles into DRUMS harmonic orbital modes:
\[
r_n^3 = n C
\]
Where \(C\) is determined by system angular momentum:
\[
C \sim \frac{L}{\rho_{sf}}
\]
This produces discrete stable radii independent of collision dynamics.
5. Angular Momentum Conservation
Total system angular momentum:
\[
L = I_{\oplus} \Omega + m_{\text{moon}} \sqrt{G M_{\oplus} r}
\]
In DRUMS, allowed states correspond to quantized solutions of:
\[
\Delta L = n \hbar_{\text{eff}}
\]
Ensuring stable Earth–Moon configuration without fine-tuned impact parameters.
6. Composition Result
- Material originates from Earth's mantle (vortex ejection)
- No external body required
- Iron deficiency arises from density stratification during ejection
7. Testable Predictions
- Other moons should lie on similar quantized orbital radii
- Isotopic composition should match parent body mantle layers
- Angular momentum ratios follow discrete allowed states
8. Conclusion
- The Moon forms via superfluid vortex ejection from Earth
- Orbital distance arises from DRUMS harmonic quantization
- Composition naturally matches Earth's mantle
- No finely tuned impact is required