Planet Sizes in the DRUMS Framework

1. Planetary Formation as Superfluid Condensation

In DRUMS, planets form as localized condensations of baryonic matter in the superfluid medium, guided by the cubic magnetic substrate:

\[ M_p = \int_{V_p} \rho_b(\mathbf{x},t) \, d^3x \]

The volume \(V_p\) is set by equilibrium between superfluid pressure and gravitational attraction.

2. Equilibrium Radius Determination

The planet radius \(R_p\) emerges from balancing self-gravity with superfluid-mediated pressure:

\[ \frac{G M_p^2}{R_p^2} \sim P_{sf} \, 4\pi R_p^2 \]

Where \(P_{sf}\) is the effective pressure from superfluid density gradients.

3. Substrate Alignment and Quantization

The cubic substrate imposes preferred radii due to phase quantization:

\[ R_p = n a \left( \frac{\rho_0}{\rho_b} \right)^{1/3}, \quad n \in \mathbb{Z} \]

Where \(a\) is the lattice constant of the substrate, \(\rho_0\) is the superfluid density, and \(\rho_b\) is the local baryon density.

4. Mass-Radius Relationship

Given the superfluid substrate effects, planet masses and radii satisfy:

\[ R_p \sim \left( \frac{3 M_p}{4 \pi \rho_b} \right)^{1/3} f(n) \]

Where \(f(n)\) accounts for substrate quantization, producing discrete preferred sizes consistent with observed planetary radii distribution.

5. Energy Considerations

Minimum energy configuration occurs when baryons condense to form stable planetary radii:

\[ E_{total} = E_{grav} + E_{sf} = - \frac{3 G M_p^2}{5 R_p} + \int_{V_p} P_{sf} \, dV \]

Stability arises when \(\partial E_{total}/\partial R_p = 0\).

6. Final Interpretation

Within the DRUMS framework, planet sizes are fully explained as:

  • Equilibrium volumes of baryonic condensates in a superfluid medium
  • Radius determined by balance of gravity and superfluid pressure
  • Discrete preferred radii imposed by cubic magnetic substrate and phase quantization
  • Energy minimization ensures stability and reproduces observed planetary size distributions