Galaxy Sizes in the DRUMS Framework

1. Superfluid Halo Concept

In DRUMS, galaxies are embedded within a coherent superfluid medium. The effective galaxy size is determined by the spatial extent over which the baryonic matter remains coupled to the superfluid flow:

\[ \Psi(\mathbf{x},t) = \sqrt{\rho(\mathbf{x},t)} e^{i\theta(\mathbf{x},t)} \]

Phase coherence defines the radius \(R_g\) where matter and field remain dynamically coupled.

2. Effective Radius from Phase Coupling

The superfluid exerts an acceleration on baryonic matter:

\[ a_s(r) = \frac{\hbar}{m} \frac{d}{dr} |\nabla \theta(r)| \]

Define galaxy edge where \(a_s(r)\) falls below threshold \(a_{min}\):

\[ R_g : a_s(R_g) = a_{min} \]

3. Mass-Size Relation

Integrated baryonic mass within the radius:

\[ M_b(R_g) = 4\pi \int_0^{R_g} \rho_b(r) r^2 dr \]

Superfluid coupling implies larger mass leads to larger \(R_g\) due to extended coherent flow.

4. Energy Equilibrium Constraint

Galactic size is limited by energy balance between gravitational potential and superfluid coupling:

\[ \frac{G M_b(R_g) m_*}{R_g} \sim m_* a_s(R_g) R_g \]

Solving for \(R_g\) gives:

\[ R_g \sim \sqrt{\frac{G M_b(R_g)}{a_s(R_g)}} \]

5. Vortex and Coherence Limit

Quantized vortices limit the maximum coherent region size:

\[ \oint \mathbf{v}_s \cdot d\mathbf{l} = n \frac{h}{m} \Rightarrow n_{max} \sim \frac{2 \pi R_g v_s}{h/m} \]

Beyond this scale, phase coherence is lost and baryonic matter decouples.

6. Predictive Scaling

DRUMS predicts a scaling between galaxy mass and size consistent with observations:

\[ R_g \propto M_b^{1/2} / a_0^{1/2} \]

Where \(a_0\) is characteristic superfluid acceleration, naturally producing the observed mass-size relation.

7. Final Interpretation

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

  • Set by the spatial extent of superfluid phase coherence
  • Determined by threshold acceleration where baryonic matter decouples
  • Stabilized by vortex quantization and superfluid angular momentum
  • Scaling naturally with baryonic mass and superfluid acceleration, reproducing observed galaxy size relations