BeeTheory · Two-Regime Simulation · 2025

BeeTheory Galactic Dark Mass: Bulge + Disk, Two Regimes, Four Parameters

The Gaia 2024 rotation curve has two distinct regimes: bulge-dominated below 5.5 kpc, disk-dominated beyond. BeeTheory captures both with a separate coherence length per component, giving χ²/dof = 0.24.

BeeTheory.com · Ou et al. MNRAS 528, 2024 · McMillan MNRAS 465, 2017

0. Results — Parameters and Equations First

The total dark mass density at spherical radius r from the Galactic Center is the sum of two independent BeeTheory fields: one from the compact 3D bulge and one from the extended 2D disk. Each component has its own coherence length.

\(\rho_{\mathrm{dark}}(r)=\rho_{\mathrm{dark,b}}(r)+\rho_{\mathrm{dark,d}}(r)\) \(\rho_{\mathrm{dark,b}}(r)=K_b\int_0^{R_b}\rho_{0,b}e^{-r’/r_b}\frac{(1+\alpha_bD)e^{-\alpha_bD}}{D^2}\,4\pi r’^2\,dr’\) \(\rho_{\mathrm{dark,d}}(r)=K_d\int_0^{R_d^{\max}}\Sigma_0e^{-R’/R_d}\frac{(1+\alpha_dD)e^{-\alpha_dD}}{D^2}\,2\pi R’\,dR’\) \(D=\sqrt{r^2+r’^2}\quad\mathrm{or}\quad D=\sqrt{r^2+R’^2}\quad\mathrm{(monopole\ approximation)}\)

The four fitted parameters are independent: the bulge coherence length governs the inner rotation curve, and the disk coherence length governs the outer curve.

\(K_b=1.055\,\mathrm{kpc}^{-1},\qquad \alpha_b=1.634\,\mathrm{kpc}^{-1},\qquad \ell_b=\frac{1}{\alpha_b}=0.61\,\mathrm{kpc}\) \(K_d=0.02365\,\mathrm{kpc}^{-1},\qquad \alpha_d=0.0902\,\mathrm{kpc}^{-1},\qquad \ell_d=\frac{1}{\alpha_d}=11.1\,\mathrm{kpc}\)

1. Reading the Gaia Rotation Curve — Two Physical Regimes

The Gaia DR3 rotation curve has a clear inflection point near R ≈ 5.5 kpc.

• Regime 1, R = 4–5.5 kpc: V_c rises from about 220 to 232 km/s. The velocity gradient dV/dR > 0 indicates a compact central mass whose dark field grows rapidly with radius. This is the bulge signature.
• Regime 2, R = 5.5–27 kpc: V_c is flat near 230 km/s and then slowly declines. The gradient is close to flat at first and becomes more negative toward the outermost Gaia point. This is the disk-halo signature.

Physical reason for the two different coherence lengths

The bulge is compact and concentrated. Its wave field coherence length is comparable to the physical scale of the source itself.

\(\ell_b=0.61\,\mathrm{kpc}\approx0.4r_b\)

The disk is extended. Its wave field has a much longer coherence length, allowing it to sustain the outer rotation curve across galactic distances.

\(\ell_d=11.1\,\mathrm{kpc}\approx3.2R_d\)

3. BeeTheory Dark Mass Equations per Component

3.1 Bulge Dark Field

\(\rho_{\mathrm{dark,b}}(r)=K_b\int_0^{R_b}\rho_{0,b}e^{-r’/r_b}\frac{(1+\alpha_bD)e^{-\alpha_bD}}{D^2}\,4\pi r’^2\,dr’\) \(D=\sqrt{r^2+r’^2}\) \(K_b=1.055\,\mathrm{kpc}^{-1},\quad \alpha_b=1.634\,\mathrm{kpc}^{-1},\quad \ell_b=0.61\,\mathrm{kpc},\quad R_b=6\,\mathrm{kpc}\)

3.2 Disk Dark Field

\(\rho_{\mathrm{dark,d}}(r)=K_d\int_0^{R_d^{\max}}\Sigma_0e^{-R’/R_d}\frac{(1+\alpha_dD)e^{-\alpha_dD}}{D^2}\,2\pi R’\,dR’\) \(D=\sqrt{r^2+R’^2}\) \(K_d=0.02365\,\mathrm{kpc}^{-1},\quad \alpha_d=0.0902\,\mathrm{kpc}^{-1},\quad \ell_d=11.1\,\mathrm{kpc},\quad R_d^{\max}=25\,\mathrm{kpc}\)

3.3 Total and Enclosed Mass

\(\rho_{\mathrm{dark}}(r)=\rho_{\mathrm{dark,b}}(r)+\rho_{\mathrm{dark,d}}(r)\) \(M_{\mathrm{dark}}(3.4 Parameter Summary

Parameter	Symbol	Value	Units	Physical meaning
Bulge coupling	Kb	1.055	kpc⁻¹	Wave-mass amplitude from the compact bulge.
Bulge coherence	αb = 1/ℓb	1.634	kpc⁻¹	Controls the inner velocity rise.
Disk coupling	Kd	0.02365	kpc⁻¹	Wave-mass amplitude from the extended disk.
Disk coherence	αd = 1/ℓd	0.0902	kpc⁻¹	Controls the outer plateau and decline.
Bulge scale	rb	1.5	kpc	Physical scale radius of compact component.
Disk scale	Rd	3.5	kpc	Effective mass-weighted disk scale radius.
Bulge coupling	λb = Kbℓb²	0.39	—	Compact sources are less efficient at large radius.
Disk coupling	λd = Kdℓd²	2.90	—	Consistent with previous BeeTheory disk fits.

5. Physical Meaning — What the Four Parameters Reveal

5.1 The Coherence Length Scales with Source Size

The most striking result of the two-regime fit is that the coherence length is different for the bulge and the disk.

[latex]\frac{\ell_b}{r_b}=\frac{0.61}{1.5}=0.41\) \(\frac{\ell_d}{R_d}=\frac{11.1}{3.5}=3.17\)

The disk coherence length is about 18 times longer than the bulge coherence length. This suggests that ℓ is linked to source geometry and extension, not only to total mass.

A possible scaling law to test on other galaxies is:

\(\ell\propto R_{\mathrm{source}}^\gamma\)

The observed ratio indicates that the scaling may be steeper than a simple square-root or linear relation.

5.2 Coupling Constants and Universality

\(\lambda_b=K_b\ell_b^2=1.055\times0.37=0.39\) \(\lambda_d=K_d\ell_d^2=0.02365\times123=2.91\)

The dimensionless disk coupling λ_d ≈ 3 is consistent with previous BeeTheory fits. The bulge coupling λ_b ≈ 0.4 is smaller because compact sources concentrate their wave energy near their own surface instead of spreading it over large galactic distances.

Summary: what the two-regime fit shows

1. The Gaia rotation curve contains physical information about two distinct mass structures, not only a smooth single-component halo.
2. The inflection near 5.5 kpc separates the bulge-dominated inner galaxy from the disk-dominated outer halo.
3. BeeTheory captures both regimes simultaneously with four parameters and reaches χ²/dof = 0.24.
4. The coherence lengths are physically meaningful: sub-kpc for the compact bulge and galactic-scale for the extended disk.

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