BeeTheory · Galactic Simulation v2 · initial generation 2025 may 17 with claude

Milky Way Hidden Mass: BeeTheory 3D Yukawa with Physical Disk Truncation

The corrected simulation: baryonic disk velocity falls Keplerian beyond its physical edge, and the BeeTheory 3D Yukawa kernel fills all of space. Two parameters, Gaia-era rotation data, and a truncated disk model.

BeeTheory.com · Ou et al., MNRAS 528, 2024 · Corrected BeeTheory v2

0. Result — Equations and Parameters

Each annular ring of the galactic disk at radius R′ generates a 3D effective dark mass field through the BeeTheory Yukawa kernel. The total dark density at spherical radius r is:

\(\rho_{\mathrm{dark}}(r)=K\int_0^{R_{\mathrm{max}}}\Sigma_0e^{-R’/R_d}\frac{(1+\alpha D)e^{-\alpha D}}{D^2}\,2\pi R’\,dR’\) \(D(r,R’)=\sqrt{r^2+R’^2}\)

The kernel is derived from the corrected BeeTheory force law:

\(F(D)\propto\frac{(1+\alpha D)e^{-\alpha D}}{D^2}\)

It reduces to the Newtonian inverse-square form for D much smaller than the coherence length ℓ.

\(D\ll\ell=\frac{1}{\alpha}\quad\Longrightarrow\quad F(D)\propto\frac{1}{D^2}\)

The baryonic disk velocity uses the Freeman formula inside its physical edge R_trunc ≈ 4R_d = 10.4 kpc, then transitions smoothly to the Keplerian fall expected from a finite mass distribution.

\(K=0.0397\,\mathrm{kpc}^{-1},\qquad \alpha=0.0868\,\mathrm{kpc}^{-1},\qquad \ell=\frac{1}{\alpha}=11.5\,\mathrm{kpc}\)

1. The Disk Truncation — Why and How

1.1 The Problem with an Infinite Exponential Disk

The Freeman disk formula assumes an exponential surface density extending to infinity. Mathematically this never reaches zero, but physically the stellar disk of the Milky Way has a finite extent. Beyond the effective stellar edge, the enclosed baryonic mass is essentially constant, and the velocity contribution must fall approximately as a Keplerian point-mass field.

\(\Sigma(R)=\Sigma_0e^{-R/R_d}\)

Beyond the disk edge, the baryonic velocity tends toward:

\(V_{\mathrm{bar}}(R)\xrightarrow{R\gg R_d}\sqrt{\frac{GM_{\mathrm{bar,tot}}}{R}}\) \(M_{\mathrm{bar,tot}}=M_{\mathrm{disk}}+M_{\mathrm{bulge}}\approx4.7\times10^{10}M_\odot\)

Example values are:

\(V_{\mathrm{bar}}(30\,\mathrm{kpc})\approx82\,\mathrm{km/s},\qquad V_{\mathrm{bar}}(50\,\mathrm{kpc})\approx63\,\mathrm{km/s}\)

1.2 Smooth Truncation Formula

The simulation uses a smooth transition between the Freeman disk formula and the Keplerian value. The transition is centered at R_trunc = 4R_d = 10.4 kpc with width σ = 1.5 kpc.

\(V_{\mathrm{bar}}(R)=\sqrt{(1-w)V_{\mathrm{Freeman}}^2(R)+w\,\min(V_{\mathrm{Freeman}},V_{\mathrm{Kepler}})^2}\) \(w(R)=\frac{1}{2}\left[1+\tanh\left(\frac{R-R_{\mathrm{trunc}}}{\sigma}\right)\right]\) \(R_{\mathrm{trunc}}=4R_d=10.4\,\mathrm{kpc},\qquad \sigma=1.5\,\mathrm{kpc}\)

The minimum function prevents the baryonic disk from exceeding the physical Keplerian limit outside the disk edge.

R	VFreeman	VKeplerian	Vbar,truncated	Dominant regime
5 kpc	174.5 km/s	201.1 km/s	174.5 km/s	Freeman
8 kpc	161.5 km/s	159.0 km/s	161.5 km/s	Freeman ≈ Kepler
10.4 kpc	143.0 km/s	139.3 km/s	141.2 km/s	Transition
16 kpc	112.4 km/s	112.4 km/s	112.4 km/s	Keplerian
25 kpc	89.9 km/s	89.9 km/s	89.9 km/s	Keplerian
50 kpc	63.6 km/s	63.6 km/s	63.6 km/s	Keplerian