Rotation curves: exponential disk + screened medium (Toy v4)

This is a short numerical note from UCM-Engine. The goal is not data fitting, but a controlled mechanism test: how the rotation-curve tail changes when a single medium parameter (screening length L) is varied while the baryonic geometry is kept realistic (an exponential disk with scale R_d).

We solve a screened Poisson equation for an effective potential Φ(r) in a 1D radial toy setup:

(1/r) d/dr ( r dΦ/dr ) − Φ/L² = S · ρ(r)

Acceleration is g(r) = −dΦ/dr, and the circular speed is defined as v(r) = sqrt(r · |g(r)|).

The key change compared to earlier “spherical source” tests is the source geometry: an effective exponential disk profile ρ(r) = ρ0 exp(−r/Rd), which gives a finite mass and a meaningful outer tail test.

We quantify the outer tail by the logarithmic slope s(r) = d ln v / d ln r, averaged over r ∈ [0.6 Rmax, 0.9 Rmax]. Interpretation:

• slope_mean ≈ 0 → flat tail (galactic-like)
• slope_mean ≈ −0.5 → Newtonian decline
• slope_mean ≪ −0.5 → rapidly declining curve

The transition is monotonic with L/Rd: for small L the curve declines fast; for L ≫ Rd the tail approaches flat.

This is achieved without dark matter and without MOND interpolation functions, with a single medium length scale L. See the full parameter set and diagnostics in the downloadable summary.

This is: a controlled numerical mechanism demonstration and a clean separation of geometry vs medium effects.

This is not: a galaxy-by-galaxy data fit, not a full 2D disk calculation, and not a claim of “solving dark matter”.

The natural next steps are a finer scan in L/Rd, then an axisymmetric 2D disk model, and comparisons to empirical relations (e.g., RAR/MOND-like phenomenology).

Download ucm_rotation_curve_1d_v4.py and run with standard Python.

python ucm_rotation_curve_1d_v4.py

The exact parameter set and the diagnostic table are in Results summary (txt).