A Velocity Coupled Radial Acceleration Ansatz for Disk-Galaxy Rotation Curves: Fits to SPARC, Bayesian Inference, and Parameter Identifiability

TL;DR

This work proposes a minimal two-parameter velocity-coupled radial acceleration (VCA) model, a phenomenological force a_VCA(r)=γ(r)v with γ(r)=v_∞/(r+r_0), to describe disk-galaxy rotation curves. It preserves circular-motion self-consistency and yields a closed-form solution v(r)=½[A(r)+√(A(r)^2+4 v_bar^2(r))] with A(r)=v_∞ r/(r+r_0), and it is benchmarked against NFW and Burkert halos using the SPARC dataset. Across 171 galaxies, VCA is typically competitive in information criteria and holds up in radial holdout predictions, though Burkert often provides the best empirical fits; identifiability is limited for many systems, with a strong v_∞–r_0 degeneracy. The VCA framework reproduces the gross radial acceleration relation and serves as a compact descriptive model to probe baryon–kinematics couplings, offering a potential dynamical pathway for future theoretical development and observational tests beyond circular rotation alone.

Abstract

Observed rotation curves of disk galaxies remain a sharp empirical probe of the relationship between baryons and dynamics. We study a minimal, explicitly \emph{phenomenological} alternative to standard halo parameterizations: an additional inward \emph{radial} acceleration proportional to the local \emph{tangential} speed, $a_{\vca}(r)=γ(r)\,v(r)$, with a saturating coupling $γ(r)=\vinf/(r+\rzero)$. Combining this ansatz with the circular-motion condition yields a quadratic equation for $v(r)$ with a closed-form physical branch.We fit this ``velocity-coupled acceleration'' (\vca) model to $N_\mathrm{gal}=171$ rotation curves from the SPARC sample using the published baryonic decompositions (gas, disk, bulge), and we compare to two commonly used two-parameter halo models (NFW and Burkert) using an identical optimization pipeline and error model.For a fiducial systematic error floor $σ_0=5~\mathrm{km\,s^{-1}}$, the \vca model is typically competitive with an NFW halo and performs comparably (though not uniformly better) than a Burkert halo in information-criterion comparisons.We further perform MCMC inference for \vca parameters, quantify posterior predictive coverage, and show that parameter posteriors exhibit a strong $\vinf$--$\rzero$ degeneracy for many galaxies; only $47/171$ galaxies yield well-identified posteriors under simple width-based criteria. We also perform a simple radial holdout cross-validation (outer 30\% of radii) and find predictive RMSE comparable to NFW and Burkert under this protocol.Finally..

Figures (15)

• Figure 1: Representative SPARC rotation curves with best-fit models and residuals (in units of $\sigma_\mathrm{eff}$; Eq. \ref{['eq:sigmaeff']}). Shown are a high-quality extended spiral (NGC3198), an intermediate case (NGC2403), and a dwarf (DDO154). The vca VCA model (red) is competitive with NFW (blue) and Burkert (green) in many cases, though performance varies galaxy-to-galaxy.
• Figure 2: AIC comparison at $\sigma_0=5~\mathrm{km\,s^{-1}}$. Left: $\Delta\mathrm{AIC}=\mathrm{AIC}_\mathrm{NFW}-\mathrm{AIC}_ \mathrm \textit{VCA }$ (positive favors vca VCA ). Right: $\Delta\mathrm{AIC}=\mathrm{AIC}_\mathrm{Bur}-\mathrm{AIC}_ \mathrm \textit{VCA }$ (positive favors vca VCA ).
• Figure 3: Radial cross-validation using the outer 30% of radii as a test set. The RMSE distributions for VCA, NFW, and Burkert are broadly similar, indicating comparable out-of-sample performance for this radial holdout protocol.
• Figure 4: Model anatomy for NGC3198. The implicit effective contribution $v_{ \mathrm \textit{VCA } }(r)$ (Eq. \ref{['eq:veff']}) rises and saturates, producing a flat outer rotation curve when combined with the baryonic component.
• Figure 5: Example vca VCA posterior (NGC2403). The $\log v_\infty$--$\log r_0$ degeneracy is visible but still yields relatively tight marginal constraints.