A Dynamical-Photometric Phase Space for Spiral Galaxies: Probing the Local Coupling Between Light and Gravity
The interplay between luminous matter distribution and the local gravitational field within disc galaxies encodes physical information beyond that captured by global scaling relations. We introduce a \emph{dynamical--photometric phase space} defined by the kinematic variable $X(R)=V(R)/R$ and the photometric variable $Y(R)=\mathrm{d}\ln I/\mathrm{d}\ln R$, placing the local gravitational scale and the logarithmic surface brightness gradient into direct pointwise correspondence at each galactocentric radius $R$. The quantity $X=V/R=\omega$ represents the angular frequency of circular motion and acts as a probe of the local mean mass density, while $Y$ measures the radial steepness of the stellar light distribution. The baryon-dominated inner disc is characterized by large negative $Y$, whereas the dark-matter-dominated outer region approaches $Y\rightarrow0$. This two-regime behaviour is described by the smooth sigmoid relation $Y=[a\ln X+b]/(1+\exp[k(X-X_{\rm trans})])$, which reduces to the logarithmic coupling $Y=a\ln X+b$ in the baryonic zone. We apply this framework to 136 late-type galaxies from the SPARC database, spanning inclinations $20^{\circ}$--$89^{\circ}$, distances $1$--$130$\,Mpc, and five decades in stellar mass. The median coefficient of determination is $R^{2}=0.930$. Statistical validation includes eight independent tests together with 5-fold cross-validation. The transition parameter $X_{\rm trans}$ identifies the onset of dark-matter dominance, corresponding to a median transition radius $R_{\rm trans}=5.40$\,kpc across the sample.