An Extended, Physically Calibrated FP for Elliptical Galaxies

TL;DR

This paper derives a virial-based extension of the Fundamental Plane for elliptical galaxies by explicitly incorporating structural non-homology via the Sérsic index $n$, stellar population effects through the mass-to-light ratio $\Upsilon_*$, and central dark matter fractions $f_{\rm DM}$. The key innovation is the introduction of a Sérsic-dependent virial coefficient $k(n)$ and a structure-informed potential energy treatment, which, together with anisotropy considerations, yields an extended FP that more accurately predicts galaxy sizes with reduced scatter compared to the classical FP. Robust statistical tests (bootstrap, VIF, RANSAC, ANOVA) show the extended FP significantly improves fit quality and stabilizes residuals, with coefficients that carry clear physical interpretations: $\log\sigma$ strengthens toward virial expectations, while $\log I_e$, $\log(M/L)$, $\log(1-f_{\rm DM})$, and $\log k(n)$ quantify the contributions of photometric concentration, stellar populations, dark matter, and structure to the FP tilt. The framework reconciles theoretical virial predictions with observed scaling relations, enabling a physically motivated decomposition of FP tilt and scatter, and provides a unified approach applicable to a wide range of ellipticals and environments. It also offers a path to test evolutionary scenarios by extending the calibration to higher redshift with future surveys and integral-field data.

Abstract

We present a physically motivated extension of the FP for elliptical galaxies, derived from the scalar virial theorem and calibrated using observational data. Starting from the basic equilibrium condition, we incorporate key physical parameters that govern galaxy structure and dynamics, namely stellar mass-to-light ratio, central dark matter fraction, and structural non-homology as traced by the Sersic profile. The resulting model retains the original dependencies on velocity dispersion and surface brightness, but introduces physically interpretable corrections that significantly improve the fit to real data. Using a large galaxy sample, we demonstrate that this extended FP achieves a higher level of accuracy than the classical form, with all parameters showing strong statistical significance. Our results indicate that the observed FP can be understood as an empirical refinement of the virial prediction, once variations in stellar populations, dark matter content, and internal structure are taken into account. This work provides a unified framework that bridges theoretical expectations with observed scaling relations in elliptical systems.

Figures (14)

• Figure 1: Comparison between the exact structural coefficient $k_{\rm exact}(n)$ and $k_{\rm approx}(n) = \tfrac{1}{2} A(n)$ as a function of the Sérsic index $n$
• Figure 2: Numerical values of $\eta(n)$ compared with the rational fitting formula
• Figure 3: Comparison of observed effective radii (log $R_{e}$) against predicted values from the extended Fundamental Plane (FP) model
• Figure 4: Comparison of observed log $R_{e}$ against predicted values from the classical Fundamental Plane (FP) model
• Figure 5: Distribution of residuals for the extended FP model