Distortion of Magnetic Fields in a Starless Core IV: Magnetic Field Scaling on Density and Mass-to-flux Ratio Distribution in FeSt 1-457

In the present study, the magnetic field scaling on density, $|B| \propto \rho^{\kappa}$, was revealed in a single starless core for the first time. The $\kappa$ index of $0.78 \pm 0.10$ was obtained toward the starless dense core FeSt 1-457 based on the analysis of the radial distribution of the polarization angle dispersion of background stars measured at the near-infrared wavelengths. The result prefers $\kappa = 2/3$ for the case of isotropic contraction, and the difference of the observed value from $\kappa = 1/2$ is 2.8 sigma. The distribution of the ratio of mass to magnetic flux was evaluated. FeSt 1-457 was found to be magnetically supercritical near the center ($\lambda \approx 2$), whereas nearly critical or slightly subcritical at the core boundary ($\lambda \approx 0.98$). Ambipolar-diffusion-regulated star formation models for the case of moderate magnetic field strength may explain the physical status of FeSt 1-457. The mass-to-flux ratio distribution for typical dense cores (critical Bonnor--Ebert sphere with central $\lambda=2$ and $\kappa=1/2$--$2/3$) was calculated and found to be magnetically critical/subcritical at the core edge, which indicates that typical dense cores are embedded in and evolve from magnetically critical/subcritical diffuse surrounding medium.