Ly$\alpha$ radiative transfer modeling for 163 MUSE Ly$\alpha$-emitting galaxies at $z=$3--6

We utilized Ly$\alpha$ radiative transfer calculations from \citet{Song2020} to investigate the properties of extended Ly$\alpha$ halos around star-forming galaxies in the \textit{Hubble} Ultra Deep Field, observed by the Multi-Unit Spectroscopic Explorer. Expanding on the work of \citet{Song2020}, which was limited to eight galaxies, we derived best-fit models for a significantly larger sample of 163 galaxies, which successfully reproduced both their Ly$\alpha$ spectra and surface brightness profiles (SBPs). These best-fit models suggest a broad medium distribution surrounding each galaxy, with low expanding velocities at large radii. This conclusion could not have been drawn from modeling either the spectrum or SBP alone, but only through simultaneous modeling of both. Our correlation analysis between observables and model parameters reveals that the spatial extent of Ly$\alpha$ halos is primarily determined by the extents of the medium and the source, while the spectral peak shift and full width at half maximum are governed mainly by optical depth, with the velocity structure of the medium playing a secondary yet non-negligible role. The fact that various correlations derived from the full set of models and those from the best-fit subset can differ significantly highlights the complex and interdependent nature of Ly$\alpha$ radiative transfer. All model parameters interact to shape the observed Ly$\alpha$ features in a non-trivial way.