Deciphering the Lyman-$\alpha$ Emission Line: Towards the Understanding
  of Galactic Properties Extracted from Ly$\alpha$ Spectra via Radiative
  Transfer Modeling
  Existing ubiquitously in the Universe with the highest luminosity, the
Lyman-$\alpha$ emission line encodes abundant physical information about the
gaseous medium it interacts with. Nevertheless, the resonant nature of
Ly$\alpha$ complicates the radiative transfer (RT) modeling of the line
profile, making the extraction of physical properties of the surrounding
gaseous medium notoriously difficult. In this paper, we revisit the problem of
deciphering the Ly$\alpha$ emission line with RT modeling. We reveal intrinsic
parameter degeneracies in the widely-used shell model in the optically thick
regime for both static and outflowing cases, which suggest the limitations of
the model. We have also explored the connection between the more physically
realistic multiphase, clumpy model and the shell model. We find that the
parameters of a ``very clumpy'' slab model and the shell model have the
following correspondences: (1) the total column density of the clumpy slab
model is equal to the HI column density of the shell model; (2) the effective
temperature of the clumpy slab model, which incorporates the clump velocity
dispersion, is equal to the effective temperature of the shell model; (3) the
average radial clump outflow velocity is equal to the shell expansion velocity;
(4) large intrinsic line widths are required in the shell model to reproduce
the wings of the clumpy slab models; (5) adding another phase of hot
inter-clump medium will increase peak separation, and the fitted shell
expansion velocity lies between the outflow velocities of two phases of gas.
Our results provide a viable solution to the major discrepancies associated
with Ly$\alpha$ fitting reported in previous literature, and emphasize the
importance of utilizing information from additional observations to break the
intrinsic degeneracies as well as interpreting the model parameters in a more
physically realistic context.
