Is Dust Cloud around $\lambda$ Orionis a Ring or a Shell, or Both?
  The dust cloud around $\lambda$ Orionis is observed to be circularly
symmetric with a large angular extent ($\approx$ 8 degrees). However, whether
the three-dimensional (3D) structure of the cloud is shell- or ring-like has
not yet been fully resolved. We study the 3D structure using a new approach
that combines a 3D Monte Carlo radiative transfer model for ultraviolet (UV)
scattered light and an inverse Abel transform, which gives a detailed 3D radial
density profile from a two-dimensional column density map of a spherically
symmetric cloud. By comparing the radiative transfer models for a spherical
shell cloud and that for a ring cloud, we find that only the shell model can
reproduce the radial profile of the scattered UV light, observed using the
S2/68 UV observation, suggesting a dust shell structure. However, the inverse
Abel transform applied to the column density data from the Pan-STARRS1 dust
reddening map results in negative values at a certain radius range of the
density profile, indicating the existence of additional, non-spherical clouds
near the nebular boundary. The additional cloud component is assumed to be of
toroidal ring shape; we subtracted from the column density to obtain a
positive, radial density profile using the inverse Abel transform. The
resulting density structure, composed of a toroidal ring and a spherical shell,
is also found to give a good fit to the UV scattered light profile. We
therefore conclude that the cloud around $\lambda$ Ori is composed of both ring
and shell structures.
