Constraining Spatial Densities of Early Ice Formation in Small Dense
  Molecular Cores from Extinction Maps
  Tracing dust in small dense molecular cores is a powerful tool to study the
conditions required for ices to form during the pre-stellar phase. To study
these environments, five molecular cores were observed: three with ongoing
low-mass star formation (B59, B335, and L483) and two starless collapsing cores
(L63 and L694-2). Deep images were taken in the infrared JHK bands with the
United Kingdom Infrared Telescope (UKIRT) WFCAM (Wide Field Camera) instrument
and IRAC channels 1 and 2 on the Spitzer Space Telescope. These five
photometric bands were used to calculate extinction along the line of sight
toward background stars. After smoothing the data, we produced high spatial
resolution extinction maps ($\sim$13-29") . The maps were then projected into
the third dimension using the AVIATOR algorithm implementing the inverse Abel
transform. The volume densities of the total hydrogen were measured along lines
of sight where ices (H$_2$O, CO, and CH$_3$OH) have previously been detected.
We find that lines of sight with pure CH$_3$OH or a mixture of CH$_3$OH with CO
have maximum volume densities above 1.0$\times$10$^5$ cm$^{-3}$. These
densities are only reached within a small fraction of each of the cores
($\sim$0.3-2.1%). CH$_3$OH presence may indicate the onset of complex organic
molecule formation within dense cores and thus we can constrain the region
where this onset can begin. The maximum volume densities toward star-forming
cores in our sample ($\sim$1.2-1.7$\times$10$^6$ cm$^{-3}$) are higher than
those toward starless cores ($\sim$3.5-9.5$\times$10$^5$ cm$^{-3}$).
