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\input{preface}
\section{\formaldehyde Mapping}
We were lucky enough to be awarded \emph{double} the time we asked for on the GBT,
allowing us to observe large areas in mapping mode.
Naturally, we picked the brightest and best-known regions for mapping studies.
In the inner galaxy, we mapped out an area $\sim50\arcmin\times20\arcmin$ centered
on the W51 massive star forming complex. This region is ideally suited to study from Arecibo,
the GBT, and the VLA, since it is at declination +14 and is one of the brightest continuum
sources in the Galactic plane. It also turns out to be the nearest proto-massive cluster
at a VLBI-parallax-measured distance of 5.1 kpc (see Chapter \verb|\ref{ch:ympc}| for a discussion
of massive proto-clusters). The simple reduction of this data is nearly complete, but
analysis has only begun.
In the outer galaxy, we targeted two regions: the Sh 255 complex in Gem OB1 and
the Sh 233-IR/IRAS 05358 complex I studied for my Comps II project. We made
small ($\sim 5\arcmin\times 5\arcmin$) maps of these objects in order to
evaluate the density profiles and determine what systematic biases may have
been present in our single-pointing observations. These outer galaxy sources
are both at $D<2$ kpc, so our resolution is $\lesssim 0.5 $pc and we therefore
have some marginal hope of discovering dense cores without diluting their
signal too badly.
\subsection{\formaldehyde maps of S233IR}
For the S233IR region, we were able to create a density map, but found the
surprising and counterintuitive result that the density was smallest at the
peak of the BGPS 1.1 mm emission. The ``envelope'' is at a nearly constant density
$n\sim10^{3.5}\percc$, but the core is either at a low density (which is effectively
ruled out on other considerations) or is significantly self-absorbed. It turns out
that \formaldehyde \twotwo \emph{emission} fills in the absorption.
Figure \verb|\ref{fig:s233irmulti}| shows the mapping results for the S233 IR region, where
`envelope' densities are measured to be $n\sim10^{3.3}-10^{3.7} \percc$, but the `core'
density is more weakly constrained to be $10^{4.5}\percc < n < 10^{5.5}\percc$ based
on the presence of \formaldehyde \twotwo emission and the absence of \oneone emission.
The lack of a direct measurement makes density profile measurement with the current
observations impossible.
The moderate densities observed in the envelopes are nonetheless an order of
magnitude higher than typical volume-averaged GMC densities
\verb|\citep{Roman-Duval2010}| as was previously noted for ordinary GMCs with
\formaldehyde detections in \verb|\citet{Ginsburg2011}|.
Perhaps most interesting is the contrast between the two BGPS sources shown
in Figure \verb|\ref{fig:s233irmulti}|. In \verb|\citet{Ginsburg2009}|, I examined primarily
S233IR, but its neighboring region G173.58+2.45 is also a well-studied proto-cluster.
Unlike S233IR, which has a B1/B2 10 \msun star that is probably still accreting,
G173.58 contains no massive stars and has an upper mass limit $M\lesssim4 \msun$
\verb|\citep{Shepherd2002}|. The \formaldehyde observations reveal that the density in this
clump is $\sim10^{3.6}$ \percc, substantially less than the expected $n\sim10^5$ \percc
in the massive-star forming S233IR.
The BGPS data show this difference as well, but less strikingly. In the BGPS
data, the inferred masses of S233IR and G173 are 840 and 190 \msun,
respectively, though lower by a factor of $\sim2$ in each when considering only
their condensed $r\lesssim0.4$ pc cores. Their densities differ by a smaller
amount using the BGPS data and assuming spherical symmetry, with peak densities
$n\sim10^{4.1}$ in G173 and $n\sim10^{4.8}$ in S233IR. The density difference
reinforces the conclusion drawn from the \formaldehyde data, but also show that its
density measuring power is greater, since the spherical symmetry assumption is known
to be flawed.
% S233IR:
% peak column: 6.09*2.08e22*(np.exp(13.01/20)-1) = 1.16e23
% h2co column: "%e" % (6.09*2.08e22*(np.exp(13.01/20)-1) * 1e-9) = 1.16e14
% total flux = 220 / 23.8 = 9.24
% mass is then 9.2 * 1.8**2 * 14.3 * (np.exp(13.01/20)-1) = 400 msun (600 by other measures; close enough)
% radius ~30 arsec ~ 0.3pc
% n ~ 6.7e4, 10^4.8
% TOTAL flux/mass, in full aperture, is 471/23.8=19.7 Jy, or 840 msun
%
% G173:
% flux ~62 Jy/bm, /ppbeam = 2.6 Jy ~ 2.6 * 1.8**2 * 14 = 120 msun
% TOTAL 107 Jy/beam /ppbeam=4.5 -> 190 msun
% radius = 40 arcsec = 0.35 pc
% density = 120 * 2e33 / (4/3*np.pi*(0.35*3.08e18)**3) / ( 2.8 * 1.67e-24 ) = 1.3e4
\Figure{figures/S233IR_multipanel}
{The S233IR / IRAS 05358+3543 region and its neighbor G173.58+2.45.
\textit{Top left:} The \formaldehyde density map covering densities
$10^2 \percc10^5$ \percc.
\textit{Top center: } The \formaldehyde \oneone absorption map.
\textit{Top right: } The \formaldehyde \twotwo absorption map.
Note the lack of absorption at the contour peak: this is probably \twotwo emission
filling in \twotwo absorption, indicating a high $n\gtrsim10^5$ \percc density.
\textit{Bottom left: } CO 3-2 peak line brightness map.
\textit{Bottom center: } The BGPS v2.0 1.1 mm emission map, with contours at 0.2, 1.0, and 3.0 Jy.
These contours are shown on all of the other maps for reference.
\textit{Bottom right: } SO $5_6-4_5$ map. This line has a very high critical density $n\sim3.5\ee{6}$ \percc
and an upper level energy $T_U=35$ K.
Its morphology, with a hole at the peak of the dust emission, backs the claim that the density is highest
in the area around the dust peak.}
{fig:s233irmulti}{0.2}{0}
% tried subtracting "envelope" from "core" and still got absorption
\subsection{W51}
The W51 survey was completed in September 2011. The data reduction process
presented unique challenges: at C-band, the entire region surveyed contains
continuum emission, so no truly suitable `off' position was found within the
survey data. Similarly, \formaldehyde is ubiquitous across the region, so it
was necessary to `mask out' the absorption lines when building an off position.
This was done by interpolating across the line-containing region with a
polynomial fit (Figure \verb|\ref{fig:h2comask}|).
\Figure{figures/a2705.20120915.b0s1g0.00000_offspectra.png}
{An example of the \formaldehyde line masking procedure for building an Off
spectrum. The line-containing regions for each polarization are shown in cyan
and purple, with the interpolated replacement in red and green.
}{fig:h2comask}{0.4}{0}
The W51 data are converted into ``optical depth'' data cubes by dividing the
integrated \formaldehyde absorption signature by the measured continuum level.
These $\tau$ cubes are then fit with the RADEX models used for other
\formaldehyde fitting. However, there are multiple velocity components in W51,
so I used a two-component (unconstrained) fit for each pixel, which is
frequently unstable but in the case of W51 looks to have produced reasonable
results. Note that there was \emph{no} \formaldehyde emission detected anywhere
in the W51 region.
A first interesting note is that a local cloud at $v_{lsr}\sim5 \kms$ is
detected in \formaldehyde \oneone across most of the cloud and not detected at
\twotwo, with $\tau_{\oneone}/\tau_{\twotwo} \gtrsim 3$, implying a
very low column
$N_{\formaldehyde}\sim10^{11.5}$ or $N_{\hh} \sim 10^{20.5}$.
The cloud is seen in \thirteenco as a very weak, diffuse feature, and in HI absorption
as a very sharp, deep (self)-absorption feature.
% This density measurement
% is consistent with observations from \verb|\citet{Ginsburg2011}| of high density in
% GMCs. However, GMCs are generally thought of as being low-density clouds, so
% this result may be surprising.
%FIGURE: mcmc column vs density
I successfully made density maps of the W51 cloud, though because the velocity
structure is quite complicated, it was necessary to fit two components to most of the
map. Two-component fits are never particularly stable, so it was necessary to
restrict the parameters being fitted, and even then the results aren't
perfectly reliable. Despite those caveats, there are some reliable fits,
particularly towards the `core' of W51 Main / W51 IRS 2. There are two
high-density components with $n\sim10^5-10^{5.5}$ \percc at different velocities evident
in Figure \verb|\ref{fig:w51h2cofits}|. The southern component, centered on W51 Main,
has $v_{LSR}\sim56-59$. The northern component, a strip going through IRS 2
and towards the west, peaks around $v_{LSR}\sim68-69$ \kms. A 10 \kms difference
between two extremely dense components, both which are necessarily in the
foreground of the HII region, is shocking (probably, anyway, unless the sound
speed is very high).
\FigureTwo{figures/W51_H2CO_2parfit_v1_densityvelocity.png}
{figures/W51_H2CO_2parfit_v2_densityvelocity.png}
{Density and velocity fits to the W51 Arecibo and GBT \formaldehyde
data cubes. The yellow regions in the top panel correspond to \oneone
detections and \twotwo nondetections, indicating upper limits $n<10^{3.8}$
(68\% confidence) or $n<10^{4.3}$ (99.7\% confidence).}
{fig:w51h2cofits}{1}
There is a large area where \oneone was detected, but \twotwo was not. Our
sensitivity allows us to place a modest upper limit on the gas density, with
$3-\sigma$ upper limits $\lesssim10^{4.3}$ \percc (but the most likely
densities are $10^2 < n < 10^4$ \percc). Figure \verb|\ref{fig:w51MCMCcompare}| shows
a particular model for a spectrum that is especially unconstrained. The
\oneone/\twotwo optical depth ratio in this object is $\sim10-20$, indicating that
the volume density must be low.
\FigureTwo{figures/MCMC_DensColplot_67_64.png}{figures/spec67_64_bestfit_MCMC.png}
{Plots demonstrating upper limit fits. The left plot shows the allowed
parameter space from MCMC sampling of the data given the RADEX model. The
right plot shows the `best-fit' model to the optical depth spectra, which is
clearly unconstrained by the relatively insensitive \twotwo\ spectrum. The
sensitivity in the \oneone line is better in large part because of brighter 6
cm background across the whole W51 region. Despite the lack of constraint on the
volume density, there is a reasonably strong constraint on the column density.}
{fig:w51MCMCcompare}{1}
The molecular gas is concentrated near, but not exactly on, the bright cm
peaks. W51 IRS2 has a massive clump of gas at 65 \kms, and W51 e2 has a
similar clump. However, e2 also seems to have a very dense ($n>10^5 \percc$)
infalling clump. The spectra, along with multicomponent fits, are shown in
Figure \verb|\ref{fig:w51hiispectra}|.
\FigureTwo{figures/W51_bestfit_spec53_49_IRS2.png}{figures/W51_bestfit_spec53_49_W51e2.png}
{Plots of the optical depth spectra centered on W51 IRS2 (left) and W51 e2, an
ultracompact HII region (right). IRS2 shows high-density gas with a slight
hint of infall, but otherwise a somewhat vanilla spectrum. W51e2 has a large,
high-density red shoulder, indicating high-density gas at the most red velocity
in the system. Because this is foreground gas, that high-density gas
\emph{must} be moving towards the \uchii region.}
{fig:w51hiispectra}{1}
\input{solobib}
\end{document}