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in the area around the dust peak.}  {fig:s233irmulti}{0.2}{0}  \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{figures_chH2CO/a2705.20120915.b0s1g0.00000_offspectra.png}  {An example of the \formaldehyde line masking procedure for building an  Off spectrum}{fig:h2comask}{1}{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.  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 only weakly  detected at \twotwo, with $\tau_{\oneone}/\tau_{\twotwo} \approx 2$, curiously  implying a fairly high density $n\sim10^{4.5}-10^5$ with a very low column  $N_{\formaldehyde}\sim10^{11.5}$ or $N_{\hh} \sim 10^{20.5}$. This measurement  is consistent with observations from \citet{Ginsburg2011} of high density in GMCs.  However, GMCs are generally thought of as being low-density clouds, so this  result may be surprising.  \input{solobib}  \end{document}  diff --git a/ch_w5.tex b/ch_w5.tex  index 8b7bdcb..c9f047f 100644  --- a/ch_w5.tex  +++ b/ch_w5.tex 
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this is because W5 is so faint at millimeter wavelengths compared to many  Galactic Plane sources.  This work is essentially a very detailed study of a star-forming region with  minimal implications for star forming theories at the moment.  \section{Introduction} 
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\usepackage{lscape}  \usepackage{grffile}  \usepackage{standalone}  %\standalonetrue \standalonetrue  \usepackage{import}  \usepackage[utf8]{inputenc}  diff --git a/w5_appendix.tex b/w5_appendix.tex  new file mode 100644  index 0000000..5003c6b  --- /dev/null  +++ b/w5_appendix.tex 
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\appendix  \onecolumn  \section{Optically Thin, LTE dipole molecule}  \label{appendix:dipole}  While many authors have solved the problem of converting CO 1-0 beam  temperatures to \hh\ column densities  \citep{garden1991,bourke1997,Cabrit1990,lada1996}, there are no examples in  the literature of a full derivation of the LTE, optically thin CO-to-\hh\  conversion process for higher rotational states. We present the full  derivation here, and quantify the systematic errors generated by various  assumptions.  We begin with the assumption of an optically thin cloud such that the radiative  transfer equation \citep[][eqn 1.9]{rohlfs} simplifies to  \begin{equation}  \label{eqn:radtrans}  \frac{dI_\nu}{\ds} = -\kappa_\nu I_\nu   \end{equation}  The absorption and stimulated emission terms yield   \begin{equation}  \label{eqn:kappa}  \kappa_\nu = \frac{h \nu_{ul} B_{ul} n_u}{c} \varphi(\nu)  -\frac{h \nu_{ul} B_{lu} n_l}{c} \varphi(\nu)  \end{equation}  where $\varphi(\nu)$ is the line shape function ($\int\varphi(\nu) \dnu \equiv  1$), $n$ is the density in the given state, $\nu$ is the frequency of the transition,  $B$ is the Einstein B coefficient, and $h$ is Planck's constant.  By assuming LTE (the Boltzmann distribution) and using Kirchoff's Law and the definition of   the Einstein A and B values, we can derive a more useful version of this equation  \begin{equation}  \kappa_\nu = \frac{c^2}{8 \pi \nu_{ul}^2} n_u A_{ul} \left[\exp\left(\frac{h \nu_{ul} }{k_B T_{ex}}\right) - 1 \right] \varphi(\nu)  \end{equation}  where $k_B$ is Boltzmann's constant.  The observable $T_B$ can be related to the optical depth, which is given by   \begin{equation}  \int \tau_\nu \dnu = \frac{c^2}{8 \pi \nu_{ul}^2} A_{ul} \left[\exp\left(\frac{h \nu_{ul} }{k_B T_{ex}}\right) - 1 \right] \int \varphi(\nu) \dnu \int n_u \ds   \end{equation}  Rearranging and converting from density to column ($\int n \ds = N$) gives an equation for the column density  of the molecule in the upper energy state of the transition:  \begin{equation}  \label{eqn:nuppertau}  N_u = \frac{8\pi \nu_{ul}^2}{c^2 A_{ul}} \left[\exp\left(\frac{h \nu_{ul} }{k_B T_{ex}}\right) - 1 \right]^{-1} \int \tau_\nu \dnu  \end{equation}  In order to relate the brightness temperature to the optical depth, at CO transition frequencies the full blackbody  formula must be used and the CMB must also be taken into account. \citet{rohlfs} equation 15.29   \begin{equation}   \label{eqn:tbrightnesscmb}  T_B(\nu) = \frac{h \nu}{k_B} \left(\left[e^{h \nu / k_B T_{ex}} - 1\right]^{-1} - \left[e^{h \nu / k_B T_{CMB}} - 1\right]^{-1} \right) (1-e^{-\tau_\nu})  \end{equation}  is rearranged to solve for $\tau_\nu$:  \begin{equation}  \label{eqn:tau}  \tau_\nu = -\ln\left[ 1 - \frac{k_B T_B}{h \nu} \left(\left[e^{h \nu / k_B T_{ex}} - 1\right]^{-1} - \left[e^{h \nu / k_B T_{CMB}} - 1\right]^{-1} \right)^{-1} \right]  \end{equation}  We convert from frequency to velocity units with $\dnu = \nu/c \dv$, and plug \eqref{eqn:tau} into \eqref{eqn:nuppertau} to get  \begin{equation}  \label{eqn:nuppernoapprox}  N_u = \frac{8\pi \nu_{ul}^3}{c^3 A_{ul}} \left[\exp\left(\frac{h \nu_{ul} }{k_B T_{ex}}\right) - 1 \right]^{-1} \int -\ln\left[ 1 - \frac{k_B T_B}{h \nu_{ul}} \left(\left[e^{h \nu_{ul} / k_B T_{ex}} - 1\right]^{-1} - \left[e^{h \nu_{ul} / k_B T_{CMB}} - 1\right]^{-1} \right)^{-1} \right] \dv  \end{equation}  which is the full LTE upper-level column density with no approximations applied.  The first term of the Taylor expansion is appropriate for $\tau<<1$ ($\ln[1+x]\approx x-\frac{x^2}{2}+\frac{x^3}{3}\ldots$)  \begin{equation}  N_u = \frac{8\pi \nu_{ul}^3}{c^3 A_{ul}} \left[\exp\left(\frac{h \nu_{ul} }{k_B T_{ex}}\right) - 1 \right]^{-1} \int \frac{k_B T_B}{h \nu_{ul}} \left(\left[e^{h \nu_{ul} / k_B T_{ex}} - 1\right]^{-1} - \left[e^{h \nu_{ul} / k_B T_{CMB}} - 1\right]^{-1} \right)^{-1} \dv  \end{equation}  which simplifies to  \begin{equation}  \label{eqn:nupper}  N_u = \frac{8\pi \nu_{ul}^2 k_B}{c^3 A_{ul} h } \frac{e^{h\nu_{ul}/k_B T_{CMB}} - 1}{e^{h\nu_{ul}/k_B T_{CMB}} - e^{h\nu_{ul}/k_B T_{ex}}} \int T_B \dv  \end{equation}  This can be converted to use $\mu_e$ \citep[0.1222 for  \twelveco; ][]{Muenter1975}, the electric dipole moment of the molecule, instead  of $A_{ul}$, using \citet{rohlfs} equation 15.20 $\left((A_{ul}=(64\pi^4)/(3 h  c^3)\right)\nu^3 \mu_{e}^2$):  \begin{equation}  \label{eqn:nuppermuju}  N_u = \frac{3 }{8 \pi^3 \mu_e^2 } \frac{k_B}{\nu_{ul}} \frac{2 J_u + 1}{J_u}   \frac{e^{h\nu_{ul}/k_B T_{cmb}} - 1}{e^{h\nu_{ul}/k_B T_{CMB}} - e^{h\nu_{ul}/k_B T_{ex}}} \int T_B \dv  \end{equation}  The total column can be derived from the column in the upper state using the partition  function and the Boltzmann distribution  \begin{equation}  n_{tot} = \sum_{J=0}^\infty n_J = n_0 \sum_{J=0}^\infty (2J+1) \exp\left(-\frac{J(J+1) B_e h}{k_B T_{ex}}\right) \label{eqn:approxpartition}\\  \end{equation}  This equation is frequently approximated using an integral  \citep[e.g.][]{Cabrit1990}, but a more accurate numerical solution using up to  thousands of rotational states is easily computed  \begin{equation}  n_J = \left[ \sum_{j=0}^{j=j_{max}} (2j+1) \exp\left(-\frac{j(j+1) B_e h}{k_B T_{ex}}\right) \right]^{-1} (2J+1) \exp\left(-\frac{J(J+1) B_e h}{k_B T_{ex}}\right)  \end{equation}  The effects of using the approximation and the full numerical solution are shown in figure \ref{fig:approx}.  %We note that there are a number ($>1$) of different values of $\mu_e$ frequently reported in the literature.  %\citet{Burrus1958} reports a Stark-effect measurement of $\mu_e = 0.112\pm0.005$ Debye. \citet{Muenter1975}  %report an improvement on this measurement, yielding $\mu_e = 0.1222$. More recently, \citet{Goorvitch1994}  %report a value for the rotationless dipole momenut $\mu_0 = 0.1101$, which is negligibly different from the   %\citet{Muenter1975} value...  \Figure{columnconversion_vs_tex_allapprox}  {The LTE, optically thin conversion factor from $T_B$ (K \kms) to N(\hh)  (\persc) assuming X$_{\twelveco}=10^{-4}$ plotted against $T_{ex}$. The  dashed line shows the effect of using the integral approximation of the   partition function \citep[e.g.][]{Cabrit1990}. It is a better  approximation away from the critical point, and is a better approximation  for higher transitions. The dotted line shows the effects of removing the   CMB term from \eqref{eqn:tbrightnesscmb}; the CMB populates the lowest two  excited states, but contributes nearly nothing to the $J=3$ state. Top (blue):  J=1-0, Middle (green): J=2-1, Bottom (red): J=3-2.}  {fig:approx}{1.0}  The CO 3-2 transition is also less likely to be in LTE than the 1-0 transition.  The critical density ($n_{cr}\equiv A_{ul}/C_{ul}$) of \twelveco\ 3-2 is 27  times higher than that for 1-0. We have run RADEX \citep{VanDerTak2007} LVG  models of CO to examine the impact of sub-thermal excitation on column  derivation. The results of the RADEX models are shown in Figure  \ref{fig:coradex}. They illustrate that, while it is quite safe to assume the  CO 1-0 transition is in LTE in most circumstances, a similar assumption is  probably invalid for the CO 3-2 transition in typical molecular cloud  environments.  \Figure{figures/CO_excitation}  {{\it Top}: The derived N(\hh) as a function of $n_{\hh}$ for $T_{B}=1$ K.  The dashed lines represent the LTE-derived $N(\hh)/T_B$ factor, which has   no density dependence and, for CO 3-2, only a weak dependence on temperature.  We assume an abundance of \twelveco\ relative to \hh\ $X_{CO} = 10^{-4}$.  {\it Bottom}: The correction factor (N(\hh)$_{RADEX}$ / N(\hh)$_{LTE}$) as  a function of $n_{\hh}$.  For $T_K=20$ K, the ``correction factor'' at $10^3$ \percc\ (typical GMC  mean volume densities) is $\sim15$, while at $10^4$ \percc\ (closer to $n_{crit}$ but  perhaps substantially higher than GMC densities) it becomes negligible. The  correction factor is also systematically lower for a higher gas kinetic  temperature.  For some densities, the ``correction factor'' dips below 1, particularly for CO  1-0. This effect is from a slight population inversion due to fast spontaneous  decay rates from the higher levels and has been noted before  \citep[e.g.][]{Goldsmith1972}.  }{fig:coradex}{1.0}  %\bibliography{column_derivation}