this is for holding javascript data
\input{preface}
\section{Introduction}
Nearly all gas in the interstellar medium is supersonically turbulent. The
properties of this turblence are essential for determining how star formation
progresses.
There are now predictive theories of star formation that include formulations
of the Initial Mass Function
\verb|\citep{Hopkins2012b,Chabrier2010a,Hennebelle2011a,Hennebelle2013a,Padoan2012a,Padoan2011b,Padoan2007a,Krumholz2005a}|.
The distribution of stellar masses depends critically on the properties of the
turbulence. It is therefore essential to measure the properties of turbulence in the
molecular clouds that produce these stars.
Recent works have used simulations to characterize the density distribution
from different driving modes of turbulence
\verb|\citep{Federrath2013a,Federrath2011a,Federrath2010a,Federrath2009a,Federrath2008a,Kritsuk2011a}|.
These works determined that there is a relation between the mode of turbulent driving and the width
of the turbulent distribution, with $\sigma_{\ln \rho} = \ln\left(1+b^2 \mathcal{M}^2 \frac{\beta}{\beta+1}\right)$,
where $\beta=2 (\mathcal{M}_A / \mathcal{M})^2 = 2 (c_s/v_A)^2$.
This equation can also be expressed in terms of the compressive mach number
$\mathcal{M}_c = b \mathcal{M}$, with $b\approx 1/3$ corresponding to
solenoidal forcing and $b = 1$ corresponding to purely compressive forcing
\verb|\citep{Federrath2010a,Konstandin2012a,Molina2012a}|.
%However, \verb|\citet{Hopkins2013a}| notes
%that the lognormal approximation of the turbulent density distribution
All of the turbulence-based theories of star formation explicitly assume a
lognormal form for the density probability distribution $P_V{\ln \rho}$ of the
gas. However, recent simulations \verb|\citep{Federrath2013a}| and theoretical work
\verb|\citep{Hopkins2013a}| have shown that the assumption of a lognormal distribution
is often very poor\footnote{The simultaneous assumption of a lognormal
mass-weighted and volume-weighted density distribution is also not
self-consistent \verb|\citep{Hopkins2013a}|. }, deviating by orders of magnitude at
the extreme of the density distributions. Since these theories all involve an
integral over the density probability distribution funcion (PDF), skew in the
lognormal distribution can drastically affect the overall star formation rate
and predicted initial mass function.
While simulations are powerful probes of wide ranges of parameter space, no
simulation is capable of including all of the physical processes and spatial
scales relevant to turbulence. Observations are required to provide additional
constraints on properties of interstellar turbulence and guide simulators
towards the most useful conditions and processes to include.
\verb|\citet{Kainulainen2013a}| and \verb|\citet{Kainulainen2012a}| provide some of the first
observational constraints on the mode of turbulent driving, finding
$b\approx0.4$, i.e. that there is a mix of solenoidal and compressive modes.
% However, these observations still attempted to characterize a lognormal
% distribution.
Formaldehyde, \formaldehyde, is a unique probe of density in molecular clouds.
Like CO, it is ubiquitous, with a nearly constant abundance wherever CO is
found \verb|\citep{Tang2013a,Mangum1993a}|. The lowest rotational transitions of
\ortho at 2 and 6 cm can be observed in absorption against the cosmic microwave
background or any bright continuum source \verb|\citep{Ginsburg2011a,Darling2012b}|.
The ratio of these lines is strongly sensitive to the local density of \hh, but
it is relatively insensitive to the local gas temperature
\verb|\citep{Wiesenfeld2013a,Troscompt2009a}|. Unlike critical density tracers, the
\formaldehyde line ratio has a direct dependence on the density that is
independent of the column density.
However, the particular property of the \formaldehyde densitometer we explore
here is its ability to trace the \emph{mass-weighted} density of the gas.
Typical density measurements from \thirteenco or dust measure the total mass
and assume a line-of-sight geometry, measuring a \emph{volume-weighted}
density, i.e. $\rho_V = M_{tot}/V_{tot}$. In contrast, the \formaldehyde
densitometer is sensitive to the density that corresponds to the most mass,
i.e. $\rho_M = \int M \rho d M / M_{tot}$. The volume- and mass- weighted
densities will vary with different drivers of turbulence, so in clouds
dominated by turbulence, if we have measurements of both, we can infer the
driver.
In \verb|\citet{Ginsburg2011a}|, we noted that the \formaldehyde densitometer revealed
volume densities much higher than expected given the cloud-average densities
from \thirteenco observations. The densities were higher even than typical
turbulence will allow. However, this argument was made on the basis of a
statistical argument; here we attempt to demonstrate that the clumps in GMCs
are of very high density in individual clouds.
\section{Observations}
We report \formaldehyde observations performed at the Arecibo Radio
Observatory\footnote{The Arecibo Observatory is part of the National Astronomy
and Ionosphere Center, which is operated by Cornell University under a
cooperative agreement with the National Science Foundation. } and the Green
Bank Telescope\footnote{ The National Radio Astronomy Observatory operates the
GBT and VLA and is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. } that will be
described in more detail in \verb|\citep{Ginsburg2011a}|, with additional data to be
published in a future work. Arecibo and the GBT have FWHM$\approx50$\arcsec
beams at the observed frequencies of 4.829 and 14.488 GHz, respectively.
Observations were carried out in a position-switched mode with 3 and 5.5\arcmin
offsets for the Arecibo and GBT observations respectively.
The Boston University / Five-College Radio Astronomy Observatory Galactic Ring
Survey \thirteenco data was also used. The BU FCRAO GRS \verb|\citep{Jackson2006a}|
is a survey of the Galactic plane in the \thirteenco\ 1-0 line with $\sim
46\arcsec$ resolution. We used reduced data cubes of the $\ell=43$ region.
\subsection{A non-star-forming molecular cloud}
% In order to detect low-column-density clouds, we must use bright background
% illumination sources at 2 and 6 cm, i.e. HII regions. There are a few dozen of
% these within the inner Galactic plane, including the sources observed in
% \verb|\citet{Ginsburg2011a}| and the majority of the bright sources in the BGPS
% \verb|\citep{Ginsburg2013b}|.
We examine the line of sight towards G43.17+0.01, also known as W49. In a
large survey, we observed two lines of sight towards W49, the second at
G43.16-0.03. Both are very bright continuum sources, and two GMCs are easily
detected in both \formaldehyde absorption and \thirteenco emission. Figure
\verb|\ref{fig:w49fullspec}| shows the spectrum dominated by W49 itself, but with
clear foreground absorption components. The continuum level subtracted from the spectra
are 73 K at 6 cm and 11 K at 2 cm for the south component, and 194 K at 6 cm
and 28 K at 2 cm for the north component.
% 2001ApJ...551..747S
\FigureTwo{figures/G43.17+0.01_H2CO_overplot_gbt9x.png}
{figures/G43.16-0.03_H2CO_overplot_gbt9x.png}
{Spectra of the \formaldehyde \oneone (black), \twotwo (red), and \thirteenco
1-0 (green) lines towards G43.17+0.01 (left) and G43.16-0.03 (right).
The \formaldehyde spectra are shown continuum-subtracted, and the \thirteenco
spectrum is offset by 1 K for clarity. The GBT \twotwo spectra are multiplied
by a factor of 9 so the smaller lines can be seen.
}{fig:w49fullspec}{1}
We focus on the ``foreground'' line at $\sim40$ \kms, since it is not
associated with the extremely massive W49 region. The cloud, known as GRSMC
43.30-0.33 \verb|\citep{Simon2001a}|, was confirmed to have no associated star
formation in that work. Additional \formaldehyde spectra of surrounding
sources that are bright at 8-1100 \um and within the \thirteenco contours of
the cloud show that they are all at the velocity of W49 and therefore are not
associated with these foreground clouds.
The \formaldehyde lines are is observed in the outskirts of the cloud, not at
the peak of the \thirteenco emission. The cloud spans $\sim0.6\degrees$, or
$\sim30$ pc at $D=2.8$ kpc \verb|\citep{Roman-Duval2009a}|. It is detected in \oneone
absorption at all 6 locations observed in \formaldehyde (Figure
\verb|\ref{fig:40kmscloud}|), but \twotwo is only detected in front of the W49 HII
region because of the higher signal-to-noise at that location. The detected
\thirteenco and \formaldehyde lines are fairly narrow, with \formaldehyde FWHM
$\sim1.3$-$2.8$ \kms and \thirteenco widths from 1.8-5.9 \kms. The \thirteenco
lines are 50\% wider than the \formaldehyde lines.
The highest \thirteenco contours are observed as a modest infrared dark cloud
in Spitzer 8 \um images, but no dust emission peaks are observed at 500 \um or
1.1 mm associated with the dark gas. This is an indication that any star
formation, if present, is weak - no massive dense clumps are present within
this cloud.
%
% Full GRSMC GLON deg GLAT deg VLSR km/s DelV km/s Rad pc Mass Msun e_ Msun nH2 cm-3 Tex K tau Sigma Msun/pc2 alpha Note RD09 _RA.icrs deg _DE.icrs deg
% 1 G043.14-00.36 043.14 -00.36 41.17 3.13 3.9 6.8e+03 2.2e+03 431.4 5.66 1.92 144.8 0.91 i RD09 287.88 +08.91
% 2 G043.04-00.11 043.04 -00.11 41.59 3.48 4.2 8.3e+03 3.2e+03 394.6 5.68 1.77 145.7 1.02 i RD09 287.61 +08.94
% 3 G043.14-00.76 043.14 -00.76 59.02 2.92 9.8 3.0e+04 1.0e+04 117.6 5.78 1.28 100.4 0.45 i RD09 288.24 +08.72
% 4 G043.49-00.71 043.49 -00.71 41.59 1.84 1.9 1.3e+03 4.6e+02 645.7 5.23 1.85 108.8 0.84 i RD09 288.35 +09.06
%
% 6.8+8.3 = 15.1 x10^3 msun
% circle is closer to 0.3 degrees, radius=14.66 pc (0.3 * 3600 * 2800 / 206265.)
% In [105]: 1.5e4 * 2e33 / (2.8*1.67e-24) / (4/3.*pi*(15*3.08e18)**3)
% Out[105]: 15.532172896708314
%
% In [106]: 1.5e4 * 2e33 / (2.8*1.67e-24) / ((2*15*3.08e18)**3)
% Out[106]: 8.132626711097554
%
% In [107]: 1.5e4 * 2e33 / (2.8*1.67e-24) / (4/3.*pi*(15*3.08e18)**3*(1*1*0.1))
% Out[107]: 155.32172896708315
%
The cloud has mass $M_{CO} = 1.5\ee{4}$ \msun in a radius $r=15$ pc, so its
mean density is $n(\hh) \approx 15$ \percc assuming spherical symmetry. If we
instead assume a cubic volume, the mean density is $n(\hh)\sim8$ \percc. For
an oblate spheroid, with minor axis $0.1\times$ the other axes, the mean
density is $n\sim150\percc$, which we regard as a conservative upper limit.
\verb|\citet{Simon2001a}| report a mass $M_{CO} = 6\ee{4} \msun$ and $r=13$ pc,
yielding a density $n(\hh)=100$ \percc, which is consistent with our estimates
but somewhat higher than measured by \verb|\citet{Roman-Duval2010a}| because of the
improved optical depth corrections in the latter work.
% It resembles, in that respect, the California molecular
% cloud. However, it is much smaller, with $M\approx8.3\ee{3}\pm3.2\ee{3} \msun$
% compared to California's $\sim10^5$.
\Figure{figures/W49_RGB_40kms_aplpy.png}
{The G43 40 \kms cloud. The background image shows Herschel SPIRE 70 \um (red),
Spitzer MIPS 24 \um (green), and Spitzer IRAC 8 \um (blue) in the background with
the \thirteenco integrated image from $v=36 \kms$ to $v=43 \kms$ at contour levels of
1, 2, and 3 K superposed in orange contours. The red and black circles
show the locations of \formaldehyde pointings, and their labels indicate the LSR velocity
of the strongest line in the spectrum. The W49 HII region is seen
behind some of the faintest \thirteenco emission that is readily associated
with this cloud. The dark swath in the 8 and 24 \um emission going through the
peak of the \thirteenco emission in the lower half of the image is likely a low
optical depth infrared dark cloud associated with this GMC.}
{fig:40kmscloud}{0.5}{0}
\section{Modeling \formaldehyde}
In order to infer densities using the \formaldehyde densitometer, we use the
low-temperature collision rates given by \verb|\citet{Troscompt2009a}| with RADEX
\verb|\citep{van-der-Tak2007a}| to build a grid of predicted line properties covering
densities from $10-10^8$ \hh \percc, temperatures from 5-50 K, column densities
$N(\ortho)$ from $10^{11}-10^{16}$ \persc, and ortho-to-para ratios from
$10^{-3}-3$.
The \formaldehyde densitometer measurements are shown in Figure \verb|\ref{fig:h2codensg43}|.
The figures show optical depth spectra, given by the equation
\begin{equation}
\tau = -\ln\left(\frac{S_\nu + 2.73}{\bar{C_\nu} + 2.73}\right)
\end{equation}
where $S_\nu$ is the spectrum (with continuum included) and $\bar{C_\nu}$ is
the measured continuum, both in Kelvins. The cosmic microwave background
temperature is added to the continuum since \formaldehyde can be seen in
absorption against it, though towards W49 it is negligible.
% G43.17
% [Param #0 DENSITY0 = 4.36419 +/- 0.0755311 Range: [1,8],
% Param #1 COLUMN0 = 12.4276 +/- 0.0417072 Range: [11,16],
% Param #2 ORTHOPARA0 = -1.25514 +/- 1.30736 Range:[-3,0.477121],
% Param #3 TEMPERATURE0 = 27.5313 +/- 18.9722 Range: [5,55],
% Param #4 CENTER0 = 39.5386 +/- 0.00108955 ,
% Param #5 WIDTH0 = 0.379159 +/- 0.000709161 Range: [0,inf)]
% stats_dict['DENSITY0']['CI'] = [9337.9885256493308, 23130.782791203092, 7697.2821104556024]
% G43.16
% [Param #0 DENSITY0 = 4.30989 +/- 0.108066 Range: [1,8],
% Param #1 COLUMN0 = 12.1953 +/- 0.0535173 Range: [11,16],
% Param #2 ORTHOPARA0 = -1.25075 +/- 1.31576 Range:[-3,0.477121],
% Param #3 TEMPERATURE0 = 28.037 +/- 19.9428 Range: [5,55],
% Param #4 CENTER0 = 40.3406 +/- 0.0102343 Range: [35,45],
% Param #5 WIDTH0 = 0.765835 +/- 0.0100109 Range: [0,inf)]
% stats_dict['DENSITY0']['CI'] = [10105.478740355829, 20412.420448321209, 11646.197755316312]
\FigureTwo{figures/G43.16-0.03_40kmscloud_MCMCfit_nolegend.png}
{figures/G43.17+0.01_40kmscloud_MCMCfit_nolegend.png}
{Optical depth spectra of the \oneone and \twotwo lines towards the two W49
lines of sight, G43.16 (left) and G43.17 (right).
The optical depth ratio falls in a regime where temperature has very little
effect and there is no degeneracy between low and high densities. }
{fig:h2codensg43}{1}
% fitted using ~/work/h2co/G43.17+0.01/fit_small_lines.py, specifically the MC40 million-long chains
% and G43.16-0.03/fit_small_lines.py
We performed line fits to both lines simultaneously using a Markov-chain
monte-carlo approach, assuming uniform priors across the modeled parameter
space and independent gaussian errors on each spectral bin. The density
measurements are very precise, with $n\approx23,000 {}^{+9300}_{-7700}$ \percc
(95\% confidence interval) and $n\approx 20,400 {}^{+12000}_{-10000}$ \percc for
G43.17+0.01 and G43.16-0.03 respectively. While this is a precise measurement
of gas density, we now need to examine exactly what gas we have measured the
density of.
%At this level of precision, the
%density measurements are dominated by systematic uncertainties in temperature and
%the ortho-to-para ratio of \hh.
%However...
% and collision rate uncertainties - which limit the accuracy to $\sim50\%$ using
% the \verb|\citet{Green1991}| rates
% \verb|\citep{Zeiger2010}|.
% The measured density is much higher than the \thirteenco-measured cloud-average
% density $n\approx 400$ \percc \citep[for cloud
% GRSMC\_G043.04-00.11;][]{Roman-Duval2010a}, with
% $n_{\formaldehyde}/n_{\thirteenco} \approx 50$. The discrepancy is worse using
% the \verb|\citet{Simon2001a}| cloud-averaged density $n\approx 100$ \percc.
% Our density measurements are about 4$\times$ higher than CO/CI LVG density
% measurements from \verb|\citet{Plume2004a}|, though those measurements rely on
% uncertain abundances and are fairly sensitive to temperature.
Since the W49 line of sight is clearly on the outskirts of the cloud, not
through its center, such a high density is unlikely to be an indication that
this line of sight corresponds to a centrally condensed density peak (e.g., a
core). The comparable density observed through two different lines of sight
separated by $\sim 2$ pc also supports this idea.
% Using
% Figure 4 of \verb|\citet{Ginsburg2011a}|, we can `turbulence-correct' the density
% measurements, but even in the most extreme case with a turbulent density
% distribution lognormal width $\sigma_s = 1.5$, the correction is only a factor
% of 2.5, reducing the discrepancy to a factor of $\sim20$.
% We should then ask, if there is gas at high density, how much is at this density?
% To address this question, we'll assume that the densities in all of the \formaldehyde
% lines of sight in the cloud are the same, and compare the \thirteenco and
% \formaldehyde derived column densities. The \oneone line robustly reflects the
% total \formaldehyde column, even though it does not constrain the density
% without a corresponding \twotwo detection.
% Comparing the integrated \formaldehyde lines to the integrated \thirteenco
% lines, the integrated \formaldehyde column densities are
% $N_{\ortho} = 2.03\ee{12} $ and $1.56\ee{12}$ \persc for G43.16
% and G43.17 respectively.
% The \thirteenco integrated spectra have brightness $T_{MB} = 2.6$ K and $1.3$ K
% for G43.16 and G43.17 respectively. Using the cloud-averaged excitation
% temperature for this cloud, $\tau_{13}=2.3$ and $0.6$ respectively, so
% \verb|\citet{Roman-Duval2010a}| equation 3 yields column densities $N_{13} = 6.2\ee{15}
% $ and $1.6\ee{15}$ \percc respectively. Assuming a \thirteenco abundance relative to \hh,
% $X_{13} = 1.8\ee{-6}$ \verb|\citep[consistent with ][]{Roman-Duval2010a}|, the
% resulting \hh column densities are 3.5\ee{21} and 9.0 \ee{20} \percc
% respectively. The abundances of \ortho relative to \thirteenco are 3.2\ee{-4}
% and 9.8\ee{-4} respectively, or relative to \hh, 5.8\ee{-10} and 1.7\ee{-9},
% which are entirely consistent with other measurements of $X_{\ortho}$.
%These
%are relatively modest column densities, with $A_V=17$ and 4.5;
%these measurements are consistent with \verb|\citet{Plume2004a}| if the different
%A_V/N(H_2) conversions are corrected.
% These measurements for a specific cloud validate the statistical argument made
% in \verb|\citet{Ginsburg2011a}|.
% However, upon closer inspection of the cloud
% morphology, the real explanation may be simple: the filling factor of gas
% within the GMC is small on large scales, not local scales. The implied volume
% filling factor from this analysis and the \verb|\citet{Ginsburg2011a}| analysis is
% $\sim10^{-2}$; the assumption of spherical symmetry is therefore extremely
% poor.
% This low filling factor has major implications for the gas: if it is in
% gravitational collapse, the free-fall times are shorter by an order of
% magnitude than usually assumed. The long lifetimes of GMCs therefore implies
% that the cloud cannot be undergoing gravitational collapse, but instead
% maintains a turbulent equilibrium. \todo{Strengthen this argument...}
%
% It also demonstrates that density-based star-formation thresholds do not
% independently predict star formation \verb|\citep{Parmentier2011a}|. Star formation
% cannot simply be driven by the free-fall time of gas, since apparently much of
% the gas above $n>10^4$ \percc is not in free-fall.
% 3c111 is in california, not 3c123
% \subsection{Comparison to 3C123 and the California Nebula}
% The radio source 3C123, an active galactic nucleus, is often used as a flux
% calibrator for radio telescopes. We used it for that purpose in our GBT
% observations, and detected the \twotwo line. \verb|\citet{Liszt1995a}| detected the
% \oneone line with the NRAO 43m telescope. The line ratio in front of 3C123 is
% approximately $\tau_{1-1}/\tau_{2-2} \approx 10$, which indicates a density
% $n\approx10^{3.6}$ \percc. This density is significantly lower than in the
% W49 40 \kms cloud, but still higher than expected in an inactive GMC
% \verb|\citep[which this is][]{Harvey2013a}|.
%
% 3c111 may have VLBA, VLA observations
% A low filling-factor may have major impact on analyses of the distribution
% functions of column density that have recently become popular
% \verb|\citep[e.g][]{Kainulainen2009}|.
\section{Turbulence and \formaldehyde}
Supersonic interstellar turbulence can be characterized by its driving mode,
Mach number $\mathcal{M}$, and magnetic field strength.
Assuming the distribution follows a lognormal distribution, defined as
\begin{equation}
\label{eqn:lognormal}
P_V(s) = \frac{1}{\sqrt{2 \pi \sigma_s^2}} \exp\left[-\frac{(s+\sigma_s^2/2)^2}{2 \sigma_s^2}\right]
\end{equation}
where the subscript $V$ indicates that this is a volumetric density
distribution function.
The width of the turbulent density distribution
is given by
\begin{equation}
\label{eqn:sigmas}
\sigma_s^2 = \ln\left(1+b^2 \mathcal{M}^2 \frac{\beta}{\beta+1}\right)
\end{equation}
where $\beta= 2 c_s^2/v_A^2 = 2 \mathcal{M}_A^2/\mathcal{M}^2$ and $b$ ranges
from 1/3 (solenoidal, divergence-free forcing) to 1 (compressive, curl-free)
forcing \verb|\citep{Federrath2010a}|. The parameter $s$ is the logarithmic
overdensity, $s\equiv\ln(\rho/\rho_0)$.
The observed \formaldehyde ratio roughly depends on the \emph{mass-weighted}
probability distribution function (as opposed to the volume-weighted
distribution function, which is typically reported in simulations). We first
examine the implications assuming a lognormal distribution for the
mass-weighted density.
% such that $p_m(s) = \rho \cdot p_s(s)$, or
% \begin{equation}
% \label{eqn:lognormal}
% p_m(s) = \frac{s}{\sqrt{2 \pi \sigma_s^2}} \exp{\left(-\frac{(\ln(\rho/\rho_0))^2}{2 \sigma_s^2}\right)}
% \end{equation}
% where we have assumed a lognormal form for $p_m(s)$.
%Other forms of the density PDF will be addressed in Section \verb|\ref{sec:simpdfs}|.
We use LVG models of the \formaldehyde lines, which are computed assuming a
fixed local density, as a starting point to model the observations of
\formaldehyde in turbulence. Starting with a fixed \emph{volume-averaged}
density $\rho_0$, we compute the observed \formaldehyde optical depth in both
the \oneone and \twotwo
line by averaging over the mass-weighted density distribution.
\begin{equation}
\label{eqn:tauintegral}
\tau(\rho_0) = \int_{-\infty}^\infty \frac{\tau_p(\rho)}{N_p} P_m(\ln \rho/\rho_0) d \ln \rho
\end{equation}
$\tau_p(\rho)/N_p$ is the optical depth \emph{per particle} at a given density, where $N_p$ is the column
density (\perkmspc) from the LVG model.
We assume a fixed abundance of \ortho relative to \hh
(i.e., the \formaldehyde perfectly traces the \hh). Figure \verb|\ref{fig:lvgsmooth}|
shows the result of this integral for an abundance of \ortho relative to \hh
$X(\ortho)=10^{-9}$, where the x-axis shows $\rho_0 = n(\hh)$ and the Y-axis
shows the observable optical depth ratio of the two \formaldehyde centimeter
lines.
%% pp removed: we don't really need to worry about the effect on the column density
%% for the theoretically computed plot since we have constraints on the column...
%which necessarily implies a higher
%column density of \ortho for the higher densities in Equation
%\verb|\ref{eqn:tauintegral}|. As long as the \formaldehyde lines are optically thin,
%this approach should yield the right \emph{ratio} of the two lines, although the
%absolute optical depths may be substantially smaller because of lower total
%\ortho columns. An example of this smoothing is shown in Figure
%\verb|\ref{fig:lvgsmooth}|.
% /Users/adam/work/h2co/pilot/plotcodes/lognormal_density_massweighted.py
% path: /Volumes/disk5/Users/adam/work/h2co/pilot/figures/models/lognormalsmooth_density_ratio_massweight_logopr0.0_abund-9.png
% cp ~/work/h2co/pilot/figures/models/lognormalsmooth_density_ratio_massweight_withhopkins_logopr0.0_abund-9.png figures/
% cp ~/work/h2co/pilot/figures/models/lognormalsmooth_density_ratio_massweight_withhopkins_logopr0.0_abund-9_withG43.png figures/
\Figure{figures/lognormalsmooth_density_ratio_massweight_withhopkins_logopr0.0_abund-9_withG43.png}
{The predicted \formaldehyde \oneone/\twotwo ratio as a function of volume-weighted mean
density for a fixed abundance relative to \hh $X(\ortho) = 10^{-9}$ and \hh
ortho/para ratio 1.0. The legend shows the effect of smoothing with different
lognormal mass distributions as described in Equation \verb|\ref{eqn:sigmas}|. % and \verb|\ref{eqn:lognormal}|.
The solid line, labeled LVG, shows the predicted ratio
with no smoothing (i.e., a $\delta$-function density distribution).
The blue errorbars show the G43.17 \formaldehyde measurement and the GSRMC
43.30-0.33 mean density.
}
{fig:lvgsmooth}{0.5}{0}
\subsection{Turbulence and GRSMC 43.30-0.33}
We use the density measurements in GSRMC 43.30-0.33 to infer properties of that
cloud's density distribution.
We measure the abundances of \ortho relative to \thirteenco,
$X(\ortho/\thirteenco) = 3.2\ee{-4}$ and 9.8\ee{-4} for G43.16 and G43.17
respectively, or relative to \hh, 5.8\ee{-10} and 1.7\ee{-9}, which are
entirely consistent with other measurements of $X_{\ortho}$ and allow us to use
Figure \verb|\ref{fig:lvgsmooth}| for this analysis. The observed formaldehyde line
ratio $\tau_{1-1}/\tau_{2-2} \sim 6$, while the volume averaged mean density of the cloud $8
\lesssim \rho_0 < 150$.
Assuming a temperature $T=10$ K, consistent with both the \formaldehyde and CO
observations \verb|\citep{Plume2004a}|, the sound speed in molecular gas is $c_s=0.19$
\kms. The observed line FWHM in G43.17 is 0.95 \kms for \formaldehyde and 1.7 \kms for
\thirteenco 1-0, so the Mach number of the turbulence is $\mathcal{M} \approx
5.1-9.1$.
If we assume the density distribution is lognormal, we can determine the values
of the `compressibility coefficient' $b$ from Equation \verb|\ref{eqn:sigmas}|.
Assuming the thermal dominates the magnetic pressure ($\beta>>1$),
the allowed values of $\sigma_s$ given the line-width based limits on
$\mathcal{M}$ range from 1.8-2.1 for $b=1$ and 1.2-1.5 for $b=1/3$. If
magnetic pressure is significant, the allowed values of $\sigma_s$ drop.
Given that the observed mean cloud density is $n(\hh)\lesssim10^2 \percc$, Figure
\verb|\ref{fig:lvgsmooth}| shows that only the most extreme values of $\sigma_s$ can
explain the mean density. Even if the cloud is extremely oblate, e.g. with a
line-of-sight axis $0.1\times$ the plane-of-sky axes, $\sigma_s > 1.5$ is
required.
In order to achieve a self-consistent mass and volume PDF, we use the
\verb|\citet{Hopkins2013a}| distribution with the fitted relation $T = 0.25 \ln
(1+0.25 \sigma_s^4 (1+T)^{-6}$.
Using the $\sigma_s=2.5$ distribution, which is just consistent with the
observations, $T=0.29$, and based on \verb|\citet{Hopkins2013a}| Figure 3, the
compressive Mach number $\mathcal{M}_c ~ 20 T \approx 5.8$. Compared to the
mach number restrictions from the line width, this $\mathcal{M}_C$ implies a
compressive-to-total ratio $b > 0.6$.
% In [74]: hopkins_pdf.T_of_sigma(2.5)
% Out[74]: 0.2906836447265763
%
% In [75]: hopkins_pdf.T_of_sigma(2.5) * 20
% Out[75]: 5.813672894531527
%
% In [76]: hopkins_pdf.T_of_sigma(2.0)
% Out[76]: 0.22018342653110817
%
% In [77]: hopkins_pdf.T_of_sigma(2.0) * 20
% Out[77]: 4.403668530622164
The restrictions on $\sigma_s$ using either assumed density distribution are
strong indications that compressive forcing must be a significant, if not
dominant, mode in this molecular cloud.
% Since magnetic fields have the
% opposite effect of compressive turbulence on the density distribution, magnetic
% fields cannot explain the observations.
% If magnetic fields are in balance with
% or dominate thermal pressure in this cloud, $\beta\gtrsim2/3$, the forcing must
% be predominantly compressive, with $b>1/3$.
% Crutcher & others seem not to have detected Zeeman splitting in this cloud
All of the systematic uncertainties tend to require a \emph{greater} $b$
value. Temperatures in GMCs are typically 10-20 K: warmer temperatures
increase the sound speed, decrease the Mach number, and therefore decrease
$\sigma_s$. Stronger (i.e. non-negligible) magnetic fields decrease
$\sigma_s$.
\Figure{figures/lognormalsmooth_density_distributions_sigma2.0.png}
{Example volume- and mass-weighted density distributions with $\sigma_s=2.0$.
The vertical dashed lines show $\rho = 15$ and $\rho=10^4$, approximately
corresponding to the volume-averaged mean density of GRSMC 43.30 and the
\formaldehyde-derived density}
{fig:distributions}{0.5}{0}
% For G43.16-0.03:
% {'std': 0.11326554809612598, 'med': 4.3331511700465288, 'CI': [9368.5009376639355, 21535.312095032383, 11915.556844413139], 'quantiles': {2.5: 4.08517676736491, 97.5: 4.5244074037392616, 75: 4.4038773064784564, 84.134474606854297: 4.4358245264528868, 50: 4.3331511700465288, 15.865525393145708: 4.2122132860734167, 2.2750131948179195: 4.078075142949765, 25: 4.2563141956721111, 97.724986805182084: 4.5273461301744797}, 'mad': 0.10889113037076946, 'mean': 4.3251999406282646, 'logCI': [0.24797440268161886, 4.3331511700465288, 0.19125623369273281]}
%
% \subsection{Simulated PDFs}
% \label{sec:simpdfs}
% Real turbulent PDFs are not truly lognormal, though often they are
% well-approximated as lognormals. We have used some of the PDFs from
% \verb|\citet{Federrath2012a}| to perform additional smoothing and determine
% whether deviations from lognormal can explain the observed density contrasts.
%
% To perform the smoothing, we converted the simulation's volume-weighted PDF to
% a mass-weighted PDF using Equation \verb|\ref{eqn:lognormal}| and used an identical
% PDF shape for each mean density (i.e., we kept the shape of the PDF the same
% but changed its mean for use in Equation \verb|\ref{eqn:tauintegral}|). Results of this process
% are shown in Figure \verb|\ref{fig:rescalepdfs}|.
% ~/work/h2co/simulations/federrath_pdfs.py
% cp ~/work/h2co/simulations/VolumeVsMassWeighting.png figures/
%\Figure{figures/VolumeVsMassWeighting.png}
%{Mass-weighted mean density vs volume-weighted mean density for a variety of
%turbulent distributions. The black dashed line shows the $\rho_M = \rho_V$
%relation. The other lines show the relationship between $\rho_M$ and $\rho_V$
%for different lognormal widths $\sigma_s = \sigma_{\ln \rho}$ and values of $T$
%in the \verb|\citet{Hopkins2013a}| distribution. The measured densities for the cloud
%GRSMC 43.30-0.33 are shown with conservative error bars; the vertical bars show
%the 95\% confidence interval for the \formaldehyde density, while the
%horizontal bars show the range $8-150$ \percc, the full range of geometrically
%allowed volume-averaged densities. However, the \formaldehyde density measurement
%is not strictly a mass-weighted density (see Equation \verb|\ref{eqn:tauintegral}|),
%so the positions of the data in this figure are somewhat misleading.
%}{fig:volvsmass}{0.5}{0}
\section{Conclusions}
We demonstrate the use of a novel method of inferring the shape of the density
probability distribution in a molecular cloud using \formaldehyde densitometry
in conjunction with \thirteenco-based estimates of total cloud mass.
Our data show evidence for compressively driven turbulence in a
non-star-forming giant molecular cloud. Such high compression in a fairly
typical GMC indicates that compressive driving is probably a common feature of
all molecular clouds.
\input{solobib}
\end{document}