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\begin{document}

\title{Cataloguing Molecular Cloud Populations in Galaxy M100}


\author{Natalie Hervieux}
\affil{Affiliation not available}

  
\author{Erik Rosolowsky}
\affil{Affiliation not available}
  



\date{\today}

\maketitle

\selectlanguage{english}
\begin{abstract}
We compare the properties of giant molecular associations in the galaxy Messier 100 (M100) with those of the less massive giant molecular clouds in the Milky Way and Local Group, while also observing how those properties change within M100 itself. From this analysis of cloud mass, radius, and velocity dispersion, we determine that the clouds are in or near virial equilibrium and that their properties are consistent with the underlying trends for the Milky Way. We find differences between nuclear, arm and inter-arm M100 populations, such as the nuclear clouds being the most massive and turbulent, and arm and inter-arm populations having differently shaped mass distributions from one another. Through the analysis of velocity gradients, cloud motion can be attributed to turbulence rather than large scale shearing motion. This is supported by our comparison with turbulence regulated star formation models. Finally, we calculate ISM depletion times to see how quickly clouds turn gas into stars and found that clouds form stars more efficiently if they are turbulent or dense. %
\end{abstract}%



\email{nhervieu@ualberta.ca}

\maketitle

\section{INTRODUCTION}
Molecular clouds are cold, dense regions of interstellar medium where gravity is able to overcome gas pressure, enabling them to be the sole location of star formation. An improved understanding of the structure of molecular clouds will provide insight into star formation. Here, we catalogue the molecular clouds in the spiral galaxy Messier 100 (M100) of the Virgo cluster to determine whether they follow the behaviours found in previous studies \citep{Solomon_1987} \citep{Fukui_2010}. At masses greater than $10^5 M_{\odot}$, molecular clouds fall into the range of Giant Molecular Clouds (GMCs). However, due to M100's distance of 14.3 Mpc from earth, and the large resolution element of our observations compared to the size of typical GMCs, we are looking at complexes of GMCs called Giant Molecular Associations (GMAs). We determine whether traditional scalings extend to these larger, more massive regions. We also analyze how cloud properties differ between populations within the galaxy. We seek to understand these molecular cloud properties to gain insight into their motion, evolution, and ability to form stars.





\section{DATA}

This study is done using new-millimetre-wave observations from the Atacama Large Millimeter/submillimeter Array (ALMA) interferometer in Chile. Although similar research has been done on other galaxies, ALMA provides a particularly well resolved data set, allowing us to resolve the centre and width of spectral lines, and thus measure the clouds' radial velocities and magnitudes of internal motions.

Although molecular gas is comprised mainly of molecular hydrogen ($H_2$), at GMCs' typically low temperatures of 10K, $H_2$ does not emit radiation. Instead, we obtained our spectral data from the next most abundant gas: carbon monoxide molecules's J=1-0 emission. 

The ALMA observations provided us with data cubes that form a map of the galaxy where each position is associated with a frequency spectrum. Using the Doppler shift and distance to the galaxy, we transformed the data cubes into position-position-velocity (PPV) cubes, which we then used to identify molecular clouds and their properties. Fig. \ref{xypos} displays the maximum temperature map of M100 in sky coordinates, where each point is the brightest value in the spectrum at that position in the data cube. The bright carbon monoxide (CO) regions make the nucleus and spiral arms clearly visible and allow us to identify the GMAs.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/galaxy/galaxy}
\caption{{\label{xypos} ALMA CO(1-0) maximum temperature map displaying a visual of nuclear, arm, and inter-arm populations of GMAs, which we manually selected using this figure. Brighter regions correspond to higher CO intensity and white points correspond to the GMA locations used, as identified by CPROPS.%
}}
\end{center}
\end{figure}

\section{RESULTS AND ANALYSIS}
\subsection{GMA PROPERTIES}
We derived the following cloud properties using the python packages CPROPS and cpropstoo created by \citet{Rosolowsky_2006}. Using moment methods, we derived size, line width, and flux directly from the PPV cubes' emission spectras and then calculated radius, FWHM line width, and CO luminosity from those quantities, respectively. From the CO luminosity and the empirical X-factor relationship \cite{Bolatto_2013}, we found the luminous mass of $H_2$.

To begin our analysis, we recreated the Larson's Laws plots, originally found in \citet{Larson_1981} and subsequently in \citet{Solomon_1987} and \citet{Fukui_2010} for GMCs in the Milky Way Galaxy and  Local Group of galaxies, respectively, to test their scaling relationships for our M100 data. We identified the nuclear, arm and inter-arm GMAs manually using their positions in Fig.~\ref{xypos} and considered each population separately.

In Fig.~\ref{mvir_mlum}, we plotted the luminous masses, $M_{\mathrm{lum}}$, derived from the luminosity of CO, against the virial masses, $M_{\mathrm{vir}}$, inferred from the FWHM line width and cloud radius \citep{Rosolowsky_2006}. The scaling line is the expected one to one ratio for clouds in virial equilibrium. Fig.~\ref{mvir_mlum} shows that the luminous masses are generally higher than the virial masses and that the nuclear clouds are, on average, the most massive, followed by arm and then inter-arm clouds. These results are confirmed in the first two columns of Table \ref{means}, which present the geometric mean of luminous mass, $\overline{M}_\mathrm{lum}$, and the corresponding standard error, $SE(M_\mathrm{lum})$.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/MlumMvir-matplotlib/MlumMvir-matplotlib}
\caption{{\label{mvir_mlum} Virial mass, $M_{\mathrm{vir}}$, compared to the luminous mass, $M_{\mathrm{lum}}$, of the GMAs. The data are consistent with the fit line, representing the expected one to one ratio for virial equilibrium. Blue triangles represent nuclear clouds, which are the most massive, green circles represent inter-arm clouds, which are the least massive, and purple diamonds represent arm clouds.%
}}
\end{center}
\end{figure}

Fig.~\ref{r_sigma} shows luminous mass as a function of radius, $R$, for each GMA, with the expected fit line as found in \citet{Solomon_1987} for GMCs in the Milky Way galaxy:

\begin{equation}
M_{\mathrm{lum}} = 540R^2 (M_{\odot}).
\label{eqn1}
\end{equation}

The nuclear GMAs are more massive for their sizes compared to equivalently sized disk clouds. While the inter-arm GMAs are less massive than similarly sized arm GMAs.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/MlumRad-matplotlib/MlumRad-matplotlib}
\caption{{\label{r_sigma} Luminous mass, $M_{\mathrm{lum}}$, as a function of GMA radius, $R$. Expected scaling is from Eq.~\ref{eqn1}. Blue triangles represent nuclear clouds, which are the most massive, green circles represent inter-arm clouds, which are the least massive, and purple diamonds represent arm clouds.%
}}
\end{center}
\end{figure}


In Fig.~\ref{mlum_r}, we plotted the radii of GMAs against their velocity dispersions, $\sigma$, which represent the root mean squared internal velocity of each cloud caused by the increased difference in velocity for more spatially separated gas parcels. Here, the fit for the Milky Way scales with Eq.~\ref{eqn2} from \citet{Solomon_1987}, which they attribute to virial equilibrium rather than Kolmogorov turbulence as suggested by \citet{Larson_1981}. We see basic correlation with that fit, despite large spread in velocity dispersion.

\begin{equation}
\sigma^2 = \frac{\sqrt{\mathrm{\pi}}} {3.4} R.
\label{eqn2}
\end{equation}

Fig.~\ref{mlum_r} shows that turbulence increases with radius and that the nuclear GMAs are more turbulent than disk GMAs of similar radii. There are no differences between the turbulent properties of the arm and inter-arm GMAs at any radius.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/LwRad-matplotlib/LwRad-matplotlib}
\caption{{\label{mlum_r} Velocity dispersion, $\sigma$, as a function of GMA radius, $R$, with expected scaling from Eq.~\ref{eqn2} for GMCs in the Milky Way. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds. Nuclear GMAs have higher turbulence than disk GMAs.%
}}
\end{center}
\end{figure}

Using the python package {\sc galaxies} created by Dr. Rosolowsky, we calculated the GMAs' positions relative to the galaxy's centre. Fig.~\ref{rad_sig} shows these galactocentric radii, $R_\mathrm{gal}$, plotted against the turbulent line widths on a 1pc scale, $\sigma_0$, that we calculated with Eq.~\ref{sigma_o} to normalize $\sigma$ for size. Here, the distinction between nuclear and disc GMA populations is clear with the cut off at 1kpc from the centre.   

\begin{equation}
\sigma_0 = \frac{\sigma}{\sqrt{\frac{R}{1\ \mathrm{pc}}}}.
\label{sigma_o}
\end{equation}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/sigma0-Rgal-matplotlib1/sigma0-Rgal-matplotlib}
\caption{{\label{rad_sig} Turbulent line width on a 1 pc scale, $\sigma_0$, as a function of the GMA's distance from the galactic centre, $R_\mathrm{gal}$. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds. Turbulence is highest in nuclear clouds and the same, on average, for arm and inter-arm populations.%
}}
\end{center}
\end{figure}

Fig.~\ref{rad_turb} is a plot of turbulent line width on a 1 pc scale as a function of surface density, $\Sigma$, described by Eq.~\ref{density}. 

\begin{equation}
\Sigma = \frac{M_{\mathrm{lum}}}{\mathrm{\pi} R^2}.
\label{density}
\end{equation}

From Fig.~\ref{rad_sig} and Fig.~\ref{rad_turb}, and confirmed in Table \ref{means}, which presents the geometric mean of turbulent line width on a 1 pc scale, $\overline{\sigma_0}$, and the corresponding standard error, $SE(\sigma_0)$, nuclear clouds tend to be more turbulent and dense than disk clouds, but there is no statistical difference in the average turbulence of arm versus inter-arm populations. Though, the inter-arm clouds do tend to have lower surface densities than both arm and nuclear clouds as shown in Fig.~\ref{r_sigma} and Fig.~\ref{rad_turb}. Note that we ignored arm datum with abnormally large values of $\Sigma > 10^3~M_{\odot}\ \mathrm{pc}^{-2}$ after finding that their surface densities were an order of magnitude smaller when calculated with area derived radii rather than $\sigma$ derived radii. Though the turbulent properties are changing among GMAs, the agreement with a single line suggests that the GMAs are all virialized and thus their virial mass is consistent with their luminous mass.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Sigma0-Mden-matplotlib/Sigma0-Mden-matplotlib}
\caption{{\label{rad_turb} Turbulent line width on a 1 pc scale, $\sigma_0$, as a function of surface density, $\Sigma$. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds. Nuclear clouds are the most turbulent. The linear agreement suggests that the clouds of all populations are virialized.%
}}
\end{center}
\end{figure}

To compare the angular momentum of the arm and inter-arm populations, we plotted luminous mass as a function of velocity gradient, $\nabla v$, in Fig.~\ref{vgrad}, where $\nabla v$ is generally a signature of both turbulence and large scale shearing motions within the galaxy. Note that we ignored clouds with balanced emissions and thus negligible velocity gradients. The geometric means, $\overline{\nabla v}$, and standard errors, $SE({\nabla v})$, of $\nabla v$ in Table \ref{means}, show that $\overline{\nabla v}$ is highest in the nuclear population. Even though $\overline{\nabla v}$ is slightly higher in the spiral arms than in the inter-arm region, it is not statistically significant, and thus we conclude that the velocity gradient is due to turbulent motion rather than shear. This is supported by the fairly tight correlation between $\sigma_0$ and $\nabla v$ in Fig.~\ref{sig_velgrad}.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Mlum-vgrad-matplotlib-zoomin2/Mlum-vgrad-matplotlib-zoomin2}
\caption{{\label{vgrad} Luminous mass, $M_\mathrm{lum}$ as a function of velocity gradient, $\nabla v$. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds. Velocity gradient is highest in the nuclear clouds, but there is no clear separation between the arm and inter-arm populations, suggesting that turbulent motion dominates over shear motion.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/sigma0-vgrad-matplotlib/sigma0-vgrad-matplotlib}
\caption{{\label{sig_velgrad} Turbulent line width on a 1 pc scale, $\sigma_0$, as a function of velocity gradient, $\nabla v$.  Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds. The tight correlation for both arm and inter-arm GMAs supports the conclusion that turbulent motion dominates rather than galactic shear motion.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{table*}
\centering
    \begin{tabular}{ c c c c c c c }
        $\mathrm{Clouds}$ & $\overline{M}_{\mathrm{lum}}\ (M_{\odot})$ & $SE(M_{\mathrm{lum}}) (M_{\odot})$ & $\overline{\sigma_0}\ ({\mathrm{km\ s^{-1}}})$ & $SE(\sigma_0)\ ({\mathrm{km\ s^{-1}}})$ & $\overline{\nabla v}\ (\mathrm{km\ s}^{-1}\ \mathrm{pc}^{-1})$ & $SE(\nabla v)\ (\mathrm{km\ s}^{-1}\ \mathrm{pc}^{-1})$  \\
        \midrule
        All & 7.26 & 0.04 & -0.36 & 0.02 & -1.79 & 0.03 \\
        Nuclear & 8.25 & 0.09 & 0.03 & 0.06 & -1.57 & 0.13 \\
        Arm & 7.39 & 0.04 & -0.39 & 0.02 & -1.77 & 0.03 \\
        Interarm & 6.90 & 0.05 & -0.44 & 0.03 & -1.85 & 0.04 \\
        \bottomrule
        \end{tabular}
        \caption{{The geometric mean and standard error of the logarithms of luminous mass, $log_{10}(M_\mathrm{lum})$, turbulent line width, $log_{10}(\sigma_0)$, and velocity gradient, $log_{10}(\nabla v)$, for the combined data set and each GMA population treated separately. \label{means} }}
\end{table*}



\subsection{MASS DISTRIBUTIONS}
To better understand the origins of the GMAs' structures, we studied their mass distributions to find the parameters of the powerlaw fit to the data. The theoretical cumulative mass function, $N(>M)$, representing the number of clouds more massive than a given mass, $M$, is expressed by Eq.~\ref{massdist} where $\beta$ is a normalization constant, $\alpha$ is the index, $M_{\mathrm{max}}$ is the maximum cloud mass, and $N_{\mathrm{max}}$ is the number of clouds near that mass \citep{Rosolowsky_2005}.

\begin{equation}
 N(>M) = \frac{\beta M_{\odot}}{\alpha +1} \left(\frac{M}{M_{\odot}}\right)^\alpha = \frac{N_\mathrm{max}}{M_\mathrm{max}} \left[ \left( \frac{M}{M_\mathrm{max}} \right) ^{\alpha +1}-1 \right] .
\label{massdist}
\end{equation}

The index $\alpha$ is of particular interest as it illustrates how the mass is distributed across the range of masses and suggests what drives fragmentation. For pure gravitational fragmentation, we expect an index of -2, but turbulence and galactic dynamics alter that value.

Created using the python package {\sc powerlaw}, Fig.~\ref{powerlaw} shows the powerlaw and truncated powerlaw fits for each of our populations, where the powerlaw fit has a distribution function in the form of Eq.~\ref{powerlawdist}, and the truncated powerlaw uses an exponential cutoff as in Eq.~\ref{truncpowerlawdist}, with index $\lambda$. Note that we have manually set the minimum mass included in the fits to $10^7~M_{\odot}$ based on the noise in the data sets and the CPROPS algorithm. 

\begin{equation}
p(x) = x^{\alpha}
\label{powerlawdist}
\end{equation}

\begin{equation}
p(x) = x^{\alpha}\mathrm{e}^{-\lambda x}
\label{truncpowerlawdist}
\end{equation}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/powerlaw/powerlaw-combinedp}
\caption{{\label{powerlaw}
Cumulative mass function of data for: \textbf{a)} the combined data set, \textbf{b)} arm and inter-arm GMAs, \textbf{c)} Nuclear GMAs.%
}}
\end{center}
\end{figure}

Table~\ref{alpha_table} presents the powerlaw and truncated powerlaw indices, $\alpha$ and $\alpha_\mathrm{trunc}$, respectively; the log of the likelihood ratio, $R$, that the truncated or nontruncated distribution is better, where $R$ being greater than zero means that the data are more consistent with the nontruncated powerlaw and $R$ being smaller than zero means the data are more consistent with the truncated powerlaw; the significance of the result, $p$, where a value close to zero means that the difference between the two distributions is significant; and the truncation mass, $M_\mathrm{trunc}$, defined as the reciprocal of $\lambda$. We see that this truncation mass falls off with radius, as does the number of clouds. \citet{Rosolowsky_2005} showed that for GMCs the powerlaw index, $\alpha$, varies from -2.5 to -1.5. Our results for M100 GMAs fit in this expected range for lower mass systems, but the index for the nuclear population and all of the truncated indices are lower than expected. The mass distribution of the nuclear clouds is very flat, although we note that this population is not well represented by a power law fit because it contains only a small number of $10^8M_{\odot}$ GMAs. We see a clear difference between the distributions of arm and inter-arm populations, where the index for inter-arm is notably quite steep. This implies that most of the mass in the inter-arm region is contained in low-mass clouds, and in high-mass clouds for the arm region \cite{Rosolowsky_2005}.\selectlanguage{english}
\begin{table}
\centering
\begin{tabular}{cccccc}
$\mathrm{Clouds}$ & $\alpha$ & $\alpha_\mathrm{trunc}$ & $R$ & $p$ & $M_\mathrm{trunc} ~(10^7M_{\odot})$ \\
\midrule
All & -1.83 & -1.44 & -4.80 & 0.002 & 30 \\
Nuclear & -1.39 & -1.00 & -5.11 & 0.001 & 63 \\
Disk & -1.97 & -1.56 & -3.18 & 0.012 & 21 \\
Arm & -1.83 & -1.34 & -3.51 & 0.008 & 21 \\
Inter-arm & -2.37 & -1.00 & -2.86 & 0.017 & 3 \\
\bottomrule
\end{tabular}
\caption{{Results from Fig.~\ref{powerlaw} powerlaw fit. Listed are the indices for truncated ($\alpha_\mathrm{trunc}$) and non-truncated ($\alpha$) distributions, likelihood ratios of truncated and non-truncated distributions ($R$), result significances ($p$), and the truncation mass ($M_\mathrm{trunc}$), considering the total data and the sub-populations separately.\label{alpha_table}}}
\end{table}




We graphically determined the area associated with each cloud population, finding area to be $\mathrm{99.07~kpc^2}$ for the spiral arms, $\mathrm{106.47~kpc^2}$ for the inter-arm region, and $\mathrm{4.21~kpc^2}$ for the nuclear region. Fig.~\ref{norm_mass} shows the mass distributions with the number of clouds in each population normalized by their respective areas, denoted by $N_\mathrm{norm}$. While the nuclear distribution maintains its flat shape with a steep drop at its bulk of $10^8 M_\odot$ clouds, we also see that the nuclear clouds have higher number density and characteristic masses than disk clouds. The arm and inter-arm distributions are also clearly different from one another, with the inter-arm clouds having a lower average number density and a steeper mass distribution than arm clouds. The inter-arm region, therefore, carries more of its mass in lower mass stars compared to the spiral arms.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/massdist-all-norm1/massdist-all-norm}
\caption{{\label{norm_mass} Cumulative mass distribution of each GMA population normalized by their areas.%
}}
\end{center}
\end{figure}

\subsection{STAR FORMATION}
Using star formation rates (SFR) measured in 500pc regions within the galaxy provided by collaborator Adam Leroy, we calculated the average SFR for each GMA in our catalog. We then calculated the molecular interstellar medium (ISM) depletion time, $\tau_\mathrm{H2}$, which is the time needed to convert all of the region's gas into stars. Fig.~\ref{tdep1} and Fig.~\ref{tdep2} show $\tau_\mathrm{H2}$ as a function of GMA surface density and velocity dispersion on a 1pc scale, respectively.  $\tau_\mathrm{H2}$ decreases with increasing $\Sigma$ and $\sigma_0$. From Table.~\ref{tdep_means}, which presents the geometric mean and standard error of $\tau_\mathrm{H2}$, we see that the ISM depletion time is much shorter for nuclear clouds than for disk clouds, and slightly lower for arm clouds than for inter-arm clouds. This all makes physical sense as turbulent clouds have more efficient star formation because the turbulence forces more of the cloud to high densities where gravity can dominate.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tdep-Mden-matplotlib/Tdep-Mden-matplotlib}
\caption{{\label{tdep1} Molecular ISM depletion time, $\tau_{\mathrm{H2}}$, as a function of surface density, $\Sigma$. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tdep-sigma0-matplotlib/Tdep-sigma0-matplotlib}
\caption{{\label{tdep2} Molecular ISM depletion time, $\tau_{\mathrm{H2}}$, as a function of velocity dispersion on a 1 pc scale, $\sigma_0$. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds.%
}}
\end{center}
\end{figure}

We compared this with the cloud depletion time, $\tau_\mathrm{GMA}$, calculated using the GMAs' surface densities, $\Sigma$, cloud radii, $R$, and the star formation surface densities. Fig.~\ref{tdep3} and Fig.~\ref{tdep4} display the results, and Table \ref{tdep_means} also presents the average cloud depletion times and standard errors for each population. Here $\tau_\mathrm{GMA}$ increases with $\Sigma$ and $\sigma_0$ and, though there is little statistical difference in average between populations, $\tau_\mathrm{GMA}$ also increases from nuclear to arm to inter-arm clouds.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tdep2-Mden-matplotlib/Tdep2-Mden-matplotlib}
\caption{{\label{tdep3}
 Cloud depletion time, $\tau_{\mathrm{GMA}}$, as a function of surface density, $\Sigma$. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tdep2-sigma0-matplotlib/Tdep2-sigma0-matplotlib}
\caption{{\label{tdep4}  Depletion time, $\tau_{\mathrm{GMA}}$, as a function of velocity dispersion on a 1pc scale, $\sigma_0$. Blue triangles represent nuclear clouds, green circles represent inter-arm clouds and purple diamonds represent arm clouds.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{table}
\centering

\begin{tabular}{ccccc}
    $\mathrm{Clouds}$ & $\overline{\tau}_{H2}\ (yrs)$ & $SE(\tau_{H2})\ (yrs)$ & $\overline{\tau}_{GMA}\ (yrs)$ & $SE(\tau_{GMA})\ (yrs)$ \\
    \midrule
    All & 9.56 & 0.01 & 9.97 & 0.03 \\
    Nuclear & 9.20 & 0.03 & 9.73 & 0.15 \\
    Arm & 9.53 & 0.02 & 9.97 & 0.04 \\
    Interarm & 9.64 & 0.02 & 10.02 & 0.04 \\
    \bottomrule
    \end{tabular}
\caption{{The geometric mean and corresponding standard error of the logarithms of molecular interstellar medium depletion time, $log_{10}(\tau_{H2})$, and cloud depletion time, $log_{10}(\tau_{GMA})$ for each population of GMAs.\label{tdep_means}}}
\end{table}

\section{DISCUSSION}

We can examine the \citet{Krumholz_2005} model for the star formation in these systems which predicts that the star formation efficiency $(\propto \tau_{\mathrm{H2}}^{-1})$ should scale like
\begin{equation}
\mathrm{SFE} = \epsilon_{\mathrm{SF}} \frac{(\mathcal{M}/100)^{-0.32}}{\tau_{\mathrm{ff}}} \left(\frac{\alpha_{\mathrm{vir}}}{1.3}\right)^{-0.68}
\end{equation}
where $\epsilon_{\mathrm{SF}}$ is the star formation efficiency per free-fall time in the system (=0.014) and $\tau_{\mathrm{ff}}$ is the free-fall time: 
\begin{equation}
\tau_{\mathrm{ff}} = \sqrt{\frac{3\pi}{32 G \bar{\rho}}}
\end{equation}
where $\bar{\rho}$ is the average mass density in the cloud.

In Fig.~\ref{SFE}, we examine the relationship between the ISM depletion times, the corresponding star formation efficiencies, $SFE$, and the changing cloud populations. Here $SFE$, defined by Eq.~\ref{sfeeqn}, is a quantity proportional to the Krumholz-Mckee SFE.

\begin{equation}
\mathrm{SFE} \propto \sigma ^{-0.32} \left( \frac{M_\mathrm{lum}}{R^3} \right) ^{0.5}.
\label{sfeeqn}
\end{equation}

There is some separation between the three populations, including differences between the arm and inter-arm populations as well as between the nucleus and the remainder. Thus, while there is not a clear correlation between depletion time and any one parameter in the study, the {\it combination} of parameters that is thought to shape star formation efficiency on the large scales does show good correlation. Therefore, the clouds here in M100 seem to be consistent with a turbulence-regulated star formation model.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/KM/KM2}
\caption{{The interstellar medium depletion time, $\tau_\mathrm{H2}$, plotted against a quantity proportional to the Krumholz-McKee star formation efficiency, ($\sigma_v^{(-0.32)} (M/R^3)^{(0.5)}$), on a logarithmic scale. The small points are the nuclear clouds. The unfilled, large points are the arm clouds and the filled large points are the inter-arm clouds. The separation between the populations suggests that GMAs are different in the arm vs. inter-arm and they drive a change in the star formation efficiency in the rough direction expected by theory.  
\label{SFE}%
}}
\end{center}
\end{figure}

\section{CONCLUSION}
Through our analysis, we showed that the GMAs in the galaxy M100 follow the same basic behaviours as the less massive GMCs in the Milky Way \citep{Solomon_1987} and in the Local Group of galaxies \cite{Fukui_2010}. Such behaviours include both luminous mass and velocity dispersion increasing with cloud radius and the virial mass scaling with the luminous mass of CO, demonstrating that the clouds are virialized. There are clear differences between GMA populations when we separate the galaxy into nuclear, arm and inter-arm regions. Compared to the disk clouds, the nuclear clouds have the highest masses and thus surface densities, and have higher velocity dispersions. From the cumulative mass distributions, our indices were consistent with the expected range of -2.5 to -1.5 \citep{Rosolowsky_2005}, though we saw a steeper index for inter-arm clouds ($\alpha = -2.37$) than for arm clouds ($\alpha = -1.83$), suggesting that the inter-arm population carries more of its mass in lower mass clouds. The agreement of the mean velocity gradients for the arm and inter-arm clouds and the tight correlation of turbulent line width on a one parsec scale with velocity gradient points to turbulent motion dominating over galactic shear motion. The ISM depletion time decreases with increasing cloud surface density and turbulent line width on a one parsec scale, demonstrating that more turbulent clouds form stars more efficiently and denser clouds form stars more efficiently. It is hard to sort out causality, but this is consistent with expectations from turbulence regulated star formation models like \citet{Krumholz_2005}.

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