Erik Rosolowsky edited subsection_Mass_distributions_label_sec__.tex  about 8 years ago

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\subsection{Mass distributions}  \label{sec:mspec}  Figure \ref{fig:XXX} plots the cumulative mass distribution functions, normalized by the survey area for five different radial bins through the galaxy. We present two representations of these data. The normalization to equal area bins mimics the binning used in \citet{Adamo_2015} to facilitate comparison with that work. That binning divides the galaxy into regions with equal area in the original optical data. The data set we analyze here does not span the full range of angles around the galaxy, though the areas are still significant fractions of the galaxy (see Figure \ref{})  We binned the clouds by equal surface area on the disk as in \citet{Adamo_2015}, and each bin examined the complementary  cumulative mass function: distribution function (CCDF):  \begin{equation}   N(>M)=\int^\infty_M N(>M)=1-\frac{1}{M(\infty)}\int^\infty_M  \frac{dN}{dM}dM \end{equation}  The We fit three models to the CCDF using the  {\sc powerlaw} fit function package by \cite{Alstott_2014}. The general CCDF form  is \begin{equation}   \frac{dN}{dM} = M^{\alpha} \exp\left(-\frac{M}{M_c}\right) M^{\beta} \exp\left(-\frac{M}{M_c}\right),  \end{equation}  which is a power-law mass distribution with an exponential truncation. The cutoff (truncation) mass in the distribution is $M_c$ and the index of the distribution is $\beta$. We consider a pure power-law distribution, letting $M_c\to\infty$ and we also consider a Schechter function, constraining $\beta=-2$ and optimizing for $M_{c}$. The pure power-law function has been traditionally considered in previous on molecular cloud mass distributions \citep{Solomon_1987,Rosolowsky_2005a}. In all cases, we limit the fits to clouds with masses $M>3\times 10^{5}~M_{\odot}$ as determined by the minimum mass of clouds admitted by our cataloging procedure. The {\sc powerlaw} package optimizes the fits to the data using the procedure described in \citet{Clauset_2009}, which uses a maximum likelihood framework and a Kolmogorov-Smirnoff test to assess goodness-of-fit. Of note, the method provides likelihoods for favoring one functional representation over another.  where $N(>M)$ is the number of clouds above a certain mass $M$, and $\alpha$ is the index. The latter expression represents the truncated power law case where $M_c$ is the maximum mass we are looking for. Each bin was fit by two distributions: an ordinary power law and a truncated power law (\ref{fig:massdist}). A Schechter function with index -2 was also examined for it's suitability to the data. The distributions were constrained by a minimum mass of $3\times 10^5$ M$_\odot$ above which there is a stable fit for the whole galaxy.   The loglikelihood ratio R, for the ordinary power law over the truncated version, and it's significance p indicate that there is an upper truncation mass for all bins (\ref{table:properties}). Table \ref{table:properties} also shows the index $\alpha_{GMC}$, the truncation mass $M_{c,GMC}$, the truncation mass of the Schechter function $M_{s,GMC}$, the mass of the largest cloud, the mass of the fifth largest cloud, the stellar cluster index $\alpha_{cluster}$ and the stellar cluster truncation mass $M_{c, cluster}$.  $\alpha_{GMC}$ for the truncated power law changes below the cutoff mass, and it increases outward in the galaxy similar to the stellar cluster indices.