deletions | additions       diff --git a/section_Analysis_We_examined_the__.tex b/section_Analysis_We_examined_the__.tex  index b4d4c00..2365e19 100644  --- a/section_Analysis_We_examined_the__.tex  +++ b/section_Analysis_We_examined_the__.tex 
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\section{Analysis}  We examined the GMC properties as in Solomon et al. (1987). M83 clouds exhibit similar relationships for velocity line width to radius, which confirms our assumption of the clouds being in virial equilibrium (Figure 1). As well, the virial mass to luminous mass and luminous mass to radius relationships are also consistent with previous findings (Figures 2 and 3). We then binned the clouds by equal surface area on the disk as in Adamo et al. (2015), and for each bin examined the cumulative mass function:  \begin{center} $N(>M)=\int^\infty_M \begin{equation}   N(>M)=\int^\infty_M  \frac{dN}{dM}dM=\frac{\beta M_\odot}{\alpha+1}\left(\frac{M}{M_\odot}\right)^\alpha=\frac{N_{max}}{M_{max}}\left[\left(\frac{M}{M_{max}}\right)^{\alpha+1}-1\right]$ \end{center} M_\odot}{\alpha+1}\left(\frac{M}{M_\odot}\right)^\alpha=\frac{N_{max}}{M_{max}}\left[\left(\frac{M}{M_{max}}\right)^{\alpha+1}-1\right]   \end{equation}  where $N(>M)$ is the number of clouds above a certain mass $M$, $\beta$ is a normalization constant and $\alpha$ is the index. The latter expression represents the truncated power law case where $M_{max}$ is the maximum mass we are looking for. Using the ‘powerlaw’ package in Python, each bin was fitted by two distributions: an ordinary power law and a truncated power law (example in Figure 4). The distributions were constrained by a minimum mass of 2496477 M$_\odot$ above which there is a stable fit for the whole galaxy. ‘Powerlaw’ also returned the index α for the power law that best described the data.