\subsubsection{The Cloud}
The mass within the cloud and the shell is estimated by calculating the mass within projected regions using the column density map. As discussed in \S4.1, we adopt the 2MASS extinction map in our mass calculation. We then calculate the mass column density from the equation (\citet{Arce_2001}):

\begin{equation}
\Sigma = \mu\,N_{H_2} = \mu\,(9.4\times10^{20})A_V\;\text{g/cm}^2
\end{equation}

$\mu$ is the average mass per particle and is $\sim$ 2.72 times mass of a hydrogen atom (a.m.u.) for molecular clouds in our Milky Way (ref?). We then obtain the total mass from $M = \Sigma\times \text{Area}$.

In the case of the Ophiuchus, we first calculate the threshold extinction value used to determine the edge of the cloud by calculating the average extinction in places with no obvious far-infrared emission. This threshold value is $\sim$ 0.93 mag. ($\sim$ $8.7 \times 10^{20}$ cm$^{-2}$) on our 2MASS extinction map. We then select the projected region of our cloud to include places where the extinction is larger than 0.93 mag. This threshold value is then subtracted from the extinction within the selected region. The total mass of the Ophiuchus cloud is estimated to be $\sim$ 6000 M$_{\sun}$.

The turbulent velocity of the cloud is estimated using the $^{12}$CO (1-0) line emission. Since the line profile in a molecular cloud at this scale varies from one part of the cloud to another, it is unsuitable to fit the profile and to calculate the velocity dispersion from the fit. A natural way to calculate the velocity dispersion is the second central moment:

\begin{equation}
\sigma_v = \sqrt{\left(\frac{\int{I(\mathbf{p}, v)\,(v(\mathbf{p}, v)-v_{cent}(\mathbf{p}))^2\,\text{d}(\mathbf{p}, v)}}{\int{I(\mathbf{p}, v)}\,\text{d}(\mathbf{p}, v)}\right)}\;\;\;\text{,}
\end{equation}

where $I(\mathbf{p}, v)$ and $v(\mathbf{p}, v)$ are the intensity and the velocity at each position $\mathbf{p}$ and velocity $v$, respectively. $v_{cent}(\mathbf{p})$ is the velocity centroid and is calculated from:

\begin{equation}
v_{cent} = \frac{\int{I(\mathbf{p}, v)\,v(\mathbf{p}, v)\,\text{d}(\mathbf{p}, v)}}{\int{I(\mathbf{p}, v)\,\text{d}(\mathbf{p}, v)}}
\end{equation}

This is also the first moment, which can be understood as the mean velocity weighted by the intensity at each velocity. In the data cube obtained at the FCRAO, the velocity resolution is 0.21 km/s, smaller than the typical thermal linewidth in molecular clouds ($\sim$ 1 km/s). Thus, we can use the discrete sum to calculate the integrals above without losing any information.

The average of the second central moment in the Ophiuchus cloud is $\sim$ 0.97 km/s. The rms velocity of the turbulence is $\sqrt{3}$ times the velocity dispersion and is thus $\sim$ 1.68 km/s on average over the cloud. To calculate the total momentum and energy in the turbulence, we match the maps of the second moment and the 2MASS/NICER extinction. We then calculate the momentum/energy within each pixel by using the extinction and the second moment at the pixel. For regions outside the coverage of the FCRAO observations, we adopt the mean second moment (0.97 km/s) for the turbulent velocity. Summing up the energy $E_i = 0.5\,M_i\,v_{turb,\,i}^2$ over the entire cloud gives us an estimate of the total momentum and energy in the turbulence. These are 9900 M$_{\sun}$ km/s and $8.82\times10^{46}$ erg, respectively.
