deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 40fd024..44f783c 100644  --- a/untitled.tex  +++ b/untitled.tex 
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The isothermal equation of state $P=(c^{\rm{iso}}_{s})^2 \rho$ gives us that $c^{\rm{iso}}_{s} = \sqrt{RT}$. So we can write  \begin{equation}  \int{\frac{dP}{\rho}} = \frac{(c^{\rm{ad}}_{s})^2 }{\gamma -1} = \frac{\gamma (c^{\rm{iso}}_{s})^2 }{\gamma -1} \quad \rm{Adiabatic} \ \rm{Flow}  \label{dP_ad}  \end{equation}  \begin{equation}  \int{\frac{dP}{\rho}} = (c^{\rm{iso}}_{s})^2 \ln{\frac{\rho}{\rho_0}} \quad \rm{Isothermal} \ \rm{Flow}  \label{dP_iso}  \end{equation}  \section{Hydrostatic Balance} 
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If we define the disk Mach number as the ratio of the isothermal sound speed to the keplerian orbital velocity $v_K=\sqrt{GM/r}$, then we have an expression for the Mach number in terms of disk sound speed for hydrostatic equilibrium in teh vertical direction,  \begin{equation}  \mathcal{M} = \frac{H}{r} = \frac{v_K}{c^{\rm{iso}}_s}  \end{equation} Now we can rewrite