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\end{equation}  Now we can rewrite (\ref{dP_ad}) and (\ref{dP_iso}) in terms of the disk Mach number  \begin{equation}  \int{\frac{dP}{\rho}} = \frac{\gamma}{\gamma -1} \frac{\mathcal{M}^2 }{v^2_K} \quad \frac{v^2_K}{\mathcal{M}^2}\quad  \rm{Adiabatic} \ \rm{Flow} \end{equation}  \begin{equation}  \int{\frac{dP}{\rho}} = \frac{\mathcal{M}^2}{v^2_K} \frac{v^2_K}{\mathcal{M}^2}  \ln{\frac{\rho}{\rho_0}} \quad \rm{Isothermal} \ \rm{Flow} \end{equation}