$(10$ points$)$ Rayleigh Flow: Derive a closed$-$form analytical expression for the Mach number in terms of the ratio of the total temperature to its diabatic sonic counterpart $($i$.$e$.,\frac{T_0}{T_0^{**}}).$ Clearly identify the subsonic and supersonic solutions, with proper justification. Also, determine the ranges of $\frac{T_0}{T_0^{**}}$ in which your equations are valid.

$\frac{T_o}{T_o^{**}}=\frac{(+1)M^2}{(1+M^2{)}^2}[2+(-1)M^2]$

consider gamma as $1\mathrm{.}4$ and To $/$ To$*$ as k$.$ k is a constant