critical point, one can see that equation (\[integ eq\]) becomes infinity. Thus, in order to obtain a finite solution, $d\theta/dx$ must vanish at the critical point. From equations (\[P\]) and (\[integ eq\]), the pressure gradient becomes finite as follows: $$\begin{aligned}
\left( \frac{dP}{dx} \right)_{\mathrm{crit}} &=& - \left( \frac{2{\cal R}j^2}{\sqrt{M^2-4{\cal R}j^2T}} 
        \frac{dT}{dx}\right)_\mathrm{crit} \\
        &=& - \left( \rho{\cal L}_\mathrm{crit} \right)\sqrt{\frac{\rho_\mathrm{crit}}{P_\mathrm{crit}}}\nonumber
        \label{}\end{aligned}$$ The pressure at the critical point can be derived from the momentum flux conservation. At the CNM $(x=-\infty)$, since the velocity is still subsonic because of its high density, the momentum flux becomes $$M=\rho_\mathrm{c}v_\mathrm{c}^2 + P_\mathrm{c}\simeq P_\mathrm{c}.$$ On the other hand, from equation (\[P\]), the pressure at the critical point is given by $P_\mathrm{crit}=M/2$. Therefore, the pressure at the critical point can be expressed by $P_\mathrm{c}$ as follows: $$P_\mathrm{crit}\simeq P_\mathrm{c}/2.
        \label{P crit}$$ The density at the critical point is given by $$\rho_\mathrm{crit}=j^2/P_\mathrm{crit}\simeq 2j^2/P_\mathrm{c}.
        \label{rho crit}$$ One can see that $P_\mathrm{crit}$ for $j=500j_\mathrm{ZP}$ and 1000$j_\mathrm{ZP}$ is roughly equal to $P_\mathrm{c}/2=2000k_\mathrm{B}$ K cm$^{-3}$, and its density increases for larger mass flux as seen in equation (\[rho crit\]). Around the critical point, since $dT/dx=d\theta/dx/\kappa=0$, the trajectories in the $(n,P)$ plane approach constant temperature lines of $T=T_\mathrm{crit}$ in figure \[fig:large\_flux\_sonic\].

We investigate the topological properties at 