{\cal R}/(\gamma-1)$ is the specific heat at constant pressure. From equations (\[eos\]), (\[const\]), and (\[const mom\]), density, pressure, and velocity can be expressed by using temperature, the mass flux, and momentum flux as follows: $$\rho = \frac{1}{2{\cal R}T}\left( M + \sqrt{M^2 - 4 {\cal R}j^2 T } \right),
        \label{rho}$$ $$v = \frac{1}{2j}\left( M - \sqrt{M^2 - 4{\cal R} j^2 T}  \right),        
        \label{v}$$ $$P = \frac{1}{2}\left( M + \sqrt{M^2 - 4 {\cal R}j^2 T } \right).        
        \label{P}$$ For convenience, instead of $T$, we introduce new variable $\theta$ defined by $d\theta=\kappa dT$. Using $\theta$ and equations (\[rho\])-(\[P\]), equation (\[integ eq0\]) can be rewritten as $$\frac{d^2\theta}{dx^2}  = \frac{ {\cal R}j}{2\kappa(\theta)}\left[ \frac{\gamma+1}{\gamma-1} 
        + \frac{M}{\sqrt{M^2-4{\cal R}j^2T(\theta)}}\right] \frac{d\theta}{dx}
        +\rho{\cal L}.
        \label{integ eq}$$

@SI01 and @IIK06 described a detailed numerical method to solve equation (\[integ eq\]). Since equation (\[integ eq\]) is the second order differential equation, we need two boundary conditions. At $x=-\infty$, we consider a uniform CNM with thermal equilibrium, ${\cal L}(\rho_\mathrm{c},P_\mathrm{c})=0$, where physical variables in the CNM are denoted by the subscript of ‘c’. The boundary conditions are given by $$\theta(x=-\infty)=\theta_\mathrm{c},\;\;\mathrm{and}\;\;\;\frac{d\theta}{dx}\Bigr|_{x=-\infty}=0.
        \label{bou}$$ We have two free parameters: $P_\mathrm{c}$ and $j$. Given $P_\mathrm{c}$, the density and temperature can be obtained from equation (\[eos\]) and the equilibrium 