m_H c}{\sigma_T} = 1.76 \times 10^{38}\left(\frac{M}{M_\odot}\right)~erg.s^{-1}$$ In this case, eq. (5) can be rewritten as $$\label{euler2}
V\frac{dV}{dr}+\frac{1}{\rho}\frac{dP}{dr}= -\frac{GM(1-\Gamma)}{r^2}$$ where $\Gamma = L/L_E$.

Assume that the gas equation of state is given by $P = K\rho^{\gamma}$ and define also the adiabatic sound velocity as $a^2 = \gamma(P/\rho)$. Under these conditions, using the mass conservation equation to express the mass density gradient, after some algebra, eq.\[euler2\] can be recast as $$\label{euler3}
V\left(1-\frac{a^2}{V^2}\right)\frac{dV}{dr} = \frac{2a^2}{r}-\frac{GM(1-\Gamma)}{r^2}$$

The ”critical” point of the flow in the usual mathematical sense corresponds to the point where both sides of eq. \[euler3\] vanish. Hence, in order to have the continuity of the flow through the critical point, two conditions must be simultaneously satisfied, namely $$\label{criticalvelocity}
V_* = a_*$$ and $$\label{criticalradius}
r_* = \frac{GM(1-\Gamma)}{2a_*^2}$$ for the critical velocity and the critical radius respectively. These relations imply that the critical and the sonic points of the flow coincide (this is not always the case) and that the luminosity radiated from inside must be [**[smaller]{}**]{} than the Eddington value in order that the critical radius be real. Note that once the critical point is surpassed, the left side of eq. \[euler3\] is negative, requiring imperatively that the right side be also negative 