spherically symmetric inflow under the influence of gravitation and radiation pressure is $$\label{euler}
V\frac{dV}{dr}+\frac{1}{\rho}\left[\frac{d(P+P_r)}{dr}\right]+\frac{GM}{r^2}=0$$ In the above equation $V$ is the radial flow velocity, $P$ and $P_r$ are respectively the gas and the radiation pressure, $\rho$ is the gas density, $G$ is the gravitational constant and $M$ is the mass of the central object. The radial gradient due to the radiation pressure is given by $$\frac{dP_r}{dr}=-\frac{1}{c}\int^{\infty}_{0}\kappa_{\nu}\phi_{\nu}d\nu = -\frac{\kappa}{c}\phi$$ where $\kappa$ is a suitable frequency average of the total absorption coefficient (including scattering) and $\phi$ is the total radiative flux. If the accreting gas envelope is highly ionized, the absorption of photons is essentially due to the Thomson scattering and, in this case $$\kappa = \frac{\sigma_T}{\mu m_H}\rho$$ where $\sigma_T = 6.65\times 10^{-25}~cm^2$ is the Thomson cross-section, $\mu$ is the mean molecular weight and $m_H$ is the proton mass. If the medium is optically thin and the radiation comes essentially from the deep inside region of the envelope, then the radiative flux in the outer regions is simply $$\phi = \frac{L}{4\pi r^2}$$ Combine eqs. (2), (3), (4) and replace into the Euler equation to obtain $$V\frac{dV}{dr}+\frac{1}{\rho}\frac{dP}{dr}= -\frac{GM}{r^2}+\frac{\sigma_T L}{4\pi\mu m_H c r^2}$$ Define now the Eddington luminosity as $$\label{eddington}
L_E = \frac{4\pi GM\mu 