or, equivalently, that $\Gamma < 1$. In other words, the spherical accretion of an optically thin envelope requires sub-Eddington conditions otherwise the inflow cannot be established. As we will see later, this requirement is weakened when the inflow geometry is modified as, for instance, in the case of an accretion disk.

As it was mentioned previously, many authors assume that the central black hole accretes mass with the envelope radiating near the Eddington limit. Since the Eddington luminosity is proportional to the mass of the black hole (see eq. 6), it results an exponential growth with a timescale $$\tau_E = \frac{\eta}{(1-\eta)}\frac{c\sigma_T}{4\pi\mu m_H G} = 3.22\times 10^8\frac{\eta}{(1-\eta)}~yr$$ where $\eta$ is the accretion efficiency. Such a short timescale in often considered as an argument to explain the presence of SMBHs at high-z. However, as we have seen above, the Eddington limit is derived under conditions in which the envelope is optically thin and the opacity is due only to the Thomson scattering. The optical depth of the envelope is given by $$\tau(r,\infty) = \int^{\infty}_r \frac{\sigma_T}{\mu m_H}\rho(r')dr' = 7.8\times 10^{-6}\left(\frac{M}{M_\odot}\right)n_{\infty}$$ and the gravitational radius was taken as a lower limit in order to obtain the numerical value on the left side of the 