A formation scenario of black hole-envelope systems --viscous hydrodynamics simulation in general relativity--
By performing a viscous hydrodynamics simulation in general relativity for super-Eddington accretion flows onto massive black holes of mass $M=10^5$--$10^7M_\odot$, we discuss a formation scenario for black hole-envelope systems. We consider the mass accretion rate of $a^3/G \approx 1.5 \times 10^{25} (a/10\,\mathrm{km\,s^{-1}})^3$\,g/s, comparable to the Eddington mass accretion rate of a $10^7M_\odot$ black hole, assuming that the gas temperature of the infalling matter is $\lesssim 10^4$\,K. Here, $a$ and $G$ denote the sound speed and gravitational constant. For the accretion flow, we set up a quasi-spherical Bondi-type flow in which radial inflow dominates over angular momentum in the distant region. It is found that (i) for low-mass black holes with $M \lesssim 10^6M_\odot$, a photon-trapped region forms in the inner region, and a significant viscous outflow driven near the polar region overcomes the ram pressure of the mass inflow, leading to an inflow-outflow structure; (ii) for massive black holes of $M \gtrsim 3 \times 10^6M_\odot$, the outflow is not launched, and a convective envelope around the black hole gradually develops; and (iii) irrespective of the black-hole mass, the mass accretion rate onto the black hole is of order 10\% of the Eddington accretion rate for reasonable values of the viscous coefficient. As the mass accretion rate onto the black holes is much lower than the mass growth rate of the envelope for low-mass black holes with $M\lesssim 10^6M_\odot$, the envelope mass is likely to increase until the total viscous heating rate exceeds the Eddington luminosity of the system, if the mass accretion rate is preserved to be high for $\gtrsim 10^8 (M/10^7M_\odot)$\,yrs.