Quasi-stars: accreting black holes inside massive envelopes

We study the structure and evolution of `quasi-stars', accreting black holes embedded within massive hydrostatic gaseous envelopes. These configurations may model the early growth of supermassive black hole seeds. The accretion rate on to the black hole adjusts so that the luminosity carried by the convective envelope equals the Eddington limit for the total mass, M_* + M_BH ~ M_*. This greatly exceeds the Eddington limit for the black hole mass alone, leading to rapid growth of the black hole. We use analytic models and numerical stellar structure calculations to study the structure and evolution of quasi-stars. We show that the photospheric temperature of the envelope scales as T_ph ~ M^-2/5_BHM^7/20_*, and decreases with time while the black hole mass increases. Once T_ph < 10⁴ K, the photospheric opacity drops precipitously and T_ph hits a limiting value, analogous to the Hayashi track for red giants and protostars, below which no hydrostatic solution for the convective envelope exists. For metal-free (Population III) opacities, this limiting temperature is approximately 4000 K. After a quasi-star reaches this limiting temperature, it is rapidly dispersed by radiation pressure. We find that black hole seeds with masses between 10³ and 10⁴M_solar could form via this mechanism in less than a few Myr.