A Lower Limit on the Halo Mass to form Supermassive Black Holes

We consider a scenario where supermassive black holes form through direct accumulation of gas at the centre of proto-galaxies. In the first stage, the accumulated gas forms a super-massive star whose core collapses when the nuclear fuel is exhausted, forming a black hole of $M_{\rm BH} \approx 100 M_{\sun}$. As the black hole starts accreting, it inflates the surrounding dense gas into an almost hydrostatic self-gravitating envelope, with at least 10-100 times the mass of the hole. We find that these “quasistars” suffer extremely high rates of mass loss through winds from their envelopes, in analogy to very massive stars such as eta-Carinae. Only for envelope masses greater than 2.8 \times 10^{5} (M_{\rm BH}/100 M_{\sun})^{9/11} is the envelope evaporation time-scale longer than the accretion time-scale of the black hole. This relation thus constitutes a “threshold growth line” above which quasistars can grow their internal black holes. Accretion rates can be 10 to 100 times the Eddington rate. The quasistars born in this “growth region” with 10^7-10^8 M_{\sun} can grow black holes with masses between 10^4$ to $10^5 M_{\sun}, before crossing the threshold growth line and dispersing their envelopes in less than $10^4$ yr. This scenario therefore predicts that massive black hole seeds can be found only in dark matter halos with total masses larger than about 10^9 M_{\sun}, which can provide sufficiently high accretion rates to form such massive quasistars.

💡 Research Summary

The paper investigates a specific pathway for the formation of super‑massive black hole (SMBH) seeds through the intermediate “quasistar” (QS) phase. In this scenario, a massive gas inflow into the centre of a proto‑galaxy first creates a super‑massive star whose core collapses into a modest black hole (BH) of order 100 M⊙. The nascent BH accretes at a hyper‑critical rate, inflating the surrounding dense gas into a massive, radiation‑pressure‑dominated envelope that is 10–100 times more massive than the BH. This configuration is termed a quasistar.

The authors construct a semi‑analytic model of the QS structure, treating the envelope as an n = 3 polytrope dominated by convection. Convection can transport a maximum luminosity (L_{\rm c,max}=4\pi r^{2}\rho c_{s}^{3}) (set by the requirement that convective cells remain sub‑sonic). The BH luminosity is assumed to be a fraction (\alpha\le1) of this limit, (L_{\rm BH}= \alpha L_{\rm c,max}). Inside a “transition radius” the envelope remains hydrostatic; beyond it, the radiative layer becomes super‑Eddington and drives a powerful wind. The wind’s mass‑loss rate is limited by the total luminosity and an effective opacity (\kappa_{\rm eff}) that is reduced relative to the Thomson value because of photon‑bubble‑type instabilities (as observed in η‑Carinae).

A key result is the derivation of a “threshold growth line” in the (M_{\rm BH})–(M_{\rm env}) plane. By equating the BH accretion timescale (t_{\rm accr}=M_{\rm BH}/\dot M_{\rm BH}) with the envelope evaporation timescale (t_{\rm evap}=M_{\rm env}/\dot M_{\rm wind}), they find that a quasistar can successfully grow its central BH only if

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