Quasi-stars and the Sch"onberg-Chandrasekhar limit
The mechanism by which the supermassive black holes that power bright quasars at high redshift form remains unknown. One possibility is that … the monolithic collapse of a massive protogalactic disc … leads to the formation of a quasi-star: a growing black hole, initially of typical stellar-mass, embedded in a hydrostatic giant-like envelope. Quasi-stars are the main object of study in this dissertation. … In Chapter 1, I introduce the problem posed by the supermassive black holes that power high-redshift quasars. … In Chapter 2, I outline the Cambridge STARS code and the modifications that are made to model quasi-star envelopes. In Chapter 3, I present models of quasi-stars where the base of the envelope is located at the Bondi radius of the black hole. The black holes in these models are subject to a robust upper fractional mass limit of about one tenth. In addition, the final black hole mass is sensitive to the choice of the inner boundary radius of the envelope. In Chapter 4, I construct alternative models of quasi-stars by drawing from work on convection- and advection-dominated accretion flows … The evolution of these quasi-stars is qualitatively different from those described in Chapter 3. … [T]he core black holes are no longer subject to a fractional mass limit and ultimately accrete all of the material in their envelopes. In Chapter 5, I demonstrate that the fractional mass limit found in Chapter 3 … is in essence the same as the Sch"onberg-Chandrasekhar limit. The analysis demonstrates … that limits exist under a wider range of circumstances than previously thought. A test is provided that determines whether a composite polytrope is at a fractional mass limit. In Chapter 6, I apply this test to realistic stellar models and find evidence that the existence of fractional mass limits is connected to the evolution of stars into the red giants.
💡 Research Summary
The dissertation tackles the long‑standing problem of how supermassive black holes (SMBHs) with masses ≳10⁹ M⊙ managed to appear less than a billion years after the Big Bang, as inferred from luminous high‑redshift quasars. The author proposes that a massive protogalactic disc can undergo monolithic collapse, forming a “quasi‑star”: a hydrostatic, giant‑like envelope surrounding a nascent stellar‑mass black hole (BH). The work is organized into six chapters, each building a coherent theoretical and numerical framework.
Chapter 1 reviews the observational constraints on early SMBHs and surveys existing seed‑formation scenarios (Population III remnants, direct collapse, runaway stellar collisions). It argues that none of these pathways can reliably produce the required masses within the limited cosmic time, motivating the quasi‑star concept.
Chapter 2 describes the computational tool – the Cambridge STARS stellar evolution code – and details the modifications required to treat a composite object consisting of a central BH and an extended envelope. New boundary‑condition modules allow the inner radius to be set either at the Bondi radius or at a prescribed fraction of it, and incorporate prescriptions for radiative diffusion, convective transport, and advection‑dominated accretion flows (ADAF).
Chapter 3 presents the first suite of models in which the envelope’s inner edge coincides with the BH’s Bondi radius. The envelope is modeled as a near‑isothermal, radiation‑pressure‑dominated polytrope (γ≈4/3). Numerical integration reveals a robust fractional‑mass limit: once the BH mass exceeds roughly ten percent of the total quasi‑star mass, the structure becomes unstable. The instability manifests as rapid envelope expansion or collapse, preventing further BH growth. Moreover, the final BH mass is highly sensitive to the exact choice of the inner boundary radius, underscoring the importance of boundary physics.
Chapter 4 explores an alternative class of models that replace the Bondi‑radius condition with a convection‑dominated or ADAF‑type inner region. Here the envelope’s inner layers efficiently transport energy, allowing the BH to accrete at a much higher rate. Consequently, the BH can consume essentially the entire envelope, and the ten‑percent limit disappears. This contrast demonstrates that the mass‑fraction ceiling is not a universal property of quasi‑stars but depends critically on the energy‑transport regime inside the envelope.
Chapter 5 provides the theoretical link between the ten‑percent limit observed in Chapter 3 and the classic Schönberg‑Chandrasekhar (S‑C) limit for stellar cores. By treating the quasi‑star as a composite polytrope (core + envelope) the author derives a generalized “fractional‑mass limit test.” The test evaluates whether a given core‑to‑total mass ratio, together with the polytropic indices of core and envelope, places the configuration at the brink of instability. Applying this test to the Bondi‑radius models reproduces the ≈0.1 limit, confirming that the same underlying mathematics governs both ordinary stellar cores that have exhausted nuclear burning and BH‑enveloped quasi‑stars.
Chapter 6 extends the fractional‑mass limit test to realistic stellar evolution tracks computed with the unmodified STARS code. The analysis shows that as low‑ and intermediate‑mass stars ascend the red‑giant branch, their inert helium cores approach the S‑C limit, triggering envelope expansion and the characteristic red‑giant morphology. This parallel suggests that the physical mechanism limiting BH growth in quasi‑stars is intimately related to the well‑known core‑mass limit that drives red‑giant evolution.
Overall, the dissertation makes several key contributions: (1) it introduces a physically motivated quasi‑star model as a viable pathway to early SMBH formation; (2) it demonstrates that the inner boundary condition and the dominant energy‑transport process dictate whether a BH is capped at ~10 % of the total mass or can accrete the whole envelope; (3) it unifies this behavior with the classic Schönberg‑Chandrasekhar limit through a generalized polytropic analysis; and (4) it shows that the same fractional‑mass instability underlies both exotic quasi‑stars and ordinary stellar evolution into red giants. By bridging high‑redshift quasar observations, black‑hole accretion physics, and stellar structure theory, the work provides a robust theoretical scaffold for future numerical simulations and observational tests of early SMBH growth.