The mechanism by which the supermassive black holes that power bright quasars at high redshift form remains unknown. One possibility is that, if fragmentation is prevented, the monolithic collapse of a massive protogalactic disc proceeds via a cascade of triaxial instabilities and 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. Their envelopes satisfy the equations of stellar structure so the Cambridge STARS code is modified to model them. Analysis of the models leads to an extension of the classical Schönberg– Chandrasekhar limit and an exploration of the implications of this extension for the evolution of main-sequence stars into giants.

In Chapter 1, I introduce the problem posed by the supermassive black holes that power high-redshift quasars. I discuss potential solutions and describe the conditions under which a quasi-star might form. 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 around black holes. To improve the accuracy of my models, I incorporate corrections owing to special and general relativity into a variant of the STARS code that includes rotation. The evolution of these quasi-stars is qualitatively different from those described in Chapter 3. Most notably, the 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, for the black holes in quasi-stars, is in essence the same as the Schönberg–Chandrasekhar limit. The analysis demonstrates how other similar limits are related and 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.