Chapter 4: Accretion Disc Dynamics – From Thin to Thick

In our journey through the cosmos, we have examined how gravity, radiation, and angular momentum interplay to govern the behavior of some of the Universe's most extreme environments. In this chapter, we turn our attention to the swirling reservoirs of matter that feed compact objects—accretion discs. These structures, found around black holes, neutron stars, and even white dwarfs, serve as the cosmic engines that power energetic phenomena across the Universe. Here, we will explore how accretion discs form and evolve, beginning with their fundamental formation and structure, then delving into the elegant simplicity of the thin disc paradigm as encapsulated by the Shakura–Sunyaev model, and finally, examining the transition to thick and slim discs that arise when the flow of matter overcomes angular momentum barriers. By weaving together historical insights, modern observational evidence, and advanced computational models, this chapter aims to provide a comprehensive and engaging exploration for the PhD-level reader.

Throughout our discussion, we will build upon concepts introduced in earlier chapters—such as the balance between radiation pressure and gravity and the importance of angular momentum in regulating astrophysical processes—thereby enriching our understanding of how accretion discs mediate energy transport and mass inflow in a variety of astrophysical settings. Conceptual diagrams, such as those depicted in Figure 1 and Figure 2, will serve as mental guides to help visualize the dynamic and sometimes turbulent nature of these systems.

4.1 Formation and Structure of Accretion Discs

The formation of an accretion disc is a natural consequence of the conservation of angular momentum. When matter falls toward a compact object, such as a black hole, it rarely falls in radially. Instead, any slight rotation in the infalling gas is amplified as the material moves closer to the gravitational center. This effect is similar to an ice skater who spins faster as they pull their arms in. The resulting structure is a disc—a flattened, rotating configuration in which the material gradually spirals inward.

At the most basic level, the process begins with a cloud of gas, perhaps the remnant of a disrupted star or the ambient material in a galactic nucleus, that is gravitationally attracted to a compact object. As the gas converges, collisions and interactions among particles dissipate energy. Because angular momentum must be conserved, the gas cannot collapse directly into the compact object but instead settles into an orbiting disc. In our minds, it is useful to picture this process as a cosmic whirlpool (conceptually depicted in Figure 1), where streams of gas swirl around a central drain. The resulting disc is not uniform; rather, it is characterized by radial gradients in temperature, density, and pressure.

Within this disc, several key processes work in tandem to govern its structure and evolution:

Energy Release: As gas spirals inward, gravitational potential energy is converted into thermal energy. This energy heats the disc, causing it to emit electromagnetic radiation. The efficiency of this conversion is one of the key metrics in understanding the brightness of accretion-powered sources. Viscous Transport: In order for matter to move inward, it must lose angular momentum. Viscosity, both molecular and turbulent, acts as a mediator, transferring angular momentum outward while allowing material to drift inward. Imagine stirring a viscous liquid such as honey: the internal friction gradually enables the fluid to spread out, and in the disc, similar frictional processes help redistribute angular momentum. Vertical Structure: Although we often think of an accretion disc as a flat, two-dimensional structure, it has a finite thickness determined by the balance between the gravitational pull toward the midplane and the pressure forces pushing outward. This vertical structure is sensitive to both gas pressure and radiation pressure and can vary significantly depending on the conditions in the disc.

The overall architecture of an accretion disc is thus determined by the interplay between these factors. In the outer regions, where the gravitational field is relatively weak, the disc tends to be cooler and thicker. Closer to the compact object, where gravitational forces are much stronger, the disc is typically hotter and can become very thin if the energy is efficiently radiated away. However, under conditions of extremely high accretion rates, even the inner disc may become thick as radiation pressure and advective processes become dominant.

To summarize the formation and structure of accretion discs, consider the following bullet points: • Conservation of angular momentum causes infalling gas to settle into a rotating, disc-like structure. • Gravitational potential energy is converted into heat as matter spirals inward, providing the primary energy source for the disc. • Viscosity, whether arising from turbulence or magnetic fields, facilitates the outward transport of angular momentum. • The disc exhibits a vertical structure determined by the competition between gravitational forces and pressure gradients, leading to variations in thickness and temperature across different radii.

Over the years, both observational evidence and advanced computer simulations have enriched our understanding of these processes. For example, high-resolution observations with instruments such as the Event Horizon Telescope have begun to resolve the structures of discs around supermassive black holes, while numerical simulations have provided detailed insights into the complex fluid dynamics and magnetohydrodynamic (MHD) processes within these discs (McKinney et al. 2012; Sadowski et al. 2016).

4.2 The Thin Disc Paradigm: Shakura–Sunyaev Model

Once the basic picture of an accretion disc is established, one of the most influential theoretical frameworks comes into focus: the thin disc paradigm as formulated by Shakura and Sunyaev in the early 1970s. Their model, which remains a cornerstone of disc theory, offers a remarkably successful description of many accretion discs observed in nature, especially those where the accretion rate is moderate and the disc remains geometrically thin.

In the thin disc model, the disc is assumed to be optically thick, meaning that photons generated within the disc are absorbed and re-emitted multiple times before escaping. This assumption, combined with the notion that the disc is geometrically slender, implies that the disc's vertical thickness is much smaller than its radial extent. As a result, the disc's structure can be effectively treated as a series of concentric rings, each radiating as a blackbody with a temperature that depends on the distance from the central object.

One of the central ideas in the Shakura–Sunyaev model is the introduction of an effective viscosity parameter—a dimensionless quantity that encapsulates the efficiency of angular momentum transport in the disc. While the exact microphysical origin of this viscosity remains a subject of ongoing research, with magnetic turbulence (such as that generated by the magnetorotational instability) playing a significant role, the parameter provides a practical way to model the complex dissipation processes within the disc. To give a simple analogy, think of the disc as a layered cake in which each layer must slide past the adjacent one; the friction between the layers determines how easily the material can flow inward.

The thin disc model has several attractive features: • It provides a clear prescription for the radial distribution of temperature and luminosity. In the simplest terms, the inner regions of the disc are much hotter and radiate more intensely than the cooler, outer regions. • The model predicts that the total energy radiated by the disc is directly linked to the rate at which gravitational potential energy is released as matter moves inward. This offers a natural explanation for the observed luminosities of many accreting systems. • The steady-state assumption of the thin disc, where the accretion rate is constant with radius, allows for relatively straightforward calculations that have been remarkably successful in matching observational data from X-ray binaries and active galactic nuclei.

In conceptual terms, imagine a series of nested rings, each glowing with its own characteristic temperature. As depicted in Figure 2, the innermost ring might radiate at ultraviolet or even X-ray wavelengths, while the outer rings emit in the optical or infrared. The integrated light from all these rings produces the observed spectrum of the disc, a spectrum that is often used to infer the physical properties of the system.

Despite its successes, the thin disc model is not without limitations. It is best suited to situations where the disc remains cool enough that radiation pressure does not dominate and where the accretion rate is low to moderate. When the inflow of matter becomes too intense, or when the effects of general relativity become significant near the innermost stable orbit, the assumptions of the thin disc model can break down. This realization has paved the way for the exploration of alternative disc models that account for higher accretion rates and more complex physics.

Key points of the thin disc paradigm can be summarized as follows: • The thin disc model assumes a geometrically slender, optically thick disc where the vertical thickness is small compared to the radial extent. • Viscosity, encapsulated in a dimensionless parameter, facilitates the outward transport of angular momentum, allowing material to spiral inward. • The model successfully describes the radial variation of temperature and luminosity, matching many observed properties of accreting systems. • Its assumptions, however, limit its applicability to systems with moderate accretion rates, necessitating the development of alternative models for more extreme regimes.

The Shakura–Sunyaev model has served as a benchmark for decades, influencing countless studies and forming the basis for more advanced theories of disc dynamics. Researchers continue to refine the model by incorporating additional physics, such as magnetic fields, relativistic effects, and time-dependent behavior, thereby extending its applicability and enhancing our understanding of disc evolution (Shakura and Sunyaev 1973; Kato et al. 2008).

4.3 Transitioning to Thick and Slim Discs: Overcoming Angular Momentum Barriers

While the thin disc paradigm provides a robust framework for understanding many accreting systems, nature often presents us with situations where the accretion rate is so high that the assumptions underlying a thin disc no longer hold. Under these conditions, the disc structure can become "puffed up," transitioning to what are broadly classified as thick discs or, in certain regimes, slim discs. In these discs, the increased pressure—both gas and radiation—along with the complex interplay of advective processes, alters the dynamics of angular momentum transport and energy release.

In high accretion rate environments, the inner regions of the disc may become dominated by radiation pressure. When this occurs, the disc can no longer efficiently radiate away the energy generated by viscous dissipation. Instead, a significant fraction of the energy is advected inward with the flow of matter. This advective process acts as a kind of energy conveyor belt, carrying heat toward the central object rather than allowing it to escape locally. The result is a disc that is geometrically thicker, as the pressure support in the vertical direction increases substantially. This scenario is often referred to as a "slim disc" regime, where the disc retains some of the characteristics of a thin disc but with a pronounced vertical expansion.

The transition from a thin to a thick or slim disc is not abrupt but occurs gradually as the accretion rate increases. One way to conceptualize this transition is to imagine the difference between a calm, gently flowing stream and a turbulent, overflowing river. In the calm stream—the thin disc—the flow is orderly and the energy is dissipated gradually. In contrast, as the flow rate increases and turbulence sets in, the river overflows its banks, analogous to how a disc becomes thick when the accretion rate surpasses a certain threshold.

Several physical processes contribute to this transition: • Radiation Pressure Dominance: As the accretion rate increases, the energy generated in the inner disc becomes so intense that radiation pressure begins to dominate over gas pressure. This increased pressure inflates the disc vertically, leading to a thicker structure. • Inefficient Cooling: In thin discs, the energy produced by viscous dissipation is efficiently radiated away. However, when the density and optical depth increase, the disc can become so opaque that photons are trapped. This trapping of radiation means that energy is advected inward rather than being emitted locally. • Angular Momentum Transport: In thick discs, the mechanisms for transporting angular momentum may change. While viscous stresses remain important, additional processes—such as magnetic torques and turbulent eddies—can become more prominent. These processes help the disc overcome the angular momentum barrier that normally hinders the inward flow of matter in a thin disc. • Advective Flows: The advective transport of energy not only modifies the vertical structure but also affects the radial temperature profile. In slim discs, the temperature distribution tends to be flatter compared to the steep gradients predicted by thin disc models. This altered temperature profile can influence the emitted spectrum and provides observational signatures that help distinguish between disc regimes.

To illustrate these points, consider a conceptual diagram (as depicted in Figure 3) showing two scenarios side by side. On one side, a thin disc is represented by a narrow, flat structure with well-defined, concentric temperature gradients. On the other side, a thick disc is shown as a more vertically extended and turbulent structure, with indications of advective flows carrying energy inward. This visual contrast underscores the physical differences between the two regimes and highlights how an increase in accretion rate can fundamentally alter the disc's structure and emission characteristics.

In practical terms, the transition from thin to thick discs has significant implications for our understanding of various astrophysical phenomena. For instance, observations of ultraluminous X-ray sources and certain active galactic nuclei reveal luminosities that appear to exceed the classical limits predicted by thin disc theory. These super-Eddington sources are thought to be powered by thick or slim disc accretion, where the enhanced vertical structure and advective processes allow the system to radiate at levels that would otherwise be prohibited by the standard thin disc model.

Key bullet points summarizing the transition include: • High accretion rates lead to increased radiation pressure, which can inflate the disc vertically. • When cooling becomes inefficient, energy is advected inward rather than being radiated away, resulting in a thick or slim disc structure. • The mechanisms of angular momentum transport may change in thicker discs, with turbulent and magnetic processes playing an enhanced role. • Observational evidence from super-Eddington sources supports the existence of these alternative disc regimes, challenging traditional thin disc models.

Advanced computational simulations have been particularly instrumental in elucidating the dynamics of thick and slim discs. By solving the equations of magnetohydrodynamics in the strong gravitational field near a compact object, researchers have uncovered rich, complex behaviors that defy the simplicity of the thin disc approximation. These simulations reveal that in addition to the expected vertical expansion, thick discs may exhibit spiral structures, large-scale turbulence, and even the formation of jets—narrow streams of matter that are ejected from the disc's surface. Such findings underscore the intricate and multifaceted nature of accretion physics in extreme environments (Abramowicz et al. 1988; Narayan and Yi 1995).

Furthermore, the study of thick and slim discs is not solely of academic interest. These regimes have profound implications for our understanding of black hole growth and feedback mechanisms in galaxies. When a supermassive black hole in a galactic nucleus accretes matter at rates that push the disc into the thick regime, the altered emission and mechanical outflows can influence the surrounding interstellar medium, potentially regulating star formation and driving galaxy evolution. In this sense, the transition from thin to thick disc dynamics is a key piece in the larger puzzle of cosmic evolution.

In summary, the dynamics of accretion discs encompass a wide spectrum of behaviors, from the orderly and well-understood thin disc regime to the chaotic and energetic thick disc state. The Shakura–Sunyaev model provides a solid foundation for understanding thin discs, yet nature frequently pushes systems beyond its limits. As accretion rates increase, the interplay of radiation pressure, inefficient cooling, and complex angular momentum transport leads to the formation of thick or slim discs, characterized by significant vertical expansion and advective energy transport. These processes not only alter the observational appearance of accreting systems but also have far-reaching consequences for the evolution of the central compact object and its host galaxy.

To encapsulate the core ideas of this chapter: • Accretion discs form as a natural outcome of angular momentum conservation in infalling gas, resulting in a rotating, disc-like structure. • The thin disc paradigm, described by the Shakura–Sunyaev model, assumes a geometrically slender, optically thick disc that radiates efficiently, with viscosity facilitating the transport of angular momentum. • When accretion rates are high, the disc can transition into a thick or slim configuration. This change is driven by dominant radiation pressure, inefficient local cooling, and modified angular momentum transport processes. • Observational and computational studies support the existence of these alternative disc regimes, which play a crucial role in powering super-Eddington sources and influencing the evolution of compact objects and their host galaxies.

As we move forward in our exploration of high-energy astrophysical processes, the insights gleaned from studying accretion disc dynamics will continue to inform our understanding of extreme environments. The progression from thin to thick discs serves as a vivid example of how slight changes in accretion rate and internal physics can lead to fundamentally different structures and behaviors. By linking theoretical models with observational evidence and advanced simulations, we are steadily uncovering the mechanisms that allow nature to transport matter and energy in the most extraordinary ways.

The study of accretion discs also offers a fertile ground for interdisciplinary research, bridging the gap between plasma physics, fluid dynamics, and general relativity. For instance, the role of magnetic fields in driving turbulence and angular momentum transport has opened up new avenues of investigation, with implications not only for astrophysics but also for laboratory plasma experiments. Similarly, understanding how energy is advected in thick discs can provide insights into other systems where radiation and matter interact in complex ways.

In conclusion, accretion disc dynamics, spanning the spectrum from thin, radiatively efficient discs to thick, advective, and turbulent flows, represent one of the most fascinating areas of astrophysical research. They encapsulate the delicate balance between gravitational forces, angular momentum, and radiative processes—a balance that ultimately governs the growth and evolution of compact objects. As observational techniques and computational models continue to advance, our understanding of these discs will undoubtedly deepen, offering ever more detailed glimpses into the energetic heart of the cosmos.