Accretion disks

In this lecture the basic theory of accretion disks is reviewed, with emphasis on aspects relevant for X-ray binaries and Cataclysmic Variables. The text gives a general introduction as well as a selective discussion of a number of more recent topics.

💡 Research Summary

The lecture paper provides a comprehensive review of the fundamental theory of accretion disks with a focus on applications to X‑ray binaries and cataclysmic variables (CVs). It begins by outlining the astrophysical context: matter with excess angular momentum spirals inward toward compact objects such as neutron stars, black holes, or white dwarfs, forming a rotating, flattened structure known as an accretion disk. The author then derives the governing equations—mass conservation, angular momentum transport, and energy balance—and simplifies them under the thin‑disk approximation (disk height H ≪ radius R). This reduction yields a one‑dimensional radial diffusion equation that describes the evolution of the surface density Σ(r, t).

A central element of the model is the prescription for viscosity. The paper adopts the classic α‑viscosity formalism introduced by Shakura and Sunyaev, where the kinematic viscosity ν is expressed as ν = α c_s H, with c_s the local sound speed and α a dimensionless parameter encapsulating the effects of turbulence and magnetic stresses. Typical values of α lie between 0.01 and 0.1, a range that is later justified by magnetorotational instability (MRI) simulations.

The thermal structure of the disk is addressed through vertical hydrostatic equilibrium and radiative transfer. By integrating the vertical energy equation, the author obtains temperature profiles that, when combined with the local black‑body approximation, produce multi‑temperature spectra. These spectra can be directly compared with observed X‑ray, UV, and optical data to infer disk parameters such as mass accretion rate Ṁ and inner radius.

A major portion of the review is devoted to the thermal‑viscous instability that arises near the hydrogen ionization temperature (~6500 K). The cooling and heating curves intersect to form an S‑shaped equilibrium curve in the Σ‑T plane. The disk can therefore exist in a hot, high‑viscosity state or a cool, low‑viscosity state, and transitions between these states generate limit‑cycle outbursts. This mechanism underpins the dwarf‑nova eruptions in CVs (the Disk Instability Model) and the state transitions observed in X‑ray binaries between hard and soft spectral states. The paper presents the time‑dependent diffusion equation, boundary conditions, and numerical techniques used to simulate these cycles.

The microscopic origin of the effective viscosity is explored through MRI. The author summarizes the Balbus‑Hawley analysis, showing that a weak vertical magnetic field in a differentially rotating, conducting fluid leads to exponential growth of perturbations, driving turbulence that transports angular momentum outward. Global MHD simulations confirm that MRI naturally yields α values consistent with those required by observations, and they reproduce the characteristic variability timescales seen in light curves.

Relativistic extensions of the thin‑disk model are also discussed. Near a black hole, general relativistic effects modify the orbital frequency, the location of the innermost stable circular orbit (ISCO), and the efficiency of energy release. The paper adopts the Novikov‑Thorne formalism, which incorporates frame‑dragging for spinning black holes and imposes a zero‑torque condition at the ISCO. These relativistic corrections are essential for interpreting the broad Fe Kα line profiles and the high‑energy tail of the spectra in black‑hole X‑ray binaries.

The review briefly covers low‑efficiency, advection‑dominated accretion flows (ADAFs), which become relevant at low mass‑accretion rates (Ṁ ≪ Ṁ_Edd). In ADAFs, most of the viscously generated heat is advected into the black hole rather than radiated, leading to hard X‑ray spectra and low radiative efficiencies. This model provides a framework for understanding low‑luminosity X‑ray binaries and low‑luminosity active galactic nuclei.

Finally, the paper surveys recent developments, including three‑dimensional global MHD simulations that couple MRI turbulence with realistic radiative transfer, and the emerging field of multi‑messenger astronomy where accretion‑disk physics is linked to gravitational‑wave observations of compact‑object mergers. The author emphasizes that upcoming observatories such as XRISM and Athena will deliver high‑resolution spectroscopy capable of testing detailed predictions of disk structure, ionization states, and relativistic effects. In summary, the lecture offers a unified roadmap from the basic steady‑state thin‑disk theory through time‑dependent instability models to cutting‑edge numerical and observational frontiers, establishing a solid foundation for both current research and future advances in accretion‑disk astrophysics.