Shakurasunyaev ModelEdit
The Shakura-Sunyaev model is a foundational analytic description of radiatively efficient, geometrically thin accretion disks around compact objects such as black holes, neutron stars, and white dwarfs. It introduces a practical prescription for viscosity to capture how angular momentum is transported outward, allowing concrete predictions for disk temperature profiles, luminosities, and spectral properties. In practice the model is often summarized through the so-called alpha-disk framework, named after the viscosity parameter alpha that encapsulates our ignorance of the microphysics driving turbulence in the disk.
Developed in the early 1970s by Shakura-Sunyaev model, the approach provides a coherent set of equations for a thin disk in vertical hydrostatic balance, with the disk height H small compared to the radial distance R. The local energy generated by viscous dissipation is assumed to be radiated away locally, yielding tractable scaling relations for observables. This framework played a pivotal role in interpreting emissions from systems ranging from X-ray binaries to active galactic nuclei, and it laid the groundwork for imagining the multi-temperature blackbody spectra characteristic of accretion disks around compact objects like black holes and neutron stars. For many systems, the total radiative output is tied to the mass accretion rate Mdot, with a characteristic radiative efficiency eta tying L to Mdot c^2, and a common benchmark being the Eddington limit that constrains how bright an accreting object can be before radiation pressure counteracts gravity.
The model remains a workhorse because it delivers transparent, testable predictions while remaining mathematically manageable. Its core idea—the alpha viscosity—provides a compact way to relate turbulent stresses to local thermodynamics, making it possible to deduce how temperature, density, and emitted spectrum vary with radius. In the simplest implementations, the effective temperature follows a radial scaling that, for much of the disk, yields a spectrum dominated by radiation from a range of temperatures, often described as a multi-temperature blackbody spectrum. The framework also makes clear the role of key physical boundaries, such as the innermost stable circular orbit for material orbiting a black hole, which sets the inner edge of the standard thin disk and influences the high-energy end of the spectrum. See, for example, discussions of Eddington luminosity and innermost stable circular orbit in connection with the model’s predictions.
Framework and core ideas
The alpha-viscosity prescription
The Shakura-Sunyaev model posits that the kinematic viscosity nu in the disk can be written as nu = alpha c_s H, where c_s is the local sound speed and H is the disk’s scale height. The dimensionless alpha parameter encapsulates our ignorance about the microphysical mechanism of angular-momentum transport, with typical applications using alpha in the range of a few thousandths up to around 0.4. The idea is that turbulent stresses scale with local pressure, so the transport is tied to the thermodynamics of the disk rather than an external input. See viscosity and alpha-disk for related concepts and historical development.
Disk structure and energy balance
In the standard thin-disk picture, the disk is in vertical hydrostatic balance, and the rate at which energy is dissipated by viscosity locally equals the radiative loss from the disk surface. This balance yields radial profiles for quantities such as surface density Sigma, temperature, and emitted flux F, which in turn determine the effective temperature T_eff(R) and the emergent spectrum. For most of the disk, the radiation is effectively diffusive, which underpins the classic multi-temperature blackbody interpretation of the spectrum. The inner regions are influenced by relativistic effects near the compact object, especially when the central body is a black hole, where the location of the innermost stable circular orbit and relativistic corrections matter for the high-energy tail of the spectrum. See multi-temperature blackbody and innermost stable circular orbit for further context.
Inner boundary conditions and relativistic context
Around black holes, the standard thin-disk formulation is augmented by the physics of strong gravity, notably the ISCO that acts as the effective inner boundary. The precise treatment of the inner boundary and relativistic corrections can alter the innermost-disk emission, influencing interpretations of high-energy observations and the inferred spin of the black hole. See innermost stable circular orbit and general relativity for related topics.
Observables and limitations
The alpha-disk framework yields predictions for how the disk’s spectrum evolves with accretion rate and radius, how luminosity scales with Mdot, and how spectral features relate to disk temperature structure. It remains a baseline model against which more sophisticated simulations and observations are measured. However, the model’s simplifying assumptions—namely a steady, axisymmetric, radiatively efficient, geometrically thin disk with a prescribed viscosity—mean it cannot capture all time-dependent phenomena, strong-radiation-pressure regimes, or nonlocal transport effects. See radiative cooling and thermal instability for discussions of some of these caveats.
Debates and developments
The mechanism of angular-momentum transport
While the alpha prescription provides a pragmatic parameterization, many in the field view it as a phenomenological shortcut rather than a fundamental microphysical description. The magnetorotational instability (magnetorotational instability) has become the leading candidate for driving disk turbulence and enabling angular-momentum transport in many accretion flows. Modern work uses global and local magnetohydrodynamic simulations, including general-relativistic treatments (GRMHD), to derive turbulence properties and effective transport coefficients from first principles. See magnetorotational instability and GRMHD for further detail.
Beyond the thin disk: high accretion rates and alternative models
In regimes where radiation pressure dominates, or when the accretion rate approaches or exceeds the Eddington limit, the thin-disk assumptions can break down. Variants such as the slim disk and the advection-dominated accretion flow (ADAF) models describe disks in which advection carries a significant fraction of the energy inward rather than radiating it locally. These alternatives, and their observational signatures, are central to contemporary debates about the applicability of the Shakura-Sunyaev framework across the full range of accretion phenomena. See slim disk and advection-dominated accretion flow.
Time variability, instabilities, and non-steady disks
Real accretion systems often exhibit variability, outbursts, and non-steady behavior that stretch the assumptions of stationary, axisymmetric models. Researchers increasingly study time-dependent versions of the disk equations and the role of magnetic fields, disk winds, and coronae in shaping variability. While the Shakura-Sunyaev model remains valuable for its clarity and analytical insight, it is routinely complemented by numerical simulations and time-dependent analyses. See time variability and wind (astrophysics) for related topics.