Elementary processes in astrophysics (NAST024)

### Shakura-Sunyaev model of disc accretion

Model of disc accretion presented in this section is one of the first models of accretion onto compact objects that was able to explain properties of quasars discovered several years before. Later on, a broad variety of models of disc-like accertion was formulated, nevertheless, the Shakura-Sunyaev solution is still viable and considered to be valid in many astrophysical systems.

Derivation of the "standard" solution is in Sections 4.3 and 4.4 of the textbook by Vladimir Karas. Some steps can be found in my hand-notes in a bit more detail. Derivations presented in these texts (as well as in many other textbooks) are based on somewhat heurisitc approach to derivation of the angular momentum transfer. Also the equation (4.25) which is the azimuthal component of the Euler equation is actually not derived from the basic form of the Euler equation. For more rigorous derivation of the hydrodynamic equations governing evolution of the standard accretion disc see another set of my hand notes which shows that these equations can be derived under the classical Naviere-Stokes approximation in cylindrical coordinates.

What to concentrate on:

• derivation of the local dissipation measure, $D\left(R\right)$
• derivation of the closed set of equations (see top of fourth page of my hand-notes) that lead to the so called "standard" solution of the steady state disc accretion. Note that for closing the set of equations, a particular prescription for opacity needs to be given. Analytic solution is possible only if either the radiation pressure or the gas pressure is taken into account. Also consider the very approximative form of some equations, e.g., the vertical component of the Euler equation
• keep in mind that several simplifying assumptions are considered in derivation of spectrum of the standard disc (e.g., isotropic black-body radiation from each point of the disc surface; Doppler shift neither due to fluid motion, nor due to the deep potential well of the central object is taken into account). The shape of the spectrum as sketched on the last page of my hand-notes can be intuitively obtained as a superposition of local black body spectra with different temperatures. In order to derive the power-law slope of the middle region, ${K}_{\mathsf{B}}{T}_{\mathsf{out}}\ll \nu \ll {K}_{\mathsf{B}}{T}_{\mathsf{in}}$, consider the shape of the integrated function – its value below ${x}_{\mathsf{in}}$ and above ${x}_{\mathsf{out}}$ respectively is small, i.e. changing the limits to zero and infinity leads to only small overestimate of the integral. The advantage of the integral from zero to infinity lies in that it does not depend on the value of frequency, $\nu$.