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,
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,
consider the shape of the integrated function – its value below
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, .