Accretion discs
Marek A. Abramowicz and Odele Straub (2014), Scholarpedia, 9(8):2408.  doi:10.4249/scholarpedia.2408  revision #144791 [link to/cite this article] 
Introduction
The observable physical quantity of radiation produced in accretion discs is the luminosity. As photons carry momentum and thus can exert pressure there is a maximum possible luminosity at which gravity is able to balance the outward pressure of radiation. The limit for a steady, spherically symmetric accretion flow is given by the Eddington luminosity,
\( L_{\rm Edd} = G M \frac{4 \pi m_H \, c}{\sigma_T} = 1.26 \times 10^{38} \; (M/M_{\odot}) \; {\rm erg/s}, \)
where G is the gravitational constant, c is the speed of light, M is the mass of the gravitating body, \(m_H\) and \(M_{\odot}\) are the hydrogen and solar mass, respectively, and \(\sigma_T\) is the Thompson crosssection. The Eddington limit is used as a unit to quantify the luminosity of an object. Since accretion discs are not spherical and often have additional stresses (see above) that can counteract the radiation pressure along with gravity, they may be brighter than this limit and radiate at superEddington luminosity.
A short summary of the basic properties of accretion discs  
protoplanetary systems 
white dwarfs (WD) in cataclysmic binaries 
black hole (BH) or neutron star (NS) binaries 
quasars and other AGN 
gamma ray burst (GRB) sources 

Basic physics  






Accreting central object  






Disc size  
disc = \(10^{11}  10^{15} \) cm 
disc = \(10^{9}  10^{10} \) cm 
disc = \(10^{6}  10^{11}\) cm 
disc = \(10^{6}  10^{11}\) cm [\(M/M_{\odot}\)] 
disc = \(10^{5}  10^{?}\) cm [\(M/M_{\odot}\)] 

Midplane temperature (inner  outer disc)  
\(T_c = 10^3  10^1\) K  \(T_ = 10^5  10^3\) K  \(T_c = 10^7  10^3\) K  \(T_c = 10^5  10^2\) K  \(T_c = 10^{10}  10^9\) K  
Luminosity  
\(L << L_{\rm Edd}\)  \(L << L_{\rm Edd}\)  \(L << L_{\rm Edd}\) \( L ~ L_{\rm Edd}\) 
\(L < L_{\rm Edd}\) \(L > L_{\rm Edd}\), \(L >> L_{\rm Edd}\) 
\(L >> L_{\rm Edd}\)  
Theoretical disc models  






Angular momentum transport  






Cooling processes  






References  
Hartmann (1998) Alexander (2008) 
Frank, King & Raine (2002) Warner (2003) 
Frank, King & Raine (2002) Remillard & McClintock (2006) 
Krolik (1998) online compilation 
Popham, Woosley & Fryer (1999) Di Matteo, Perna & Narayan (2002) 
Accretion discs in the Universe
Accretion discs in young stellar objects (YSOs)
During star formation, the central part of a dense molecular cloud collapses to a protostar with a gaseous envelope that finally settles to a rotating protoplanetary accretion disc. Sedimentation and selfgravity in such discs trigger the formation of planets and planetary systems. Protostars are heavily embedded in surrounding gas and dust and for this reason visible only in the infrared, millimetre or submillimetre wavelength bands. Most of the material that goes into forming a star is accreted through a circumstellar disc and in this process the protostellar system drives an energetic bipolar jet and outflow into its surroundings. The least evolved protostars are surrounded by remnant protoplanetary accretion discs. Based on the spectral energy distribution in the infrared and visible light, YSOs are divided into five classes (0IV), associated with their evolutionary stages. Class 0 refers to collapsing molecular clouds, protoplanetary discs exist in classes IIII, and class IV contains the zeroage mainsequence star. A related issue: the extrasolar planets 
Accretion discs in cataclysmic variables (CVs)
CVs are binary star systems consisting of a white dwarf ("primary") and a normal star ("secondary", or "companion"). Typically, the CVs have sizes comparable to the EarthMoon system, and orbital periods of a few hours. When the outer layers of the companion overflow the "Roche lobe", the companion loses matter through the first Lagrangian point \(L_1\) of the rotating binary system. When the white dwarf is only weakly magnetised, the matter forms an accretion disc around it and eventually reaches its surface. "Dwarf novae" (DN) are CVs that show outbursts lasting for about a week and separated by weeks to months of quiescence. U Gem is the prototype of dwarf novae. The brightness in the visible light of U Gem increases 100fold every 120 days or so, and returns to the original level after a week or two. The DN phenomenon is due to a specific accretion disc limitcycle instability, tidal torques, and fluctuations in the masstransfer rate from the secondary. The geometry of accretion is very different in magnetic CVs, where accretion discs are either truncated or absent and accretion occurs along the magnetic field lines. There is a solid observational evidence for accretion discs in CVs based on very accurate photometry and spectroscopy.

Accretion discs in Quasars and other active galactic nuclei (AGN)
Most galaxies have supermassive (millions to billions of solar masses) black holes at their centres (nuclei). In AGN, the black hole accretion produces radiative power that usually outshines its host galaxy. The accretion disc is surrounded by a hot corona which contains clouds of gas. Fast moving clouds close to the disc produce broad lines and slow moving clouds further away from the disc produce narrow lines in the AGN spectra. The observational appearance of an AGN may be affected by the presence of a large outer dust torus.
Property  Quasars  Seyfert Galaxies 
Radio Galaxies  Blazars  
Galaxy type  Spiral, Elliptical  Spiral  Giant Elliptical  Elliptical  
Appearance  compact, blue  compact, bright nucleus  elliptical  bright, starlike  
Maximum luminosity  1001,000 Milky Way  comparable to bright Spirals  strong radio  10,000 Milky Way  
Continuum spectrum  nonthermal  nonthermal  nonthermal  nonthermal  
Absorption lines  yes  none  yes  none  
Variability  days to weeks  days to weeks  days  hours  
Radio emission  some  weak  strong  weak  
Redshifts  z > 0.5  z ~ 0.5  z < 0.05  z ~ 0.1 
AGN are generally divided into two families, the "radio loud" and the "radio quiet", depending on whether they exhibit jets or not. In each family several types of AGN are distinguished by their emission properties. Among them, the spiral galaxies with broad and narrow emission lines (Seyfert 1) or galaxies with just narrow emission lines (Seyfert 2) and their counterparts in the radio loud family, the broad line and/or narrow line radio galaxies. The absence of broad lines in type 2 galaxies has been attributed to a partially obscured inner disc by the outer dust torus when the viewing angle is above ~60°. Since high luminosity Seyfert galaxies are preferably of type 1 the alternative scenario of a receding/advancing torus being responsible for type 1/type 2 is plausible (C. Simpson, 1998). The most luminous beacons in the universe are the radio loud and radio quiet quasars (i.e. quasistellar objects). They are observed up to highest redshifts, implying cosmological distance and gigantic energy output. Therefore, despite their name, quasistellar objects are anything but stars. However, because quasars shine at such large distances, it is not possible to resolve the bright core. Recent observations detect jets and nebulosity around some of them.
Accretion discs in Microquasars and Xray binaries
Mainsequence stars in orbit around an accreting neutron star or black hole are called neutron star binaries (NSB) or black hole binaries (BHB), respectively, and are common objects in a galaxy. Young neutron stars are often strongly magnetised so that their accretion discs are truncated by the magnetic field or do not exist at all. In these cases matter is lead by partial or total column accretion to the neutron star.
Neutron stars and black holes are more compact than white dwarfs. NSB and BHB show thus, due to their stronger gravitational potential, a much higher energy output in comparison to CVs: Their spectral energy distribution is observed up to the Xray regime. For this reason they are also called Xray binaries (XRB). Depending on the mass of the companion star, XRB are roughly divided into two categories:
 Lowmass Xray binaries, LMXB (in Figure 6, systems with coloured companion stars)
 Highmass Xray binaries, HMXB (in Figure 6, systems with white companion stars)
The HMXB family has two subclasses, (i) the soft Xray transients that can contain a NS or a BH and exhibit quasiperiodic outbursts and (ii) the pulsars that contain a NS and emit a bean of radiation along the axis of the magnetic field (not necessarily aligned with the NS rotational axis).
Many, if not all, black hole Xray binaries go through periods during which they emit relativistic twin jets that propagate along the rotational axis of the compact object. They then look like scaled down quasars and are thus often called microquasars.
Accretion discs in gamma ray bursts (GRBs)
The most energetic explosions seen in the universe are gammaray bursts. They are short, collimated flares of lowenergy \(\gamma\)rays with relativistic emission simultaneously also at longer wavelengths (e.g., Xray flashes, radio jets). Observations indicate that GRBs are cosmological and followed by slowly fading afterglows. The duration of GRB prompt emission can last from 0.01  2 seconds (short bursts) up to 2  500 seconds (long bursts) and may be explained by merging compact objects or failed supernovae (collapsars), respectively. Afterglows on the other hand are observed and monitored from a couple of days up to several years. All evidence on the origin of the inner engines (i.e., mergers, collapses, pulsars) of GRBs is deduced indirectly. Energetic requirements suggest, however, a similar configuration of the end products: the formation of a solarmass black hole surrounded by a massive debris disc (~ 0.1 M_{sun}) with a huge accretion rate. The time scale of the burst is determined by the accretion time of this disc. According to these time scales accretion discs in GRBs are most likely hyperaccreting. This means, the temperatures and densities at the required accretion rates are such, that neutrino production is switched on and the electrons are mildly relativistic and degenerate. Generally, GRBs seem to show similarities to radioloud AGN and galactic microquasars, since all these systems eject strongly collimated, more or less relativistic flows of matter and involve accretion onto a black hole.
 Piran T., 2005, "The Physics of GammaRay Bursts", Rev.Mod.Phys., 76, 1143
 GammaRay Burst Online Index
Basic physics of accretion discs
The black hole gravity
The black hole gravitational field is described by three parameters: mass \(M\ ,\) angular momentum \(J\) and charge \(Q\ .\) It is convincingly argued that the astrophysical black holes relevant for accretion discs are uncharged, \(Q = 0\ .\) They are described by the stationary and axially symmetric Kerr geometry, with the metric \(g_{\mu\nu}\) given in the spherical BoyerLindquist coordinates \(t, \phi, r, \theta\) by the explicitly known functions of the radius \(r\) and the polar angle \(\theta\ ,\) which are listed in the table below. The table also gives the contravariant form of the metric, \(g^{\mu\nu}\ ,\) defined by \(g^{\mu\beta}\,g_{\nu\beta} = \delta^{\mu}_{~\nu}\ .\) It is defined, \(\Delta = r^2  2Mr + a^2\ ,\) \(\Sigma = r^2 + a^2\cos^2\theta\ .\) The signature \((+\,\,\,)\) is used.
The mass and angular momentum have been rescaled into the \(c = G = 1\) units, \(M \rightarrow GM/c^2\ ,\) \(J \rightarrow a = J/c\ .\) For a proper black hole solution it must be \(\vert a \vert \le M\ ,\) and the metric with \(\vert a \vert > M\) corresponds to a naked singularity. The Penrose cosmic censor hypothesis (unproved) states that there are no naked singularities in the Universe.
\(g_{\mu\nu}\)   \(g^{\mu\nu}\)  
\(t\)  \(\phi\)  \(r\)  \(\theta\)  \(t\)  \(\phi\)  \(r\)  \(\theta\)  
\(t\)  \(1  2\,M\,r/\Sigma\)  \(4\,M\,a\,r\sin^2\theta/\Sigma\)  \(0\)  \(0\)  
\((r^2 + a^2)^2/\Sigma\,\Delta\) \( a^2\Delta\sin^2\theta/\Sigma\,\Delta\) 
\(2M\,\,a\,r/\Sigma\,\Delta\)  \(0\)  \(0\) 
\(\phi\)  \(4\,M\,a\,r\sin^2\theta/\Sigma\) 
\((r^2 +
a^2)\sin^2\theta\)  \(0\)  \(0\)   \(2M\,\,a\,r/\Sigma\,\Delta\) 
\(\frac{\Deltaa^2\sin^2\theta}{\Delta\Sigma\sin^2\theta}\)  \(0\)  \(0\) 
\(r\)  \(0\)  \(0\)  \(\Sigma/\Delta\)  \(0\)   \(0\)  \(0\)  \(\Delta/\Sigma\)  \(0\) 
\(\theta\)  \(0\)  \(0\)  \(0\)  \(\Sigma\)   \(0\)  \(0\)  \(0\)  \(1/\Sigma\) 
\[ \Omega = \frac{j\,g_{tt} + g_{t\phi}}{j\,g_{t\phi} + g_{\phi\phi}}, ~~~ j =\frac{\Omega\,g_{\phi\phi} + g_{t\phi}}{\Omega\,g_{t\phi} + g_{tt}}, ~~~ U_{\rm{eff}} = \frac{1}{2} \ln \left( g^{tt}  2j\,g^{t\phi} + j^2\,g^{\phi \phi}\right). \]  \((3.2)\) 
\[ \Omega_K = \frac{c^3}{GM}\left( {r_*}^{3/2} + {a_*}\right)^{1}, ~~~ \omega_R^2 = \Omega_K^2 \left( 1  6{r_*}^{1} + 8{a_*}\,{r_*}^{3/2}  3{a_*}^2\,{r_*}^2 \right), ~~~ \omega_Z^2 = \Omega_K^2 \left( 1  4{a_*}\,{r_*}^{3/2} + 3{_*a}^2\,{r_*}^{2} \right). \]  \((3.3)\) 
Stable circular Keplerian orbits exist only with radii greater than the radius of ISCO (the innermost stable circular orbit radius). All Keplerian orbits closer to the black hole than ISCO are unstable: without an extra support by nongravitational forces (i.e. pressure or magnetic field) matter cannot stay there orbiting freely, but instead it must fall down into the black hole. This strongfield property of Einstein's gravity, absent in Newton's theory, is the most important physical effect in the black hole accretion disc physics.
Dynamical, thermal and viscous processes
 Dynamical processes occur with the timescale \(t_{\rm dyn}\) (a time in which pressure force adjusts to combined gravitational and centrifugal forces).
 Thermal processes occur with the timescale \(t_{\rm the}\) (a time in which the entropy redistribution occurs due to dissipative heating and cooling processes (in particular radiation).
 Viscous processes occur with the timescale \(t_{\rm vis}\) (a time in which angular momentum distribution changes due to torque caused by dissipative stresses).
 O.M. Blaes, Physics of luminous accretion discs around black holes, Les Houches 2002 (excellent introduction to the accretion disc physics; short, but containing all relevant equations)
 Gordon Ogilvie, March 2005, lectures at Cambridge University (very pedagogical, highly recommended for students)
Dynamical processes, with the timescale \(t_{\rm dyn}\)

\[ \left[ \left(\frac{\partial U_{\rm eff}}{\partial r}\right)_j = 0 \right] \rightarrow \left[ \Omega_{\rm{K}}^2 = \Omega_{\rm{K}}^2(r) \right], ~~~\omega_r^2(r) = \left(\frac{\partial^2 U_{\rm eff}}{\partial r^2}\right)_j, ~~~\omega_z^2(r) = \left(\frac{\partial^2 U_{\rm eff}}{\partial z^2}\right)_j, \]  \((3.1)\) 
where \(j\) is the specific angular momentum, and derivatives are taken at the symmetry plane \(z = 0\ .\) Small (epicyclic) oscillations around the circular orbit \(r = r_0 = const\ ,\) \(z = 0\) are governed by \( \delta{\ddot r} + \omega^2_r\,\delta r = 0\ , \) \( \delta{\ddot z} + \omega^2_z\,\delta z = 0\ , \) with solutions \( \delta{r} \sim \exp( i\omega_r t)\ , \) \( \delta{z} \sim \exp( i\omega_z t)\ , \) which are unstable when \(\omega^2_r < 0\) or \(\omega^2_z < 0\ .\) In Newton's gravity \(U_{\rm eff} = \Phi + j^2/2r\ .\) A spherical Newtonian body has the gravitational potential \(\Phi = GM/r\ .\) Thus, in this case, \(\Omega_{\rm K}^2 = \omega_r^2 = \omega_z^2 = GM/r^3 > 0\ ,\) i.e. all slightly noncircular orbits are closed and all circular orbits are stable.
In Einstein's gravity, for a spherical body, it is \(\Omega_{\rm K}^2 = \omega_z^2 > \omega_r^2\ ,\) i.e. noncircular orbits are not closed. In addition, for circular orbits with radii smaller than \(6GM/c^2\ ,\) it is \(\omega_r^2 < 0\ ,\) which indicates the dynamical instability of these orbits. We describe this and other aspects of the black hole gravity that are most relevant to the accretion disc physics in subsection The black hole gravity of this Scholarpedia article.
Paczynski and Wiita (1980) realised that by a proper guess of an artificial Newtonian gravitational potential, \(\Phi = GM/(r  r_G)\) (with \(r_G = 2GM/c^2\)), one may accurately describe in Newton's theory the relativistic orbital motion, and in particular the existence of ISCO. Paczynski's model for the black hole gravity became a very popular tool in the accretion disc research. It is used by numerous authors in both analytic and numerical studies. Effects of special relativity have been added to Paczynski's model by Abramowicz et al.(1996), and a generalization to a rotating black hole was done by e.g. Karas and Semerak (1999). Newtonian models for rotating black holes are cumbersome and for this reason not widely used, see Abramowicz (2009).Viscous processes, with the timescale \(t_{\rm vis}\)
Despite the fact that the crucial role of accretion power in quasars and other astrophysical objects was uncovered already forty years ago by Salpeter and Zeldovich, several important aspects of the very nature of accretion discs are still puzzling. One of them is the origin of the viscous stresses. Balbus and Hawley recognised in 1991 that, most probably, viscosity is provided by turbulence, which originates from the magnetorotational instability. The instability develops when the matter in the accretion disc rotates nonrigidly in a weak magnetic field. There is still no consensus on how strong the resulting viscous stresses are and how exactly they shape the flow patterns in accretion discs. A great part of our detailed theoretical knowledge on the role of this source of turbulence in accretion disc physics comes from numerical supercomputer simulations. The simulations are rather difficult, time consuming, and hardware demanding. Due to mathematical difficulties, in analytic models one does not directly implement a (small scale) magnetohydrodynamical description, but describes the turbulence (or rather the action of a small scale viscosity of an unspecified nature) by a phenomenological "alphaviscosity prescription" introduced by Shakura and Sunyaev: the kinematic viscosity coefficient is assumed to have the form \[\nu = \alpha H V\ ,\] where \(\alpha =\,\,\)const is a free parameter, \(H\) is a length scale (usually the pressure scale), and \(V\) is a characteristic speed (usually the sound speed). There are several versions of this prescription, the most often used assumes that the viscous torque \(t_{r\phi} = \alpha P\) is proportional to a pressure (either the total, or the gas pressure). The rate of viscous dissipation of energy is\[q = \nu \Sigma (r \frac{d\Omega}{dr})^2\] 
Thermal processes, with the timescale \(t_{\rm the}\)
Gravitational and kinetic energy of matter falling onto the central object is converted by dissipation to heat. Heat is partially radiated out, partially converted to work on the disc expansion and (in the case of BH accretion) partially lost inside the hole. The efficiency of accretion disc \(\eta\) is defined by \(L = \eta {\dot M}c^2\ ,\) where \(L\) is the total luminosity (power) of the disc radiation. Sołtan gave a strong observational argument, confirmed and improved later by other authors, that the efficiency of accretion in quasars is \(\eta \approx 0.1\ .\) Note that the efficiency of thermonuclear reactions inside stars is about \(\eta \approx 0.007\). The theoretically predicted efficiency of geometrically thin and optically thick ShakuraSunyaev accretion disc around a black hole is \(\eta \ge 0.1\ .\) Thus, ShakuraSunyaev accretion discs could explain the energetics of the "central engines" of quasars, which are the most efficient steady engines known in the Universe. Other types of accretion discs models (like ADAFs and slim discs) are called the "radiatively inefficient flows" (RIFs) because they are radiatively much less efficient.
The energy budget may also include rotational energy that could be tapped from the central object. In the black hole case, this possibility was described in a seminal paper by Blandford and Znajek. The BlandfordZnajek process is an electromagnetic analogy of the wellknown Penrose process. Some of its aspects are not yet rigorously described in all relevant physical and mathematical details, and some remain controversial. It is believed that the BlandfordZnajek process may power the relativistic jets.
Analytic models of accretion discs
General remarks
Analytic models describe extreme geometries, matter supplies and optical depths
Nonlinear, coupled partial differential equations of radiative viscous hydrodynamics (or magnetohydrodynamics) that describe physics of accretion discs are too complex to be exactly solved analytically in the general case. Usually, analytic models assume that the accretion is stationary and axially symmetric. For such discs, useful approximate solutions exist in extreme cases corresponding to the following three fundamental divisions (as shown in Figure 1):

Extreme geometries: vertically "thin" and "thick" accretion discs
Extreme mass supply : "sub" and "super" Eddington accretion rates
The accretion rate is defined as the instantaneous mass flux through a spherical surface \(r =\,\,\)const inside the disc. In nonstationary accretion discs accretion rate depends on both time and location, but in stationary disc models with no substantial outflows (no strong winds) it is \[{\dot M}(r, t) = const.\] Accretion discs may be divided into two classes, depending on whether accretion rate is much smaller than, or comparable to the characteristic Eddington accretion rate, that depends only on the mass of the central accreting object \(M\ ,\) \[ {\dot M}_{\rm Edd} = \frac{L_{\rm Edd}}{\eta c^2} = 1.5 \times 10^{17} \, ({M/M_{\odot}}) \, {\rm g/s}. \] \(M_{\odot} = 2 \times 10^{33}\, {\rm g}\) denotes the mass of the Sun, and \(L_{\rm Edd}\) is the Eddington luminosity (radiation power), familiar from the theory of stellar equilibria: at the surface of a star shining at the Eddington rate, the radiation pressure force balances the gravity force (see also the introduction of this Scholarpedia article). The figure on the left shows radio maps of SS433, a wellknown Galactic object with a superEddington accretion disc. A rather common belief that a black hole cannot accrete at a rate higher than the Eddington limit is wrong. In particular, the Eddington rate is not a limit for the mass growth rate of a black hole due to accretion, \(dM/dt\ .\) It could be that \(dM/dt \gg {\dot M}_{\rm Edd}\ .\) This is relevant for modelling the cosmological evolution of black holes. 
Extreme optical depth: "opaque" and "transparent" accretion discs
Note: a more detailed discussion of the subject presented in this Section is given in Narayan & Yi (1995). Optical depth in vertical direction \(z\) is approximated by \(\tau = \Sigma \kappa H\ .\) Here \(\kappa\) is the opacity coefficient, and \(\Sigma\) is the surface density, i.e. vertically integrated density.
Opaque discs (\(\tau \gg 1\)): Such discs are not very hot, the temperature is much less than the virial temperature, \(T \ll T_{\rm{vir}} = \frac{2}{3}\frac{G M m_p}{k_B r}\ .\) Simple (and often used) analytic models approximate the flux emitted locally (at a fixed radius \(r\)) from the disc surface by the "diffusive" black body formula, \(f = f(r) = {8\sigma T^4}/{3H\tau}\ .\) In calculating spectra, the total flux from whole surface of accretion disc is (roughly) approximated by the Planck formula,
\[ F_{\nu}=4\pi \frac{\nu^3 \cos i}{c^2 d^2}\int^{r_{out}}_{r_{in}}\frac{r}{\exp[h\nu/kT(r)]1}dr\ ,\]
where \(d\) and \(i\) are the distance and inclination angle to the rotation axis, respectively, as seen by an observer. More advanced models solve (approximately) radiative transfer equation in the vertical direction, considering dependence on the radiation frequency \(\nu\ .\)
Transparent discs (\(\tau \ll 1\)): Such discs have relatively hightemperatures and lowdensities. Bremsstrahlung, synchrotron and Compton radiative processes are most relevant,
\[f = f_{\rm{br}} + f_{\rm{br,C}} + f_{\rm{syn}} + f_{\rm{syn,C}}\ .\]
They cool down the electrons in the gas much more efficiently than the ions, and therefore a temperature separation between electrons \(T_e\) and ions \(T_i\) is expected. Radiative cooling by Bremsstrahlung,
\[f^_{\rm{br}} = f_{ei} + f_{ee}\]
is given by,
\[f_{ei} = n_e n_i c \sigma_T \alpha_f m_e c^2 F_{ei}(\theta_e) \,\, {\rm erg \,\, cm^{3} \,\, s^{1}},\]
where \(F_{ei}(\theta_e<1) = 4 (\frac{2 \theta_e}{\pi^3})^{1/2} (1 + 1.781 \theta_e^{1.34})\ ;\) \(F_{ei}(\theta_e>1) = \frac{9 \theta_e}{2\pi}[\ln(1.123 \theta_e + 0.48) + 1.5]\)
and
\[f_{ee} = n_e^2 c r_e^2 \alpha_f m_e c^2 F_{ee}(\theta_e) \,\, {\rm erg \,\, cm^{3} \,\, s^{1}},\]
where \(F_{ee}(\theta_e<1) = \frac{20}{9 \pi^{1/2}}(443\pi^2)\theta_e^{3/2} (1+1.1\theta_e^2  1.25\theta_e^{5/2})\ ;\) \(F_{ee}(\theta_e>1) = 24 \theta_e [\ln(0.5616 \theta_e) + 1.28]\ .\)
A simple approximation inverse Compton scattered Bremsstrahlung emission writes
\[f_{\rm{br,C}} = 3\eta_1\{\frac{1}{3}(1\frac{x_c}{\theta_e})\frac{1}{\eta_3+1}[(\frac{1}{3})^{\eta_3+1}(\frac{1}{3\theta_e})^{\eta_3+1}]\} f_{\rm{br}}, \]
where \(n_e\) and \(n_i\) are the number densities of electrons and ions, \(\sigma_T\) is the Thomson crosssection and \(\alpha_f\) the fine structure constant, \(m_e\) and \(r_e = e^2/m_ec^2\) are the electron's mass and radius and \(c\) the speed of light. \(F_{ee}(\theta_e)\) and \(F_{ei}(\theta_e)\) are the radiation rate functions, given by the dimensionless electron temperature \(\theta_e = kT_e/m_ec^2\, \). The Compton enhancement factor \(\eta = \eta_1 + \eta_2 + ({x}/{\theta_e})^{\eta_3}\) is given by \(\eta_1 = P(A1)/(1PA)\), \(\eta_1 = 3^{\eta_3} \eta_1\) and \(\eta_3 =  1  \ln P / \ln A\ ,\) where \(x = h\nu/m_ec^2\, \) factor \(P\) is the probability that a photon scatters and \(A\) is the mean energy amplification factor by that photon and \(x_c = h\nu_c/m_e c^2\ .\) If a magnetic field \(B\) is present, there is also radiative cooling by synchrotron emission
\[f_{\rm{syn}} = \frac{2\pi}{3c^2}kT_e(r)\frac{d\nu_c^3(r)}{dr}\,, ~~ \nu_c = \frac{3 e B}{4 \pi m_e c} \theta_e^2 x_M,\]
\[f_{\rm{syn,C}} = [\eta_1  \eta_2(\frac{x_c}{\theta_e})^{\eta_3}] f_{\rm{syn}},\]
where the coefficient \(x_M\) must be numerically calculated from a relativistic Maxwellian distribution of electrons. For a more thorough study of Comptonisation see, e.g., Coppi & Blandford (1990).
Thin discs
Expansion in the "smallness" parameter. The "global" parameters of a thin disc: mass, accretion rate, viscosity
Most of the analytic and semianalytic accretion disc models assume that the disc is stationary and axially symmetric. "Thin" disc models assume in addition that the vertical extension of the disc is small in a general sense that in cylindrical coordinates \(r, z, \phi\) the disc surface is given by \(z_{\pm} = H_{\pm}(r)\) and \(\max(H/r) \equiv \epsilon \ll 1\ .\) Here \(z = 0\) describes the location of the accretion disc plane. In the spherical coordinates the plane is given by \(\cos \theta = 0\ ,\) and the condition of the small vertical extension by \(\cos \theta \ll 1\) everywhere inside the disc.
The thin disc models are based on expanding the hydrodynamic (or MHD) equations in powers of \(\epsilon\ .\) The expansion procedure is not unique, and depends on some extra physical assumptions made. It leads to equations of the general form \(A_0^i\epsilon^0 + A_1^i\epsilon^1 + A_2^i\epsilon^2 + ... = 0\ .\) For most models, the resulting set of equations, \(A^i_k = 0\ ,\) consists of a number of coupled, linear firstorder ordinary differential equations (containing \(d/dr\) derivatives) and a few nonlinear algebraic equations. Usually, the integration constants may be associated with (and calculated from) the three "global" parameters of thin disc models that are the mass of the central accreting object \(M\ ,\) the accretion rate \({\dot M}\ ,\) and the viscosity parameter \(\alpha\ .\)
The "standard thin", "ADAF" and "slim" disc equations
Equations  Unknown functions  
(01)  \(\rho\,V\frac{dV}{dr} = \rho (\Omega^2  \Omega_K^2 )\,r  \frac{dP}{dr}\)  radial balance of forces  \(\rho\)  density of matter  
(02)  \({\dot M}\left[ \frac{dU}{dr} + P\,\frac{d}{dr}\left(\frac{1}{\rho}\right)\right] = 4\pi r^2 H(\tau_{r \phi}) \frac{d\Omega}{dr} + 4\pi r\,F\)  energy conservation  \(V\)  radial velocity  
(03)  \({\dot M} = 4\pi\,r\,H\rho\,V\)  mass conservation  \(\Omega\)  angular velocity of matter  
(04)  \({\dot M}\left(j  j_{0}\right) = 4\pi\,r^2\,H(\tau_{r\phi})  X_{0}\)  angular momentum conservation  \(P\)  pressure  
(05)  \(F = \frac{acT^{4}}{[\kappa\,\rho\,H]}\)  vertical radiative transfer  \(U\)  internal energy (thermodynamics)  
(06)  \(\frac{c_s}{\Omega_K\,r} = \frac{H}{r}\)  vertical momentum balance  \(H\)  vertical thickness  
(07)  \((\tau_{r\phi}) = \rho[\alpha\,c_s\,H]r \frac{d\Omega}{dr}\)  "viscous" torque \(\alpha\)prescription  \((\tau_{R\phi})\)  viscous torque  
(08)  \(P = P(\rho, T)\)  equation of state  \(F\)  flux of radiation  
(09)  \(U = U(\rho, T)\)  equation of state (internal energy)  \(T\)  temperature  
(10)  \(c_s = c_s(\rho, T)\)  equation of state (sound speed)  \(j\)  angular momentum  
(11)  \(\kappa = \kappa (\rho, T)\)  opacity  \(\kappa\)  opacity coefficient  
(12)  \(j = \Omega\,r^2\)  angular momentum and angular velocity  \(c_s\)  sound speed 
\[F = \frac{3GM\dot{M}}{8\pi r^3} \left(1  \sqrt {\frac{r_0}{r}}\right)\]
This is the famous ShakuraSunyaev flux formula. Note also that the gravitation field of the central object enters the above Newtonian equations only through the Keplerian angular velocity \(\Omega_{\rm{K}}(r)\ .\) In the general relativistic version of (01)(12) the gravity (i.e. the spacetime curvature) enters also through components of the metric tensor \(g_{\mu\nu} = g_{\mu\nu}(r)\) the equatorial plane. The Kerr geometry version of (01)(12) was written first by Lasota (1994), and later elaborated by Abramowicz, Chen, Granath and Lasota (1996); see also Sadowski (2009). However, equations (01)(12) in their form above are often used to model the black hole accretion discs. This is possible because of a brilliant discovery by Paczynski of the Newtonian model for the black hole gravity.Particular models make several additional simplifying assumptions. For example, several models assume that \(\Omega(r) = \Omega_{\rm{K}}(r)\ ,\) with \(\Omega_{\rm{K}}(r)\) being the Keplerian angular velocity, which is known since the gravitational field of the central object is known (for a spherical body with the mass \(M\) Newton's theory yields \(\Omega_{\rm{K}}(r) = (GM/r^3)^{1/2}\)). Note, that in this case the derivative \(d\Omega/dr\) that appears in equations (02) and (07) becomes a known function of \(r\ .\) Equation (07) postulates the form of the "viscous" stress \((\tau_{r\phi})\) in terms of an ad hoc ansatz that introduces the dimensionless \(\alpha\)viscosity. Note that the quantity that appears in square bracket is called in hydrodynamics the "kinematic viscosity". The original ShakuraSunyaev ansatz postulated \((\tau_{r\phi}) = \alpha\,P\ .\) Equation (05) gives the flux of radiation in (a very rough) diffusion approximation. Note that the quantity in square brackets in this equation is the optical depth, \(\tau = [\kappa\,\rho\,H]\) in the vertical direction. The equation is valid only if \(\tau \gg 1\ ,\) and if \(\tau < 1\) nonthermal radiative processes should be considered, and equation (05) replaced by \(F = F(\rho, T)\ .\)
In equation (04), \(j_{0}, X_{0}\) are the angular momentum and the viscous torque at some undefined radius \(r_{0}\ .\) In the black hole accretion discs models, it is customary to take \(r_{0} = r_G\) (= black hole horizon radius) because the viscous torque at the horizon vanishes. Then, \(j_{0}\) is the (a priori unknown) angular momentum of matter at the horizon. With respect to first order derivatives, equations (01)(12) form a linear system that may be solved for each derivative. For \(dV/dr\) this gives, \[ \frac{dV}{dr} = \frac{N(r, \rho, V, \Omega, ...)}{V^2  C^2_s}.\] Any black hole accretion flow must be transonic, i.e. somewhere it must pass the sonic radius \(r_s\ ,\) where \(V(r_s) = c_s(r_s)\ .\) In order that \(dV/dr\) and all other derivatives are nonsingular there, it must be, \[N(r_s, \rho, V, \Omega, ...) = 0.\] The above sonic point regularity condition makes the system (01)(12) over constrained, i.e. an eigenvalue problem, with the eigenvalue being the angular momentum at the horizon, \(j_{0}\ .\)
Analytic models describe black hole accretion discs down to a certain "inner edge" \(r_{in}\) which locates close to the central accreting object. The inner edge is a theoretical concept introduced for convenience, because at \(r \approx r_{in} \approx r_s\) the accretion flow changes its character. In the case of the black hole accretion, the change goes from almost circular orbits to almost radial free fall. It is therefore convenient to separately model the two regions\[r > r_{in}\] where matter moves on circular orbits, and \(r < r_{in}\) where matter free falls. Of course, in reality the situation is more complicated, as the change of the flow character occurs smoothly in an extended region on both sides of \(r_{in}\ .\) For black hole accretion the inner disc edge lies between the marginally bound radius and the innermost stable circular orbit, \(r_{\rm{MB}} \le r_{in} \le r_{\rm{ISCO}}\,\). For very efficient ShakuraSunyaev discs, \(r_{in} \approx r_{\rm{ISCO}}\ ,\) while for RIFs \(r_{in} \approx r_{\rm{MB}}\ .\) For stellar accretion, \(r_{in}\) is located near the surface of the star and the flow there is described by a boundary layer model. For more details on the inner edge of a thin disc, see Abramowicz et al. (2010) and references quoted there.
The "standard" model: ShakuraSunyaev
Characteristic, applications, references 
Analytic formulae (solution)  
Axially symmetric, stationary, local analytic model. Explicit formulae give all physical characteristics in terms of \(M, {\dot M}, \alpha\) and \(r\ .\) It is geometrically thin in the vertical direction \(H/r < 1\ ,\) and has a disclike shape. Accretion rate is very subEddington. Opacity is very high. The gas goes down on tight spirals, approximated by circular, free (Keplerian, geodesic) orbits. For black hole and (very compact) neutron star the inner edge locates at ISCO. High luminosity, high efficiency of radiative cooling. Electromagnetic spectra is not much different from that of a sum of black bodies. Alpha viscosity prescription assumed. Diffusion approximation for radiative transfer used. It is dynamically stable. When the gas is cold and radiation pressure negligible, it is also thermally and viscously stable, otherwise it is unstable in both respects. Applications: YSOs, CVs, LMXRB, AGNs. The best known and studied theoretical model. Standard reference: Shakura, Sunyaev (1974) , one of the most often quoted papers in modern astrophysics (quotation counts). Similar ideas: Pringle, Rees (1972); LyndenBell, Pringle (1974) Fully relativistic version (Novikov, Thorne 1974); for the detailed description see: Page, Thorne (1974). Recent application to spectral fits: Shafee et al. (2006); Middleton et al. (2006). Recommended review: Pringle (1981). 
Different analytic solutions are known for cases in which the total pressure is either dominated by gas or radiation pressure, and for the opacity described either by Kramers' law, or electron scattering. However, the most important formula that gives the locally emitted flux of radiation does not depend on these physical conditions, and is universally given (in Newton's theory) by,
\(H = 1.7\times 10^8\alpha^{1/10}\dot{M}^{3/20}_{16} m_1^{3/8} r^{9/8}_{10}f^{3/5} {\rm cm}\) \(T = 1.4\times 10^4 \alpha^{1/5}\dot{M}^{3/10}_{16} m_1^{1/4} r^{3/4}_{10}f^{6/5}{\rm K}\) \(\rho = 3.1\times 10^{8}\alpha^{7/10}\dot{M}^{11/20}_{16} m_1^{5/8} r^{15/8}_{10}f^{11/5}\rm{g/cm^3}\) Here \(T\) and \(\rho\) are the midplane temperature and density respectively. \(\dot{M}_{16}\) is the accretion rate, in units of \(10^{16} \rm{g/s}\ ,\) \(m_1\) is the mass of the central accreting object in units of a solar mass, \(r_{10}\) is the radial location in the disc, in units of \(10^{10}{\rm{cm}}\ ,\) and \(f = [1(r_0/r)^{1/2}]^{1/4}\ ,\) where \(r_0\) is the inner radius of the disc. 
Specific versions (and modifications) of the ShakuraSunyaev models
Scurves and the thermalviscous instability: The standard accretion discs are known to be subject to thermalviscous instability due to the partial hydrogen ionisation. As a result of this instability, the disc cycles between two states: a hot and mostly ionised state with a large local accretion rate and a cold, neutral state with a low accretion rate. This instability was originally proposed to explain the largeamplitude luminosity variations observed in cataclysmic variables (Smak 1982; Meyer & MeyerHofmeister 1982). It is also believed that the same mechanism is responsible for the eruptions in soft Xray transients (see, e.g. Cannizzo et al. 1982; Dubus et al. 2001; and Lasota 2001 for a review). The H ionisation instability was also shown to operate in some discs around supermassive black holes in active galactic nuclei (Lin & Shields 1986; Clarke 1988; Mineshige & Shields 1990; Siemiginowska et al. 1996), but not necessarily on a global scale (see Menou & Quataert 2001; Janiuk et al. 2004). The characteristic timescales of cycle activity scale roughly with the mass of a compact object (Hatziminaoglou et al. 2001). Therefore, the observed cycle timescale of the order of years in binaries translates into thousands to millions of years in galaxies that harbour a supermassive black hole.
Warped and precessing discs: A geometrically thin, optically thick accretion disc is unstable to selfinduced warping when illuminated by a sufficiently strong central radiation source (Pringle 1996; Maloney et al. 1996; see also Petterson 1977). The instability is important for the standard (not advectively dominated) discs around neutron stars and black holes in the Xray binaries, in active galactic nuclei (Pringle 1997), and in particular in the "maser" galaxy NGC 4258 (Maloney et al. 1996). For discs around less compact objects, where efficiency is orders of magnitude smaller, steady discs are predicted to be stable. The warped discs are important in some models of the (~kilohertz) coherent oscillations (QPOs) observed in neutron star and black hole Xray binaries. In the Schwarzschild metric, the damping or excitation of gmodes with azimuthal dependence \(\exp(i\,m\phi)\) has been considered by Kato (2003, 2004, 2005, 2007), who found that m = 0, 1 modes undergo resonant amplification when interacting with a nonrotating onearmed (m = 1) stationary warp, assumed to be present in the accretion disc. Description of the warp instability here is based on Armitage & Pringle 1997. See a critical examination of the idea e.g. in Ivanov & Papaloizou 2008.
ADAFs
Characteristic, applications, references 
Radial distributions 
SubEddington accretion, very small opacity. ADAFs are cooled by advection (heat captured by moving matter) rather than by radiation. They are very radiatively inefficient, geometrically extended, similar in shape to a sphere (or a "corona") rather than a disc, and very hot (close to the virial temperature). Because of their low efficiency, ADAFs are much less luminous than the ShakuraSunyaev thin discs. ADAFs emit a powerlaw, nonthermal radiation, often with a strong Compton component. For black hole and (very compact) neutron star the inner edge at a radius smaller than ISCO. Dynamically, thermally and viscously stable. Applications: mostly LMXRB, AGNs, with good fits to observed spectra. Numerical 1.5D (vertically integrated) stationary transonic models (in Kerr): Abramowicz et al. (1996); Narayan et al. (1997); Popham, Gammie (1998) Numerical 2D nonstationary models (in PaczynskiWiita): Igumenshev et al. (1996). Most influential paper, describing a Newtonian, selfsimilar, stationary, axially symmetric analytic, model: Narayan, Yi (1994). Immediate followup by the same group: Narayan, Yi (1995); Narayan, Mahadevan (1995); Narayan et al. (1996). The idea mentioned first time: Ichimaru (1987), see also: Rees et al. (1982); Abramowicz et al. (1995). Recommended review: Narayan, McClintock (2008). 
The profiles of temperature, optical depth, ratio of scale height to radius, and the advection factor of a hot onetemperature accretion solution (solid lines). The parameters are\[M = 10\,M_{\odot}\ ,\] \({\dot M} = 10^{5}{\dot M}_{\rm{Edd}}\ ,\) \(\alpha = 0.3\ ,\) \(\beta = 0.9\ .\) The outer boundary conditions are \(R_{out} = 10^3R_s\ ,\) \(T = 10^9\)[K], \(v/c_s = 0.5\ .\) For simplicity, effects associated with outflow or convection are not taken into account. The twotemperature solutions with the same parameters and \(\delta = 0.5\) (dashed lines) and 0.01 (dotdashed lines) are also shown for comparison. Figure credit: Yuan, Taam, Xue, Cui (2006)

Specific versions (and modifications) of the ADAF models
Convection: CDAFs: The Convection Dominated Accretion Flow (CDAF) models an ADAF with convective eddies in which gas can be trapped. This leads to radial modifications of the accretion rate.
Outflows: ADIOS The Adiabatic InflowOutflow Solution (ADIOS) accounts for the loss of gas due to outflow. In this model the accretion rate decreases with decreasing radius.
Magnetized ADAFs
Slim discs
Characteristic, applications, references 
The local flux 
Nearly Eddington accretion. Large opacity. Cooled by radiation and advection, i.e. heat trapped in matter and transported towards the central accretor. Radiatively much less efficient than the standard ShakuraSunyaev discs. \(H/R\) only slightly less than one. For black hole and (very compact) neutron star the inner edge at a radius smaller than ISCO. Dynamically, thermally and viscously stable. Rotation differs (slightly but importantly) from the Keplerian one. The pressure gradient along the disc plane direction is dynamically important. Slim discs models are described by a set of ordinary differential equations, and one must explicitly solve the eigenvalue problem connected with the regularity condition at the sonic radius (which does not coincide with the ISCO).
Applications: mostly LMXRB, AGNs, ULX. For detailed spectral fits see e.g.:
Figure on the right shows the local flux of radiation for different mass accretion rates and the black hole spins. Each subplot contains six solid lines for the following mass accretion rates (in the Eddington units): 0.01 (the thickest line), 0.1, 0.3, 0.6 and 0.9 (the thinnest line). The upper panel is for a non rotating black hole (a = 0), the middle one for (a = 0.6) while the bottom one for a highly spinning black hole (a = 0.98). The black hole mass is \(9.4\,M_{\odot}\ .\) NOTE that a considerable amount of energy is radiated inside the ISCO; this is an effect of advection. Figure credit: Sadowski (2009)

KluzniakKita
Fully two dimensional analytic solution (stationary, axially symmetric) obtained through a mathematically exact expansion in the small parameter H/r of the equations of viscous hydrodynamics. Significant backflows in the midplane of the disc have been found.
Kluzniak, Kita (2000), numerical follow up: Umurhan et al. (2006).
Disc solutions
Branches IIII are thin disc solutions, branch IV is a thick disc solution. Lines correspond to fixed M, r, and \(\alpha\ .\) An example of each of the four branches is shown in a corresponding color: pink, blue, green, and yellow. The congruence of all branches has a critical point, corresponding to \( \alpha = \alpha_{crit}\ . \) In different places of the parameter space, the cooling is dominated by black body radiation, bremsstrahlung, Compton losses, pair production, or by advection, as indicated by arrows. Figure adapted from Björnsson et al. (1996). Branch I (blue): ShakuraSunyaev (gas pressure) + ShakuraSunyaev (radiation pressure) + Slim. Branch II (green): ShakuraSunyaev (gas pressure) + SLE Branch III (yellow): SLE + ADAF. Branch IV (pink): Polish doughnut (thick disc, see next section). 
The different types of thin discs may coexist radially. Microquasars display distinct spectral states. In order of increasing luminosity these are the quiescent state, low state, intermediate state, high state, and very high state. Narayan with collaborators presented a model of accretion flows around black holes that unifies most of these states. At low mass accretion rates, the inner ADAF zone in the model radiates extremely inefficiently, and the outer thin disc is restricted to large radii. The luminosity therefore is low, and this configuration is identified with the quiescent state. For larger accretion rates the radiative efficiency of the ADAF increases rapidly and the system becomes fairly luminous. The spectrum is very hard and peaks around 100 keV. This is the low state. For still greater rates, the ADAF progressively shrinks in size, the transition radius decreases, and the Xray spectrum changes continuously from hard to soft. This is the intermediate state. Finally, the inner ADAF zone disappears altogether and the thin accretion disc extends down to the marginally stable orbit. The spectrum is dominated by an ultrasoft component with a weak hard tail. This is the high state.

Thick discs
Thick discs: assumptions
For "thick discs" models of accretion discs one assumes that:
 Matter distribution is stationary and axially symmetric, i.e. matter quantities such as density \(\epsilon\) or pressure \(P\) are independent on time \(t\) and the azimuthal angle \(\phi\ .\)
 Matter moves on circular trajectories, i.e. the four velocity has the form \(u^i = [u^t, u^{\phi}, 0, 0]\ .\) The angular velocity is defined as \(\Omega = u^{\phi}/u^t\ ,\) and the angular momentum as \(\ell =  u_{\phi}/u_t\ ,\)
 \(t_{\rm dyn} \ll t_{\rm the} < t_{\rm vis}\ ,\) with \(t_{\rm dyn}\) being the dynamical timescale in which pressure force adjusts to the balance of gravitational and centrifugal forces, \(t_{\rm the}\) being the thermal timescale in which the entropy redistribution occurs due to dissipative heating and cooling processes, and \(t_{\rm vis}\) being the viscous timescale in which angular momentum distribution changes due to torque caused by dissipative stresses. Mathematically, this is equivalent to assume the stressenergy tensor in the form, \(T^i_{~\nu} = u^{\mu}\,u_{\nu}\,(P + \epsilon)  \delta^{\mu}_{~\nu}\,P\ .\)
Using this form of the stressenergy tensor, Abramowicz et al. 1978 have derived from the equilibrium condition \(\nabla_{\mu}\,T^{\mu}_{~\nu} = 0\) the relativistic "Euler" equation,
\[ \frac{\nabla_{\mu} P}{\epsilon + P} = \frac{\nabla_{\mu}\,g_{tt} + 2\Omega\,\nabla_{\mu}\,g_{t\phi} + \Omega^2\,\nabla_{\mu}\,g_{\phi\phi}}{g_{tt} + 2\Omega\,g_{t\phi} + \Omega^2\,g_{\phi\phi}} = \nabla_{\mu}\ln A + \frac{\ell\,\nabla_{\mu}\Omega}{1  \ell\,\Omega}, ~~~{\rm with}~~~A^2(r, \theta) = \frac{1}{g_{tt}(r,\theta) + 2\Omega\,g_{t\phi}(r,\theta) + \Omega^2\,g_{\phi\phi}(r,\theta)}. \]  \((3.2:1)\) 
Equipressure surfaces: analytic solution in a general case
The "equipressure" surfaces are defined by an implicit condition \(P(r, \theta) = const\ ,\) which may be solved to get the explicit form \(\theta = \theta(r)\ .\) Then, (3.2:1) implies that the function \(\theta(r)\) obeys,
\[ \frac{d\theta}{dr} =  \frac{\partial_{r}\,g_{tt} + 2\Omega\,\partial_{r}\,g_{t\phi} + \Omega^2\,\partial_{r}\,g_{\phi\phi}} {\partial_{\theta}\,g_{tt} + 2\Omega\,\partial_{\theta}\,g_{t\phi} + \Omega^2\,\partial_{\theta}\,g_{\phi\phi}} =  \frac{\partial_{r}\,g^{tt}  2\ell\,\partial_{r}\,g^{t\phi} + \ell^2\,\partial_{r}\,g^{\phi\phi}} {\partial_{\theta}\,g^{tt}  2\ell\,\partial_{\theta}\,g^{t\phi} + \ell^2\,\partial_{\theta}\,g^{\phi\phi}}. \]  \((3.2:2)\) 
If \(\ell = \ell(r, \theta)\) or, which is equivalent, \(\Omega = \Omega(r, \theta)\) are known functions, the equation (3.2:2) for the equipressure surfaces takes the form of a standard ordinary differential equation \(d\theta/dr = f(r, \theta)\) with known rhs, and it may be directly integrated. This has been done first by Jaroszynski et al. (1980) and then by several other authors. In the Figure below, we show an example from a recent paper by Lei et al. (2008) who assumed the angular momentum distribution in the form that depends on three constant parameters (\(\eta, \beta, \gamma\)),
\[ {\ell}(r, \theta) = {\ell}_0\left\{ \frac{{\ell}_K(r)}{{\ell}_0}\right\}^{\beta}(\sin\theta)^{2\gamma} ~\left({\rm for}~~ r \geq r_{\rm ISCO}\right) ,~~ {\ell}(r, \theta) = {\ell}_{\rm ISCO}(\sin\theta)^{2\gamma} ~\left({\rm for}~~ r < r_{\rm ISCO}\right). ~~{\rm Here}~~ {\ell}_0 \equiv \eta\,{\ell}_K(r_{\rm ISCO}) ~~{\rm and}~~ {\ell}_{\rm ISCO} = {\ell}_0\left\{\frac{{\ell}_K(r_{\rm ISCO})}{{\ell}_0}\right\}^{\beta}. \]  \((3.2:3)\) 





Barytropic thick discs and the von Zeipel theorem
\[ W(P) \equiv \int \frac{dP}{\epsilon(P) + P} = \ln A + \int \frac{d\Omega}{1  \Omega\,\ell(\Omega)} \equiv \ln A(r, \theta) + F(\Omega), ~~~{\rm and}~~~\ell = \ell(\Omega) =  \frac{g_{t\phi}(r, \theta) + \Omega\,g_{\phi\phi}(r, \theta)}{g_{tt}(r, \theta) + \Omega\,g_{t\phi}(r, \theta)}. \]  \((3.2:4)\) 
\[ \sin^2\theta = \frac{\ell_0}{C_0\,r^2 + 2\,G\,M\,\left(1  r_G/r\right)^{1}}. ~~~{\rm Here}~~C_0=const ~~{\rm numerates~the~equipressure~surfaces}. \]  \((3.2:5)\) 
The Roche lobe overflow
Fig. 5 Taken from Abramowicz et al. (1980) (a) At the location \(r = r_{in}\ ,\) called the "cusp", angular momentum in the disc equals the Keplerian one, \(\ell_{\rm disc}(r_{in}) = \ell_K(r_{in})\ .\) For \(r > r_{in}\) it is \(\ell_{\rm disc}(r_{in}) > \ell_K(r_{in})\) and \(d\ell/dr > 0\ ,\) and for \(r < r_{in}\) it is \(\ell_{\rm disc}(r_{in}) < \ell_K(r_{in})\) and \(d\ell/dr \approx 0\ .\) (b) The particular equipotential surface \(W = W_{in}\ ,\) called the "Roche lobe", crosses itself at the cusp. For \(r \gg r_{in}\) the surface of the disc (\(P = 0\)) coincides with the equipotential \(W = W_{S}\ .\) (c) The nonzero potential difference \(\Delta W = W_S  W_{in}\) implies that no equilibrium is possible at radii around and smaller than the cusp. Instead, there will be dynamical mass loss from the disc with the accretion rate that for a polytropic fluid \(P = K\rho^{1+1/n}\) equals (Kozlowski et al. 1978, Abramowicz 1985)
The mass loss (3.2:5) induced by the Roche lobe overflow selfregulates the accretion rate in the innermost part of all types of accretion discs (thin, slim, ADAF, thick) around black holes and sufficiently compact neutron stars. This selfregulated overflow has several important consequences:
 
Fig. 6 Taken from Igumenshchev and Beloborodov (1997). The analytic formula (3.2:5) is very accurate, as a comparison with the numerical simulations shows. (a) The numerically calculated equipressure structure close to the central black hole is remarkably similar to that calculated analytically. In particular, there is obviously a cusp there. (b) In a qualitative agreement with the analytic model, the radial fluid velocity (described by arrows) is small far away from the cusp \(r \gg r_{in}\ ,\) but large (no equilibrium) at \(r \approx r_{in}\ .\) (c) Even more impressive is the excellent quantitative agreement of the predictions of the analytic formula (3.2:5), represented by lines, with results of numerical simulations, represented by points (circles, squares and triangles).

SuperEddington luminosity of radiation pressure supported thick discs ("Polish doughnuts")
\[ L_{\rm Edd} = L_{\rm Planck}\frac{\Sigma_{\rm grav}}{\Sigma_{\rm rad}} = \frac{4\pi\,G\,M\,m_P\,c}{\sigma_T} = 1.4 \times 10^{38}\left( \frac{M}{M_{\odot}}\right) [{\rm erg}/{\rm sec}]. \]  \((3.2:7)\) 
A "Polish doughnut" is a radiation pressure supported thick accretion discs around a central black hole. Polish doughnuts have toroidal shapes, resembling a large sphere (\(r \gg r_G\)) with a pair of empty narrow funnels along the rotation axis. The total luminosity of a Polish doughnut may be approximated as \(L/L_{\rm Edd} \equiv \lambda \approx \log (r/r_G)\ .\) The logarithm here is of a crucial importance. It prevents astrophysically realistic doughnuts (i.e. with \(r < 10^6\,r_G\ ,\)) to have highly superEddington luminosities. The theory predicts for such "realistic" fat tori slightly superEddington total (isotropic) luminosities \(\lambda \le 7\ .\) However, because the funnels have solid angles \(\Theta^2 \sim r_G/r\ ,\) radiation in the funnels may be, in principle, collimated to highly superEddington values \(\lambda \sim r/r_G \gg 1\ .\) This simple estimate agrees with a more detailed modelling of the Polish doughnuts radiation field by Sikora (1981) and Madau (1988) who obtained \(\lambda \ge 10^2\) for discs with \(r/r_G \sim 10^2\ .\) A typical value that follows from observational estimates for the nonblazar active galactic nuclei, e.g. by Czerny & Elvis (1987) is \(\lambda \sim 10\ ,\) but of course for blazars and other similar sources (e.g. for ULXs, if they are powered by stellar mass black holes, as argued e.g. by King (2008), it must be \(\lambda \gg 10\ .\) Fig. 7 (Credit: Madau (1988)) shows radiation from a Polish doughnut seen at different inclination angles, inclination \(0^{\circ}\) corresponds to line of sight along the funnel axis. 
\[ {\dot M}_{\rm Edd} = \frac{L_{\rm Edd}}{c^2} = 1.5 \times 10^{17}\left( \frac{M}{M_{\odot}}\right) [{\rm g}/{\rm sec}], ~~~{\dot m} = \frac{\dot M}{{\dot M}_{\rm Edd}}. \]  \((3.2:8)\) 
 Radiation pressure supported black hole thick accretion discs ("Polish doughnuts") have typically superEddington luminosities \(\lambda > 1\ .\)
 These discs have very small accretion efficiency and therefore must have highly superEddington accretion rates \({\dot m} \gg 1\ .\)
 SuperEddington accretion does not necessarily imply strong outflows but is often accompanied by them.
Variability, instability, oscillation
One of the key features observed in accretion discs is the strong and often chaotic time variability in various wavebands. At shorter wavelengths the observed variability in a given source has in general a larger amplitude and a shorter timescale. Variability in different wavebands can be correlated. Different kinds of time variabilities have been observed in accretion discs, some may originate from instabilities in the disc, others may be caused by an ensemble of oscillations and/or waves in the innermost disc region.
Disc instabilities
Thermalviscous instability and the limit cycle
When, at a fixed radius, the \(\dot{M}\Sigma\) relation for a thin (slim) disc is shaped as an Scurve, with the upper and lower branches being stable, and the middle branch being thermally and viscously unstable, a limit cycle behaviour may occur when the mass supply \(\dot{M}_0\) is in the unstable range. Such a schematic picture, is useful in understanding the basic reason for the disc instability and the limit cycle behaviour, but it can be misleading, because although the instability is local the limit cycle is a global process and quite often the resulting local disc behaviour does not correspond to the simple, schematic Scurve diagram.
Figure 1: Schematic Scurve showing the (local) limitcycle behaviour in accretion discs. (Figure credits: Kato et al., 1998)  Figure 2: Theoretical twoalpha Disc Instability Model (DIM) to explain the limitcycle behaviour seen in cataclysmic variables. (Figure credits: Lasota, 2001)  Figure 3: Scheme of a dwarf nova outburst in cataclysmic variables. (Figure credits: ???) 
Cataclysmic variables, low accretion rates: The instability is related to partial hydrogen ionisation, which implies that convection may become the dominant mode of energy transport. The resulting limit cycle is a main ingredient for explaining the dwarf nova outbursts. However, other mutually related, effects also play a role. Particularly important are the varying masstransfer rate from the secondary, (self) illumination of the secondary and the disc, tidal instabilities. For the model to work, it is necessary to assume that \(\alpha\) on the upper branch is significantly larger than on the lower branch.
Xray binaries, high (around Eddington) accretion rates: Theory predicts that the instability should occur for radiation pressure supported discs with luminosities in excess of $L > 0.01L_{Edd}$ and with the standard ShakuraSunyaev alphaviscosity prescription in which the stress is proportional to the total pressure. Other, nonstandard, viscosity prescriptions in particular the one suggested by Merloni in which the total stress is proportional to a geometrical mean of total and gas pressure, are consistent with thermal stability (see e.g. Ciesielski et al., 2012 for a discussion). Observations argue against the existence of this instability. Stellarmass black hole sources cross this limit both during their rise to peak luminosity and on their decline to quiescence, showing no symptoms of unstable behaviour. Instead, observations suggest that discs in black hole Xray binaries are stable up to at least \(L \approx 0.5L_{Edd}\) (Done et al. 2004).
The issue is still not solved. The most recent numerical simulations by YanFei Jiang et al. (2013) indicate thermal instability, but on another type than the classic ShakuraSunyaev one.
Disc oscillations
Timing measurements (e.g. by the Rossi Xray Timing satellite RXTE or the Xray MultiMirror Mission XMMNewton) have produced power density spectra of the fluctuations in the light flux of binaries and showed low and high frequency peaks in the PSD, known as quasiperiodic oscillations (QPOs). They were first noticed in dwarf novae, i.e., erupting cataclysmic variables, as incoherent pulses with time scales of 30170 s along with coherent periodic oscillations with a period of 20 s, the so called dwarf nova oscillations (DNOs).
In general, oscillations are the result of restoring forces acting on perturbations. For instance, if one perturbs a fluid element radially inwards, it conserves its own angular momentum and will be rotating too slow for its new location. Centripetal forces consequently push it outwards again. These kind of inertial oscillations are the so called epicyclic oscillations. Disc oscillations may be axisymmetric (e.g. m=0, corresponding to the fundamental mode) or nonaxisymmetric (e.g. m=1, corresponding to the first overtone). The field of studying the temporal behaviour of discs by means of oscillations is called discoseismology.
Thin discs
One can then express the Eulerian perturbations of all physical quantities through a single function \(\delta W \propto \delta p/\rho\) which satisfies a secondorder partial differential equation. Since the accretion disc is considered to be stationary and axisymmetric, the angular and time dependencies are factored out as \(\delta W = W(r,z)e^{i(m\phi  \sigma t)}\ ,\) where the eigenfrequency \(\sigma(r,z) = \omega  m\Omega\) and \(m\) is the azimuthal wave number. It is assumed that the variation of oscillation modes in radial direction is much stronger than in vertical direction. The resulting two (separated) partial differential equations for the functional amplitude \(W(r,z) = W_r(r) W_y(r,y)\) are given by
\(\frac{d^2W_r}{dr^2}\frac{1}{(\omega^2\omega_r^2)}[\frac{d}{dr}(\omega^2\omega_r^2)]\frac{dW_r}{dr} +\alpha^2(\omega^2\omega_r^2)(1\frac{\Psi}{\tilde{\omega}^2})W_r = 0\) and
\((1y^2)\frac{d^2W_y}{dy^2}2gy\frac{dW_y}{dy}+2g\tilde{\omega}^2[1(1\frac{\Psi}{\tilde{\omega}^2})(1y^2)]W_y = 0\ .\)
The radial eigenfunction, \(W_r\ ,\) varies fast with \(r\) and the vertical eigenfunction, \(W_z\ ,\) varies slowly with \(r\ .\) The radial and vertical epicyclic frequencies are given by \(\omega_r(r)\) and \(\omega_{\theta}(r)\ ,\) respectively, \(\tilde{\omega}(r) = \omega(r)/\omega_{\theta}(r)\) and \(\alpha(r) = \frac{dt}{d\tau}\frac{\sqrt{g_{rr}}}{c_s(r,0)}\ ,\) using coefficients of the Kerr metric in BoyerLindquist coordinates. \(\Psi\) is the eigenvalue of the (WKB) separation function. The radial boundary conditions depend on the type of mode and its capture zone (see below). Oscillations in accretion discs are studied by means of \(\Psi(r,\sigma)\) with the angular mode number \(m\ ,\) the vertical and radial mode numbers (number of nodes in the corresponding eigenfunction) \(j\) and \(n\ ,\) respectively.
Figure 1: Schematic picture showing trapped axisymmetric gmodes with n=1 in the region between \(r_1\) and \(r_2\ ,\) below the radial epicyclic mode \(\kappa(r)=\omega_r(r)\ .\) pmodes only exist above the Keplerian frequency \(\Omega_K\ .\) (Figure credits: Kato et al., 1998)  Figure 2: Schematic picture showing the dependence of three characteristic disc frequencies on the spin \(a\) of the black hole. (Figure credits: Wagoner, 1999) 
Classification:
A mode oscillates in the radial range outside the inner disc, \(r > r_i\ ,\) where
\( (\omega^2  \omega_r^2) (1  \frac{\Psi}{\tilde{\omega}^2}) > 0\).
The two points \(r=r_{\pm}(m, a, \sigma)\) refer to the location where the Lindblad resonances occur.
 pmodes are inertialacoustic waves in horizontal direction that have pressure as their main restoring force. They are defined by \(\Psi < \tilde{\omega}^2\) and are trapped where \(\omega^2 > \omega_r^2\) in two zones between the inner (\(r_i\)) and outer (\(r_o\)) radius of the disc. The inner pmodes are trapped between the inner disc edge and the radial epicyclic frequency at \(r_i < r < r_ \) where the gas already is accreted rapidly. The outer pmodes occur at \(r_+ < r < r_o\ .\) The latter, shown in Figure 1, produce the stronger luminosity modulation. In the corotating frame these modes appear at frequencies slightly higher that the radial epicyclic frequency. The (inertialacoustic) pmode oscillation has no node (m=0) and is in this sense the fundamental mode of disc oscillation. There are vertical pmodes which are verticalacoustic waves that occur at higher overtones (\(m \geq 2 \))
 cmodes are corrugation or surface waves in vertical direction and have the vertical component of the gravitational force due to the central object as restoring force. They are defined by \(\Psi = \tilde{\omega}^2\ .\) They are nonradial (\(m=1\)) and vertically incompressible modes that appear near the inner disc edge and precess slowly around the rotational axis. The cmodes are controlled by the radial dependence of the vertical epicyclic frequency. In the corotating frame these modes appear at highest frequencies.
 gmodes are wave modes where horizontal and vertical motion is coupled and gravity is the main restoring force. They are defined by \(\Psi > \tilde{\omega}^2\ .\) They are trapped where \(\omega^2 < \omega_r^2\) in the zone \(r_ < r < r_+\) given by the radial dependence of \(\omega_r\ ,\) i.e., gmodes are gravitationally captured in the cavity of the radial epicyclic frequency and are thus the most robust among the modes. Since this is the region of the temperature maximum of the disc, gmodes are expected to be observed best. In the corotating frame these modes appear at low frequencies. In Figure 1 \(r_1=r_\) and \(r_2=r_+\) and in figure 2 the spin dependency is plotted for \(m=0\ .\)
 fmodes usually refer to surface gravity modes (see gmodes) in stars. Occasionally, nonaxisymmetric (m=1) modes in discs are referred to as fmodes or tiltmodes.
All modes have frequencies that scale \(\propto 1/M\ .\) Upon the introduction of a small viscosity (\(\nu \propto \alpha, \,\, \alpha << 1\)) most modes grow at a dynamical timescale such that the disc should become unstable.
 Wagoner, R., 1999, "Relativistic Diskoseismology", PhR, 311, 259
 Kato S., 2001, "Basic Properties of ThinDisk Oscillations", PASJ, 53, 1
 Stella & Vietri, 1998, "LenseThirring Precession and Quasiperiodic Oscillations in LowMass XRay Binaries", APJ, 492, 59
Thick discs
Geometrically thick discs, i.e. stationary tori, always allow axisymmetric, incompressible modes corresponding to global oscillations of the entire torus at radial (\(\sigma=\omega_r\)) and vertical (\(\sigma=\omega_{\theta}\)) epicyclic frequencies. Other possible modes, provided \(m=0\ ,\) are basically acoustic (p), surface gravity (g) and internal inertial (c) modes and can be found by solving the relativistic PapaloizouPringle equation
\( \frac{1}{(g)^{1/2}}\left\{\partial_{\mu}\left[(g)^{1/2}g^{\mu\nu}f^n\partial_{\nu}W\right]\right\}  \left(m^2g^{\phi\phi}  2m\omega g^{t\phi} + \omega^2 g^{tt}\right)f^n W = \frac{2nA(\bar{\omega}m\bar{\Omega})^2}{\beta^2 r^2_0}f^{n1}W \)
together with the boundary condition that the Lagrangian perturbation in pressure at the unperturbed surface (\(f=0\)) vanishes\[ \Delta p = (\delta p + \xi^{\alpha}\nabla_{\alpha}p) = 0 \]
Classification:
 \(\times\)modes are surface gravity modes (k=2) derived from an eigenfunction \(W = a xy\ ,\) for some constant \(a\ ,\) which is odd in \(x\) and \(y\) and results in two modes.
\( \bar{\sigma}_0^2 = \frac{1}{2}\{\omega_r^2+\omega_{\theta}^2 \pm [(\omega_r^2+\omega_{\theta}^2)^2+4\kappa_0^2\omega_{\theta}^2]^{1/2}\}, \)
where \(\bar{\sigma}_0^2=\sigma_0/\Omega_0\) taken at the location \(r_0\) of the pressure maximum in the torus centre (hence the index 0). \(\kappa_0^2 = \frac{\mathcal{E}_0^2}{l_0 A_0^2}(\frac{g^{tt}_{,r}l_0 g^{t\phi}_{,r}}{g_{rr}}\frac{dl}{dr})\) is the squared frequency of the inertial oscillation in the fluid due to an angular momentum gradient. For constant angular momentum distribution this term vanishes. The positive square root of which gives the xmode that is a surface gravity mode (Figure 8). The negative square root gives a purely incompressible inertial (c) mode whose poloidal velocity field represents a circulation around the pressure maximum.
 breathing and \(+\)modes are derived from an eigenfunction \(W = a + bx + cy\ ,\) for some constants \(a,b,c\ .\) The resulting eigenfrequencies are (for \(b=c=0\)) the zero corotation frequency mode as well as
\( \bar{\sigma}_0^2 = \frac{1}{2n}\{(2n+1)(\omega_r^2+\omega_{\theta}^2)(n+1)\kappa_0^2 \pm [ ((2n+1)(\omega_{\theta}^2\omega_r^2)^2 + (n+1)\kappa_0^2)^2 + 4(\omega_r^2  \kappa_0^2) \omega_{\theta}^2 ]^{1/2} \}, \)
 breathing  modes have frequencies corresponding to the upper sign in the above equation. The torus cross section contracts and expands (Figure 6). Breathing modes are comparable to acoustic modes (k=0, j=1) in the incompressible Newtonian limit for \(l=const.\ ,\) while in the Keplerian limit the mode frequency becomes that of a vertical acoustic wave.
 \(+\)modes have frequencies corresponding to the lower sign. In the incompressible \(n\rightarrow 0\) limit they are comparable to (k=2) gravity modes.
To nonaxisymmetric oscillation modes, however, accretion tori are dynamically unstable. The instability, discovered by Papaloizou & Pringle (1984), affects all nonaccreting torus configurations, and most violently tori with a constant angular momentum distribution.
Whether or not hydrodynamical oscillation modes may survive such global instabilities or the presence of a weak magnetic field (MRI turbulence), is subject of current, numerical investigations.
 Papaloizou, J., Pringle, J., 1984, "The dynamical stability of differentially rotating discs with constant specific angular momentum", MNRAS, 208,721
 Blaes O. et al, 2006, "Oscillation modes of relativistic slender tori", MNRAS, 369, 1235
Spectra
Theory versus observations: the electromagnetic spectra
Theoretical models predict both line and continuous electromagnetic spectra of accretion discs. Comparison with observations shows a rather good general agreement. However, several details are not satisfactory fitted and others are fixed by empirical ad hoc assumptions. Generally, the radiative transfer through the disc is not treated as accurately as in the case of stellar atmospheres. The two most acute difficulties that one faces in the accretion disc case are: (a) the disc geometry calls for a 3D treatment of the problem, (b) the viscous energy generation rate is not known as a function of the position in the disc. Fundamental papers that discuss methods to deal with these difficulties; see e.g. Hubeny et al. (2001) and Davis et al. (2005). In the black hole and neutron star accretion discs, the fact that radiation moves in a highly curved space (spacetime) further complicates the problem of calculating the spectra. Several "raytracing" methods of calculating photon trajectories along their geodesics from the source to a distant observer's screen have been developed to deal with this issue; see e.g. Vincent et al. (2011) and JeanAlain Marck (who was a pioneer of modern raytracing methods).
The protoplanetary discs appear in the spectra of young stellar objects as infrared excess on top of a stellar blackbody signature. Depending on the stage of the system, the excess hump can dominate (class I) the emission profile or is barely notable (class III). Protoplanetary discs contain a large amount of dust, e.g., silicates, minerals or water, whose chemical composition and geometrical shape show up as a forest of line features in the spectrum and greatly influence the opacity and evolution of the disc (Dullemond & Dominik, 2005).
The Cataclysmic Variables (CVs) with accretion discs (i.e., the WD is not or only weakly magnetised) can be of very different types. The bright dwarf novae discs are controlled by a temporary enhancement of the rate of mass transfer. The underluminous and cold dust or debris discs around WD are the result of tidally disrupted asteroids or planetoids. Generally, discs of CVs are divided into two states:
 the high state (high luminosity, active masstransfer)
 the low state (low luminosity, quiescent)
The Xray Galactic binaries (BHBs) display a number of distinct spectral states (Zdziarski & Gierlinski 2004; Remillard & McClintock, 2006; Done et al., 2007):
 the very high state (very luminous)
 the thermal state (disc dominated, luminous, high/soft state),
 the intermediate state
 the powerlaw state (slightly less luminous, low/hard state with strong variability) ,
 the quiescent state (very underluminous).
The thermal state is well described by the thin disc model and a multicolour disc (MCD) blackbody model (e.g., diskbb, Mitsuda et al. 1984; Zimmerman et al. 2005; available in XSPEC, Arnaud et al. 1996). Recently, fully relativistic versions of the MCD model for arbitrary BH spin (kerrbb, Li et al. 2005; bhspec, Davis & Hubeny 2006) have been developed, based on the relativistic thin disc model of Novikov & Thorne (1973). These models provide excellent fits to the Xray spectra of BHBs in the thermal and intermediate state.
Examples of continuous spectra
ShakuraSunyaev disc spectraThe BeppoSAX observation of LMC X3 in the thermal (high/soft) state in comparison with the best ﬁt BHSPEC models (for inclination i = 67°, distance D = 52 kpc and viscosity prescription \(\alpha = 0.01\)). The total model is given by the green curve, BHSPEC (red, longdashed curve) and COMPTT (violet, shortdashed curve) are plotted model components. The unabsorbed BHSPEC model is shown by the orange, solid curve. Figure credits: Davis et al. (2006) 
ADAF spectra
The observed lowstate spectrum of the XRay Nova XTE J1118+480 (crosses, triangles and squares) compared with the two spectra calculated with the ADAF model (red and green lines). Figure taken from Esin et al. (2001) 
Slim discs spectraThe soft hump and hard Xray spectrum of RE J1034+396. The data are taken from Puchnarewicz et al. (2001). The dashed, solid and dotted lines represent slim disc models with \({\dot M}/{\dot M}_{Edd} =\) 5, 10, and 20 respectively. In all cases the central black hole has the mass \(2.25 \times 10^6 M_{sun}\ .\) The fits include Comptonization. Figure taken from Wang & Netzer (2003). 
Ion tori spectraImpact of spin on an ion torus spectrum: a = 0 (solid blue), 0.5M (dashed red) or 0.9M (dotted green). The open circles and the "bow tie" mark observational data of Sgr A*. Note that this is merely a comparison: the spectra do not represent best fits. Figure taken from Straub et al. (2012) 
Examples of line spectra
Figure 1: Schematic profile of the fluorescent iron line. (Figure credits: Müller & Camenzind, 2004)  Figure 2: Observed Fe K alpha line in the Seyfert 1 galaxy MCG 30615. (Figure credits: Vaughan & Fabian, 2003) 
Accretion discs often show all kinds of emission and absorption lines. The most prominent line is the fluorescent emission line of neutral or mildly ionised iron. It occurs in the Xray band (depending on the degree of ionisation between 67 keV), i.e., at energies that can only be created close to a central compact object. The fluorescent Fe line is thus used to probe the vicinity of compact objects. The Xray line photons are produced when Xrays from the hot inner flow (corona) irradiate the underlying cool, weakly ionised disc. The iron in the disc absorbs the hard Xrays whereby an electron gets excited from the ground state (Kshell) to a higher energy level. After a while it assumes again the ground state by releasing the excess energy as a photon. The strongest line is the Fe \(K_{\alpha}\) line at ~6.4 keV. Due to nonrelativistic (Doppler) and special relativistic (beaming) and general relativistic effects (gravitational redshift) the line is not a simple Gaussian spike but a skewed and asymmetric profile.
Examples of images
The reason why there are so far no direct observations of the immediate environment of black holes is that observed from Earth they have a very small apparent size in the sky. The largest of all is the supermassive black hole in the centre of our galaxy, Sagittarius A*: The apparent diameter of its event horizon is only 53 micro arcseconds. Such dimensions have long been out of reach for astronomical instruments. However, a new generation of interferometers will change that in the coming years (first observations in 20152020). The Event Horizon Telescope (EHT), operating in the submillimetre radio range will perform Very Large Baseline Interferometry (VLBI) using a network of antennas distributed over the whole globe. Only those photons of an accretion disc which escape the black hole environment are observable. In an image, a black hole is thus silhouetted dark against the radiant accretion flow. The black hole silhouette itself is confined by a ring of photons that originate from immediately outside the horizon (second order image of the accretion flow) and is calculated by integrating the photon trajectories through spacetime from the observer towards the black hole. This method is called raytracing. The image and silhouette that emerge from this calculation and that may be observed by EHT in the near future are rich in information about the properties of the accretion flow and in particular of the black hole. Figure 12 shows images of a toroidal accretion flow (ion torus) around a supermassive Kerr black hole located at the galactic centre as seen by an observer on Earth. Top left: black hole spin \(a_*=0.5\), torus inclination towards the line of sight inc=80°, torus size \(\lambda=0.3\). Top right: The same as in top left but with \(a_*=0.9\). Bottom left: The same as in top left but with \(\lambda = 0.7\). Bottom right: The same as in top left but with inc=40°.

References
On line monographs and lecture courses on the accretion discs theory (highly recommended for students)
Original (often cited) discovery papers and influential reviews (available online)
 Abramowicz M.A., Chen X., Kato S., Lasota J.P., Regev O., 1995, Thermal equilibria of accretion disks, Astrophys. J., 438, L3
 Abramowicz M.A., Czerny B., Lasota J.P., Szuszkiewicz E., 1988, Slim accretion disks, Astrophys. J., 332, 646
 Abramowicz M.A., Jaroszynski M., Sikora M., 1978, Relativistic, accreting disks, Astron. Astrophys., 63, 221
 Balbus S.A., Hawley J.F., 1991, A powerful local shear instability in weakly magnetized disks, Astrophys. J., 376, 214
 Ichimaru S., 1977, Bimodal behavior of accretion disks. Theory and application to Cyg X1 transitions, Astrophys. J., 214, 840
 Jaroszynski M., Abramowicz M.A., Paczynski B., 1980, Supercritical Accretion Disks, Acta Astron., 30, 1
 LyndenBell D., Pringle J.E., 1974, Evolution of viscous discs and the origin of nebular variables, Month. Not. Roy. astron. Soc., 168, 603
 Narayan R., Yi I., 1994, Advectiondominated accretion: A selfsimilar solution, Astrophys. J., 428, L13
 Narayan R., Yi I., 1995, Advectiondominated Accretion: Underfed Black Holes and Neutron Stars, Astrophys. J., 452, 710
 Nayaran R., Yi I., Mahadevan R., 1995, Explaining the Spectrum of SgrA* with a Model of an Accreting BlackHole, Nature, 374, 623
 Pringle J.E., 1981, Accretion discs in astrophysics, Ann. Rev. Astron. Astrophys., 19, 137
 Pringle J.E., Rees M.J., 1972, Accretion Disc Models for Compact XRay Sources, Astron. Astrophys., 21, 1
 Rees M.J., Phinney E.S., Begelman M.C., Blandford R.D., 1982, Ionsupported tori and the origin of radio jets, Nature, 295, 17
 Salpeter E.E., 1964, Accretion of Interstellar Matter by Massive Objects, Astrophys. J., 140, 796
 Shakura N.I., Sunyaev R.A., 1973, Black holes in binary systems. Observational appearance, Astron. Astrophys., 24, 337
 Zeldovich Ya.B., Novikov, I. D., 1964, Dokl. Akad. Nauk SSSR, 158, 311 (non available online)
Monographs and textbooks (non available online)
 Abramowicz M.A., Björnsson G., Pringle J.E., 1999, Theory of Black Hole Accretion Discs, Cambridge University Press, Cambridge
 Frank J., King A., Raine D.J., 2002, Accretion Power in Astrophysics: Third Edition, Cambridge University Press, Cambridge,
 Kato S., Fukue J., Mineshige S., 1998, Blackhole accretion disks, Kyoto University Press, Kyoto