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 \(\ell =  u_{\phi}/u_t\ ,\)
 \(t_{dyn} \ll t_{the} <t_{vis}\ ,\) with \(t_{dyn}\) being the dynamical timescale in which pressure force adjusts to the balance of gravitational and centrifugal forces, \(t_{the}\) being the thermal timescale in which the entropy redistribution occurs due to dissipative heating and cooling processes, and \(t_{vis}\) 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,
\[
\frac{\nabla_{\mu} P}{\epsilon + P} = \frac{\nabla_{\mu}\,g_{tt} + 2\Omega\,\nabla_{\mu}\,g_{t\phi} + \Omega^2\,\nabla_{\mu}\,g_{\phi\phi}}{g_{tt} + 2\Omega\,g_{t\phi} + \Omega^2\,g_{\phi\phi}} = \nabla_{\mu}\ln A + \frac{\ell\,\nabla_{\mu}\Omega}{1  \ell\,\Omega},
~~~{\rm with}~~~A^2(r, \theta) = \frac{1}{g_{tt}(r,\theta) + 2\Omega\,g_{t\phi}(r,\theta) + \Omega^2\,g_{\phi\phi}(r,\theta)}.
\]
 \((3.1.1)\) 
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 (\(\eta, \beta, \gamma\)),
\[
{\ell}(r, \theta) = {\ell}_0\left\{ \frac{{\ell}_K(r)}{{\ell}_0}\right\}^{\beta}(\sin\theta)^{2\gamma} ~\left({\rm for}~~ r \geq r_{ms}\right)
,~~
{\ell}(r, \theta) = {\ell}_{ms}(\sin\theta)^{2\gamma} ~\left({\rm for}~~ r < r_{ms}\right).
~~{\rm Here}~~ {\ell}_0 \equiv \eta\,{\ell}_K(r_{ms}) ~~{\rm and}~~ {\ell}_{ms} = {\ell}_0\left\{\frac{{\ell}_K(r_{ms})}{{\ell}_0}\right\}^{\beta}.
\]
 \((3.2.3)\) 
Fig. 1 Equipressure surfaces for a non rotating black hole, and with angular momentum distribution (3.2.3) with \(\gamma = 0.00\ ,\) \(\eta = 1.085\ ,\) \(\beta = 0.00\ .\) Angular momentum is constant everywhere in space.


Fig. 2 Equipressure surfaces for a non rotating black hole, and with angular momentum distribution (3.2.3) with \(\gamma = 0.50\ ,\) \(\eta = 1.085\ ,\) \(\beta = 0.00\ .\) Angular momentum is constant radially, but it changes along
the polar angle. 

Fig. 3 Equipressure surfaces for a non rotating black hole, and with angular momentum distribution (3.2.3) with \(\gamma = 0.18\ ,\) \(\eta = 1.085\ ,\) \(\beta = 0.9\ ,\) compared with global MHD numerical simulations by
Fragile et al., 2007, time averaged over a period corresponding to the orbital period at \(25r_G\) (colorcoded). 

Fig. 4 Equipressure surfaces for a non rotating black hole, and with angular momentum distribution (3.2.3) with \(\gamma = 0.00\ ,\) \(\eta = 1.085\ ,\) \(\beta = 0.99\ .\) Angular momentum does not depend on the polar angle, its radial distribution is almost Keplerian. 

Barytropic thick discs and the von Zeipel theorem
From equation (3.1.1) one proves (see e.g. Abramowicz et al. 1978) that for barytropic fluids \(\epsilon = \epsilon(P)\ ,\) the surfaces of constant angular velocity and of constant angular momentum coincide, i.e. \(\ell = \ell(\Omega)\ .\) This is often called "the von Zeipel theorem". The equation of state \(\epsilon = \epsilon(P)\) and the rotation law \(\ell = \ell(\Omega)\) are independent and may be separately assumed. When they are known, the analytic solution is given by,
\[
W(P) \equiv \int \frac{dP}{\epsilon(P) + P} = \ln A + \int \frac{d\Omega}{1  \Omega\,\ell(\Omega)} \equiv \ln A(r, \theta) + F(\Omega),
~~~{\rm and}~~~\ell = \ell(\Omega) =
 \frac{g_{t\phi}(r, \theta) + \Omega\,g_{\phi\phi}(r, \theta)}{g_{tt}(r, \theta) + \Omega\,g_{t\phi}(r, \theta)}.
\]
 \((3.2.4)\) 
The functions \(W = W(P)\ ,\) \(F = F(\Omega)\) and \(\Omega = \Omega(r, \theta)\) are explicitly known, and therefore one knows explicitly location of the equipressure surfaces \(P = P(r, \theta)\ .\) In the special (but important) case \(\ell = \ell_0 = const\ ,\) the function \(\theta = \theta(r)\) that gives the location of equipressure surfaces \(P(r, \theta) = const\) is given explicitly (for a nonrotating black hole)
\[
\sin^2\theta = \frac{\ell_0}{C_0\,r^2 + 2\,G\,M\,\left(1  r_G/r\right)^{1}}. ~~~{\rm Here}~~C_0=const ~~{\rm numerates~the~equipressure~surfaces}.
\]
 \((3.2.5)\) 
Putting \(r_G = 0\) in (3.2.5), one recovers the Newtonian formula for \(\ell = \ell_0 = const\) discs in the \(G\,M/r\) potential.
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, \(\ell_{disc}(r_{in}) = \ell_K(r_{in})\ .\) For \(r > r_{in}\) it is \(\ell_{disc}(r_{in}) > \ell_K(r_{in})\) and \(d\ell/dr > 0\ ,\) and for \(r < r_{in}\) it is \(\ell_{disc}(r_{in}) < \ell_K(r_{in})\) 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 disk with the accretion rate that for a polytropic fluid \(P = K\rho^{1+1/n}\) equals ( Kozlowski et al. 1978, Abramowicz 1985)
\[
{\dot M}_{in} = (2\pi)^{3/2}\frac{\Gamma(n + 3/2)}{(1 + 1/n)^n (1 + 1/2)^{n + 3/2}\Gamma(n + 3)}K^{n}\frac{r_{in}}{\Omega_K(r_{in})}\Delta W^{n + 1},
~~{\rm with}~~ \Gamma(m) ~~{\rm being~the~Euler~gamma~function}.
\]
 \((3.2.6)\) 
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 disks (thin, slim, adafs, thick) around black holes and sufficiently compact neutron stars. This selfregulated overflow has several important consequences:
 It locally stabilizes accretion discs against thermal and viscous instabilities (Abramowicz, 1981) and globally against the Papaloizou and Pringle instability (Blaes, 1987)
 The amount of overflow, and therefore \({\dot M}_{in}\ ,\) may be modulated by global discs oscillations. In the case of neutron stars, this leads to a modulation of the luminosity of the boundary layer at the neutron star surface. Although oscillations originate in the disk, the are observed in radiation that comes from the boundary layer. This is relevant for the observed neutron star quasi periodic oscillation (QPO) (Horak et al., 2007).
 The "runaway" instability occurs when the mass exchange between accretion disc and black hole causes the cusp to move deeper into the disc, increasing the mass lost rate \({\dot M}_{in}\ .\) This effect was suggested by Abramowicz et al. (1983), and studied by several other authors (e.g. Daigne and Font, 2004 or Montero et al., 2008). It may determine the lifetime of a massive torus around a black hole, which is relevant to some models of gamma ray bursts.


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 luminosities of radiation pressure supported thick discs ("Polish doughnuts")
The maximal energy available from an object with the mass \(M\) (and gravitational radius \(R_G = GM/c^2\)) is \(E_{max} = M\,c^2\ .\) The minimal time in which this energy may be liberated is \(t_{min} = R_G/c\ .\) Thus, the maximal power \(L_{max} = E_{max}/t_{min} = c^5/G \equiv L_{Planck} = 10^{58}\,[{\rm erg/sec}] = 10^{52}\,[{\rm Watts}]\ .\) An object with mass \(M\) has the "gravitational crosssection" \(\Sigma_{grav} = 4\,\pi\,R^2_G\ .\) If radiation interacts with matter by electron scattering (with the Thomson cross section \(\sigma_T = 8\pi\,e^4/3\,m^2_e\)), its "radiation cross section" is \(\Sigma_{rad} = (M/m_P)\sigma_T\ .\) M. Sikora noticed (unpublished) that the upper limit for the radiative power of an object in which gravity and radiation pressure are in equilibrium is given by the Planck power and the object gravitational and radiative cross sections,
\[
(\,\rm{Eddington~luminosity}\,) \equiv L_{Edd} = L_{Planck}\frac{\Sigma_{grav}}{\Sigma_{rad}} = \frac{4\pi\,G\,M\,m_P\,c}{\sigma_T}
= 1.4 \times 10^{38}\left( \frac{M}{M_{sun}}\right) [{\rm erg}/{\rm sec}].
\]
 \((3.2.7)\) 
  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 \(L/L_{Edd} \equiv \lambda \approx \log (R/R_G)\ .\) The logarithm here is of a crucial importance. It prevents astrophysically realistic doughnuts (i.e. with \(R < 10^6\,R_G\ ,\) say) 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 modeling 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 and 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. 
\[
(\,\rm{Eddington~accretion~rate}\,) \equiv {\dot M}_{Edd} = \frac{L_{Edd}}{c^2}
= 1.5 \times 10^{17}\left( \frac{M}{M_{sun}}\right) [{\rm g}/{\rm sec}], ~~~{\dot m} = \frac{\dot M}{{\dot M}_{Edd}}.
\]
 \((3.2.8)\) 
Theoretical predictions about superEddington accretion rates:
 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.
The assumption that \({\dot m} = 1\) is the upper limit for the growth rate of the seed black holes, often adopted in the context of the cosmic structure formation, is false.
