# Accretion discs/Analytic models of accretion discs

## Two major classes of analytic models: "thin" and "thick" accretion discs

Non-linear, coupled partial differential equations of radiative hydrodynamics and magnetohydrodynamics that describe physics of accretion discs are too complex to be exactly solved analytically in the general case. Some useful approximate solutions exist in two important limits of stationary and axially symmetric "thin" and "thick" accretion discs. Thin discs locate at the symmetry plane of the central accreting body and have everywhere their thickens across the plane much smaller than the distance from the center along the plane, $$H \ll R\ .$$ This means that their structure depends only on $$R$$ and may be described by ordinary differential equations. Thick discs have toroidal shapes with $$H \approx R\ .$$ One assumes for thick discs that $$t_{dyn} \ll t_{the} <t_{vis}\ ,$$ 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. Detailed models of thin and thick discs are described in the sub-sections of this Scholarpedia article,
 3.1. Thin discs3.2. Thick discs
In the case of thin discs, the analytic solution is possible because the adopted simplifying assumptions concerning mostly mathematics (symmetries, $$H \ll R$$) reduce the problem to a set of linear, ordinary differential equations. In the case of thick discs, the analytic solution is possible because simplifying assumptions concerning mostly physics ($$t_{dyn} \ll t_{the} <t_{vis}$$) reduce the problem to a directly integrable one.

## The analytic model of the external gravitational field

Most of the accretion discs types (except proto-planetary and GRB ones) have a negligible self-gravity: the external gravity of the central accreting object dominates. The external gravity is important in shaping several crucial aspects of the internal physics of accretion discs, including their characteristic frequencies (that are connected to several important timescales) and their size (inner and outer radius). The most fundamental gravity's characteristic frequencies are the Keplerian orbital frequency $$\Omega_K\ ,$$ the radial epicyclic frequency $$\omega_R\ ,$$ and the vertical epicyclic frequency $$\omega_Z\ .$$ They are directly relevant for motion of free particles and also play a role for determining equilibria and stability of rotating fluids. In both Newton's and Einstein's gravity the three frequencies are derived from the effective potential $$U_{eff}(R, j)\ ,$$ and given by the same formulae,

 $\left[ \left(\frac{\partial U_{eff}}{\partial R}\right)_j = 0\right] \rightarrow \left[ \Omega_K^2 = \Omega_K^2(R) \right], ~~~\omega_R^2(R) = \left(\frac{\partial^2 U_{eff}}{\partial R^2}\right)_j, ~~~\omega_Z^2(R) = \left(\frac{\partial^2 U_{eff}}{\partial Z^2}\right)_j,$ $$(3.1)$$

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 $$U_{eff} = \Phi + j^2/2R\ .$$ A spherical Newtonian body has the gravitational potential $$\Phi = -GM/R\ .$$ Thus, in this case, $$\Omega_K^2 = \omega_R^2 = \omega_Z^2 = GM/R^3 > 0\ ,$$ i.e. all slightly non-circular orbits are closed and all circular orbits are stable. In Einstein's gravity, for a spherical body it is $$\Omega_K^2 = \omega_Z^2 > \omega_R^2\ ,$$ i.e. non-circular 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.

## Einstein's theory description ot the black hole gravity (the analytic Kerr metric)

In the black hole gravity, the stable circular Keplerian orbits exist only with radii greater than the radius of ISCO, or the innermost stable circular orbit radius. All Keplerian orbits closer to the black hole than ISCO are unstable: without an extra support by non-gravitational forces (i.e. pressure or magnetic field) matter cannot stay there orbiting freely, but instead it must fall down into the black hole. This strong-field property of Einstein's gravity, absent in Newton's theory, is the most important physical effect in the black hole accretion disc physics.

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 Boyer-Lindquist 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.
 $$g_{\mu\nu}$$ $$g^{\mu\nu}$$ $$t$$ $$\phi$$ $$r$$ $$\theta$$ $$t$$ $$\phi$$ $$r$$ $$\theta$$ $$t$$ $$1 - 2\,M\,r/\Sigma$$ $$4\,M\,a\,r\sin^2\theta/\Sigma$$ $$0$$ $$0$$ $$(r^2 + a^2)^2/\Sigma\,\Delta$$$$- a^2\Delta\sin^2\theta/\Sigma\,\Delta$$ $$2M\,\,a\,r/\Sigma\,\Delta$$ $$0$$ $$0$$ $$\phi$$ $$4\,M\,a\,r\sin^2\theta/\Sigma$$ $$-(r^2 + a^2)\sin^2\theta$$$$-2\,M\,a^2r\sin^4\theta/\Sigma$$ $$0$$ $$0$$ $$2M\,\,a\,r/\Sigma\,\Delta$$ $$-\frac{\Delta-a^2\sin^2\theta}{\Delta\Sigma\sin^2\theta}$$ $$0$$ $$0$$ $$r$$ $$0$$ $$0$$ $$-\Sigma/\Delta$$ $$0$$ $$0$$ $$0$$ $$-\Delta/\Sigma$$ $$0$$ $$\theta$$ $$0$$ $$0$$ $$0$$ $$-\Sigma$$ $$0$$ $$0$$ $$0$$ $$-1/\Sigma$$

In any stationary and axially symmetric spacetime, and in particular in the Kerr geometry, for matter rotating on circular orbits with four velocity $$u^{\nu} = (u^t, u^{\phi})$$ it is $$\Omega = u^{\phi}/u^t$$ and $$j = - u_{\phi}/u_t\ ,$$ from which (and $$u^{\nu}\,u_{\nu} = 1$$) it follows that,

 $\Omega = -\frac{j\,g_{tt} + g_{t\phi}}{j\,g_{t\phi} + g_{\phi\phi}}, ~~~ j =-\frac{\Omega\,g_{\phi\phi} + g_{t\phi}}{\Omega\,g_{t\phi} + g_{tt}}, ~~~ U_{eff} = -\frac{1}{2} \ln \left( g^{tt} - 2j\,g^{t\phi} + j^2\,g^{\phi \phi}\right).$ $$(3.2)$$

From equations (3.1), (3.2) and $$dR^2 = g_{rr}dr^2\ ,$$ $$dZ^2 = g_{\theta\theta}d\theta^2\ ,$$ one derives that the Keplerian frequency $$\Omega_K$$ and the two epicyclic frequencies (radial $$\omega_R$$ and vertical $$\omega_Z$$) equal

 $\Omega_K = \frac{c^3}{GM}\left( {r_*}^{3/2} + {a_*}\right)^{-1}, ~~~ \omega_R^2 = \Omega_K^2 \left( 1 - 6{r_*}^{-1} + 8{a_*}\,{r_*}^{-3/2} - 3{a_*}^2\,{r_*}^2 \right), ~~~ \omega_Z^2 = \Omega_K^2 \left( 1 - 4{a_*}\,{r_*}^{-3/2} + 3{_*a}^2\,{r_*}^{-2} \right).$ $$(3.3)$$

Here the dimensionless $${r_*}$$ and $$\vert{a_*}\vert \le 1$$ are defined by $${r_*} = rc^2/GM\ ,$$ $${a_*} = Jc/GM^2\ .$$ In the strong gravity, i.e. for $${r_*} \sim 1\ ,$$ the three frequencies scale as $$1/M\ .$$ The radial epicyclic oscillations, $$\delta r(t) \sim \exp(- i\,\omega_R\,t)\ ,$$ become dynamically unstable at $$r < ISCO\ ,$$ because there $$\omega_R^2(r) < 0\ .$$

## Paczynski's analytic Newtonian model for the black hole gravity

Paczynski and Wiita (1980) realized 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 Karas and Semerak (1999). Newtonian models for rotating black holes are far less practical and for this reason not widely used.