Analytic models of accretion discs

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Author: Dr. Marek A. Abramowicz, Physics Department, Göteborg University, Sweden and N. Copernicus Astronomical Center, PAN, Warsaw, Poland
Author: Miss Odele Straub, N. Copernicus Astronomical Center PAN, Warsaw, Poland

1 A few definitions
2 Assumptions adopted in analytic models of accretion discs
3 A Newtonian black hole model: the Paczynski-Wiita potential
4 The inner edge, and the inner boundary condition
5 A catalog of the analytic and semi-analytic accretion disc models
6 Analytic models on the accretion rate vs. surface density plane
- 6.1 The four branches of analytic models of accretion discs
7 Predictions of the analytic models: spectra
- 7.1 Continuous electromagnetic spectra
- 7.2 Line spectra
8 Predictions of the analytic models: variability
- 8.1 Long term variability (limit cycles)
- 8.2 Short term variability (oscillations, stochastic variability)

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A few definitions

$h = H/r$	dimensionless vertical thickness
${\dot m} = {\dot M}/{\dot M}_{Edd}$	dimensionless accretion rate, ${\dot M}_{Edd} =1.5 \times 10^{17}\,({M /M_0})\,[{\rm g}/{\rm sec}]$ , and $M_0 = 2 \times 10^{33}\,$ [g] is the mass of the Sun
$q = Q^{adv}/Q^+$	dimensionless (importance of) advective cooling $Q^{adv} =$ advective cooling rate, $Q^{+} =$ viscous heating rate
$\alpha$	dimensionless alpha viscosity coefficient
$\tau= \int_{0}^{H} \kappa \rho dz$	dimensionless optical depth $\rho =$ density, $\kappa =$ opacity.
$\Sigma$	dimensional [g/cm²] surface density $\Sigma = \int_{0}^{H} \rho dz.$ .

=HERE A FIGURE ILLUSTRATING THESE QUANTITIES=

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Assumptions adopted in analytic models of accretion discs

The present understanding of accretion discs rests in its major part on studies of analytic models that assume stationary and axially symmetric accretion. In particular, this assumption was explicitly made in all serious comparisons between theory and observations by a detailed spectral fitting to observed continuous spectra and line profiles.

Analytic models usually assume in addition that $t_{dyn} \ll Min(t_{the}, t_{vis})$ . Here $t_{dyn}$ is the dynamical timescale in which pressure force adjusts to the balance of gravitational and centrifugal forces, $t_{the}$ is the thermal timescale in which the entropy redistribution occurs due to dissipative heating and cooling processes, and $t_{vis}$ is the viscous timescale in which angular momentum distribution changes due to torque caused by dissipative stresses.

However, it is still unknown whether it is physically legitimate to make all these assumptions and suppose that in some "averaged" sense accretion flows may be (approximately) described in terms of stationary and axially symmetric dynamical equilibria. While observations seem to suggest that many real astrophysical sources experience periods in which this may be quite reasonable, several authors point out that results of recent numerical simulations indicate that the MRI and other instabilities could make the accretion flows genuinely non-steady and non-symmetric, and that the very concept of separate timescales may be questionable in the sense that locally it could be $t_{dyn} \approx t_{the} \approx t_{vis})$ .

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A Newtonian black hole model: the Paczynski-Wiita potential

Figure 1: Keplerian and epicyclic frequencies in the Paczynski-Wiita potential (solid) and in the Schwarzschild geometry (dashed).

For free particles, both Newton's and Einstein's orbital dynamics are described by the same principle: the orbital Keplerian frequency follows from the first derivative, and the epicyclic frequencies follow from the second derivatives of the effective potential. Thus (as Paczynski realized), by a proper guess of an artificial Newtonian potential, one should be able to accurately describe in Newton's theory the relativistic orbital motion. Paczynski's own guess proved to be the simplest and most practical, $\Phi = -GM/(r - 2r_G)$ with $r_G = GM/c^2$ . It is used by numerous authors.

B. Paczynski and P.J. Wiita Thick accretion disks and supercritical luminosities A&A 88, 23-31, 1980

M.A. Abramowicz, A.M. Beloborodov, X.-M. Chen, I.V. Igumenshchev Special relativity and the pseudo-Newtonian potential A&A, 313 334, 1996

V. Karas and O. Semerak Pseudo-Newtonian models of a rotating black hole field A&A 343, 325-332, 1999

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The inner edge, and the inner boundary condition

Analytic models describe 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 $r \approx r_{in}$ 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, $r_{MB} \le r_{in} \le r_{ISCO}$ . For very efficient Shakura-Sunyaev discs, $r_{in} \approx r_{ISCO}$ , while for RIFs $r_{in} \approx r_{MB}$ . For stellar accretion, $r_{in}$ is located near the surface of the star and the flow there is described by a boundary layer model.

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A catalog of the analytic and semi-analytic accretion disc models

Model's name	Short characteristic	Links to on-line references
Shakura-Sunyaev =standard= =thin disc=	Axially symmetric, stationary, local analytic model. Explicit formulae give all physical characteristics in terms of M, M-dot, alpha and R. Geometrically thin in the vertical direction (H/R < 1), has a disc-like shape). Accretion rate very sub-Eddington. Opacity 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 at ISCO. High luminosity, high efficiency of radiative cooling. Electromagnetic spectra not much different from that of a sum of black bodies. Alpha viscosity prescription. Diffusion approximation for radiative transfer. Dynamically stable. When the gas is cold and radiation pressure negligible also thermally and viscously stable, otherwise 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); Lynden-Bell, 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).
Shakura-Sunyaev =modifications=	Thin discs: strong opacity variation with temperature	????
	Thin discs: warps	????
	Thin discs: self-gravity	????
Adafs	Sub-Eddington 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 Shakura-Sunyaev thin discs. Adafs emit a power-law, non-thermal 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 non-stationary models (in Paczynski-Wiita): Igumenshev et al. (1996).	Most influential paper, describing a Newtonian, self-similar, stationary, axially symmetric analytic, model: Narayan, Yi (1994). Immediate follow-up 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).
Slim	Nearly Eddington accretion. Large opacity. Cooled by radiation and strong advection. Radiatively less efficient than Shakura-Sunyaev. H/R 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. Applications: mostly LMXRB, AGNs, with good fits to observed spectra. ??? ??? ???	??? ??? ???
Kluzniak-Kita	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 disk have been found.	Kluzniak, Kita (2000), numerical follow up: Umurhan et al. (2006).

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Analytic models on the accretion rate vs. surface density plane

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The four branches of analytic models of accretion discs

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).