Figure 4

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"Foundations of Black Hole Accretion Disk Theory"

Marek A. Abramowicz and P. Chris Fragile

	Abstract
1	Introduction
2	Three Destinations in Kerr’s Strong Gravity
	2.1	The event horizon
	2.2	The ergosphere
	2.3	ISCO: the orbit of marginal stability
	2.4	The Paczyński–Wiita potential
	2.5	Summary: characteristic radii and frequencies
3	Matter Description: General Principles
	3.1	The fluid part
	3.2	The stress part
	3.3	The Maxwell part
	3.4	The radiation part
4	Thick Disks, Polish Doughnuts, & Magnetized Tori
	4.1	Polish doughnuts
	4.2	Magnetized Tori
5	Thin Disks
	5.1	Equations in the Kerr geometry
	5.2	The eigenvalue problem
	5.3	Solutions: Shakura–Sunyaev & Novikov–Thorne
6	Slim Disks
7	Advection-Dominated Accretion Flows (ADAFs)
8	Stability
	8.1	Hydrodynamic stability
	8.2	Magneto-rotational instability (MRI)
	8.3	Thermal and viscous instability
9	Oscillations
	9.1	Dynamical oscillations of thick disks
	9.2	Diskoseismology: oscillations of thin disks
10	Relativistic Jets
11	Numerical Simulations
	11.1	Numerical techniques
	11.2	Matter description in simulations
	11.3	Polish doughnuts (thick) disks in simulations
	11.4	Novikov–Thorne (thin) disks in simulations
	11.5	ADAFs in simulations
	11.6	Oscillations in simulations
	11.7	Jets in simulations
	11.8	Highly magnetized accretion in simulations
12	Selected Astrophysical Applications
	12.1	Measurements of black-hole mass and spin
	12.2	Black hole vs. neutron star accretion disks
	12.3	Black-hole accretion disk spectral states
	12.4	Quasi-Periodic Oscillations (QPOs)
	12.5	The case of Sgr A*
13	Concluding Remarks
	Acknowledgements
	References
	Footnotes
	Figures
	Tables

Figure 4: In equilibrium, the equipressure surfaces should coincide with the surfaces shown by the solid lines in the right panel. Note the Roche lobe, self-crossing at the cusp. The cusp and the center, both located at the equatorial plane $𝜃 = π∕2$ , are circles on which the pressure gradient vanishes. Thus, the (constant) angular momentum of matter equals the Keplerian angular momentum at these two circles, $ℓ = ℓ (r ) = ℓ (r ) 0 K cusp K center$ , as shown in the upper left panel. In this figure $W$ refers to the effective potential. Image reproduced by permission from [98], copyright by RAS.